by sajjad naja - university of toronto t-space€¦ · sajjad naja doctor of philosophy graduate...

Essays in Dynamic Pricing of Multiple Substitutable Products

by

Sajjad Najafi

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

c© Copyright 2016 by Sajjad Najafi

Abstract

Essays in Dynamic Pricing of Multiple Substitutable Products

Sajjad Najafi

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2016

I study the dynamic pricing problem of a firm selling limited inventory of multiple differentiated

products over a finite planning horizon, where the firm wishes to maximize the expected revenue. I

formulate the firm’s optimization problem as a Markov decision process and investigate the pricing

problem in the presence of a variety of operational settings. First, I integrate consumer’s sequential

search behavior into the pricing problem. The consumer inspects products one at a time by incurring

non-zero search cost, and makes decision by comparing the utility of the best product so far versus the

reservation utility, a threshold at which the consumer is indifferent between continuation and stopping

of the sequential search. The firm aims at maximizing the expected revenue by offering the products

in the right sequence and at the right prices. I analytically derive the optimal prices in each period.

I show that under some condition it is optimal to present products in the descending order of quality.

Second, I address a problem in which the firm is subject to a set of sales volume constraints required

to be satisfied at different time points along the sales horizon. Due to stochastic nature of sales, I

incorporate a risk measure that allows the firm to manage the total sales while the expected revenue is

maximized. I formulate the problem as a chance-constrained dynamic programming and show that the

Karush-Kuhn-Tucker conditions are not only necessary but also sufficient for the optimal price. Third,

I assimilate consumer’s consideration sets to the dynamic pricing problem. When to make a purchase

decision, consumers use a two-stage decision making process, i.e., consumers constitute a consideration

set including a subset of the available products using a screening rule (e.g., brands, quality, and budget),

and they only evaluate the products in the consideration set using a utility comparison process and

opt for the product with the maximum utility. I show that the first-order condition is sufficient for the

optimal price of products if consumers apply a quality-based screening rule.

ii

To my dear parents and wife. I could not have done this without your endless love and

unmitigated support.

iii

Acknowledgements

First and foremost I would like to express my sincere appreciations to my supervisor, Chi-Guhn

Lee. I am greatly thankful to him for giving me a huge amount of flexibility in exploring and defining

the research topics. I do really appreciate his continuous support, patience, generous guidance, and

thoughtful advice throughout my studies.

My supervisory committee members, Timothy C. Y. Chan and Ming Hu, immensely contributed to

refining my research direction. I would like to acknowledge their tremendous enthusiasm for conducting

me, and I am eternally grateful for all their help, suggestions, motivation, and encouragement. I am

hugely indebted to Steven Nahmias and Sami Najafi-Asadolahi. It was a great opportunity collabo-

rating with them. They supported me and were always willing to help. I would also like to gratefully

acknowledge the external thesis appraisers, Hyoduk Shin and Daniel M. Frances, for reviewing my thesis

and providing me with constructive comments and interesting future works. I would also like to thank

my friends, Vahid Roshanaei, Kimia Ghobadi, Shuvomoy Das Gupta, for their thoughts and fruitful

discussions.

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Contents

1 Introduction 1

2 Dynamic Pricing Under Consumer’s Sequential Search 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Consumer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Sequential Search and Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Reservation Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.3 Purchase Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Firm Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Optimal Sequencing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Optimal Pricing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 Horizontally Differentiated Products . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.2 Decreasing Reservation Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.3 Stochastic and Arbitrary Search Sequences . . . . . . . . . . . . . . . . . . . 24

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.8 Additional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Dynamic Pricing Under Sales Milestone Constraints 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 The Firm’s Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

v

3.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Probabilistic Constraints and Feasibility . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Lagrangian Relaxation Approach and Optimality Conditions . . . . . . . . . . . . 48

3.3.3 Optimal Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Structural Properties of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Lagrangian Multiplier and Optimal Price Properties . . . . . . . . . . . . . . . . . 53

3.4.2 Inventory Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.1 Practical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6.1 Expected Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6.2 Pricing Policy with Revenue Constraint . . . . . . . . . . . . . . . . . . . . . . . . 62

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Dynamic Pricing Under Consumer’s Consideration Sets 72

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 The Firm Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Conclusion 91

Bibliography 93

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List of Tables

3.1 The CCD inventory states’ changes over time . . . . . . . . . . . . . . . . . . . . . . . . . 59

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List of Figures

2.1 Summary of the steps that consumers take during their sequential search. . . . . . . . . . 11

2.2 The firm’s optimal revenue for different search sequences m = (k, h, l) where product k is

inspected first, then product h, and finally product l) . . . . . . . . . . . . . . . . . . . . . 15

2.3 The impact of the remaining time on the insepected products size where only product 2

is inspected (i.e., V2t(x)) and both products 1 and 2 are inspected (i.e., V1t(x)) . . . . . . 17

2.4 Optimal price of product 1 with various inventory levels where the search sequence is

m = (1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


m = (1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


m = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


m = (2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 Optimal prices of products 1 to 4 for x=(3,1,1,1). . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Optimal prices of products 1 to 4 x=(1,3,1,1). . . . . . . . . . . . . . . . . . . . . . . . . . 21



2.12 Optimal revenue with different inventory states where product’s 1 inventory level increases 35




2.16 Optimal price of product 1 when its inventory increases. . . . . . . . . . . . . . . . . . . . 37

2.17 Optimal price of product 2 when its inventory increases. . . . . . . . . . . . . . . . . . . . 37

2.18 Optimal revenue v.s. remaining periods for different values of rj and x = (2, 2). . . . . . . 37

2.19 Optimal price of product 1 v.s. different consumers’ reservation utilities for x = (2, 2). . . 38

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2.20 Optimal price of product 2 v.s. different consumers’ reservation utilities for x = (2, 2). . . 38

3.1 Optimal price of product 1 with various inventory levels where the initial inventory is

x = (3, 2) and at least 2 products should be sold by the end of horizon . . . . . . . . . . . 53


x = (3, 2) and at least 2 products should be sold by the end of horizon . . . . . . . . . . . 53

3.3 Probability of meeting the milestone constraint of selling at least 2 products by the end

of horizon with the initial inventory level x = (3, 2) . . . . . . . . . . . . . . . . . . . . . . 54


x = (3, 3) and at least 3 products should be sold by t = 3 . . . . . . . . . . . . . . . . . . 55


x = (3, 3) and at least 3 products should be sold by t = 3 . . . . . . . . . . . . . . . . . . 55

3.6 Revenue with different inventory states x . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 The regression model to estimate the condominium units’ quality levels . . . . . . . . . . 57

3.8 Comparison of the CCD practiced prices with the optimal prices generated by our model

given the CCD actual states’ changes over time . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9 Comparison of the CCD practiced prices with the optimal prices generated by our model

where the products have a fixed inventory of one . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 Comparison of the CCD actual revenue and the expected revenue generated by our model

with adjusting factors 1.2 and 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.11 Price comparison given the CCD inventory states’ changes where at least 7 units of prod-

ucts should be sold by t = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.12 Comparison of revenue generated by the unconstrained model and that of the constrained

model with different probability threshold δ for x = (3, 3) . . . . . . . . . . . . . . . . . . 61

3.13 Comparison of revenue generated by the unconstrained model and that of the constrained

model with different sales requirement ξ for x = (3, 3) . . . . . . . . . . . . . . . . . . . . 61

4.1 Optimal price of product 1 with various inventory levels for quality-based consumers where

Ω∼ U [0, 25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Optimal price of product 2 with various inventory levels for quality-based consumers where

Ω∼ U [0, 25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Revenue with different inventory states for quality-based consumers where Ω∼ U [0, 25] . . 84

4.4 Optimal price of product 1 with various inventory levels for budget-based consumers where

p ∼ U [0, 25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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4.5 Optimal price of product 2 with various inventory levels for budget-based consumers where

p ∼ U [0, 25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.6 Revenue with different inventory states for budget-based consumers where p ∼ U [0, 25] . . 86

4.7 Optimal price of product 1 where Ω,p ∼ U [0, 25], with that of no consideration set model . 87

4.8 Optimal price of product 2 where Ω,p ∼ U [0, 25], with that of no consideration set model . 87

4.9 Revenue comparison of quality and budget-based models with the no consideration set

model where Ω,p ∼ U [0, 25], and x = (2, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.10 Revenue generated for quality and budget-based models where the upper bound l changes

in Ω, p ∼ U [0, l] for x = (2, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

x

Chapter 1

Introduction

“Pricing right is the fastest and most effective way for managers to increase profits” (Marn et al. 2003).

It is now generally believed that the ability to determine the optimal products’ prices dynamically

based on the remaining inventory of products and time is a key to firms’ revenue increase. A study by

McKinsey (Marn et al. 2003) indicates that a price improvement of 1% can generate an 8% improvement

in operating profits. We focus on a common firm-consumer setting in which a firm sells a set of multiple

substitutable products over a finite selling season. This research is concerned with how pricing and the

operational decisions of a firm can change in the presence of a variety of the well-established consumers’

choice behavior models in economics, marketing, and psychology literature. We pursue the following

three objectives:

First, we study a firm offering a line of vertically differentiated products with fixed initial inventory

over a finite sales season. Consumers arrive at the firm randomly and inspect products sequentially until

they find a product to purchase (if any). Each consumer incurs a positive cost to inspect a product and

hence may stop the sequential search without inspecting all the available items. Upon inspection, the

utility of the product is known to the consumer, who then decides whether to continue the search by

comparing the utility of the best product observed thus far (i.e., the product giving the highest utility)

with his reservation utility, i.e., the maximum utility he might gain if he continues the search. The

firm wishes to maximize the total expected revenue from the existing inventories by offering the best

sequence in which to show the products to the consumers, as well as the best price for each product at

each time. We formulate the firm’s optimization problem as a Markov decision process and obtain the

products’ purchase probabilities, and their optimal prices, given their remaining inventories and time.

We show that it is optimal to present products in a descending order of quality if the reservation utility

is stationary or increasing. Surprisingly, we show that the optimal price can increase as time approaches

1

Chapter 1. Introduction 2

the end of horizon, which is contrary to the conventional finding in the well-established literature on

dynamic pricing. As the products are perishable, their prices are typically expected to decrease over

time. However, as in the sequential search, a consumer may not be able to see all products, it can result

in the opposite price impact.

Second, we incorporate sales volume constraints into a dynamic pricing problem. We consider a firm

selling limited inventory of substitutable products over a finite planning horizon. The firm is subject to

a set of sales volume constraints required to be satisfied at different time points along the sales horizon.

The firm aims at maximizing the expected revenue while satisfying the constraints. Due to stochastic

nature of demand, the firm specifies a maximum level of risk of not satisfying the constraints, allowing the

firm to manage the total sales, while the expected revenue is maximized. We formulate the problem as a

chance-constrained dynamic programming. We show that the Karush-Kuhn-Tucker (KKT) conditions

are not only necessary but also sufficient for the optimal price, enabling us to find novel theoretical

results. We derive closed-form solution for the optimal price of products. We show counter-intuitively

that it is possible for the optimal revenue of a firm to decrease in products’ inventory. Moreover, we

analytically show that if there is a sufficiently long time until the milestone, then joint penetration-

skimming pricing strategy is optimal. The proposed model is specifically suitable for applications in

which the achievement of sales targets plays a crucial role for managers. Two main applications of this

study are (i) real estate industry, where developers aim at finding a pricing policy to maximize the

expected revenue and sell a minimum acceptable number of units to qualify for construction loan, and

(ii) penetration pricing policy in which firms intend to achieve a predetermined market share. We also

implement our proposed pricing model for a leading condominium developer in Canada, and show a

significant improvement in profitability.

Third, we consider a firm offering a line of vertically differentiated products with limited inventory

over a finite selling season. The firm’s goal is to maximize its expected revenue by correctly pricing

the products over the selling season as a function of the existing inventory level and time. Customers

randomly arrive at firm and use the following two-stage decision making process to make a choice: (i)

consumers constitute a consideration/choice set including a subset of all the available products using a

screening rule (e.g., based on brands, quality, budget, etc), and (ii) they merely evaluate the products

considered in the choice set using a utility comparison process and opt for the product with the maximum

utility (if any). We formulate the dynamic pricing problem as a discrete-time Markov decision process.

Assuming a linear utility for the consumers, we first derive the probability that a consideration set

is chosen and then we find the purchase probability of products. Finally, we examine the structural

properties of the firm’s revenue function and pricing decisions.

Chapter 2

Dynamic Pricing Under Consumer’s

Sequential Search

2.1 Introduction

Consumers often spend considerable time and effort studying alternatives before purchasing a prod-

uct. Assuming a positive search cost, the consumers face both a search problem (i.e., comparing the

alternatives), and an optimal stopping problem. Search theory, pioneered by Stigler (1961), has been

studied in both macroeconomic and microeconomic contexts. For example, macroeconomists have used

search theory to model frictional unemployment resulting from a job search whereas microeconomists

have applied it to investigate consumers’ choice behavior among a set of firms offering a certain product.

In the operations and marketing contexts, search theory is focused on the consumers seeking the

option with the maximum utility alternative among a set of discrete alternatives (Ratchford 2009). If

the search is sequential, after each inspection, the consumer decides whether to stop or inspect another

item. The decision depends on the search cost, which arises from the time and effort needed to assess

the quality and price of a product, and compare them with those of other products. The emergence of

the internet has substantially reduced search costs, but they can still be significant. For example, Hann

and Terwiesch (2003) estimated a cost of approximately $5 per search on the internet. As a result, a

consumer may inspect only a subset of available products before purchasing a product. In their study of

camcorders sold on Amazon.com, Kim et al. (2010) showed that the median (average) search set contains

11 (14) products, with about 40% of consumers stopping their search at fewer than five products (out

of a possible 90 total products). As such, it would seem that the common assumption in the revenue

3

Chapter 2. Dynamic Pricing Under Consumer’s Sequential Search 4

management literature that consumers evaluate all available products (at the same time) is unrealistic.

In addition, extensive literature suggests that consumers not only search sequentially, but also tend to

inspect products in the order presented to them (see, e.g., Armstrong and Zhou 2011, Zhou 2011, Rhodes

2011, Armstrong et al. 2009). For example, Granka et al. (2004) showed that consumers tend to inspect

products from the top to the bottom of a web page. Another example is the assisted search, in which the

seller assists consumers during the search process and tries to influence their search sequence, whenever

possible, in order to maximize profit. An assisted search is common in airline seat selection, sales of

luxury products, and real estate sales (e.g., Dellaert and Haubl 2012).

In this paper, we focus on a general operational setting in which a firm offers a line of vertically

differentiated perishable products with limited inventories over a finite sales season. Products are said

to be vertically differentiated if they can be ordered according to their objective quality. With equal

prices, consumers always prefer products of higher quality, but higher quality products generally have

higher prices, leaving the consumers to make a trade-off among these competing factors. Consumers

arrive at the firm randomly and inspect products sequentially until they find a product to purchase (if

any). Each consumer incurs a positive cost to inspect a product and hence may stop the sequential

search without inspecting all the available items. Upon inspection, the utility of the product is known to

the consumer, who then decides whether to continue the search. The firm wishes to maximize the total

expected revenue from the existing inventories by determining the optimal sequence in which to show the

products to the consumers, as well as the optimal price of each product at each time. We formulate the

firm’s optimization problem as a Markov decision process and analytically derive the products’ purchase

probabilities, and their optimal prices given their remaining inventories and time.

This paper makes four main contributions as follows: First, we are the first to explicitly take into

account consumers’ sequential search in a firm’s operational decisions such as the optimal dynamic

pricing and sequencing. An important aspect of this study is the effect of possibility that the consumers

may not inspect all the available products, which leads to many counterintuitive insights. Second, based

on this model, we analytically show that it is optimal for a firm to sequence products in a decreasing

order of product quality. This result is consistent with recent experimental studies (see, e.g., Suk et al.

2012), which studied the influence of products’ presentation order on the consumers’ purchase decision.

Third, we derive a closed-form solution for the optimal products’ prices in each planning period and

for each inventory state. Four, we demonstrate that a perishable product’s optimal price can increase

over time. This is a result, which is in stark contrast to the common expectation by the classic dynamic

pricing models (see, e.g., Zhao and Zheng 2000 and Gallego and van Ryzin 1994). As the products

are perishable, their prices are typically expected to decrease over time. However, as in the sequential


search, a consumer may not be able to see all the products, it can result in the opposite price behavior.

Based on these results, we caution that it is critical to consider consumers’ sequential search behavior,

when designing dynamic pricing and sequencing algorithms; otherwise, the suggested solutions can be

significantly suboptimal.

The rest of the paper is organized as follows. Section 2.2 provides an overview of the relevant

literature; followed by Section 2.3, which describes the model formulation. Section 2.4 discusses the

firm’s sequencing and dynamic pricing problems, while Section 2.5 presents some extensions to the

dynamic pricing with the sequential search problem. Section 2.6 concludes the paper and presents

directions for future research.

2.2 Related Literature

The dynamic pricing problem has been one of the most studied in the literature of revenue management.

We classify finite horizon dynamic pricing literature into three categories: single product, multi-product,

and pricing along with other operational decisions. In the single product category, Bitran and Mond-

schein (1993) and Gallego and van Ryzin (1994) developed a dynamic pricing model over a finite selling

season and showed many structural properties of the optimal pricing policy. Developing a partially

observed Markov decision process framework, Aviv and Pazgal (2005) studied a dynamic pricing prob-

lem and provided a heuristic pricing policy. Popescu and Wu (2007) studied a pricing problem where

consumers are assumed to be reference dependent and loss averse. Ahn et al. (2007) considered a pric-

ing problem in which consumers decide to purchase a product based not only on the current price,

but also on the product’s price in past and future periods. den Boer and Zwart (2015) addressed a

dynamic pricing problem with parametric uncertainty on the demand distribution. Papanastasiou and

Savva (2016) investigated how the strategic interaction between dynamic pricing and a forward-looking

consumer population is affected in the presence of social learning. Comprehensive literature reviews on

the dynamic pricing are available in Ozer and Phillips (2012), Phillips (2005), Talluri and van Ryzin

(2004a), Bitran and Caldentey (2003), Elmaghraby and Keskinocak (2003), and McGill and van Ryzin

(1999).

In the multi-product pricing context, Gallego and van Ryzin (1997) developed heuristic approaches

to optimally pricing multiple products over a finite horizon and showed that the heuristic approaches

are asymptotically optimal. Talluri and van Ryzin (2004b) and Zhang and Cooper (2005) considered

a customer’s choice model in the booking limit policies for airline revenue management. Dong et al.

(2009) used the multinomial logit choice model in the dynamic pricing of horizontally differentiated


products. Levin et al. (2009) addressed a dynamic pricing model for oligopolistic firms selling multiple

products to strategic customers who adjust their purchasing time. Akcay et al. (2010) considered the

dynamic pricing problem of multiple substitutable products under consumers’ choice set. den Boer

(2014) studied a multi-product dynamic pricing problem with infinite inventories where the demand for

each product depends on the offered price and some parameters unknown to the firm. Chen et al. (2016)

investigated a multi-product and multi-resource revenue management problem and developed a set of

heuristics providing minimal and flexible price adjustments. Li and Jain (2016) addressed the impact of

behavior-based pricing (i.e., price discrimination among consumers based on preferences revealed from

their purchase histories) on the firm’s revenue and the social welfare. Du et al. (2016) considered a multi-

product pricing problem in which consumers choose to buy a product based on its total consumption in

the market.

There is a wide range of literature on dynamic pricing in the presence of other operational decisions.

Federgruen and Heching (1999) and Chen et al. (2011) integrated pricing and inventory replenishment

policies of a single product to maximize the total profit. Maglaras and Meissner (2006) explored the

relationship between dynamic pricing and capacity control for a firm utilizing a single resource to manu-

facture multiple products. Bitran et al. (2005) extended the budget-constrained choice model in Hauser

and Urban (1986) to a continuous-time model and determined the optimal prices of a vertical assortment.

Aydin and Porteus (2008) determined the optimal prices and inventory policies of multiple products in a

given assortment in a newsvendor model. In the supply chain context, Chu et al. (2013) investigate both

pricing and capacity decisions. Lu et al. (2014) proposed a quantity-based pricing strategy in which,

at each time period, inventory replenishment, unit selling price, and the quantity-discount price are ob-

tained. Borgs et al. (2014) studied the pricing problem of a service firm where capacity levels vary over

time and all paying customers are guaranteed to receive the service. Federgruen and Hu (2015) studied

a general price competition model for a set of substitutable products and ascertained the equilibrium

prices, product assortment, and sales volumes. Alptekinoglu and Semple (2016) investigated the use of

a canonical version of a discrete choice model in joint pricing and assortment planning.

Our paper is unique among all the above-mentioned streams of literature in two respects. First,

unlike all the mentioned works, we consider consumers’ sequential search behavior in the dynamic pricing

problem, which leads to several significantly different insights. Second, unlike these works, we study the

optimal products’ sequence that the firm wishes to present to the consumers to maximize its revenue.


2.3 Consumer Model

We consider a firm selling n vertically substitutable products to consumers over a finite selling season

(e.g., different hotel rooms to be sold by a certain date through Booking.com). At the end of the selling

season, the products are perished (e.g., hotel rooms, or airline seats). We use I = 1, 2, . . . , n, n+ 1 to

denote the entire set of the available products where product (n + 1) is a virtual product representing

the consumer’s no-purchase option. The firm starts the selling season with the initial inventory ki for

product i and is unable to replenish inventory during the selling season. Each unit of product i has a

fixed quality ψi with ψl > ψk, for all l < k ∈ I. The selling season is divided into T periods, each

indexed by t ∈ T, T − 1, · · · , 0. The time index runs backwards in time. Thus, time t = T is the start

and t = 0 is the end of the selling season. We assume that in each period there is at most one consumer’s

arrival, and each consumer requires at most one unit of inventory (see, e.g., Du et al. 2016, Akcay et al.

2010, Suh 2010, and Talluri and van Ryzin 2004b for a similar setting).

2.3.1 Sequential Search and Utility

A consumer arrives at the firm at time t with probability λt and inspects the products one at a time

sequentially according to a given search sequence. The consumer incurs a cost (or loss of utility) c > 0

for inspecting an additional product in the form of the time and mental efforts required to inspect the

product’s features. As a result, the consumer may decide to stop the search before inspecting all the

products. At the end of the search, the consumer either chooses one of the inspected products, or leaves

the store without any purchase. Upon inspection of product k at time t, its quality, ψk, and price, pkt,

are revealed to the consumer; and hence, he learns product k’s utility ukt as:

ukt = νψk − pkt, k ∈ I, (2.1)

where ν is the consumer’s heterogeneous type (or taste), which is a uniform random variable over [0, 1].

We assume that the no-purchase decision yields a zero utility, i.e., un+1,t = 0, 0 ≤ t ≤ T .

Two comments are warranted with respect to the way we model the utility function (2.1) as well as the

search sequence used by the consumers. First, our way of modeling the utility function (2.1), is consistent

with several works in the management science, marketing, and economics literature (see, e.g., Bhargava

and Choudhary 2008, Berry and Pakes 2007, Hensher and Greene 2003, Train 2003, Wauthy 1996, Berry

1994, Roberts and Lilien 1993, Anderson et al. 1992, Caplin and Nalebuff 1991, Tirole 1988, Bresnahan

1987). In particular, several works in these streams of literature consider a uniform distribution for ν


in order to analytically capture the heterogeneity among consumer types1 (see, e.g., Akcay et al. 2010,

Bresnahan 1987). Second, regarding the search sequence, although in reality consumers can choose

any search sequence, in the rest of this paper, for analytical clarity, we assume that they inspect the

products in a descending order of quality (i.e., they first observe product 1, then product 2, and so on).

Nevertheless, in Section 2.5.3, we show that the optimal products’ prices obtained in this paper hold

for any arbitrary search sequence chosen by the consumers (see Proposition 6). In addition, in Section

2.4.1, we show that, under certain conditions, observing the products in a descending order of quality is

optimal for the firm’s revenue (see Proposition 1). Thus, whenever possible, the firm optimally induces

the consumers to inspect the products from the highest to the lowest quality levels.

At each stage of the search process, the consumer makes a trade-off between the search cost c > 0 and

the potential extra utility gained by continuing the search. More specifically, having inspected product

j, the consumer faces two decisions: (i) which of the products observed thus far is deemed to be the best

(comparison decision); and (ii) whether the consumer should continue or stop the search given the search

cost c and the observed utilities u1t, u2t, . . . , ujt (stopping decision). In the comparison decision, the

consumer updates his most desired product by comparing the utility of the currently inspected product,

ujt, with the maximum utility of all the previously inspected products umaxj−1,t = maxu1t, u2t, . . . , ujt.

That is,

umaxjt = maxujt, umax

j−1,t, j = 2, . . . , n, (2.2)

where umax1t = u1t. In the stopping decision, the consumer compares the search cost c with the expected

increase in maximum observed utility if he continues the search one step further. Indeed, Rothschild

(1974) showed that consumer’s one-step look-ahead policy is optimal. That is, the consumer stops the

search if c exceeds the expected gain in the maximum observed utility due to inspecting one additional

product. Let the utility of the unobserved product (j + 1) be a random variable from the arbitrary

distribution Fj+1 on the domain [Umin, Umax] for the given values Umin < Umax. The values Umin and

Umax may be viewed as the worst and best utilities that the consumer expects to see by inspecting a

product. Then, for any observed value umaxjt , the consumer decides to continue the search as long as

c <

∫ Umax

umaxjt

(U − umaxjt )dFj+1(U), (2.3)

where the right-hand side indicates the expected utility-increase due to inspecting product (j + 1) (for

the detailed proof of (2.3), see Rothschild 1974).

1For example, product k may have a high quality, but the consumer may not like it; alternatively, it may have a lowquality, but the consumer may generally like its concept.


2.3.2 Reservation Utility

As can be seen from Equation (2.3), the left side is constant while the right side is non-increasing in the

number of inspected products, j. Thus, there exists a cut-off utility rj such that when umaxjt = rj both

sides become equal:

c =

∫ Umax

rj

(U − rj)dFj+1(U). (2.4)

That is, the consumer becomes indifferent between the decision to stop or continue the search. Following

Rothschild (1974), we call this cut-off value the consumer’s reservation utility for product j. Intuitively,

rj is the maximum utility that the consumer expects to see by continuing the search. Thus, when

umaxjt ≥ rj , the consumer stops, as he does not expect to see any utility higher than umax

jt if he continues

the search.

Two different ways exist in the literature for deriving the consumer’s reservation utility, rj , based on

how the sequential search is conducted. The most common search types considered in the literature are

the static search, and the adaptive search (e.g., Kundisch 2012, Kohn and Shavell 1974). In the static

search the consumers start the search with a prior belief about the products’ utility distribution. By

definition, there is no learning involved at any stage of this type of search. In the adaptive search, the

consumers learn from the utilities of the products inspected so far, by updating their belief about the

products’ utility distribution at each stage of the product search. We refer to the reservation utilities

associated with the static and adaptive searches as (i) stationary reservation utility, and (ii) adaptive

reservation utility, respectively.

(i) Stationary (static) reservation utility The static search is widely considered in the economics

and marketing literature (see, e.g., Moraga-Gonzalez and Petrikaite 2013, Armstrong et al. 2009, Wolin-

sky 1986, Kohn and Shavell 1974). The main reason for the wide consideration of the static search is that

in several actual search settings, the consumer cannot learn and update the products’ utility distribution

meaningfully due to the substantial variability in the products’ observed utilities. In order to obtain

the reservation utility for this type of search, consistent with several recent works, we assume the utility

distribution of product (j + 1) uninspected thus far, Fj+1, is uniform over the range [Umin, Umax] (see,

e.g., Moraga-Gonzalez and Petrikaite 2013, Hagiu and Jullien 2011, Zhou 2011, Haubl et al. 2010, Arm-

strong et al. 2009, and Wolinsky 1986 for a similar assumption). Thus, replacing Fj+1(U) = U−Umin

Umax−Umin

in (2.4) and solving for rj , we can find the stationary reservation utility, which is given by Lemma 1.

Lemma 1. Let the consumer’s belief regarding the distribution of the uninspected products’ utilities in


the market be uniform over [Umin, Umax]. Then the consumer’s reservation utility rj is given by

rj= Umax −√

2c(Umax − Umin), j = 1, 2, ..., n. (2.5)

(ii) Adaptive reservation utility In order to obtain the reservation utility for the adaptive search,

we incorporate a learning process into the reservation utility’s structure by modeling the sequential search

process as a Bayesian learning with a uniform prior distribution over [Umin, Umax] (see, e.g., Haubl et al.

2010 and Bikhchandani and Sharma 1996)2. The following lemma gives the updated utility distribution

after inspecting j products as well as the consumer’s reservation utility at stage j.

Lemma 2. Assume that consumers’ belief regarding the distribution of uninspected products’ utilities

in the market is updated using Bayesian learning with a uniform prior on [Umin, Umax]. In addition,

suppose the consumers have inspected products 1, 2, ..., and j. Then their updated belief about the utility

density function for product j + 1 is

fj+1(U) =dFj+1(U)

dU=

W

(Umax − Umin)W + j, (2.6)

where W > 0 is the weight of the uniform prior at the beginning of the search in the sense that a higher

W corresponds to less updating of the prior during the search. In addition, the consumer’s reservation

utility at stage j is

rj = Umax −√

2cW (Umax − Umin) + j

W. (2.7)

The consumer stops the search as the utility of the best product observed thus far exceeds the

potential maximum utility he can see if he continues the search (i.e., umaxjt ≥ rj), and purchases the best

product with the utility umaxjt . We note that (2.5) is a special case of (2.7) in which the consumer either:

(i) considers an overly high weight on the prior (i.e., W → ∞), or (ii) is overly uncertain about the

possible utilities unobserved yet, even after inspecting j products (i.e., (Umax − Umin) >> j). In both

cases, (2.7) reduces to (2.5), which implies that updating (learning) has little impact on the consumer’s

reservation utility. Figure 2.1 shows the summary of the steps that consumers take during their sequential

search.

In addition, some recent works in the literature suggest that the search cost might vary during the

2We note that, although the consumer’s real utility is derived from the random linear function given by (2.1), theconsumer still needs considerable time and effort to learn the product’s price and the quality information, as well asrealize how far he likes it (his type). For this reason, the works in the economics and marketing literature, which studysequential search mostly assume that, at the start of the sequential search, the consumer does not really know anythingabout the products’ utilities except for the utility of the best and worst products that might be available for purchase.Thus, the consumer starts with a uniform prior distribution, but may update and modify it along the search, with eachutility observed by a new product inspection. Nevertheless, the linear utility (2.1) is used by the firm to predict consumers’utilities before their arrivals, as the firm, unlike the consumers, knows all the price and quality values beforehand.


Figure 2.1: Summary of the steps that consumers take during their sequential search.

search process (see, e.g., Koulayev 2014, Yao and Mela 2011, Wildenbeest 2011, Kim et al. 2010,Mehta

et al. 2003). In particular, as a consumer inspects more products, he tends to become more skillful in

searching and appraising the products. Thus, the search cost tends to decrease (e.g., Carlin and Ederer

2012). Related to these works, we can extend our sequential search model to take into account a different

search cost, cj , for each product j. Then in that case, we can easily show that the relations (2.5) and

(2.7) still remain the same, with the only difference that c is replaced with cj+1 in them. The next

proposition shows that, if after each inspection, the decrease in the search cost cj is sufficiently large,

then rj becomes increasing in the number of the searched products.

Proposition 1. The consumer’s reservation utility rj is increasing in j (i.e., rj+1 ≥ rj) if

cj+1

cj≤ 1− 1

W (Umax − Umin) + j, j = 1, 2, . . . , n− 1. (2.8)

As previously mentioned, upon inspecting product j, the consumer stops the search to buy product

k (1 ≤ k ≤ j) if k is both the best product observed thus far (i.e., umaxjt = ukt), and its utility exceeds

rj (i.e., ukt ≥ rj). That is:

(umaxjt = ukt

)and

(umaxjt ≥ rj

), j = 1, 2, . . . , n. (2.9)

The next proposition shows that if rj is stationary (as in the static search) or increasing in j, then the


consumer only buys product j, if he stops the search right after observing it (i.e., umaxjt = ujt). In other

words, the consumer does not apply the so-called recall option (i.e., he does not purchase any of the

previously observed products).

Proposition 2. If rj is stationary or increasing in j, then the optimal decision of a consumer who is

inspecting product j is either to continue the search or buy product j.

We note that, in addition to the case where the consumers do not apply the recall option due to

having a stationary or increasing reservation utility, in reality, there are two other scenarios where

the recall option is not applied by the consumers (even when the reservation utility is not stationary

or increasing). The first scenario is when a seller might use special sales techniques to prohibit the

recall option (Armstrong and Zhou 2010, Kohn and Shavell 1974). For example, a common method for

encouraging quick sales is an exploding offer where the consumers are forced to decide quickly whether

to buy or not. Specifically, the seller refuses to sell to the consumers, unless they buy the currently

observed product (Cialdini 2000). The second scenario is when the seller allows the recall option, but

consumers might not be able to return to the previously inspected products. This scenario frequently

occurs when the demand for products is overly high but supply is limited (e.g., searching for products

on Boxing Day, or during Black Friday Sales).

2.3.3 Purchase Probability

Let pjt = (p1t, p2t, ..., pjt) denote the price vector of the j products observed thus far at time t. In

addition, let ρk(pjt) be the probability that a consumer buys product k (1 ≤ k ≤ j) after inspecting j

products given the price vector pjt. In the rest of this section, we focus on the case where rj is stationary

or increasing. However, in Section 2.5.2, we consider the alternative case where rj can become decreasing

as well. We know that when rj is stationary or increasing, the consumer does not apply the recall option

(Proposition 2). Therefore, at stage j, any previously inspected product k < j has no chance of being

purchased. Hence,

ρk(pjt) = 0, k = 1, 2, ..., j − 1, (2.10)

and the consumer stops and buys j only if ujt ≥ rj . Thus, using (2.1), we can characterize the consumer’s

purchase probability ρj(pjt) as:

ρj(pjt) = P(ujt ≥ rj) = Pv ≥ rj + pjt

ψj

= 1− pjt + rj

ψj, j = 1, 2, . . . , n. (2.11)


From (2.11), we note that the purchase probability ρj(pjt) is only a function of pjt and not any of

the other prices p1t, . . . , pj−1t (i.e., ρj(pjt) = ρj(pjt)). This property is important as it simplifies the

complexity of the optimal dynamic pricing and sequencing problem (detailed later). We assume that

pjt ≤ ψj − rj to ensure that purchase probability in (2.11) is always positive (i.e., ρj(pjt) ≥ 0). In

addition, as when a product has no inventory (i.e., xj = 0), it has a zero purchase chance, we set its

price pjt = ψj − rj so that ρj(pjt) = 0.

2.4 The Firm Optimization Problem

As mentioned before, we assume that the products are inspected in a descending order of quality.

However, in section 2.4.1, we show that this sequence is indeed optimal for the firm’s profitability. The

firm wishes to maximize the total expected revenue over a finite horizon by optimally sequencing and

pricing products given the inventory level x = (x1, x2, · · · , xn). Let Vjt(x) denote the optimal expected

revenue given the inventory state x, from products j, j+ 1, . . . , n, and from periods t, t− 1, . . . , 1. When

the reservation utility is stationary or increasing in j, the optimality equations are formulated as follows:

Vjt(x) =

maxp1tλtρ1(p1t)(p1t + V1t−1(x− e1)) + λt(1− ρ1(p1t))V2t(x) + (1− λt)V1t−1(x) , if j = 1,

maxpjtρj(pjt)(pjt + V1,t−1(x− ej)) + (1− ρj(pjt))Vj+1,t(x) , if 2 ≤ j ≤ n− 1,

maxpnt

ρn(pnt)(pnt + V1,t−1(x− en)) + (1− ρn(pnt))V1,t−1(x) , if j = n,

(2.12)

where V1t(x) (i.e., the first line of the optimality equations given in (2.12)) is the seller’s total expected

revenue. In addition, the boundary conditions are as follows:

V10(x) = 0, ∀ x, (2.13)

V1t(0) = 0, ∀ t = T, · · · , 1. (2.14)

The optimality equations (2.12) are given in three scenarios: (1) A consumer may arrive in period t

with probability λt. Upon arrival, the consumer inspects product 1. If no consumer arrives in period t

(with probability (1 − λt)), the seller will be carried over to period (t − 1) with the expected revenue

V1t−1(x); (2) The consumer, who is already in the store and has just inspected products 1 to (j − 1),

evaluates product j (2 ≤ j ≤ n− 1); and finally (3) The consumer, who is already in the store and has

just inspected products 1 to (n − 1), evaluates product n. In each of these three scenarios, the related

optimality equation addresses two possibilities: (i) the consumer buys product j with probability ρj(pjt)


or (ii) he does not buy with probability (1 − ρj(pjt)), and considers product (j + 1). The inventory

levels are adjusted accordingly. Notice that for the scenario j = n, the consumer will leave the store

rather than inspecting an additional product. The boundary condition (2.13) ensures that the perishable

products are salvaged at zero at the end of the selling horizon, and (2.14) is self-explanatory. Next, we

address the optimal sequencing problem and its properties, and in Section 2.4.2, we find the optimal

dynamic pricing strategy.

2.4.1 Optimal Sequencing Strategy

Thus far, we have assumed that the consumers inspect the products in a decreasing order of quality. This

assumption was originally made for analytical convenience. In this section, we show that this sequence

of product presentation is indeed optimal for the firm. The next theorem shows this result.

Theorem 1 (OPTIMAL SEQUENCE). Let rj be stationary or increasing in j. Then displaying the

products in a decreasing order of quality is optimal for the firm.

Theorem 1 is intuitive because the firm wishes to maximize its total revenue, while higher-quality

products have higher prices. Thus, the firm can increase its revenue by selling higher-quality products

first. In addition, as the consumer might stop the search after observing only a subset of the products,

those products inspected earlier naturally have a higher chance of being purchased.

An important benefit of Theorem 1 is simplifying the analysis of the firm’s optimization problem:

As the pricing and the sequencing decisions are intertwined (because if the search sequence changes,

then the optimality equations in (2.12) change, which would change the optimal prices accordingly), we

have to optimize the price and sequence of products simultaneously, which is a very difficult task both

analytically and computationally. However, using Theorem 1, we can get rid of the optimal sequencing

problem (by merely considering the optimal sequence), and focus only on finding the optimal prices. We

illustrate Theorem 1’s optimal sequencing result through the following example.

Example 1. Let the firm offer three products 1, 2, and 3 with qualities ψ1 = 10, ψ2 = 7, and ψ3 = 5,

respectively. In addition, let λt = 0.8, c = 0.2, Umax = 2, and Umin = −0.5. All the values are chosen

only for the purpose of illustration. Given these values, the stationary reservation utility becomes rj = 1.

We denote a possible search sequence with the vector m = (k, h, l) (i.e., product k is inspected first,

then product h, and finally product l). Figure 2.2 shows the firm’s optimal revenue over time, when the

products are searched using different sequences. As can be seen from the figure, the sequence m = (1, 2, 3)

(i.e., products inspected in the descending order of quality) is optimal for the firm as it generates the

highest revenue. Meanwhile, the minimum generated revenue belongs to the sequence m = (3, 2, 1) (i.e.,


3 4 56

7

8

9

10

11

12

Remaining time

Optim

al r

eve

nue V

1t(x

)

m = (1,3,2)

m = (2,3,1)

m = (2,1,3)

m = (3,1,2)

m* = (1,2,3)

m = (3,2,1)

Figure 2.2: The firm’s optimal revenue for different search sequences m = (k, h, l) where product k isinspected first, then product h, and finally product l)

searching in the increasing order of quality).

Practical Evidence We have also checked whether real firms use the sequencing strategy proposed

by Theorem 1. Here, we mention two practical examples where firms are advised to present their products

in a descending order of quality.

The first example is the work by Suk et al. (2012), who examined the influence of products’ sequence

on customers’ choices and the firm’s profitability. Suk et al. (2012) realized that sellers can influence

consumers to choose a more expensive option if they present products in a descending price and quality

order. They conducted a field study for eight consecutive weeks in a pub that offered bottled beers to

consumers. Suk et al. (2012) created two versions of the beer menu: one with beers in an increasing price

(quality3) order, and one with exactly the same beers but in a decreasing price (quality) order. Over the

eight-week period, the price menus were frequently alternated. The authors concluded that the average

price and profitability of the purchased beers were higher when the menu was arranged in the decreasing

price (quality) order. In addition, the authors investigated the validity of this result by running the

experiment for a variety of other product types (e.g., hotel rooms), and in various research settings

(field, lab, and online), concluding that in all cases considered, offering the products in a descending

price (quality) order tended to be more profitable. While Suk et al. (2012) suggested this result might be

due to consumers’ loss averse behavior, Theorem 1 provides a theoretical explanation based on dynamic

pricing with the consumer’s sequential search.

The second example relates to the recommendation made by Kissmetrics, a San Francisco-based

3The higher-quality beers had higher prices.


customer analytics firm4. Kissmetrics recommends that sellers show their products in a descending

order of quality by stating that if the first price that consumers see is very expensive, they may be

more delighted to see the less expensive price that follows. While Kissmetrics uses price-anchoring as

a possible psychological reason for why firms benefit from showing the products in a decreasing order

of quality, our theoretical result based on dynamic pricing with consumers’ sequential search, provides

an alternative explanation by showing that sequencing the products in a decreasing order of quality,

maximizes the firm’s expected revenue.

The next proposition gives the results for the firm’s optimal revenue when there is a long time

remaining until the end of the sales horizon.

Proposition 3. If rj is stationary or increasing in j and t is sufficiently large, then Vlt(x) = Vkt(x).

To show this result, we note that, when t → ∞, then the price of product j approaches the upper

bound value p∗jt = ψj − rj , for which ρj(p∗jt) = 0. Thus, when t→∞, the optimality equations given by

(2.12) are simplified to:

Vjt(x) =

λtV2t(x) + (1− λt)V1t−1(x), if j = 1,

Vj+1t(x), if j = 2, · · · , n− 1,

V1t−1(x), if j = n.

(2.15)

From (2.15), it is clear to see that, when t→∞, Vjt(x) = Vj+1t(x) = V1t−1(x), which implies Vlt(x) =

Vkt(x) for all k, l ∈ I .

The result of Proposition 3 is twofold: It says that when there is a sufficiently long time until the end

of the sales horizon, then the firm’s optimal revenue will no longer depend on (i) the number of products

a consumer might observe before making a purchase decision, and (ii) the sequence he might choose to see

the products. The reason for these two observations is that, no matter how many products a consumer

inspects or what search sequence he chooses, the firm’s expected optimal revenue is unchanged as it has

enough time to sell its products (i.e., today, tomorrow and yesterday are all equivalent for the firm for

selling its products). We illustrate these results through the following example.

Example 2. We let the firm offer two products 1 and 2 with the quality levels (ψ1, ψ2) = (10, 6),

and (x1, x2) = (2, 2). In addition, we let the values of λt, Umax, Umin, and c be the same as those in

Example 1. As before, we assume the reservation utility is stationary. Figure 2.3 compares the firm’s

optimal revenue when j = 1 (a consumer is inspecting product 1) with its optimal revenue when j = 2

4www.kissmetrics.com


1 2 3 4 5 6 7 8 90

5

10

15

Remaining time

Optim

al r

eve

nue V

jt(x)

V2t

(2,2)

V1t

(2,2)

Figure 2.3: The impact of the remaining time on the insepected products size where only product 2 isinspected (i.e., V2t(x)) and both products 1 and 2 are inspected (i.e., V1t(x))

(a consumer has already inspected product 1 and is now inspecting product 2) over the remaining time.

From the figure, we can see that V1t(2, 2) and V2t(2, 2) converge to each other as the remaining time

increases. Thus, when there is a long time until the end of the sales horizon, it does not matter much

whether a consumer sees only product 1 or both products 1 and 2.

2.4.2 Optimal Pricing Strategy

In this section, we address the firm’s optimal dynamic pricing policy with the optimal search sequence

given by Theorem 1. Note that generally obtaining the firm’s optimal expected revenue, V1t(x), in (2.12)

is a very difficult task because V1t(x) becomes an (n+1)st degree multi-variable polynomial function the

optimization of which (even for the simplest case of n = 1) is NP-hard (Floudas and Visweswaran 1995).

However, when the reservation utility is stationary or increasing, the complexity of the optimization

problem is substantially reduced as it can be solved using a sequence of nested backward inductions.

In other words, we backward induct the optimality equations from the last period t = 1 toward t = T

and in each period from j = n to j = 1. The reason that the nested backward induction works well

is that, as we will show in the proof of Proposition 2, when rj is stationary or increasing, the firm’s

revenue function in (2.12) would have a nested optimal substructure in both time and products. Thus,

the optimal expected revenue when a consumer inspects product j, Vjt(x), can be obtained efficiently

from the optimal expected revenue when he inspects the subsequent product, Vj+1t(x), and the expected

revenues at time (t− 1).


Theorem 2 (OPTIMAL PRICE). If rj is stationary or increasing in j, then:

(i) Vjt(x) is strictly concave in pjt. That is,∂2Vjt(x)

∂p2jt< 0.

(ii) The optimal price of product j at time t with the inventory level xj > 0 is:

p∗jt(x) =

1

2(ψj + Vj+1t(x)− V1t−1(x− ej)− rj), 1 ≤ j ≤ n− 1,

1

2(ψj + V1t−1(x)− V1t−1(x− ej)− rj), j = n.

(2.16)

An interesting observation about the optimal price function is its equivalence to the obtained price

for a market in which all consumers have a deterministic valuation vd = ρj(pjt(x)) = 1−pjt(x)+rjψj

.

In that deterministic market, when a consumer buys product j his utility is ujt = ψjvd − pjt(x) =

ψj(1−pjt(x)+rjψj

) − pjt(x). Thus, the price is obtained such that the consumer’s purchase utility is

equal to product j’s marginal value Vj+1t(x) − V1t−1(x − ej). That is, ψj(1−pjt(x)+rjψj

) − pjt(x) =

Vj+1t(x)− V1t−1(x− ej), which gives the same price as in (2.16). This observation is surprising as the

two markets are considerably different. However, the observation implies that it is like the firm optimally

getting rid of the randomness in ν, by assuming that all consumers have the same deterministic valuation

vd = ρj(pjt(x)).

The other observation is that, as can be seen from (2.16), there is a negative relationship between

product j’s optimal price, p∗jt(x), and the consumer’s reservation utility, rj . The reason is that a higher rj

means that the consumer expects to gain a greater utility by continuing the search; thus, the probability

of stopping and buying product j is decreased. In response to the reduced purchase chance, the firm

optimally lowers the price.

Next, we note that when a product is perishable, its price is typically expected to decrease as the

time approaches the end of the sales season (e.g., Akcay et al. 2010, Zhao and Zheng 2000, Bitran

and Mondschein 1997, Gallego and van Ryzin 1994, Bitran and Mondschein 1993). However, the next

proposition shows that, with consumers’ sequential search, it is possible for the optimal price of a

perishable product to increase over time.

Proposition 4 (OPTIMAL PRICE CAN INCREASE OVER TIME). Let ∆t[Vk(x)] be the change in

the firm’s value function of product k between times t and t− 1, that is, ∆t[Vk(x)] = Vkt(x)− Vkt−1(x).

Then product j’s optimal price is decreasing in t, i.e., p∗jt(x) ≤ p∗jt−1(x), if:

∆t[Vj+1(x)] ≤ ∆t−1[V1(x− ej)]. (2.17)

The price increase behavior is interesting because it goes against the typical expectation by the classic

dynamic pricing models, which show the optimal price decreases over time. The reason for this behavior


1 2 3 4 5 6 7 8 9

5

5.5

6

6.5

7

Remaining time

Optim

al p

rice

of pro

duct

1

x = (4,2)

x = (3,1)

x = (2,1)

x = (1,2)

x = (4,1)

Figure 2.4: Optimal price of product 1 withvarious inventory levels where the search se-quence is m = (1, 2)

1 2 3 4 5 6 7 8 9

2.6

2.8

3

3.2

3.4

3.6

3.8

Remaining time

Optim

al P

rice

of pro

duct

2

x = (4,1)

x = (3,1)

x = (2,1)

x = (1,2)

x = (4,2)


is that, as the search is sequential, the consumers might stop before seeing all the available products.

Therefore, the firm should make a trade-off between two opposing forces: (i) the expected revenue-

increase by selling product j at time t (i.e., Vj+1t(x)−V1t−1(x−ej)) and (ii) the expected revenue-increase

by not selling product j at time t and deferring it to one period later (i.e., Vj+1t−1(x)− V1t−2(x− ej)).

If the firm expects a higher revenue-increase by keeping product j and encouraging the consumers to see

the other products, then it optimally increases its price to discourage them from stopping and buying the

product. In other words, the firm does not sell product j (by appropriately raising its price), because

it expects to gain a higher revenue from the subsequent products. Note that the consumer will be

endogenously forced to continue to check the next product if the current product is priced very highly.

Thus, if the price of a specific product goes up, the utility of that product decreases and, accordingly,

the chance that the product’s utility exceeds the customer’s reservation utility decreases. Hence, the

consumer becomes more willing to continue the search. We illustrate this behavior through the following

two examples.

Example 3. Consider a firm selling two products over a horizon of T = 9 periods. The values of

ψ1, ψ2, λt, Umax, Umin, and c are the same as in Example 1, and the search is stationary. Figure 2.4 and

Figure 2.5 show the optimal prices of the products when the search sequence is m1=(1, 2), i.e., product

1 is observed first. From the figures, we can see that the optimal price of product 2 decreases as the

remaining periods decrease from 9 to 1, while the optimal price of product 1 can decrease or increase

depending on its inventory state and the remaining time. Figures 2.6 and 2.7 show the optimal prices


1 2 3 4 5 6 7 8 9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

Remaining time

Optim

al p

rice

of pro

duct

1

p1t* (3,2)

p1t* (2,1)

p1t* (3,1)


1 2 3 4 5 6 7 8 9

4.5

5

5.5

6

6.5

7

Remaining time

Optim

al p

rice

of pro

duct

2

p2t* (2,1)

p2t* (3,1)

p2t* (3,2)


of the products when the search sequence is altered to m2 = (2, 1), i.e., product 2 is observed first. As

can be seen from the figures, product 2’s optimal price increases toward the end of the horizon.

Example 4. Suppose that a firm offers four products with the quality levels ψ1 = 19, ψ2 = 13, ψ3 = 10,

and ψ4 = 6 over T = 6 periods. The initial inventory states are x = (3, 1, 1, 1), (1, 3, 1, 1), (1, 1, 3, 1)

and (1, 1, 1, 3). The values of λt, Umax, Umin, and c are the same as in Example 1, and the search is

stationary. Figures 2.8-2.11 show the optimal prices of the products with respect to the remaining time.

As can be seen from the figures, the price of the product with the highest initial inventory (unlike other

products) starts to increase strictly near the end of the horizon (except product 4 in Figure 2.11, whose

price is almost fixed). In addition, we have provided an additional example in the Appendix of this

paper (Extra Supplement Section), where we study the firm’s revenue improvement with an increase in

the products’ initial inventories. There, we show that the firm’s revenue-increase is gradually decreasing

from the first to the last product in the search sequence. In other words, the pricing of earlier products

in the search sequence has a greater impact on the revenue because the consumers are less likely to

observe the subsequent products.

Practical Evidence Next, we present practical evidence concerning the counter-intuitive price

behavior stated in Proposition 4. McKinsey & Company published a report in 2010 (Baker et al. 2010)

investigating products’ pricing across their life cycles. This report emphasized the correlations among

prices in multiple products’ pricing and concluded that firms must make pricing decisions in the context

of their broader products portfolio, because when they have multiple differentiated products, a price


1 2 3 4 52

4

6

8

10

12

Remaining time

Op

tima

l Price

pjt* (x

)

p3t* (x)

p1t* (x)

p2t* (x)

p4t* (x)

Figure 2.8: Optimal prices of products 1 to 4for x=(3,1,1,1).

1 2 3 4 52

4

6

8

10

12

14

Remaining time

Op

tima

l Price

pjt* (x

)

p1t* (x)

p2t* (x)

p4t* (x)

p3t* (x)

Figure 2.9: Optimal prices of products 1 to 4x=(1,3,1,1).

move for one product can have important implications for those of others. In particular, Baker et al.

(2010) stated that the late life of a product may be an opportune time to raise rather than lower prices.

The report suggested that in some cases the consumers might be more willing to purchase a product in

its late life. Our result in Proposition 4 provides a different explanation for this possible price increase

based on the firm’s dynamic pricing and the consumers’ sequential search.

2.5 Extensions

The basic dynamic pricing model can be extended to several directions. In this section, we discuss three

extensions, leaving the rest for future research.

2.5.1 Horizontally Differentiated Products

In this section, we address the pricing problem of the horizontally differentiated products in the presence

of consumers’ sequential search. Horizontally differentiated products are those that do not differ in

quality but merely appeal to different types of consumers (e.g., ice cream with different flavors). To

study this problem, following several works (see, e.g., Berry and Pakes 2007, Anderson et al. 1992), we

express the utility of product j at time t using the following linear function:

ujt = νψj + σηj − pjt, j ∈ I,


1 2 3 4 52

4

6

8

10

12

14

Remaining time

Op

tima

l Price

pjt* (x

)p

1t* (x)

p4t* (x)

p2t* (x)

p3t* (x)


1 2 3 4 52

4

6

8

10

12

14

Remaining time

Op

tima

l Price

pjt* (x

)

p2t* (x)

p3t* (x)

p1t* (x)

p4t* (x)


where the parameters ν (0 ≤ ν ≤ 1) and σ(≥ 0) are the consumers’ mean valuation weights on product

j’s quality ψj and their personal preferences (tastes) for it, ηj , respectively. We assume that ηj is a

random variable with a Gumbel distribution with parameters (µ, β), β > 0 (see, e.g., van Ryzin and

Vulcano 2015, Mishra et al. 2014, Farias et al. 2013, Aksoy-Pierson et al. 2013, and Akcay et al. 2010 for a

similar setting). Let G(ηj) = exp(− exp(− (y−µ)β )) and g(ηj) denote the Gumbel cumulative distribution

and the probability density function of ηj , respectively. In addition, we assume that the consumers have

a stationary or increasing reservation utility. Then, by Proposition 2, the probability that a consumer

who is inspecting product j stops the search and buys product k ≤ j, ρk(pjt), is:

ρk(pjt) = P(ukt ≥ rj) =

1−G

(γkσ

)= 1− exp

(− exp

(µβ −

γjσβ

)), 1 ≤ k = j ≤ n,

0, ∀k < j,

(2.18)

where γk = rj−νψk+pkt. The next proposition shows that the value function Vjt(x) with ρk(pjt) given

by (2.18) is quasi-concave, which leads to the optimal prices.

Theorem 3 (OPTIMAL PRICE). Let the products be horizontally differentiated and rj be stationary

or increasing in j. Then, the following results hold:

(i) The value function Vjt(x) is quasi-concave.


(ii) The optimal price of product j at time t given xj > 0 is

p∗jt(x) =

A+ Vj+1t(x)− V1t−1(x− ej), 1 ≤ j ≤ n− 1,

A+ V1t−1(x)− V1t−1(x− ej), j = n,

where A is a constant defined as: A = σβ exp(γjσβ −

µβ

)(exp

(exp(µ− γj

σβ ))− 1).

As can be seen from the proposition, the optimal price p∗jt(x) for the horizontally differentiated

products, still has a similar structure as that of vertically differentiated products.

2.5.2 Decreasing Reservation Utility

In this section, we extend the dynamic pricing problem to consider decreasing reservation utilities. We

note that, with a decreasing reservation utility, the purchase probability ρk(pjt) becomes:

ρkt(pjt) =

P ujt ≥ rj , k = j = 1,

P

(ukt = umaxjt ) and (ukt ≥ rj)

, 1 ≤ k < j ≤ n,

P

(ukt = umaxjt ) and (ukt ≥ rj)

, 2 ≤ k = j ≤ n.

(2.19)

Proposition 5 provides the explicit solution for ρk(pjt).

Proposition 5. Let the reservation utility rj be decreasing in j. Then, when a consumer inspects

product j at time t, the probability that he stops the search and buys product k (1 ≤ k ≤ j) is ρk(pjt) =

P(vkjt ≤ ν ≤ vkjt

)= vkjt − vkjt in which vkjt and vkjt are given by

(vkjt, v

kjt

)=

(rj + pktψk

, 1

), k = j = 1,(

max∀l∈Sj :l>k

pkt − pltψk − ψl

,rjt + pktψk

, min∀l∈Sj :l<k


), 1 ≤ k < j ≤ n,(

rj + pktψk

, min∀l∈Sj :l<k


), 2 ≤ k = j ≤ n,

(2.20)

where Sj is the set of inspected products thus far (i.e., Sj = 1, 2, ..., j).

In addition, with a decreasing reservation utility and the search sequence (1, 2, . . . , n), the optimality


equations in (2.12) become

Vjt(x) =

maxp∈P1(x)

λtρ1(p1t)(p1t + V1t−1(x− e1)) + λt(1− ρ1(p1t))V2t(x) + (1− λt)V1t−1(x) , (j = 1),

maxp∈Pj(x)

j∑

k=1

ρk(pjt)(pkt + V1t−1(x− ek)) + (1−∑jk=1 ρk(pjt))Vj+1t(x)

, (2 ≤ j ≤ n− 1),

maxp∈Pn(x)

n∑k=1

ρk(pnt)(pkt + V1t−1(x− ek)) + (1−∑nk=1 ρk(pnt))V1t−1(x)

, (j = n),

(2.21)

where Pj(x) = (p1, p2, . . . , pj) ≥ 0 : ρi(p1, p2, . . . , pi, . . . , pj) = 0 if xi = 0, 1 ≤ i ≤ j. The optimality

Equations (2.21) can be explained similarly to (2.12). The only difference here is that, as the consumer’s

reservation utility is decreasing in j, then upon stopping immediately after inspecting product j, he

might purchase one of the previously observed products k < j (i.e., the recall option applies). Thus,

unlike (2.12), here the optimal revenue Vjt(x) is the sum of the expected revenues of all the products

k ≤ j. As the consumer can buy any of the previously inspected products, we can no longer use the

nested backward induction approach as before. Thus, the dynamic programming algorithm becomes

overly complex, making the analytical solutions hard to find.

2.5.3 Stochastic and Arbitrary Search Sequences

Thus far, we have assumed that the the products are inspected in a descending order of product quality.

In this section, we relax this assumption by incorporating a random search sequence to the dynamic

pricing problem. To this ends, we consider the consumer choosing an arbitrary search sequence in which

either (i) the firm knows the consumer’s chosen sequence (using, e.g., by the assisted search), or (ii) the

firm does not know it. Let Ω be the set of all possible sequences of the n products that a consumer

might choose. Let a = (a1, a2, ..., an) ∈ Ω, al 6= ak (1 ≤ l < k ≤ n) be a search sequence chosen by

consumers, where ai ∈ I is the ith product in the consumer’s chosen sequence. Let V aa1t(x) be the firm’s

expected revenue given the inventory state x at time t when a consumer is about to inspect the first

product based on his chosen sequence a ∈ Ω. Then, if the reservation utility is stationary or increasing,

the Bellman equations in (2.12) for the firm’s expected revenue V aajt(x) becomes:

V aajt(x) =

λtρa1(pa1t)(pa1t + Vt−1(x− ea1)) + λt(1− ρa1(pa1t))Va2t(x) + (1− λt)Vt−1(x), if j = 1,

ρaj (pajt)(pajt + Vt−1(x− eaj )) + (1− ρaj (pajt))Vaj+1t(x), if j = 2, . . . , n− 1,

ρan(pant)(pant + Vt−1(x− ean)) + (1− ρan(pant))Vt−1(x), if j = n,

(2.22)


where in this formula, pajt = (pa1t, pa2t, . . . , pajt) is the price vector, and ρaj (pajt) is the probability

that the consumer buys product aj ∈ a, given by

ρaj (pajt) = P(v ≥rj + pajt

ψaj) = 1−

pajt + rj

ψaj, aj ∈ a. (2.23)

Proposition 6 provides the optimal price of the products when the search sequence is selected by the

consumers, however the firms knows the chosen sequence.

Proposition 6 (OPTIMAL PRICE WITH ARBITRARY SEQUENCE). For any arbitrary search

sequence a ∈ Ω chosen by the consumers, if the firm knows a, and the reservation utility rj is stationary

or increasing in j, then the optimal price of products with positive inventory is expressed as

p∗ajt(x) =

1

2(ψaj + Vaj+1t(x)− Va1t−1(x− eaj )− rj), 1 ≤ j ≤ n− 1,

1

2(ψaj + Va1t−1(x)− Va1t−1(x− eaj )− rj), j = n.

(2.24)

Now, if the firm does not know the consumer’s chosen sequence, then we assume that it still knows

P(a), ∀a ∈Ω, the probability distribution of the possible chosen search sequence. Then, the Bellman

equation for the firm’s expected revenue Vt(x) becomes:

Vt(x) = maxpt∈Pn(x)

∑∀a∈Ω

P(a)V aa1t(x), (2.25)

in which V aajt(x), j = 1, 2, ..., n,, and ρaj (pajt) are given by (2.22) and (2.23), respectively.

2.6 Conclusions

We consider a firm selling vertically differentiated products over a finite horizon. Each consumer arrives

at the firm with a known probability and inspects products sequentially until he finds a product to

purchase or leaves the firm without making any purchase after inspecting all available products. As

a utility maximizer, the consumer will have to compare the expected improvement in his utility with

the loss of utility associated with inspection (search cost). As a result, he might stop the search before

inspecting all the products. Likewise, the firm needs to solve the revenue maximization problem given

the consumer’s search behavior. Consequently, the firm wishes to find the optimal sequence, where it

should present (display) the products as well as the optimal price of each product at each time (given

all the products’ inventory states). We show that the optimal sequence of products’ presentation is

the descending order of quality. This result is important as it significantly reduces the computational


complexity of the simultaneous optimal pricing and sequencing problem. Moreover, we consider the

search scenario in which the consumers decide the order of the search.

We determine the optimal prices analytically, which enables us to derive further interesting man-

agerial insights. In particular, an interesting result that we show is that it is possible for the optimal

price of a perishable product to increase over time (or decrease in the number of periods to go), which

is contrary to the typical finding in the well-established literature on dynamic pricing. To illustrate how

our model can be used as a building block for more complicated settings, we extend it to the case where

the firm offers horizontally differentiated products, and where consumers might choose random search

sequences unknown to the firm.

The research presented in this chapter has limitations. First, if the sequencing is done by consumers

and unknown to the firm, the complexity of the optimal pricing problem becomes an issue for deriving

analytical results (see Section 2.5.3). Second, the stationary or increasing reservation utility is a critical

assumption to reduce the complexity of the optimization problem. Although this assumption holds true

in several real search settings, it is unlikely to be a good universal assumption for all search scenarios.

In this paper, we retained it mostly for analytical tractability (see Section 2.5.2). Third, in this paper,

we consider dynamic pricing of perishable products. Studying a firm selling a set of non-perishable

products would be an interesting direction. Finally, we mostly focused on either vertically or horizontally

differentiated products. Exploring the problem when the consumers may search a mixture of vertically

and horizontally differentiated products is an interesting direction.

We hope that the modeling approach in this chapter can serve as a basis for many promising research

directions beyond this work, and, in doing so, stimulate future research on the dynamic pricing and

revenue management in the presence of consumers’ sequential search in operations management and

management science.

2.7 Proofs

Proof of Lemma 1

Replacing the uniform utility distribution fj = 1Umax−Umin

, j = 1, 2, ..., n in the reservation utility’s

relation (given by (2.4) in the chapter) gives:

c =

∫ Umax

rj

U − rjUmax − Umin

dU =(Umax − rj)2

2(Umax − Umin). (A.2.1)

Solving (A.2.1) for rj gives the result.

Proof of Lemma 2


Our proof for this lemma is mostly based on the proof of a similar result by Haubl et al. (2010). For

the sake of clarity, we first derive the updating rule for a discrete distribution, and then we move to

consider the continuous utility distribution. As for the discrete distribution, we let a product’s possible

utility be a value from the finite discrete set U = u1, u2, ..., um with uj < uj+1 for j = 1, ...,m − 1.

Consistent with several works in Economics and Marketing literature (e.g., Bikhchandani and Sharma

1996 and Haubl et al. 2010), we assume that the probability distribution of these utilities is multinomial

with the unknown success probabilities π1, π2, ..., πm ≥ 0, and∑mj=1 πj = 1 (πj is the chance of observing

uj). To estimate the probability distribution, the consumer assigns a prior weight σj ≥ 0 to each utility

uj (i.e., σj may be viewed as the frequency that the consumer expects to observe occurrence of uj , among

all other possible utility values, within a certain period). Thus, his prior belief about the probability

πl (1 ≤ l ≤ m) is: πl = σl∑mj=1 σj

. After observing the utility, ul, the weights related to the posterior

distribution are updated to σ1, σ2, ..., σl + 1, ..., σm (all frequencies are the same but ul is observed an

additional time). Now, if the consumer observes j products the posterior belief about the probability

πk (1 ≤ k ≤ m) of the so-far-unobserved utility uk is πk = σk∑mj=1 σj+j . We can extend this model to a

continuous distribution setting. Specifically, let the set of possible utilities be in the continuous range

[Umin, Umax] and the consumer assign an equal weight W to each possible utility outcome. (As in the

discrete case, W can be viewed as the frequency that the consumer expects to see for each utility within

a certain period.) A higher W means that each utility outcome has been observed (before) with a higher

frequency. So the prior distribution is more accurate, and needs less updating or change after a new

product utility is observed. Thus, the consumer’s prior density for the utilities of the products in the

market when he wants to inspect the first product is

f1(U) =W∫ Umax

UminWdu

=1

Umax − Umin, Umin ≤ U ≤ Umax. (A.2.2)

After inspecting the first product, the weight (frequency) related to its utility is updated to W + 1 from

W . Thus, the consumer’s posterior (Bayesian updated) utility distribution (to be used for the second

product) after observing one product, f2(u), becomes:

f2(U) =W∫ Umax

UminWdu+ 1

=W

(Umax − Umin)W + 1, Umin ≤ U ≤ Umax. (A.2.3)

Using a similar approach, it is easy to see that the updated probability density function after observing


j products, fj+1(U), becomes:

fj+1(U) =W

(Umax − Umin)W + j, Umin ≤ U ≤ Umax. (A.2.4)

In other words, fj+1(U) is the consumer’s updated belief about the possible utility given by the (j+1)st

product. Replacing (A.2.4) in the relation (2.4) in the paper, gives:

c =

∫ Umax

rj

(U − rj)fj+1(U)dU =W (rj − Umax)2

2((Umax − Umin)W + j). (A.2.5)

Thus, solving for rj gives rj = Umax −√

2cW (Umax−Umin)+jW , and the proof follows.

Proof of Proposition 1

To have the reservation utility to be increasing, the inequality rj ≥ rj−1 ∀j = 2, . . . , n must hold.

Considering the case where the search costs (loss of utilities) vary, in Equation (2.7), the search cost c

must be replaced by cj+1, which results in

rj = Umax −√

2cj+1W (Umax − Umin) + j

W, (A.2.6)

and

rj−1 = Umax −√

2cjW (Umax − Umin) + j − 1

W. (A.2.7)

Thus, rj ≥ rj−1 ifcj+1

cj≤ Φ, j = 1, 2, . . . , n− 1, where, Φ = 1− 1/(W (Umax−Umin) + j), and the proof

follows.


Suppose the search optimally terminates after inspecting product j, and the best product to choose

is k ∈ 1, 2, · · · , j − 1. That is, Umaxjt = Ukt and Ukt ≥ rj . However, as rj ≥ rk, thus Ukt ≥ rk, which

means that the search should have optimally terminated after inspecting product k, which contradicts

the assumption.

Proof of Theorem 1

Let V at (x) denote the total expected revenue where products are presented to the customers with

the sequence a at time t. We consider two products l and k and derive the condition under which it is

optimal to present product k prior to product l. In general, the sequence a1 = (. . . , k, l, s, . . .) ∈ Ω is

preferred to the sequence a2 = (. . . , l, k, s, . . .) ∈ Ω (with the reverse sequence of presenting products l

and k and keeping the position of the other products in the sequence constant), if the corresponding total

expected revenue of a1 is greater than that of a1; therefore, if we have the following then we conclude


that it is optimal to present product k before product l:

V a1t (x)− V a2

t (x) ≥ 0. (A.2.8)

To satisfy Inequality (A.2.8), it is sufficient to have:

V a1t (x)− V a2

t (x) ≥ 0, (A.2.9)

where, a1 = (k, l, s, . . .) and a2 = (l, k, s, . . .). Following Inequality (A.2.9), we have

V a1t (x)− V a2

t (x) =ρk(pkt)(pkt + Vt−1(x− ek)) + (1− ρk(pkt))(ρl(plt)

(plt + Vt−1(x− el)) + (1− ρl(plt))Vst(x))−

ρl(plt)(plt + Vt−1(x− el)) + (1− ρl(plt))(ρk(pkt)

(pkt + Vt−1(x− ek)) + (1− ρk(pkt))Vst(x)),

(A.2.10)

where, Vst(x) denotes the expected revenue from product s and onward (and from time t − 1 onward)

when the products are sorted and priced optimally given inventory state x. Plugging ρj(pjt) = 1− pjt+rjψj

in (A.2.10) and simplifying it, we find:

≤0︷︸︸︷(pkt − ψk + rj)

≤0︷︸︸︷(plt − ψl + rj)(pkt − plt + Vt−1(x− ek)− Vt−1(x− el))

ψkψl︸︷︷︸>0

≥ 0. (A.2.11)

Positivity of ρj(pjt) confirms (pkt − ψk + rj) ≤ 0 and (plt − ψl + rj) ≤ 0. Therefore, (A.2.11) will be

greater than or equal to zero if the following holds:

pkt + Vt−1(x− ek) ≥ plt + Vt−1(x− el). (A.2.12)

From the optimality Equations (2.12) the values pkt+ Vt−1(x−ek) and plt+ Vt−1(x−el) are the revenue

coefficients associated with products k and l. It is clear to see that pkt+ Vt−1(x−ek) ≥ plt+ Vt−1(x−el)

if and only if ψk > ψl. Thus, it is optimal to present a higher-quality product before a lower-quality

product. Thus, it is optimal for the seller to present the products in the descending order of quality.


We first show that if t → ∞, Vlt(x) = Vkt(x) for all k, l ∈ I. If t → ∞, then the firm optimally

decides to increase its products’ prices to the maximum possible values, i.e., p∗jt = ψj − rj , ∀j ∈ I.


Hence, limt→∞ ρj(p∗jt) = 0. Considering the optimality Equations (2.12), we have

limt→∞

Vjt(x) =

λtρ1(p∗1t)(p

∗1t + V1t−1(x− e1)) + λt(1− ρ1(p∗1t))V2t(x) + (1− λt)V1t−1(x), if j = 1,

ρj(p∗jt)(p

∗jt + V1t−1(x− ej)) + (1− ρj(p∗jt))Vj+1t(x), if j = 2, . . . , n− 1

ρn(p∗nt)(p∗nt + V1t−1(x− en)) + (1− ρn(p∗nt))V1t−1(x), if j = n,

(A.2.13)

which results in the following:

limt→∞

Vjt(x) =

λtV2t(x) + (1− λt)V1t−1(x), if j = 1,

Vj+1t(x), if j = 2, · · · , n− 1

V1t−1(x), if j = n.

(A.2.14)

Invoking (A.2.14) and using backward induction, we have:

limt→∞

Vjt(x) = V1t−1(x), ∀j = 2, . . . , n. (A.2.15)

Thus far, we have shown that limt→∞ Vlt(x) = limt→∞ Vkt(x) = V1t−1(x), ∀l, k ≥ 2. So, we need only

to show that limt→∞ V1t(x) = V1t−1(x). To do so, substituting V2t(x) by V1t−1(x) in λtV2t(x) + (1 −

λt)V1t−1(x) in Equation (A.2.14), the result is equal to V1t−1(x), and limt→∞ Vjt(x) = V1t−1(x),∀j.

Therefore, Vlt(x) = Vkt(x) for all k, l ∈ I when t→∞. Likewise, following the above procedure we can

also prove that Vlt(x) = Vkt(x) = V1t−1(x) for all k, l ∈ I.

Proof of Theorem 2

(i) We first show that if rj is stationary or increasing, then the total expected revenue function Vjt(x)

has a nested optimality substructure property. If the optimal solution of a problem can be constructed

from the optimal solutions of its sub-problems, the problem is said to have optimality substructure.

Here, we prove that Vjt(x) has a nested optimality substructure, since to find the optimal value of

Vjt(x), one implicitly has to solve the sub-problems starting from the time and product later than t and

j, respectively. We note that Vjt(x) has an optimal substructure if and only if (a) it is monotonic and

(b) all the sub-problems Vj+1t(x), . . . , Vnt(x) are independent of one another. We show that Vjt(x) has

the properties (a) and (b), as follows:

(a) First, as for the monotonicity property, we show that Vjt(x) is monotone with respect to time, t,


and product j. Taking the derivative of Vjt(x) with respect to Vj+1t(x) gives:

∂Vjt(x)

∂Vj+1t(x)=

λtpjt + rjψj

≥ 0 if j = 1,

pjt + rjψj

≥ 0 if j = 2, · · · , n,(A.2.16)

From the purchase probability ρj(pjt) = 1− pjt+rjψj

we observe thatpjt+rjψj

≥ 0. Thus we have,∂Vjt(x)∂Vj+1t(x) ≥

0, which implies the monotonicity of Vjt(x) with respect to Vj+1t(x) (i.e., Vjt(x) gains a maximum

increase if Vj+1t(x) is maximized). Next, we show the monotonicity with respect to time. Taking the

derivative of Vjt(x) with respect to V1t−1(x), we find:

∂Vjt(x)

∂V1t−1(x)=

(1− λt) ≥ 0 if j = 1,

0 if j = 2, ..., n− 1,

pjt + rjψj

≥ 0 if j = n.

(A.2.17)

As∂Vjt(x)∂V1t−1(x) ≥ 0, therefore Vjt(x) is monotonic with respect to time.

(b) For the independence, note that ρj(pjt) is a function of only pjt but not p1t, . . . , pj−1t. Thus, we

have ρj(pjt) = ρj(pjt), and the independence of the sub-problems follows. Next, we prove that the

expected total revenue function is strictly concave. To do so, we note that given the nested optimality

sub-structure property, at the stage of computing Vjt(x), we have already obtained the optimal values

of V1,t−1(x− ej), V1t−1(x), and Vj+1t(x). Hence, to prove that Vjt(x) is strictly concave, we only need

to show that the second derivative of Vjt(x) with respect to pjt is negative. However, it is easy to see

that∂2Vjt(x)

∂p2jt= λt

(−2ψj

)< 0, which implies the strictly concavity of Vjt(x). (ii) Given the concavity of

the revenue function, the optimal solution satisfies the First Order Condition as follows:

∂Vjt(x)

∂pjt=

λt

(1− pjt + rj

ψj− pjt + V1t−1(x− ej)

ψj+Vj+1t(x)

ψj

)= 0, if j = 1,(

1− pjt + rjψj

− pjt + V1t−1(x− ej)

ψj+Vj+1t(x)

ψj

)= 0, if j = 2, ..., n− 1,(

1− pjt + rjψj

− pjt + V1t−1(x− ej)

ψj+V1t−1(x)

ψj

)= 0, if j = n.

(A.2.18)

Solving the system of equations (A.2.18) for pjt(x), assuming xj > 0, gives:

p∗jt(x) =

1

2(ψj + Vj+1t(x)− V1t−1(x− ej)− rj), if 1 ≤ j ≤ n− 1,

1

2(ψj + V1t−1(x)− V1t−1(x− ej)− rj), if j = n.

(A.2.19)

In addition, if xj = 0 for any product j, then the optimal price is the one that satisfies ρj(pjt) = 0 (the


seller wishes to set the price so that it’s purchase probability becomes zero). Thus, we find

ρj(pjt) = 1− pjt + rjψj

= 0, (A.2.20)

and accordingly, p∗jt(x) = ψj − rj if xj = 0.


The optimal price of product j is decreasing in t if p∗jt(x) ≤ p∗jt−1(x). Considering the optimal price

formula from Proposition 2, the optimal price of product j is decreasing in t if and only if Vj+1t(x) −

V1t−1(x− ej) is decreasing in t. Therefore, the following must hold:

Vj+1t(x)− V1t−1(x− ej) ≤ Vj+1t−1(x)− V1t−2(x− ej). (A.2.21)

Manipulating the inequality, the optimal price will be decreasing if and only if:

∆t[Vj+1(x)] ≤ ∆t−1[V1(x− ej)], (A.2.22)

where ∆t[Vk(x)] = Vkt(x)− Vkt−1(x).

Proof of Theorem 3

(i) Note that, as the quasi-concavity proof procedure of the expected revenue functions is the same

for all j ∈ I, for the sake of space saving we only prove this property for j = 1. We know that Vjt(x) is

twice differentiable. The expected total revenue Vjt(x) is quasi-concave, if we have:

∂Vjt(x)

∂pjt= 0 and

∂2Vjt(x)

∂p2jt

≤ 0. (A.2.23)

First, taking the first order derivative with respect to Vjt(x), we find:

∂Vjt(x)

∂pjt= λt

(− 1

σβexp (L) exp (− exp (L))

)(pjt + V1t−1(x− ej))

+ (1− exp (− exp (L))) +

(1


)Vj+1t(x)

= λt

(− 1


)(pjt + V1t−1(x− ej)− Vj+1t(x))

+ 1− exp (− exp (L)) = 0, j = 1,

(A.2.24)

where, L =(µ− γj

σ

)/β and γj = rj−νψj+pjt. From (A.2.24), we can find p∗jt(x) (the price that satisfy


(A.2.24)) as:

p∗jt(x) =

A+ Vj+1t(x)− V1t−1(x− ej) if xj > 0 , 1 ≤ j ≤ n− 1,

A+ V1t−1(x)− V1t−1(x− ej) if xj > 0 , j = n,(A.2.25)

in which A is expressed as follows:

A = σβ exp

(γjσβ− µ

β

)(exp

(exp(µ− γj

σβ)

)− 1

). (A.2.26)

Now, following (A.2.23), we show that given pjt = p∗jt, we have∂2Vjt(x)

∂p2jt≤ 0. To do so, we note that

from (A.2.24), we can find the second order derivative as:

∂2Vjt(x)

∂p2jt

= λtexp (L) (exp (− exp (L)))

(σβ)2(1− exp (L))(pjt + V1t−1(x− ej)− Vj+1t(x))

− 2

σβexp (L) (exp (− exp (L))) , j = 1.

(A.2.27)

Substituting (A.2.25) in (A.2.27), we find:

∂2Vk,t(x)

∂p2kt

= λtexp(L)(exp(− exp(L)))

(σβ)2

((1− exp(L))(σβ exp(−L))(exp(exp(L))− 1)

)− 2

σβexp(L)

(exp(− exp(L))

)≤ 0,

(A.2.28)

where, L is the L in which pjt has been replaced with p∗jt. Simplifying the Inequality (A.2.28) yields:

(exp(−L)− 1

)(exp(exp(L))− 1

)≤ 2. (A.2.29)

As the range of the left hand side of the Inequality (A.2.29) is (−∞, 1), (A.2.29) holds for any value of

L. Thus, given∂Vjt(x)∂pjt

= 0, we always have∂2Vjt(x)

∂p2jt≤ 0. (ii) Given part (i), it is clear to see that the

local maximum of Vj,t(x) is also a global maximum and that the First Order Condition is a sufficient

condition for the optimal price. Therefore, if xj > 0, then the optimal price is given by (A.2.25) and

(A.2.26). In the case where xj = 0, then the optimal price is the price such that the probability of

purchasing product j becomes zero. This follows:

ρj(pjt) = 1− exp(− exp

(µ− γj

σ

)/β)

= 0, (A.2.30)

and, p∗jt(x) = +∞ for any j such that xj = 0.



The purchase probability of products ρk(pj) is given by

ρk(pjt) = P(vkjt ≤ ν ≤ vkjt

)= vkjt − vkjt, (A.2.31)

where,

1. If k = j = 1, there would not be any comparison decision (as no other product has been inspected

so far). Therefore, the probability that product k = j = 1 is bought is ρk(pjt) = Pukt ≥ rj.

Plugging the utility function in (2.1) we have

(vkjt, vkjt) =

(rj + pktψk

, 1

). (A.2.32)

2. If 1 ≤ k < j ≤ n, then the probability that a product k is chosen would be ρk(pjt) = P(ukt =

umaxjt ) ∧ (ukt ≥ rj). Using the utility function given by (2.1) we have

ρk(pjt) = P

(ν(ψk − ψl)≥ pkt − plt,∀l ∈ Sj) ∧ (ν ≥ rj + pktψk

)

, (A.2.33)

where Sj = 1, 2, ..., j is the set of all the products inspected so far. Note that (ψk − ψl) > 0

∀l ∈ Sj : l > k, and (ψk − ψl) < 0 ∀l ∈ Sj : l < k. Therefore, from (A.2.33) we have

(vkjt, vkjt) =

(max

∀l∈Sj :l>k


,rjt + pktψk

, min∀l∈Sj :l<k


). (A.2.34)

3. Similarly, if 2 ≤ k = j ≤ n, the purchase probability is ρk(pjt) = P(ukt = umaxjt ) ∧ (ukt ≥ rj).

As ∀l ∈ Sj : ψl < ψk, then (A.2.34) reduces to the following:

(vkjt, vkjt) =

(rjt + pktψk

, min∀l∈Sj :l<k


). (A.2.35)

The Equation (A.2.31) along with (A.2.32), (A.2.34), and (A.2.35) constitute the purchase probability

formulas in the following compact form:

(vkjt, v

kjt

)=

(rj + pktψk

, 1

), if k = j = 1,(

max∀l∈Sj :l>k


,rjt + pktψk

, min∀l∈Sj :l<k


), if 1 ≤ k < j ≤ n,(

rj + pktψk

, min∀l∈Sj :l<k


), if 2 ≤ k = j ≤ n,

(A.2.36)

and the proof follows.


1 2 3 4 52

6

10

14

18

22

26

Remaining Time

Op

tima

l re

ven

ue

v1

t(x)

x = (2,1,1,1)

x = (1,1,1,1)

x = (0,1,1,1)

x = (3,1,1,1)

Figure 2.12: Optimal revenue with different in-ventory states where product’s 1 inventory levelincreases

1 2 3 4 54

6

8

10

12

14

16

18

20

22

Remaining time

Op

tima

l re

ven

ue

v1

t(x)

x = (1,0,1,1)

x = (1,2,1,1)

x = (1,3,1,1)

x = (1,1,1,1)


Proof of Proposition 6 The proof is similar to the proof of Proposition 2. Replacing the descending

search sequence with a = (a1, a2, ..., an), the proof follows.

2.8 Additional Examples

Example 1. Consider Example 4 and increase the inventory level of each of the four products gradually.

The impacts of the changing inventory on the total revenue are shown in Figures 2.12-2.15.

Example 2. Consider Example 4 and increase the inventory level of either product 1 or 2 gradually.

Then, as can be seen in Figures 2.16 and 2.17 the optimal price of the product decreases as its inventory

is increased. This observation is intuitive as there is an inverse relationship between price and quantity.

Example 3. In this example we examine the impact of the reservation utilities on the products’ optimal

prices as well as the firm’s optimal expected revenue. Let the firm offer two products with the qualities

ψ1 = 19, ψ2 = 13 over seven periods. The initial inventory level and the consumers’ arrival probability

at each period are set to be x = (2, 2) and λt = 0.8, respectively. We consider the following different

stationary reservation utilities rj = 1, 2.5, 4 and 7. Figures 2.19 and 2.20 illustrate the optimal prices

of products 1 and 2 when the consumers’ reservation utility changes. As can be seen from the figures,

there is decreasing relationship between the products’ optimal prices and the consumers’ reservation

utility. This numerical observation is consistent with the analytical result given by Proposition 2 where

the optimal price p∗jt(x) is decreasing in the reservation utility rj . In addition, Figure 2.18 shows the

optimal expected revenue for various reservation utility levels. As can be seen from the figure, there is a


1 2 3 4 54

6

8

10

12

14

16

18

20

22

Remaining time

Op

tima

l re

ven

ue

v1

t(x)

x = (1,1,1,1)

x = (1,1,0,1)

The optimal revenues of x =(1,1,3,1), x =(1,1,2,1) nearly coincide.

x = (1,1,2,1)

x = (1,1,3,1)


1 2 3 4 54

6

8

10

12

14

16

18

20

22

Remaining time

Op

tima

l re

ven

ue

v1

t(x)

x = (1,1,1,3)

x = (1,1,1,2)

x = (1,1,1,1)

The optimal revenues of x =(1,1,1,3), x =(1,1,1,2) and x = (1,1,1,1) nearly coincide.

x = (1,1,1,0)


decreasing relationship between the consumers’ reservation utilities and the firm’s corresponding optimal

expected revenue. This behavior is somewhat intuitive because as the consumers have more incentive

to continue the search, they stop with a lower chance. Thus, the firm can only sell cheaper products,

which are less profitable.


1 2 3 4 510

11

12

13

Remaining time

Op

tima

l Price

of

pro

du

ct 1

x = (1,1,1,1)

x = (2,1,1,1)

x = (3,1,1,1)

x = (4,1,1,1)

Figure 2.16: Optimal price of product 1 whenits inventory increases.

1 2 3 4 5

6.8

7.3

7.8

8.3

8.8

Remaining time

Op

tima

l Price

of

pro

du

ct 2

x = (1,1,1,1)

x = (1,2,1,1)

x = (1,3,1,1)

x = (1,4,1,1)

Figure 2.17: Optimal price of product 2 whenits inventory increases.

1 2 3 4 5 6 70

5

10

15

20

25

30

Remaining time

Op

tima

l re

ven

ue

V1

t(x)

r = 2.5

r = 4

r = 7

r = 1

Figure 2.18: Optimal revenue v.s. remainingperiods for different values of rj and x = (2, 2).


1 2 3 4 5 6 76

7

8

9

10

11

12

13

Remaning time

Opt

imal

pric

e of

pro

duct

1

r = 2.5

r = 1

r = 4

r = 7

Figure 2.19: Optimal price of product 1 v.s.different consumers’ reservation utilities forx = (2, 2).

1 2 3 4 5 6 73

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Remaning time

Op

tima

l price

of

pro

du

ct 2 r = 2.5

r = 1

r = 4

r = 7

Figure 2.20: Optimal price of product 2 v.s.different consumers’ reservation utilities forx = (2, 2).

Chapter 3

Dynamic Pricing Under Sales

Milestone Constraints

3.1 Introduction

The necessity of standard metrics becomes crucial as marketing and sales processes are intertwined

(Clayton et al. 2013). Larry Norman, former president of Financial Services Group, part of AEGON

USA operating companies, says “We need to be metrics-driven and have metrics in place that track both

sales and marketing performance”. Number of sale is one of the most commonly used sales metrics in

appraising marketing success (Clayton et al. 2013). Ambler et al. (2004) empirically study the main

metrics employed by companies in the UK to measure Marketing success. The authors show that the

first two prominent marketing metrics for firms are (i) profitability and (ii) sales value and/or volume.

Determination of the best metrics as well as implementation of the best strategy to achieve the

metrics-driven goals is the fundamental challenge to measure marketing performance (Ambler 2000). A

minimum-expected marketing performance level is determined by companies, the realization of which is

continuously monitored and ensured by adopting dynamic optimal strategies. We study the aforemen-

tioned influential marketing metrics (i.e., profitability and sales volume) and develop a pricing platform

to simultaneously optimize both revenue and sales volume targets.

The sales volume targets (milestones) might be exogenously (e.g., by financial institutes) or endoge-

nously (e.g., by sales managers) imposed for various reasons or by various stakeholders. In what follows,

we address a set of examples in which both revenue and sales volume targets play critical roles: (i)

Penetration pricing strategy in which firms intend to enlarge market share or initiate word-of-mouth by

39

Chapter 3. Dynamic Pricing Under Sales Milestone Constraints 40

charging a low price (Tellis 1986, Dean 1976). The notion of low price is subjective in that the literature

has been indeterminate to ascertain what qualifies as low or high price. Recently, Spann et al. (2015)

attempted to empirically determine how low the price of a product should be in the case a firm aims

at practicing penetration pricing strategy in digital camera market. While penetration pricing is widely

adopted by many firms (Spann et al. 2015), to the best of our knowledge, there is no study to analytically

address the optimal price of products when a firm opts this strategy; (ii) Real estate industry in which

units of various configurations in a residential building are priced. According to Besbes and Maglaras

(2012), the development of such buildings involves financial institute who often imposes sales milestones

to protect its investment, and the developer should meet such milestones to qualify for downstream

installments of the construction loan; and (iii) Financial institutions and real estate investment trusts

that have gained ownership of portfolios of distressed real estate assets. The firms often wish to monetize

the assets and want to manage the process by imposing milestones in terms of the number of units to

be sold or the inventory absorption speed (Besbes and Maglaras 2012).

Motivated by the aforesaid applications, we study the problem of dynamic pricing of multiple sub-

stitutable products with sales milestone constraints. We formulate the problem as a Markov decision

process (MDP) in the presence of constraints in which the probability of selling a given number of prod-

ucts before a milestone should be greater than or equal to a target value. Customers randomly arrive

at the firm and evaluate the available products using a linear random utility and choose the product

maximizing their utility. The firm’s goal is to find a pricing policy, ensuring the achievement of the

predetermined minimum sales and the maximization of its revenue during selling horizon. The pricing

decision is highly complicated due largely to the following reasons: (i) The firm has to simultaneously

price a set of multiple substitutable products, the pricing of which is further complicated by the existence

of cross-correlation of demands among the products. Therefore, setting products’ prices for multiple sub-

stitutable products requires an explicit recognition of the underlying customer choice behavior; and (ii)

Due to the inherent stochasticity of the nature of the sales’ milestone constraints, the firm needs to solve

a chance-constrained MDP problem.

The contribution of this paper is manifold. (i) We incorporate a set of probabilistic sales constraints

into a dynamic pricing problem. We manage to transform the chance constraint to a deterministic

constraint reducing the complexity of the problem significantly; (ii) We show that not only are KKT

conditions necessary but also sufficient for the determination of the optimal price. We derive a closed-

form solution for the optimal prices of products as a function of remaining inventory and time; (iii) We

show that if the selling season is sufficiently large, joint skimming1-penetration pricing policy is optimal.

1A pricing strategy is said to be skimming if a high introduction price is charged, and is subsequently lowered (Dean1976).


We show that there is a threshold before which the firm should optimally practice skimming pricing

policy even if it initially sets the pricing policy to be penetration pricing policy; (iv) We show counter-

intuitively that it is possible for the optimal revenue of a firm to decrease in products’ inventory; and (v)

Practically, we implement the optimal pricing strategy for one of the leading condominium developers

in Canada using real data.

3.1.1 Literature Review

This research is mainly related to two lines of research in the literature; (1) joint pricing of multiple

substitutable products; and (2) dynamic pricing in the presence of constraints. We present a literature

review of the two research streams as follows.

Given initial inventories of items and a planning horizon over which sales are allowed, we are concerned

with the problem of dynamically pricing the items to maximize the total expected revenue. The papers

by McGill and van Ryzin (1999), Elmaghraby and Keskinocak (2003), Bitran and Caldentey (2003), and

the recent books by Ozer and Phillips (2012), Talluri and van Ryzin (2004a) and Phillips (2005) provide

comprehensive surveys of the literature in revenue management. We focus on models that consider

dynamic pricing with no replenishment opportunities.

In the studies that develop pricing models for multiple products, pricing decisions are linked because

of joint capacity constraints and/or demand correlations. Given starting inventories of components,

Gallego and van Ryzin (1997) model the problem of determining the price for multiple products over

a finite selling horizon. Since their model is difficult to solve, they develop heuristics based on the

deterministic solution to the problem and show that these are asymptotically optimal. Karaesmen and

van Ryzin (2004) consider the substitutability of inventories to determine overbooking limits in a two-

period model. In a case where a firm uses a single resource to produce multiple products, Maglaras and

Meissner (2006) explore the relation between dynamic pricing and capacity control and show that the

dynamic pricing problem in Gallego and van Ryzin (1997) and the capacity control approach (e.g., Lee

and Hersh 1993) can be reduced to a common formulation. Talluri and van Ryzin (2004b) and Zhang

and Cooper (2005) model customer choice behavior explicitly when considering booking limit (capacity

control) policies for airline revenue management. Focusing on a single-leg yield management problem

with exogenous fares, Talluri and van Ryzin (2004b) model how customers choose from multiple fare

products in determining the booking limits for various fare classes. Zhang and Cooper (2005) extend their

model and consider capacity control for parallel flights. For an airline revenue management application

that considers parallel flights and customer choice, Zhang and Cooper (2009) model a pricing control

problem. They acknowledge the complexity of the DP, construct heuristics, and test performance using


a numerical study. Dong et al. (2009) examine both the initial inventory and subsequent dynamic

pricing decisions of horizontally differentiated products with a multinomial logit model of customer

choice. Akcay et al. (2010) consider a dynamic pricing problem of multiple substitutable products

with consumers’ choice over a finite sales horizon. den Boer (2014) investigates a dynamic pricing

problem where the demand for each product depends on the price and some parameters unknown to the

firm. Federgruen and Hu (2015) examine a general price competition model for a set of substitutable

products and ascertain the equilibrium prices, product assortment, and sales volumes. Chen et al.

(2016) investigate a multi-product and multi-resource revenue management problem and developed a

set of heuristics providing minimal and flexible price adjustments.

Prior studies most relevant to dynamic pricing in the presence of constraints are those on dynamic

pricing by Feng and Xiao (1999, 2000), Zhao and Zheng (2000) and Levin et al. (2008). In particular,

Feng and Xiao (1999) introduce a risk factor in a dynamic pricing framework. They consider a model

with two predetermined prices, and instead of looking at the expected revenue alone, use an objective

function that reflects changes in the revenue variance as a result of price changes. Levin et al. (2008)

develop a dynamic pricing problem while imposing to reach a final revenue target with a given probability

when the demand characteristics are known. Chen et al. (2007) use utility theory to develop a general

framework for incorporating risk aversion into multi-period inventory models as well as multi-period

models that coordinate inventory and pricing strategies. Besbes and Maglaras (2012) consider dynamic

pricing of a single type product where the problem is subject to a set of financial milestone constraints on

the revenues and sales. Other references on risk-averse inventory models include Agrawal and Seshadri

(2000), and Eeckhoudt et al. (1995). Integration of risk attitudes into dynamic pricing has also been

recently addressed with different approaches by Lim and Shanthikumar (2007) and Feng and Xiao (2004).

3.2 The Model

A risk-neutral seller2 tries to maximize the revenue from inventories of n product types, which are

vertically differentiated. Let J=1, 2, . . . , n, n+1 be the set of all substitutable products. We consider

dummy product (n + 1) to represent the no-purchase option, meaning that a customer who opts the

dummy product does not buy any product from the set J . Inventories of product j have a fixed quality

index ψj across the selling season (i.e., deterioration of products’ quality is not taken into account).

Having considered products to be vertically differentiated, we assume that products in J are sorted in

a decreasing order of quality, i.e., ψj > ψv for all j < v. Let x = (x1, x2, · · · , xn) be the products’

2In section 3.6.1, we consider the case where the seller is risk-averse.


inventory state, where xj ∈ 0, 1, 2, · · · , kj and kj is the initial inventory of product j ∈ J . Similar

to Talluri and van Ryzin (2004b), Akcay et al. (2010), and Suh (2010), we consider a finite selling horizon

and divide the selling season into T backward labeled time periods, T, T − 1, · · · , 1, each indexed by t.

Each time period t is assumed to be short enough so that at most one customer arrives leading to at most

one unit purchase. It is additionally assumed that neither replenishment nor backlog is permitted. We

further consider sales milestone constraints imposing the seller to achieve sales levels by certain times.

The sales milestone constraints are defined as follows:

Definition 1 (SALES MILESTONE CONSTRAINTS). There is a set of milestones at which the

seller is required to achieve the cumulative sales volume ξs by time s ∈ τ1, τ2, . . . , τm, where T > τm >

τm−1 > . . . > τ2 > τ1 > 0, and ξ0 is the terminal goal at the the end of selling horizon. Obviously, v > w

implies ξv < ξw.

Due to the stochastic nature of sales, it is impossible to guarantee that the sales constraints are met

with certainty. Instead, there is a probability value which is designated by decision maker with which

he intends to achieve the sales targets. We use δs ∈ [0, 1) to denote the given probability threshold of

meeting the sale targets. Hence, for a combination of state and time periods left to the milestone s, we

have the following conditional probabilistic constraint:

P Ns − ξs ≥ 0| xt = x ≥ δs, for s = argminτi<tt− τ1, t− τ2, . . . , t− τm, (3.1)

where xt and Ns denote the state at time t and the number of products sold before or at time s,

respectively. A policy that fulfills the probabilistic Constraint (3.1) is called δs-feasible. We note that

depending on δs, the set of feasible policies may be empty. Constraints of type (3.1) can be either hard

or soft constraints. Hard constraints refer to those that are mandatory and required to be satisfied.

Namely, if the constraint is not met, then the problem is infeasible and the selling process stops. An

example of hard constraint can be the exogenous constraints imposing by financial institutes in real

state industry. The reason is that a failure in satisfying the constraints will result in forfeiting the

construction loan. We refer to soft constraints as those that might not be fully satisfied, in which

case the objective function is penalized by some constant cost Cs ∈ [0,+∞). For instance, endogenous

constraints planned by managers to maintain the firm’s market share in the penetration pricing strategy

can be a soft constraint. The cost Cs can be viewed as the maximum cost associated with not satisfying

the desired level of sales ξs. In this paper, we focus on hard constraint type and assume all the milestone

constraints are hard. However, the model and the proposed solution technique can simply be applied

for soft constraint variant or a combination of both.


Customer’s utility Customers face a discrete choice among the set of (n+1) products with varying

qualities and prices. To model discrete choice behavior of customers we use the linear random utility

model (LRUM), extensively used in the economics literature (see, e.g., Bhargava and Choudhary 2008,

Berry and Pakes 2007, Hensher and Greene 2003, Train 2003, Wauthy 1996, Berry 1994, Roberts and

Lilien 1993, Anderson et al. 1992, Caplin and Nalebuff 1991, Tirole 1988, Bresnahan 1987). Let pt =

(p1t, p2t, . . . , pnt) denote the price vector whose j-th entry indicates price of product j at time period t.

If a customer purchases product j, he receives a certain level of utility Ujt. The LRUM for product j is

presented as follows:

Ujt = νψj − pjt, ∀ j ∈ J , (3.2)

where ν is a random number representing the weight (or value) that a customer assigns to quality relative

to price. We assume that ν is a uniformly distributed random variable between [0, 1] (see, e.g., Akcay

et al. 2010 and Bresnahan 1987 for a similar assumption), and pn+1,t ≡ 0, ψn+1 ≡ 0, ∀t. Suppose that

F denotes the cumulative probability distribution of ν.

Customers’ probability of purchase We assume that customers are risk-neutral and utility max-

imizer, i.e., they purchase a product that generates the highest utility among all the products. We note

that customers may opt not to buy any product if (Ujt < 0 ∀j ∈ J ). Let ρj(pt) be the probability that

a customer chooses product j in period t given the vector of price pt. Hence, ρj(pt) can be expressed by

ρj(pt) = P (Ujt ≥ Uit,∀ i 6= j) . (3.3)

Following the utility function introduced by (3.2) and (3.3), the probability that a customer purchases

product j is

ρj(pt) = P (ν(ψj − ψi) ≥ pjt − pit, ∀ i 6= j).

Note that (ψj − ψi) < 0 ∀i < j, and (ψj − ψi) > 0 ∀i > j. Therefore, a customer purchases product j if

and only if (ν ≤ min

∀i∈J :i<j

pit − pjtψi − ψj

and ν ≥ max

∀i∈J :i>j

pjt − pitψj − ψi

).

With vertically differentiated products, only products that are adjacent in the quality index are sub-

stitutable, i.e., assuming sorted products based on their qualities, product (j − 1) and (j + 1) are


substitutable for product j (see, Berry 1994 and Berry and Pakes 2007). Hence, ρj(pt) is given by

ρj(pt) = P (Ujt ≥ Uit, ∀i = j − 1, j + 1)

= P (Ωjt ≤ ν ≤ Ωj−1,t) = F (Ωj−1,t)− F (Ωjt), j = 1, . . . , n,

where Ωjt shows the cut off points defined as:

Ωjt =pjt − pj+1,t

ψj − ψj+1, j = 1, . . . , n.

For consistency of notation, we let Ω0t = +∞, ∀ 0 ≤ t ≤ T . Hence, the purchase probabilities can be

expressed as follows:

ρj(pt) =

P (Ω1t ≤ ν ≤ +∞) = 1− Ω1t, j = 1;

P (Ωjt ≤ ν ≤ Ωj−1,t) = Ωj−1,t − Ωjt, j = 2, · · · , n;

1−∑nk=1 ρj(pt) = Ωnt, j = n+ 1.

(3.4)

In case of zero inventory, we set the price such that the purchase probability becomes zero (i.e., ρj(pt) =

0, ∀j for xj = 0).

3.2.1 The Firm’s Objective

Given the inventory level x and the remaining periods to the end of horizon t, the seller determines the

optimal price vector p∗t to maximize the revenue while satisfying the sales Constraint (3.1). We refer to

this problem as the constrained dynamic pricing problem (CDPP). Let λt denote the probability that a

customer arrives at period t. We define Vt(x) as the optimal expected revenue from period t to the end

of the season, given the inventory level x. The expected revenue function Vt(x) is expressed by

Vt(x) = maxpt

n∑j=1

λt ρj(pt) (pjt + Vt−1(x− ej)) + λt ρn+1(pt) Vt−1(x)

+ (1− λt) Vt−1(x).

As∑n+1j=1 ρj(pt) = 1, then Vt(x) can be rewritten as

Vt(x) = Vt−1(x) + maxpt

n∑j=1

λt ρj(pt)(pjt −∆xjVt−1(x)

) , (3.5)


with the boundary conditions

V0(x) = 0, ∀ x,

Vt(0) = 0, ∀ t = T, · · · , 1,

where ∆xjVt(x) = Vt(x)−Vt(x− ej) is defined as marginal revenue of inventory, ej is an n-dimensional

vector with its j-th component equal to 1 and the others 0, and 0 is an n-dimensional vector of zeros.

∆xjVt(x) is the maximum expected loss if the seller has one less item of product j given inventory level x

in period t. The firm will sell the products in successive milestones and therefore, revenue maximization

is the the long-term objective of the seller. On the other hand, achieving the sales’ threshold ξs represents

a secondary (short-term) objective. Obviously, if δs changes, the optimal solution of the problem will

differ and develop an efficient frontier. That is, a set of optimal solutions that makes the highest expected

return for a predefined threshold of not satisfying the constraint.

3.3 Solution Methodology

In Section 3.3.1, we develop a method to transform the chance constraint given by (3.1) to recursive

equations consistent with the Bellman Equation (3.2.1) so that it can be calculated as a part of backward

induction. Finally, the optimality conditions and the optimal pricing policy are discussed in Sections

3.3.2 and 3.3.3, respectively.

3.3.1 Probabilistic Constraints and Feasibility

To solve the CDPP, we apply the Lagrangian relaxation approach. To do so, the form of the probabilistic

Constraint (3.1) has to be consistent with the objective function (3.5) (i.e., the addition of the current

period’s revenue and the revenue corresponding to the terminal states). As a result, we reformulate

P Ns − ξs ≥ 0 |xt = x as an expected total value over a binary immediate reward. We firstly define

desirable states set Πs as a set of states stated by:

Πs∆=

x |n∑j=1

kj −n∑j=1

xj ≥ ξs

.

The desirable states set Πs is defined as a set at which the seller stops worrying about the constraints

as soon as reaches a state in Πs3. Therefore, the states of Πc

s are the states that the seller avoids at

3If the seller has to sell a specific level of a particular product, the definition of the desirable states set is changed to

Πs∆= x | kj − xjt ≥ ξsj , ∀j, where ξsj is the minimum number of the j-th product type that should be sold up to time


milestone s. In what follows, we redefine P Ns − ξs ≥ 0 |xt = x to find the probability of meeting one

of the desirable states x ∈ Πs before or at milestone s.

Definition 2 (PROBABILITY OF SATISFYING THE CONSTRAINTS). Let xt−1, xt−2, . . . , xs+1, xs

represent the stochastic sequence of states to milestone s, when the current state is xt. We define the

probability of satisfying milestone s starting with inventory level x at time t as:

µπt,s(x) = P Ns − ξs ≥ 0 |xt = x

= P (∃ l ∈ t− 1, t− 2, · · · , s+ 1, s : xl ∈ Πs| xt = x) ,

where π is the corresponding pricing policy to the states and time periods.

Clearly, if x ∈ Πs, then µπt,s(x) = 1. In addition, if (∑nj=1 xj − ξs > t − s) for a state x ∈ Πc

s, then

µπt,s(x) = 0. This describes the situation where the number of remaining periods is less than the minimum

time required to achieve the desirable states. Knowing that µπt,s(x) represent P Ns − ξs ≥ 0 |xt = x in

Constraint (3.1), we will consider the following as our constraint hereinafter,

µπt,s(x) ≥ δs. (3.6)

Next, we try to present µπt,s(x) as a total expected value function. We set a binary immediate reward

function rt(x) as follows:

rt(x) =

1, if t = s and x ∈ Πs;

0, otherwise,

where rt(x) = 1 if the seller reaches a desirable state at time t = s. If rt−1(xt−1), . . . , rs+1(xs+1), rs(xs)

shows the stochastic sequence of the reward function, then∑t−1l=s rl(xl) follows a Bernoulli distribution,

since if the system enters a desirable state, the sequence of immediate rewards rt(xl) appears exactly

once the reward of 1 and otherwise, zeros only. We aim to find the probability of satisfying the constraint,

i.e., P∑t−1

l=s rl(xl) = 1| xt = x

. We know that for a Bernoulli distribution, the probability of success

(i.e.,∑t−1l=s rl(xl) = 1) is equal to its expected value which results in the following

µπt,s(x) = E

[t−1∑l=s

rl(xl)| xt = x

]. (3.7)

For a pricing decision pt made by the seller, the probability that the state xt = x changes to xt−1 = x−ej

would be equal to λρj(pt). Now, following Equation (3.7) and similar to the calculation of the Equation

s. Note that the solution procedure remains unchanged even with this new definition of the desirable states.


(3.5), µπt,s(x) can be explicitly computed as

µπt,s(x) =

n∑j=1

λ ρj(pt)(rt−1(x− ej)−∆xj

µπt−1,s(x))+ µπt−1,s(x), (3.8)

with the boundary condition µπs−1,s(x) = 0 ∀x, where,

∆tµπt,s(x) = µπt,s(x)− µπt−1,s(x),

∆xjµπt,s(x) = µπt,s(x)− µπt,s(x− ej).

Here, ∆tµπt,s(x) represents the maximum expected reduction of µπt,s(x), with inventory level x at period

t, if the firm has one additional selling period (marginal risk of time). Similarly, ∆xjµπt,s(x) is the

maximum expected reduction of µπt,s(x), if the firm has one less unit of product j to sell (marginal risk

of inventory). We will discuss properties of µπt,s(x) in Section 3.4.

3.3.2 Lagrangian Relaxation Approach and Optimality Conditions

The CDPP can be expressed as follows:

maximizept

Vt(x) = Vt−1(x) +

n∑j=1

λt ρj(pt)(pjt −∆xjVt−1(x)

),

subject to µπt,s(x) ≥ δs, for s = argminτi<tt− τ1, t− τ2, . . . , t− τm,

V0(x) = 0, ∀ x,

Vt(0) = 0, ∀ t = T, · · · , 1.

We apply Lagrangian relaxation approach to reformulate the CDPP into an unconstrained model. This

approach has been widely used in constrained MDPs and dynamic programming (e.g., see Bellman

1956, Altman 1998, 1999, Williams et al. 2007, Topaloglu 2009, Guo and Piunovskiy 2011, Philpott

et al. 2013, and Brown and Smith 2014). The Lagrangian function is the sum of the original revenue

to be maximized and the cost of violating the constraints to be minimized, weighted by some constant

βt which is called Lagrangian multiplier. Hence, for each fixed βt, we are faced with a conventional

unconstrained revenue management problem, and we can obtain the optimal pricing policies through

well-known dynamic programming techniques. We synthesize the objective function and the constraint


given in (3.5) and (3.6), and form the Lagrangian function as follows:

Φ(x) ≡ Φ(x,pt, βt) = Vt−1(x)+

n∑j=1

λt ρj(pt)(pjt −∆xj

Vt−1(x))− βt(δs − µπt,s(x))

, (3.9)

where βt is a non-negative real number. If [δs − µπt,s(x)]+, we inflict a penalty on the revenue function,

i.e., βt[δs − µπt,s(x)]+, where [a]+ = maxa, 0. To optimize Φ(x), we first investigate the structural

properties of the Lagrangian function in (3.9). The proofs of all results including the following theorem

are deferred to Section 3.8.

Theorem 4 (CONCAVITY). The Lagrangian function Φ(x) is concave in pt(x). That is, the Hessian

matrix ∇2Φ(x) ∈ Rn×n is negative semidefinite, where

[∇2Φ(x)]ij =∂2Φ

∂pjt∂pit(x) ∀i, j = 1, 2, · · · , n.

Concavity of the Lagrangian function does not guarantee strong duality, in general. There are

additional conditions on the Lagrangian function, beyond concavity, under which strong duality holds.

These conditions are called constraint qualifications. One of the constraint qualifications is Slater’s

condition. Slater’s condition holds, if the problem is strictly feasible, i.e., µπt,s(x) > δs. Slater’s theorem

states that strong duality holds, if Slater’s condition holds and the problem is concave (Boyd and

Vandenberghe 2004). However, Theorem 5 expresses that under some circumstances, strict feasibility

can be replaced by feasibility in Slater’s condition.

Theorem 5. (Boyd and Vandenberghe 2004) In a convex/concave optimization problem with affine

constraints, Slater’s condition holds if and only if the problem is feasible.

Lemma 3 (STRONG DUALITY SUFFICIENT CONDITION). If the CDPP is feasible, Slater’s con-

dition holds.

Based on Theorem 4 and Lemma 3, for any given state x and time period t, Karush-Kuhn-Tucker

(KKT) conditions are the necessary and sufficient conditions for a price vector pt to be optimal. KKT

conditions are as follows:

(a)∂Φ(x)

∂pjt= 0,∀ j = 1, 2, . . . , n;

(b) βt(δs − µπt,s(x)

)= 0;

(c) µπt,s(x) ≥ δs;

(d) βt ≥ 0,


where,

∂Φ(x)

∂pjt=

n∑k=1

λt∂ρk(pt)

∂pjt(pkt −∆xk

Vt−1(x)) + λtρj(pt)

+ βt

n∑j=1

λt∂ρk(pt)

∂pjt

(rt−1(x− ej)−∆xj

µπt−1,s(x)).

In fact, the first-order condition, complementary slackness, primal and dual feasibility are expressed by

conditions (a) to (d), respectively. We consider these conditions simultaneously and find the optimal

pricing policy in the following section. We note that if there is no price vector so as to satisfy the

conditions, then the problem would be infeasible.

3.3.3 Optimal Price

The conditions (a) and (b) form a system of equations that the optimal price vector pt should satisfy. We

begin with the complementarity condition, i.e., condition (b): either βt = 0 or δ = µπt,s(x). If δ = µπt,s(x),

then the matrix form of the first-order condition, i.e., condition (a), can be written as follows:

λ

(∂ρ(pt)

∂pt

)(pt −∆xVt−1(x) + βt(rt−1(x− ej)−∆xµ

πt−1,s(x))) + λρ(pt) = 0. (3.10)

Multiplying Equation (3.10) by(

1λ

) (∂ρ(pt)∂pt

)−1

, we have:

pt =− ρ(pt)

(∂ρ(pt)

∂pt

)−1

+ ∆xVt−1(x)− βt(rt−1(x− ej)−∆xµ

πt−1,s(x)

), (3.11)

where ∂ρ(pt)/∂pt is the Jacobian matrix of ρ(pt) = (ρ1(pt), ρ2(pt), · · · , ρn(pt)) with ∂ρi(pt)/∂pjt as its

elements (i, j). In addition, ∆xVt−1(x), rt−1(x−ej) and ∆xµπt−1,s(x) are marginal revenue of inventory,

the immediate reward, and marginal risk of inventory vectors, respectively:

∆xVt−1(x) = (∆x1Vt−1(x),∆x2

Vt−1(x), · · · ,∆xnVt−1(x)) ,

rt−1(x− ej) = (rt−1(x− e1), rt−1(x− e2), . . . , rt−1(x− en)) ,

∆xµπt−1,s(x) =

(∆x1

µπt−1,s(x),∆x2µπt−1,s(x), · · · ,∆xn

µπt−1,s(x)).

The following proposition presents the optimal price vector for each state x at each time period t.

Proposition 7 (OPTIMAL PRICE). For any given state x at period t, if the problem is feasible the


optimal price pjt is given by

pjt(x) =1

2(ψj + ∆xjVt−1(x)− βt(rt−1(x− ej)−∆xjµ

πt−1,s(x))), j = 1, 2, . . . , n. (3.12)

If βt = 0, then the optimal price is

pjt(x) =1

2(ψj + ∆xj

Vt−1(x)), j = 1, 2, . . . , n. (3.13)

To obtain the optimal value of the Lagrangian multiplier, βt, we plug the optimal price (3.12), given

by the Proposition 7, into the equation δ = µπt (x). The optimal βt is

βt(x) =λMtHt − δs + µπt−1,s(x)

λKtHt, (3.14)

where Mt, Ht and Kt are vectors of dimension 1× n, n× 1 and 1× n, respectively, the j-th element of

which are as follows:

H(j)t = rt−1(x− ej)−∆xjµ

πt−1,s(x), j = 1, · · · , n,

M(j)t =

1

2−

∆xjVt−1(x)−∆xj+1

Vt−1(x)

2(ψj − ψj+1), j = 1;

∆xj−1Vt−1(x)−∆xj

Vt−1(x)

2(ψj−1 − ψj)−

∆xjVt−1(x)−∆xj+1

Vt−1(x)

2(ψj − ψj+1), j = 2, · · · , n− 1;

∆xj−1Vt−1(x)−∆xjVt−1(x)

2(ψj−1 − ψj)−

∆xjVt−1(x)

2ψj, j = n,

K(j)t =

−rt−1(x− ej)− rt−1(x− ej+1)−∆xj

µπt−1,s(x) + ∆xj+1µπt−1,s(x)

2(ψj − ψj+1), j = 1;

rt−1(x− ej−1)− rt−1(x− ej)−∆xj−1µπt−1,s(x) + ∆xj

µπt−1,s(x)

2(ψj−1 − ψj)

−rt−1(x− ej)− rt−1(x− ej+1)−∆xjµ

πt−1,s(x) + ∆xj+1µ

πt−1,s(x)

2(ψj − ψj+1), j = 2, · · · , n− 1;

rt−1(x− ej−1)− rt−1(x− ej)−∆xj−1µπt−1,s(x) + ∆xj

µπt−1,s(x)

2(ψj−1 − ψj)

−rt−1(x− ej)−∆xj

µπt−1,s(x)

2ψj, j = n.

Note that if xj = 0 for some js, then to find βt the following system of equation is solved:

δs − µπt,s(x) = 0;

ρj(pt) = 0, ∀j : xj = 0;

pjt(x) = 12 (ψj + ∆xjVt−1(x)− βt(rt−1(x− ej)−∆xjµ

πt−1,s(x))), ∀j : xj 6= 0.

(3.15)


3.4 Structural Properties of the Model

In the following sections, the properties of the CDPP are investigated. We assume there is a single sales

milestone constraint at the end of the sales horizon and so we remove subscript s. We first address the

behaviors of ∆xjµπt (x) and ∆tµ

πt (x) (Theorem 6). Then, the properties of the Lagrangian multiplier βt

is explained (Theorem 8). Furthermore, we study inventory sensitivity of the optimal revenue (Theorem

7). Finally, we establish the properties of the optimal price (Corollary 1).

Theorem 6. Let state x be a feasible state at time period k, i.e., there exists a price pk so that

µπk (x) ≥ δs, then the marginal risk value has the following properties:

(i) ∆xjµπt (x) is non-decreasing in t ≥ k.

(ii) ∆tµπk (x) is non-increasing in t ≥ k.

Part (i) of Theorem (6) states that as the remaining time increases, product j’s marginal risk of

inventory (weakly) increases. Intuitively, it can be interpreted that the effect of one less inventory of

product j on the violation of the constraint has a non-increasing behavior. Then the influence of product

j inventory on risk increment does not go up. Mathematically, the crucial parameter for ∆xjµπt (x) value

is the number of periods left to the milestone. Therefore, as the remaining time periods become larger,

the probability of satisfying the constraint for state x converges to the corresponding value of xt − ej .

In addition, since µπt (x) − µπt (x − ej) ≤ 0, so, convergence of µπt (x) and µπt (x − ej) generates a larger

value for ∆xjµπt (x). Part (ii) pertains to the time contribution to the decrease of the risk of not meeting

the constraint.

Example 5. A firm offers two types of products whose qualities are [ψ1, ψ2] = [25, 18] over T = 10

time periods. Customers arrive with the probability λt = 0.9 at each period t. The initial inventory of

the products is x = (k1, k2) = (3, 2), and the firm has to sell at least 2 units of inventories by the end

of horizon with δ = 0.7. Figures 3.1 and 3.2 show the optimal price of the products for different states.

The solid and dashed lines indicate the price trend for desirable and undesirable states, respectively. An

interesting observation is the sharp increase in the price of both products well after time periods 1 and 3

for the inventory states x = (3, 1) and x = (3, 2), respectively, indicating the firm’s appetite for setting

high prices after sales milestone is guaranteed (periods 1 and 3 highlight periods, coinciding profound

price decline for milestone satisfaction). We further consider a numerical illustration (see Example 6)

where the sales milestone is prior to the end of horizon. Moreover, Figure 3.3 shows the probability

of satisfying the sales constraint. Clearly, the probability of satisfying the constraint for the desirable

states is one. However, for the undesirable states x = (3, 2) and x = (3, 1), the probability is increasing

in the remaining time and approaches to 1 as time goes to infinity.


1 2 3 4 5 6 7 8 9 106

8

10

12

14

16

18

20

Remaining time

Opt

imal

pric

e of

pro

duct

1

x = (2,1)

x = (1,1)

x = (3,2)

x = (3,1)

Figure 3.1: Optimal price of product 1 withvarious inventory levels where the initial inven-tory is x = (3, 2) and at least 2 products shouldbe sold by the end of horizon

1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12

13

14

Remaining time

Opt

imal

pric

e of

pro

duct

2

x = (2,1)

x = (1,1)

x = (3,1)

x = (3,2)

Figure 3.2: Optimal price of product 2 withvarious inventory levels where the initial inven-tory is x = (3, 2) and at least 2 products shouldbe sold by the end of horizon

3.4.1 Lagrangian Multiplier and Optimal Price Properties

In the Lagrangian approach, βt corresponds to the shadow price. In our context, βt represents the price

of project security, where it is the revenue increment associated with the decrease in the predetermined

threshold of probability δ by one unit.

Proposition 8 (JOINT SKIMMING-PENETRATION STRATEGY IS OPTIMAL). The Lagrangian

multiplier βt has the following properties:

(i) if x ∈ Π, then βt(x) = 0 ∀ t.

(ii) if βk(x) = 0 for any k ≥ s, then βt(x) = 0 ∀t ≥ k.

βt = 0 ∀x ∈ Π, since µπt (x) = 1 (the seller has already met the requirement) and for any value of

δs we have µπt > δs. Therefore, the CDPP is changed to an unconstrained problem leading to a zero

Lagrangian multiplier. When x ∈ Π, then βt may penalize the prices based on δs. As time approaches

the milestone, meeting the constraint becomes more important than the revenue. Therefore, the model

may decide to have a larger modification to the price. On the other hand, time periods are decoupled

by a period l such that βt = 0 for all t ≥ l. It means that as soon as the model ensures that the seller

can satisfy the constraint in the next period, it only focuses on the revenue function. Accordingly, the

revenue loss due to the consideration of constraint would be zero. Therefore, the period l acts as pricing

strategy decoupling point so as prior to l the seller merely aims to maximize the revenue (i.e., skimming

pricing) and after which the sale constraint is binding and the seller’s priority is to satisfy the sales


1 2 3 4 5 6 7 8 9 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Remaining time

µ t(x)

x = (1,1) & x = (2,1)

x = (3,2)

x = (3,1)

Figure 3.3: Probability of meeting the milestone constraint of selling at least 2 products by the end ofhorizon with the initial inventory level x = (3, 2)

milestone constraint (i.e., penetration pricing).

Example 6. Assume that a firm offers two types of products whose qualities are [ψ1, ψ2] = [13, 9] and

T = 10. Let λt = 0.9 and initial inventory x = (k1, k2) = (3, 3). Assume that the seller is required

to sell three products (ξ = 3) with δ = 0.6 over seven periods (τ = 3) from the beginning of the sale

horizon. Figures 3.4 and 3.5 show the optimal price of products before and after the milestone (shown

at period 3). As it can be seen in Figures 3.4 and 3.5, if the seller cannot meet the constraint until

period t = 3 and stay at an undesirable state after the milestone (i.e., t < 3), there would not be any

feasible price. However, in the case where the seller could sell the minimum requirement up to t = 3

and achieve a desirable state, the seller can continue selling the products through the end of horizon.

It is noted that for the inventory states x = (2, 2) and x = (3, 3), βt = 0 for t > 4 and t > 6 (i.e.,

the milestone is guaranteed to be met), respectively. Practically, the firm adopts a penetration pricing

strategy before periods 5 and 7 to ensure the realization of its predetermined milestone, and modifies its

pricing strategy to skimming when the firm has no concern over milestone feasibility to solely concentrate

on the profitability.

3.4.2 Inventory Sensitivity

We investigate the behavior of the optimal expected return with respect to the inventory of product j.

The following theorem provides an insight into a time period before which the firm might generate less

revenue while it has more inventory. In other words, marginal revenue of inventory of product j can be

negative, implying that one more unit of product j decreases the total expected revenue.


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

11

Remaining time

Opt

imal

pric

e of

pro

duct

1

x = (1,1)

x = (2,2)

x = (3,3)

x = (2,1)Milestone

Figure 3.4: Optimal price of product 1 withvarious inventory levels where the initial inven-tory is x = (3, 3) and at least 3 products shouldbe sold by t = 3

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Remaining time

Opt

imal

pric

e of

pro

duct

2

Milestonex = (2,1)

x = (1,1)

x = (3,3)

x = (2,2)

Figure 3.5: Optimal price of product 2 withvarious inventory levels where the initial inven-tory is x = (3, 3) and at least 3 products shouldbe sold by t = 3

Theorem 7 (HIGHER INVENTORY LOWER REVENUE). For any given state x ∈ Π, if T is large

enough, then there exists some time period t∗(x) such that

Vt(x) ≤ Vt(x− ei) for t < t∗(x),

Vt(x) > Vt(x− ei) for t ≥ t∗(x).

Time period t∗(x) is an equilibrium point for the seller. The existence of the time point t∗(x) from

Theorem 7 helps the seller realizes the minimum time periods required to not only satisfy the constraint

but also generate relatively more revenue for one more inventory of product j. For instance, in real estate

industry this point is of paramount importance for both lender and developer. Knowing t∗(x) assists

both the lender and developer to negotiate and agree upon a mutually beneficial milestone. Setting a

milestone τ larger than corresponding t∗(x) to the initial inventory assures the developer that every

single unit of inventory would have a positive marginal revenue. Moreover, the t∗(x) decomposes the

time horizon into two intervals before which the marginal value of inventory is positive and after which

the marginal value of inventory is negative. This happens if what the seller loses from lowering the price

is less than what he gains from selling additional units, i.e. ∆xiVt∗(x) < C× [1−µt∗(x)], where C is the

cost associated with not satisfying the constraint. To avoid having negative marginal value of inventory,

the developer should attempt to be in one of the desirable states Π before he gets to the period t∗(x).

The time period t∗(x) is a monotonic function with respect to the j-th product inventory xj , meaning


1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

Remaining time

Optim

al R

evenue V

t(x)

x = (2,1)

x = (3,2)

x = (3,1)

x = (1,1)

Figure 3.6: Revenue with different inventory states x

the more units of inventory the higher the t∗(x).

Example 7. Next, we compare the revenue generated by different inventory states over time. Consider

Example (5). Figure 3.6 shows the optimal revenue associated with different states. An interesting

observation is that the firm generates less revenue given higher inventory. For example, inventory states

x = (3, 2) generates less revenue than other inventory levels before time period 3, whereas its revenue

exceeds inventory x = (1, 1), (2, 1), and (3, 1) after times periods 4, 5, and 7, respectively.

We know that pjt(x) carries the structural properties of ∆xjVt(x), ∆xj

µπt (x) and βt. Akcay et al.

(2010) shows that ∆xjVt(x) is non-decreasing in t. Then we can show properties of the optimal solution

derived from the aforesaid theorems and propositions by Corollary 1.

Corollary 1. The optimal price pjt(x)

(i) is non-decreasing in t.

(ii) is strictly decreasing in j.

(iii) is non-increasing in xj.

Corollary 1 indicates that the optimal price is a monotonic function with respect to the time periods

(t), quality of products (ψj), and products inventories (xj). According to the optimal price properties

stated in Corollary 1, we can conclude the following proposition in regard to ρj(pt).

Proposition 9. At optimality, the probability of selling the j-th product, ρj(pt), is a non-increasing

function in t.


0 50 100 150 200 2501

2

3

4

5

6

7

8x 10

5

Units

Pric

e

Estimated qualityActual price

Figure 3.7: The regression model to estimate the condominium units’ quality levels

Since the optimal price is a non-decreasing function of t, the seller set the prices more aggressively

and it results in lower probability of making a purchase or equivalently higher probability of no-purchase

ρn+1(pt).

3.5 Numerical Illustration

3.5.1 Practical Application

We perform a case study on a leading Canadian condominium developer (CDD) using our proposed

dynamic pricing method. The following outlines the dataset, the methodology, and the results obtained

by running the dynamic pricing technique for CCD.

Dataset The dataset provided by CCD pertains to a previously launched project in the city of

Toronto, Ontario, Canada in 2003, featuring characteristics of each individual condominium unit (in the

building) as well as the corresponding pricing history (i.e., offered prices). The characteristics contain

sold date, floor, design, view, and square footage of the units in the building.

Large-Scale Regression Model As can be seen in Proposition 7, the optimal price is a func-

tion of the quality of products. In order to estimate the quality level for each condominium unit, a

regression model is developed. The quality is directly related to the characteristics of the condominium


0 50 100 150 200 250 300 3501

2

3

4

5

6

7

8

9

x 105

Remaining time (day)

p jt* (x)

Model optimal price

p2t

(x)

p1t

(x)

p3t

(x)

The CCD actual price

Figure 3.8: Comparison of the CCD practicedprices with the optimal prices generated by ourmodel given the CCD actual states’ changesover time

50 100 150 200 250 300 3501

2

3

4

5

6

7

8

9

x 105


p jt* (x)

p2t

(x)

p1t

(x)

Model optimal price

The CCD actual price

p3t

(x)

Figure 3.9: Comparison of the CCD practicedprices with the optimal prices generated by ourmodel where the products have a fixed inven-tory of one

units, outlined previously. To construct the regression model, we consider floor, view, design, and their

interactions (square footage is not factored as it is redundant with design) in the following form:

Q = aF + bV + cD + dFV + eFD + fVD + gFVD,

where, F , V, and D are the floor number, the view code, and the design code, respectively, and a

combination of these letters signifies the interaction terms among them. In what follows, we provide the

parameters derived from the regression analysis: a = 5830.5, b = 27509.29, c = 18363.53, d = −1493.12,

e = −151.64, f = −2366.78, and g = 124.86. As observed in Figure 3.7, the regression line (blue lines)

does match the actual prices quite accurately. In particular, each floor is distinct, in the form of diagonal

lines, on the regression model. However, it is noted that some points are below the actual prices, which

is unacceptable as we aim at finding the maximum possible prices of the units (i.e., the units’ quality

levels). In order to resolve this issue, each regression price is further multiplied by a constant factor.

The analysis shows that a constant factor of about 1.2 to 1.3 is ideal for shifting all the regression points

at or above the actual price points.

Results To illustrate the pricing behavior of the condominium units, we consider the following two

cases:

(i) Each condominium is treated as a unique unit and the initial inventory of the units are therefore is


Table 3.1: The CCD inventory states’ changes over time

Time StatesDay 1-26 (1,1,1,1,1,1,1,1,1,1)Day 27-32 (1,1,1,1,1,1,1,0,1,1)Day 33 (1,1,1,0,1,1,1,0,1,1)Day 34-52 (1,1,1,0,0,1,1,0,1,1)Day 53-201 (0,1,1,0,0,1,1,0,1,1)Day 202-274 (0,1,1,0,0,1,1,0,1,0)Day 275-293 (0,0,1,0,0,1,1,0,1,0)Day 294-360 (0,0,1,0,0,1,0,0,1,0)

set to one, i.e., x = (1, 1, 1, . . . , 1). There are ten types of units 1, 2, 3, . . . , 10 with the estimated quality

levels expressed in thousands (ψ1, ψ2, . . . , ψ10) = (906, 650, 386, 365, 362, 347, 325, 279, 228, 208). Other

parameters are set to be T = 360, λt = 0.8, δ = 0.7, ξ = 7, and τ = 0. Figure 3.8 and 3.9show the pricing

evolution for the three highest quality products. The figure demonstrates the actual prices used by CCD

and the optimal prices generated by our model. Specifically, Figure 3.8 shows the optimal price of the

units given the actual change in the inventory state of CCD given in Table 3.1, while Figure 3.9 indicates

the optimal price when the inventory state does not change and is assumed to be the initial inventory

state (i.e., all products have an inventory of one). It is also clear that CCD has opted a somewhat static

pricing strategy to sell its products (i.e., a constant price over the selling season). As can be seen in

Figure 3.8, the optimal price of products increases when CCD sells a unit. For example, product type 8

is sold after 27 days from the beginning of the horizon given Table 3.1; Therefore, the optimal prices rise

at that time period. Of particular interest is that except the highest quality product, all the dynamic

pricing graphs (prices generated by our model) intercept the prices set out by CCD at some point in

time. This implies that if the unit can be sold at the CCD price during the planning horizon, then it

will also be sold certainly using the dynamic pricing scheme. Thus, our dynamic pricing method should

perform at least as well as the static pricing implemented by CCD. However, if a customer decides to

purchase the unit before the point at which the dynamic price is equivalent to the static price, then the

dynamic price will generate more revenue.

Furthermore, Figure 3.10 shows the expected revenue generated by our pricing model and the actual

revenue earned by CCD throughout the planning horizon for a full inventory case. From Figure 3.10, it

can be seen that the expected revenue generated from the model is significantly higher than the actual

revenue given by the CCD pricing scheme. This remains the case for both the 1.2 and 1.3 factor until

approximately the 300 day mark, when the two lines cross with the actual revenue. Next, we consider

the second case.

(ii) Considering each condominium unit as an exclusive unit type makes the dynamic pricing problem


0 50 100 150 200 250 300 3501.5

2

2.5

3

3.5

4x 10

6


Re

ve

nu

e

Model optimal expected revenue(factor 1.2)

Actual revenue

Model optimal expected revenue(factor 1.3)

Figure 3.10: Comparison of the CCD actualrevenue and the expected revenue generated byour model with adjusting factors 1.2 and 1.3

10 20 30 40 50 600

100

200

300

400

500

600

700

800

900

Remaining time (weaks)

p jt* (x)

Milestone

p1t* (x)

p2t* (x)

p4t* (x)

p3t* (x)

Figure 3.11: Price comparison given the CCDinventory states’ changes where at least 7 unitsof products should be sold by t = 12

intractable as the number of units increases. The reason is that the dimension of the inventory state

enlarges with the number of units where each unit has an inventory of one. The same issue also influences

the computational complexity of our model given the number of periods in the selling horizon. Therefore,

we need to reduce the dimension of inventory states and time periods to tackle computational complexity.

To this end, we categorize the units with similar quality levels into one unit type and change the time

periods from days to weeks. Accordingly, we consider the initial inventory to be x = (1, 1, 5, 3) with

quality levels (ψ1, ψ2, ψ3, ψ4) = (906, 650, 357, 238). The other parameters of the problem are set to be

T = 64, λt = 0.8, δ = 0.7, ξ = 7, and τ = 12. Figure 3.11 shows the optimal pricing of products given

the actual state change. The figure clearly demonstrates the fact that unit prices increase as the firm

satisfies the sales constraint which is intuitive from a practical standpoint.

3.5.2 Sensitivity Analysis

We compare the optimal revenue generated by a model in which no sales constraint is imposed (i.e., the

unconstrained model) with that of the one with sales milestone constraints (i.e., the CDPP). Two cases

are examined in which the loss in revenue due to tighter sales constraint in terms of (i) the probability

threshold δ, and (ii) the minimum required sales ξ are explored. Assume that there is a firm selling two

products whose quality levels are ψ1 = 13 and ψ1 = 9 over a sales season with T = 8, and λt = 0.9.

Figure 3.12 illustrates the optimal revenue of the firm under different scenarios. The solid line indicates

the optimal expected revenue associated with the unconstrained model, while the dashed lines are related


1 2 3 4 5 6 7 8

2

4

6

8

10

12

14

16

18

20

22

Remaining time

Op

tim

al re

ve

nu

e V

t(x)

No sales constraint

δ = 0.75

δ = 0.85δ = 0.65

Figure 3.12: Comparison of revenue generatedby the unconstrained model and that of theconstrained model with different probabilitythreshold δ for x = (3, 3)

1 2 3 4 5 6 7 8

2

4

6

8

10

12

14

16

18

20

22

Remaining time

Opt

imal

reve

nue

V t(x)

No sales constraint

ξ = 3

ξ = 2

ξ = 1

Figure 3.13: Comparison of revenue generatedby the unconstrained model and that of theconstrained model with different sales require-ment ξ for x = (3, 3)

to the case where the firm requires to sell a minimum number of 2 products by the end of the horizon

and the probability thresholds can be δ = 0.85, 0.75, and 0.65. The constrained models are not feasible

at period t = 1 for the entire constrained models. As expected, the higher the value of δ is, the larger the

revenue loss will be. The figure also shows that the revenue of all the models converge when the firm has

sufficient time to the milestone. In addition, Figure 3.13 describes a similar case where the probability

threshold is fixed at δ = 0.75 for all the constrained models (dashed lines), while the minimum sales

requirements by the end of the horizon are ξ = 1, 2, and 3. As it can be seen from the figure, the

constrained models become feasible at different time periods (period 1, 2, and 3, respectively), and

similar to the first scenario (Figure 3.12) all converge to the unconstrained model as time approaches

the end of sales season.

3.6 Extensions

In this section, we extend the basic dynamic pricing model. We discuss two extensions, leaving the rest

for future research.

3.6.1 Expected Utility Maximization

Thus far, we assumed that the seller is risk-neutral while from many empirical investigations it is known

that individuals might be risk-averse and the degree of risk-aversion differs among individuals (Kahneman


and Tversky 1979)4. Unlike the CDPP model, where the seller’s control leverage on the sale process is

δs; in the case of risk-averse seller, the control leverage is the degree of risk-aversion γ. We define the

utility function of the seller as follows:

Definition 3 (UTILITY FUNCTION). Θ(Λ) is defined as the utility of a revenue Λ, where Θ is strictly

increasing (dΘdΛ > 0) and concave (d

2Θd2Λ < 0).

The risk-aversion level of the decision maker depends on the concavity of the utility function. Namely,

the higher the concavity of Θ, the higher the risk-aversion level will be. We consider an exponential

utility function which is widely used in management science and economics literature (see e.g., Brockett

and Golden 1987, Bell and Fishburn 2001, Lim and Shanthikumar 2007, Tsetlin and Winkler 2009,

Diecidue et al. 2009, Cai and Kou 2011) as follows:

Θ(Λ) := 1− e−γΛ.

The objective of seller is to maximize the expected total utility as follows:

ft(x) = maxpt

E[Θ(Λ)]

= maxpt

n∑j=1

λ ρj(pt)(

1− e−γ(pjt+$t−1(x−ej)))

+ λ ρn+1(pt)(

1− e−γ$t−1(x))

+ (1− λ)(

1− e−γ$t−1(x))

,

(3.16)

where

$t(x) =

n∑j=1

λt ρj(pt)(pjt −∆xj

$t−1(x))+$t−1(x).

Note that, the only difference between $t(x) and Vt(x) is that $t(x) is not necessarily the maximum

revenue and is a random variable of pt. Simplifying Equation (3.16), it results in the following:

ft(x) = maxpt

n∑j=1

λ ρj(pt)(e−γ$t−1(x) − e−γ(pjt+$t−1(x−ej))

)− e−γ$t−1(x)

+ 1. (3.17)

3.6.2 Pricing Policy with Revenue Constraint

In reality there might be problems with financial milestone constraint, in the sense that, at each milestone

the seller requires to achieve a certain amount of revenue as well as sale.

4A person is said to be risk-averse if he always prefers to receive a fixed payment to a random payment of equal expectedvalue (Dumas and Allaz 1996).


Definition 4 (REVENUE CONSTRAINTS). The seller is required to achieve the cumulative revenue

ζs by time s ∈ τ1, τ2, . . . , τm. Obviously, v > w implies ζv < ζw.

Similar to the works of Levin et al. (2008) and Xu and Mannor (2011), we utilize discrete price sets

for each of the products. The admissible prices set of product j is denoted by ωj = pj1 , . . . , pjLj, where

Lj represents the number of pricing levels for product j and pjmin= pj1 < pj2 < . . . < pjLj

= pjmax.

For any given states and time periods, the seller needs to choose a price in ωj to construct a price menu

pt. The set of all possible price menus at each state is denoted by Ω ⊂ Rn. The available price menus

are defined such that pt = (pl1t, pl2t, . . . , plnt) ∈ Ω, where lj ∈ 1, 2, . . . , Lj. Therefore we deal with a

countable action space and the number of admissible price menus is∏nj=1 Lj . Knowing that the prices

belong to a predetermined discrete set, the revenue R is also discrete. So R ∈ 0, · · · ,x>I pmax, where

x>I is the transpose of the initial inventory vector and pmax is the price vector at which the price of each

product is set on its maximum allowable price. Since we aim to keep track of revenue for meeting the

revenue constraints, we define the state of the system at time t by the inventory vector and the set of

possible actual accumulated rewards as pairs in the finite set as follows:

Πs∆=

(x, R) : x ∈ 0, 1, . . . , kj, R ∈ 0, . . . ,x>T pmax. (3.18)

The states with q = (0, R) are considered as absorbing states. This state augmentation is a common

approach that has been applied by Levin et al. (2008) and Xu and Mannor (2011). Given the state

(x, R), let Vt(x, R) denote the optimal expected revenue from period t to the end of the season. So, the

new objective function is as follows:

Vt(x, R) = Vt−1(x, R) + maxpt

n∑j=1

λt ρj(pt)[pjt + Vt−1(x− ej , R+ pjt)− Vt−1(x, R)

] . (3.19)

The objective function is subject to the sales and revenue constraints and boundary conditions. To

solve this problem, we follow the CDPP solution method. The only difference is in the definition of the

desirable states. So we have the following as the desirable sets:

Πs =:

Q |n∑j=1

kj −n∑j=1

xj ≥ ξs and R ≥ ζr

. (3.20)

Now, we have to accept an abuse of notation in differentiability of the objective function (3.19) with

respect to pjs. Therefore, if we consider a set of admissible prices Ω that are close enough together, this

method does not affect the optimal price tremendously and the approximated price derived from the


first order condition is close enough to the optimal price.

3.7 Conclusions

We present a new joint multiple products dynamic pricing model that allows the firms to control the risk

that the total sales becomes lower than an acceptable level while the expected revenue is maximized.

The model permits the decision makers to establish a balance between maximization of expected total

revenues and the risk of losing a certain market share.

Customers randomly arrive and evaluate the available products in the store, and choose the most

desirable product by comparing the products’ utility. We formulate the problem as a chance-constrained

Markov decision process in which the firm wishes to obtain the optimal products’ prices simultaneously

to maximize the expected profit, and achieve the sales constraints. We manage to demonstrate that the

KKT conditions are necessary and sufficient for the optimal price, and derive the closed-form solution for

the optimal price. This paper also explores a detailed analysis of the structural properties of the model

and the optimal pricing policy. Of particular interest is the following: (i) Joint penetration-skimming

pricing strategy is optimal, namely, firms which intend to implement the penetration pricing policy may

optimally end up the implementation of skimming pricing policy, if there is sufficiently large periods

to the milestone. In other words, there is a decoupling point before which skimming pricing policy

is optimal, and after which penetration pricing policy is optimal; (ii) Due to the existence of the sales

constraints, it is possible for firms to generate lower revenue from higher inventory level. We also present

a model incorporating decision makers’ degree of risk aversion.

To demonstrate how our model can be used in real cases, we implement our proposed pricing strategy

for a leading Canadian condominium developer (CCD). We compare the result generated by our model

with the pricing policy of CCD, and show a significant improvement in the firm’s profitability.

We suggest the following as the research’s potential future works: (i) Incorporating different con-

sumers’ choice behavior models may change the optimal price properties. For example, customers’

sequential search can be integrated with the basic pricing model; (ii) Considering a revenue constraint

can also be interesting, as there are many cases in which firms sell a set of substitutable products, and

have both revenue and sales milestone constraints; and (iii) The problem can be modeled as an infinite

horizon MDP, as firms may have a line of non-perishable products with sales milestone constraints.


3.8 Proofs

Proof of Theorem 4.

The Lagrangian function, Φ(x), is concave in pt(x), if its corresponding Hessian matrix is negative

semi definite (NSD). Based on the formula of Φ(x) provided by Equation (3.9), the first order derivative

would be as follows: For j = 1,

∂Φ(x)

∂p1t= λt

[(1− p1t − p2t

ψ1 − ψ2

)+

(1

ψ1 − ψ2

)((p2t + rt−1(x− e2)−∆x2

µπt−1(x)−∆x2Vt−1(x))

− (p1t + rt−1(x− e1)−∆x1µπt−1(x)−∆x1Vt−1(x)))

],

for 2 ≤ j ≤ n− 1, we will have

∂Φ(x)

∂pjt= λt

[(pj−1,t − pjtψj−1 − ψj

− pjt − pj+1,t

ψj − ψj+1

)+

(1

ψj−1 − ψj

)((pj−1,t + rt−1(x− ej−1)

−∆xj−1µπt−1(x)−∆xj−1Vt−1(x))− (pjt + rt−1(x− ej)−∆xjµ

πt−1(x)−∆xjVt−1(x)))

+

(1

ψj − ψj+1

)((pj+1,t + rt−1(x− ej+1)−∆xj+1µ

πt−1(x)−∆xj+1Vt−1(x))

− (pjt + rt−1(x− ej)−∆xjµπt−1(x)−∆xj

Vt−1(x)))

],

and finally, for j = n,

∂Φ(x)

∂pnt= λt

[(pn−1,t − pntψn−1 − ψn

− pntψn

)+

(1

ψn−1 − ψn

)((pn−1,t + rs(x− en−1)

−∆xn−1µπt−1(x)−∆xn−1

Vt−1(x))− (pnt + rt−1(x− en)−∆xnµπt−1(x)−∆xn

Vt−1(x)))

−(

1

ψn

)(pnt + rt−1(x− en)−∆xn

µπt−1(x)−∆xnVt−1(x))

].

Now, according to the calculated first order derivative we can constitute the Hessian matrix as follows:

∂2Φ(x)

∂2pt=

−2ψ1−ψ2

2ψ1−ψ2

0 0 0 · · · 0

2ψ1−ψ2

[−2

ψ1−ψ2+ −2

ψ2−ψ3

]2

ψ2−ψ30 0 · · · 0

0 2ψ2−ψ3

[−2

ψ2−ψ3+ −2

ψ3−ψ4

]2

ψ3−ψ40 · · · 0

......

.... . .

. . .. . .

...

0 0 · · · 0 0 2ψn−1−ψn

[−2

ψn−1−ψn+ −2

ψn

]

.

(A.3.1)

The Hessian matrix (A.3.1) is a symmetric tridiagonal matrix. A symmetric tridiagonal matrix is

negative semi definite if the following conditions hold:

(a) The matrix is diagonally dominant,


(b) The Diagonal entries are all non-positive.

Condition (a) holds since the magnitude of the main diagonal is greater than or equal to the sum of

the off-diagonal entries and Condition (b) is trivial. Hence, the Hessian matrix (A.3.1) is NSD and the

Lagrangian function is concave in pt(x).

Proof of Lemma 3.

Slater’s condition holds, if there exists a pt(x) so that:

µπt,s(x) > δs. (A.3.2)

Such a point is sometimes called strictly feasible, since the inequality constraints hold with strict in-

equalities. Slater’s theorem states that strong duality holds, if Slater’s condition holds (and the problem

is convex). Condition (A.3.2) can be refined for the affine inequality constraints, where it does not need

to hold with strict inequality. The refined Slater’s condition reduces to feasibility when the constraints

are all linear inequalities (Boyd and Vandenberghe 2004). Hence, since µπt,s(x) is an affine function over

pt(x), then Slater’s condition holds if the CDPP is feasible.

Proof of Proposition 7.

Starting form Equation (3.11),

pt = −ρ(pt)

(∂ρ(pt)

∂pt

)−1

+ ∆xVt−1(x)− βt(rt−1(x− ej)−∆xµ

πt−1,s(x)

),

we require to form the Jacobian matrix(∂ρ(pt)∂pt

). Taking the first-order derivative of Equation (3.4)

gives the Jacobian matirix as follows:

∂ρ(pt)

∂pt=

−1ψ1−ψ2

1ψ1−ψ2

0 0 · · · 0

1ψ1−ψ2

−(ψ1−ψ3)(ψ1−ψ2)(ψ2−ψ3)

1ψ2−ψ3

0 · · · 0

......

. . .. . .

......

0 0 · · · 0 1ψn−1−ψn

−ψn−1

(ψn−1−ψn)ψn

, (A.3.3)


and the inverse of which is represented as

(∂ρ(pt)

∂pt

)−1

= −

ψ1 ψ2 · · · ψn−1 ψn

ψ2 ψ2 · · · ψn−1 ψn...

......

......

ψn−1 ψn−1 · · · ψn−1 ψn

ψn ψn ψn ψn ψn

.

Substituting(∂ρj(pt)

∂pt

)−1

= (ψj , ψj , . . . , ψj , ψj+1, . . . , ψn)>

and ρ(pt) = (1− Ω1t,Ω1t − Ω2t, . . . ,Ωn−1,t − Ωnt)

into Equation (3.11), provides the price of product j at time t as follows:

pjt = ψj − pjt + ∆xjVt−1(x)− βt(rt−1(x− ej)−∆xjµ

πt−1,s(x)

). (A.3.4)

Manipulating the Equation (A.3.4), the proof follows.

Proof of Theorem 6.

Proof of (i). We exploit the following properties of composition of multivariate functions to prove

the theorem. Assume that y : Rn → R and wj : Rn → R, for j = 1, 2, · · · , n. Defining composition

of y w(x) = y(w1(x), w2(x), . . . , wn(x)), with x = (x1, x2, · · · , xn), following are two properties of

composition of multivariate functions: (1) If y is non-increasing in each of its arguments and wj is non-

decreasing in each of its arguments, then the composite function y w is non-increasing in each of its

arguments; and (2) If y is non-increasing in each of its arguments and wj is non-increasing in each of its

arguments, then the composite function y w is non-decreasing in each of its arguments. Next, we show

that ∆tµt+1(x + ej) ≥ ∆tµt+1(x), for j = 1, 2, · · · , n. From proposition 7, first let the optimal price be

pjt = 12

(ψj + ∆xj

Vt−1(x)− βt(rt−1(x− ej)−∆xj

µπt−1,s(x)))

. Therefore

∆xjµπt−1,s(x) =

1

βt(2pjt − ψj −∆xjVt−1(x) + βtrt−1(x− ej)).

Plugging ∆xjµπt−1,s(x) into ∆tµ

πt (x) formula, let

y(pt(x)) =λtβt

n∑j=1

ρj(pt(x))(−2pjt + ψj + ∆xjVt−1(x))

,where µπt (x) = y(pt(x)) + µπt−1,s(x) and then y(pt+1(x)) = ∆tµt+1(x) = µt+1(x)− µπt (x). We take the


first derivative to see the behavior of y(pt(x)) over pjt.

∂y(pt(x))

∂p1t(x)=λtβt

[(−1

ψ1 − ψ2

)(−2p1t(x) + ψ1 + ∆x1Vt−1(x)) +

(1− p1t − p2t

ψ1 − ψ2

)(−2)

+

(1

ψ1 − ψ2

)(−2p2t(x) + ψ2 + ∆x2

Vt−1(x))

]=−2λtβt

(1− p1t − p2t

ψ1 − ψ2

)≤ 0,

∂y(pt(x))

∂pkt(x)=λtβt

[(1

ψk−1 − ψk

)(−2pk−1,t(x) + ψk−1 + ∆xk−1

Vt−1(x)) +

(pk−1,t − pktψk−1 − ψk

− pkt − pk+1,t

ψk − ψk+1

)(−2)

+

(−1

ψk−1 − ψk− 1

ψk − ψk+1

)(−2pkt(x) + ψk + ∆xk

Vt−1(x)) +

(1

ψk − ψk+1

)(−2pk+1,t(x) + ψk+1

+ ∆xk+1Vt−1(x))

]=−2λtβt

(pk−1,t − pktψk−1 − ψk

− pkt − pk+1,t

ψk − ψk+1

)≤ 0, k = 2, · · · , n− 1,

∂y(pt(x))

∂pnt(x)=λtβt

[(1

ψn−1 − ψn

)(−2pn−1,t(x) + ψn−1 + ∆xn−1Vt−1(x))

+

(pn−1,t − pntψn−1 − ψn

− pntψn

)(−2) +

(−1

ψn−1 − ψn− 1

ψn

)(−2pnt(x) + ψn + ∆xnVt−1(x))

]

=λtβt

[(−1

ψn

)(−2pnt(x) + ψn + ∆xn

Vt−1(x))− 2

(pn−1,t − pntψn−1 − ψn

− pntψn

)]≤ 0,

(A.3.5)

where equality of

−2p1,t(x) + ψ1 + ∆x1Vt−1(x) = −2p2,t(x) + ψ2 + ∆x2

Vt−1(x) = · · · = −2pn,t(x) + ψn + ∆xnVt−1(x),

comes from equality of

rxt(xt − e1)−∆x1

µπt−1,s(x) = rxt(xt − e2)−∆x2

µπt−1,s(x) = · · · = rxt(xt − en)−∆xn

µπt−1,s(x),

and non-negativity of(

1− p1t−p2tψ1−ψ2

),(pk−1,t−pkt

ψk−1−ψk− pkt−pk+1,t

ψk−ψk+1

)and

(pn−1,t−pnt

ψn−1−ψn− pnt

ψn

)confirms by the

permissible price set imposed by (3.4), and non-negativity of (−2pnt(x) + ψn + ∆xnVt−1(x)) verifies

through non-negativity of rxt(xt − en) −∆xn

µπt−1,s(x). Hence, we require to show that the composite

function y(pt+1(x)) is increasing in xj , j = 1, 2, · · · , n. Based on (A.3.5), we know that y(pt(x))

is a decreasing function of each of its arguments. Through property (2) of the composition of the

multivariate functions, y(pt+1(x)) would be increasing in xj if pk,t+1(x) is decreasing in xj for all k,

that is pk,t+1(x + ej) ≤ pk,t+1(x) for k, j = 1, 2, · · · , n. On the other hand, we know by corollary that

pk,t(x) is decreasing in xj for all k. As a result, y(pt+1(x)) = ∆tµt+1(x) is a non-decreasing function in

xj or equivalently, ∆xjµt+1(x) is a non-decreasing function in t. So far, we proved the theorem when


pjt = 12

(ψj + ∆xjVt−1(x)− βt

(rt−1(x− ej)−∆xjµ

πt−1,s(x)

)). Now, let pjt = 1

2

(ψj + ∆xjVt−1(x)

).

Following the same procedure as before, we will calculate ∂y(pt(x))∂pkt(x) for k = 1, 2, · · · , n,

∂y(pt(x))

∂pkt(x)=

0 ≤ 0, j = 1, 2, · · · , n− 1;(−λt

ψn

)(rxt(xt − en)−∆xnµ

πt−1,s(x)) ≤ 0, j = n.

(A.3.6)

Then y(pt(x)) is again non-increasing in all its argument. Since pk,t(x) is decreasing in xj for all k,

therefore y(pt+1(x)) = ∆tµt+1(x) is a non-decreasing function in xj or equivalently, ∆xjµt+1(x) is a

non-decreasing function in t.

Proof of (ii). We need to prove ∆tµπt (x) ≥ ∆tµ

πt+1(x). Similar to the way followed in part (a), let

y(pt(x)) = ∆tµπt (x) and

∆tµπt (x) =

λtβt

n∑j=1

ρj(pt(x))(−2pjt + ψj + ∆xjVt−1(x))

pt(x) is increasing in t in each of its arguments. It guarantees that ∆tµ

πt (x) = y(pt(x)) is de-

creasing in t through property (1) of the composition of the multivariate functions. Now, if pjt =

12

(ψj + ∆xj

Vt−1(x)), Equation (A.3.6) holds and since the new pjt is increasing in t in each of its

arguments as well, then ∆tµπt (x) would be non-increasing function in t.


Proof of (i). According to the complementary slackness condition, βt(δs − µπt,s(x)

)= 0. We know

that if x ∈ Π, then (δs − µπt,s(x) 6= 0). As a result, βt = 0. Another word, the dual variable βt takes a

positive value when the constraint is binding. When the system is in one of the desirable states, it means

that the constraint would not be binding (it has already satisfied the constraint). Therefore, βt = 0 if

x ∈ Π.

Proof of (ii). As soon as a state becomes feasible, the probabilistic constraint would not be a binding

constraint for next periods. Then, if the there is a sufficiently large number of time period for the

milestone, there exists a decoupling period k such that the dual variable gets zero and remains zero to

the end of horizon.

Proof of Theorem 7.

Let’s assume t = 1, then merely those states are likely to achieve the desirable states set Π which

(x− ei) ∈ Π. Therefore, at period one the seller cannot generate any revenue through the states so that

(x−ei) /∈ Π. Clearly, the marginal revenue of inventories of any product type for the states (x−ei) /∈ Π

would be negative. On the other hand, as time goes by ∆xiVt(x) approaches to the quality of product


i, ψi which is positive. Akcay et al. (2010) proves that the marginal revenue of inventory ∆xiVt(x) is

a non-decreasing function of t. Hence, since we proved that there are positive and negative values for

∆xiVt(x), then there exists some t∗(x) such that ∆xi

Vt(x) ≤ 0 for t ≤ t∗(x) and ∆xiVt(x) > 0 for

t > t∗(x).

Proof of Corollary 1.

Proof of (i). To analyze the behavior of pjt over time, we should investigate the components of the

optimal price. Considering the following three facts about the components, we can simply prove that

pjt is a non-decreasing function in t.

(1) rt−1(x− ej)−∆xj

µπt−1,s(x) ≥ 0,

(2) βt ≥ 0 and is non-increasing function of t,

(3) ∆xjVt−1(x) is non-decreasing function of t.

Proof of (ii). The following reasons indicate that pjt is strictly decreasing in j. ∆xjVt−1(x) >

∆xj−1Vt−1(x), then the optimal price of the products with higher quality is always grater than the

lower quality products.

(1) ψj > ψj−1,

(2) ∆xjVt−1(x) > ∆xkVt−1(x) ∀ k > j,

(3) βt(rt−1(x− ej)−∆xjµπt−1,s(x)) = βt(rxt(xt − ej−1)−∆xj−1µ

πt−1,s(x)).

Proof of (iii). We need to show that pt(x) ≥ pt(x + ej). The following properties of the components

of pt(x) will indicates that the optimal price is non-increasing in xj .

(1) ∆xjVt−1(x) ≥ ∆xjVt−1(x + ej)

(2) βt has a decoupling point.


According to Corollary 1, price is a is a non-decreasing function of t. On the other hand,as time

goes by ∆xjVt−1(x) approaches to ψj and so according to the optimal price formula, the optimal price

approaches to ψj . Then the cut off points

Ωjt =pjt − pj+1,t

ψj − ψj+1


approaches to 1 when t becomes larger and larger. Therefore it implies that the probability of the jth

product purchase is a decreasing function of t.

Chapter 4

Dynamic Pricing Under Consumer’s

Consideration Sets

4.1 Introduction

Since the pioneering work of Howard and Sheth (1969), the notion of consideration sets has received

substantial attention in marketing and economics. The basic postulate is that when to make a purchase

decision, consumers consider only a subset of all the offered products as potential purchase options.

The resulting choice set is called consumers’ consideration set as opposed to the set of all available

products called consumers’ primary choice set. Consumers then inspect/evaluate the alternatives in the

consideration set and choose the most desirable product. Incorporating consumers’ consideration set

into the choice models is crucial as it has been shown that if consideration set formation is not taken

into account in models of choice, it leads to the underestimation of the impact of marketing control

variables (see e.g., Bronnenberg and Vanhonacker 1996 and Chiang et al. 1999).

Consumers employ a variety of mechanisms to screen the products and form their choice sets. The

applied methods may be the result of previous learning, information processing constraints, or solving

some previous constrained optimization problem (Gilbride and Allenby 2004). A wide spectrum of papers

(see e.g., Tversky 1972, Hauser and Wernerfelt 1990, Shapiro et al. 1997, Chakravarti and Janiszewski

2003, Hauser et al. 2010, and Trinh 2014) investigate how consumers construct their consideration sets.

In this chapter, we consider two prominent screening rules: (i) quality-based screening rule in which

consumers merely consider the products whose qualities (or brand) are above a predetermined threshold

(see, e.g., Kardes et al. 1993, Chakravarti and Janiszewski 2003, Gilbride and Allenby 2004, Erdem and

72

Chapter 4. Dynamic Pricing Under Consumer’s Consideration Sets 73

Swait 2004, Gilbride and Allenby 2006, Arora et al. 2011), and (ii) budget-based screening rule in which

the consumers decide to take a product into account if the product’s price does not exceed a certain

budget (or price) limit (see, e.g., Simonson et al. 1993, Divine 1995, Mehta et al. 2003, Gilbride and

Allenby 2004, Gilbride and Allenby 2006, Swait and Tulin 2007, Allenby et al. 2007, Arora et al. 2011).

Consideration set constitution is a fundamental step of prechoice decision making processes (Alba

et al. 1991, Ratneshwar and Shocker 1991). Nevertheless, works in dynamic pricing literature conven-

tionally assume that consumer evaluate all the available products and choose the one maximizing his

utility. Kim et al. (2010) in a study of camcorders sold on Amazon.com show that the median (average)

customers’ search set contains 11 (14) products, with about 40% of consumers stopping their search

at fewer than five products, out of a possible 90 total products. Therefore, it would seem that the

common assumption in revenue management studies that the consumer evaluates all available products

is unrealistic. One of our contributions is the inclusion of consumers’ consideration sets into the pricing

context.

In this paper, we consider a firm offering a line of vertically differentiated products with limited

inventory over a finite selling season1. The objective is to maximize the expected revenue by correctly

pricing the products over the selling season as a function of the existing inventory level and time.

Consumers randomly arrive at the firm and use the following two-stage decision making process to

make a choice: (i) consumers constitute a consideration/choice set including a subset of all the products

using a screening rule (e.g., brands, quality, and budget), and (ii) they evaluate the products in the

choice set to opt for the product with the maximum utility (if any). We formulate the dynamic pricing

problem as a discrete-time Markov decision process. Assuming a linear utility function, we first derive

the probability that a consideration set is chosen and then we find the purchase probability of products

for both quality and budget-based consumers. Finally, we examine the structural properties of the firm’s

revenue function and pricing decisions.

The contributions of this chapter are threefold. (i) We develop a model integrating probabilistic

consumers’ consideration sets into a well-established pricing problem; (ii) We investigate how pricing

decisions are affected when consumers’ consideration sets are taken into account in the consumers’

decision making process; and (iii) We show that the first order condition is necessary and sufficient for

setting the optimal price. We then find the optimal price of each product as a function of remaining

inventories and time periods to the end of horizon.

The rest of the paper is organized as follows. The next section addresses the related literature.

Section 4.2 provides the model formulation. Section 4.3 discusses the firm’s dynamic pricing problems.

1Products are said to be vertically differentiated if they can be ordered according to their objective quality. All productprices being equal, consumers always prefer products of higher quality.


Finally, Section 4.4 concludes the paper and presents directions for future research.

4.1.1 Literature Review

The theory of consumers’ consideration sets has been well-studied in economics and marketing. We

first classify the studies into two categories. In the first category, studies are developed to explain the

theoretical concepts of consideration set formations. In the second category, papers generally constitutes

of research developing specific models, usually grounded upon existing consideration set theories, to

estimate the consumers’ consideration sets or the size of the consideration sets. We then address the

papers in multi-product pricing literature.

Tversky (1972) developed the elimination by aspects (EBA) model to describe the process by which

consumers use the specific attribute levels of choice alternatives to alleviate the cognitive burden of

decision making. To also address the same process, Payne (1976) developed a two-stage screening

rule model with the first stage being EBA model (to lessen cognitive burden) and the second stage

being additive difference model (to evaluate comprehensively). Later on, Hauser and Wernerfelt (1990)

provided a model where the expected utility of consumption must exceed the cognitive cost of evaluation

for an alternative to be considered. Andrews and Srinivasan (1995) studied the dynamics of consideration

sets as consumers encounter information inside and outside of the store. Fotheringham (1988) examined

the possible existence of hierarchical decision-making in the choice of retail outlets through investigating

the fuzzy spatial choice sets. Shapiro et al. (1997) researched the effect of incidental ad exposures on the

consumer’s consideration set formation. Haubl and Trifts (2000) studied the impact of online assisting

interactive tools on both the quality and efficiency of purchase decisions formations. Mehta et al. (2003)

developed a structural model in which consumers’ optimal trade-offs between incurring search cost

and the potential benefits arising from price search are explored. Chakravarti and Janiszewski (2003)

investigated the influence of macro-level motives on the consumers’ employment of different types of

consideration set screening strategies. Erdem and Swait (2004) examined the role of brand credibility

on brand choice, and found that brand credibility increases the probability that a brand is included in

the consideration set. Irwin and Naylor (2009) addressed the influence of ethics, as an attribute, on the

consideration set formation in various circumstances. Hauser et al. (2010) developed the disjunctions-

of-conjunctions (DOC) decisions rules that are able to generalize decision models such as disjunctive,

conjunctive and lexicographic rules.

Desarbo and Jedidi (1995) advanced a scaling methodology to spatially represent preference intensity

and probability of each membership of consumers consideration sets. Rangaswamy and Wu (2003)

developed a two-stage choice model based upon the fuzzy set theory. The model reflects consideration


set formation process and offers insight into the influence of internal and external information acquisition

on consideration set formation. To assess consumers’ use of screening rules as part of a discrete choice

model, Gilbride and Allenby (2004) proposed a model that accommodates conjunctive, disjunctive, and

compensatory screening rules. Gilbride and Allenby (2006) introduced Bayesian methods for estimating

two behavioral models that eliminate alternatives using specific attribute levels. van Nierop et al.

(2010) proposed a model to capture unobserved consideration from discrete choice data. Arora et al.

(2011) developed a model that investigates both compensatory and non-compensatory aspects of the

joint decision process on dyadic choice. Liu and Dukes (2013) developed a modeling framework for

analyzing consumers’ considerations in two dimensions: within and across firms. Trinh (2014) proposed

a stochastic model that combines Poisson distribution with the lognormal distribution to determine the

possibility distribution of individual consumer’s consideration set sizes.

In multi-product pricing, Gallego and van Ryzin (1997) advanced heuristic approaches to price mul-

tiple products optimally in the context of a finite horizon, and exhibited such heuristic approaches are

asymptotically optimal. Customer’s choice model was considered by Talluri and van Ryzin (2004b) for

airline revenue management’s booking limit policies. Dong et al. (2009) utilized the multinomial logit

choice model in dynamic pricing of horizontally differentiated products. Akcay et al. (2010) examined

dynamic pricing problem in the presence of consumers’ choice set where the firm sells a set of multiple

substitutable products. Levin et al. (2009) created a dynamic pricing model for oligopolistic firms who

sell a set of differentiated perishable goods to strategic customers whose purchasing time are adjusting.

den Boer (2014) reviewed a multi-product dynamic pricing problem with infinite inventories where de-

mands for each product depends upon the proposed price and upon unknown parameters. Chen et al.

(2016) studied a multi-product and multi-resource revenue management problem and created heuris-

tics that provide minimal and flexible price adjustment during the time horizon. Li and Jain (2016)

contemplated the behavior-based pricing’s impact (i.e., price discrimination among consumers based on

preferences revealed from their purchase histories) on social welfare and the firm’s revenue. Du et al.

(2016) investigated a multiple-good pricing problem where customers’ purchasing decisions are based

upon the particular product’s total consumption in the market. Comprehensive literature reviews on the

dynamic pricing are available in Ozer and Phillips (2012), McGill and van Ryzin (1999), Elmaghraby and

Keskinocak (2003), Bitran and Caldentey (2003), Talluri and van Ryzin (2004a), and Phillips (2005).

Moreover, there is a wide spectrum of literature on dynamic pricing involving operational decisions.

Federgruen and Heching (1999) took into account both pricing and inventory replenishment policies

of a single product to maximize the total profit. Later on, Chen et al. (2011) integrated inventory

and pricing decisions of a firm where price adjustment is costly. Maglaras and Meissner (2006) in-


vestigated the relationship between dynamic pricing and capacity control for a firm utilizing a single

resource to manufacture multiple products. Aydin and Porteus (2008) examined the optimal prices and

inventory policies of multiple products in a given assortment in a newsvendor model. Brotcorne et al.

(2008) integrated design decisions into a pricing problem of a service firm. Lu et al. (2014) proposed a

quantity-based pricing strategy such that each time period inventory replenishment, unit selling price,

and the quantity-discount price are obtained. Borgs et al. (2014) investigated a service firm’s pricing

problem where capacity levels change over time and all customers are guaranteed to receive the service.

Federgruen and Hu (2015) studied a general price competition model for a set of substitutable products

and ascertained the equilibrium prices, product assortment, and sales volumes. Alptekinoglu and Sem-

ple (2016) investigated the utilization of a discrete choice model’s canonical version in joint pricing and

assortment planning. Our paper is unique among all the above-mentioned streams of literature in the

following aspect: Unlike all the aforesaid works, we consider consumers’ consideration set behavior in

the dynamic pricing problem.

4.2 The Model

We first introduce the firm’s problem. Then, we model the consumers’ behavior, followed by deriving

closed-form probability distributions of the two-stage purchase decisions.

The firm We consider a firm selling a line of vertically differentiated products, i.e., N = 1, 2, . . . , n

each indexed by j, over a finite selling horizon. We also consider a dummy product (n+1) as the no-buy

option. We indicate the quality of product j by Ωj where, Ωs > Ωk ∀s < k, and Ωn+1 ≡ 0. The

products’ quality does not deteriorate and have a fixed level over the selling season. We consider a

discrete time horizon composed of T time periods, each indexed by t, where t is labeled backward. The

firm commences the sales season with a certain level of inventory and is unable to replenish that during

the season. Let the n-vector x = (x1, x2, . . . , xn) show the inventory state whose j-th element is the

inventory of product j. The firm’s objective is to maximize its revenue by optimally choosing the price

vector pt = (p1t, p2t, . . . , pnt) given time, inventory state, and the consumers’ behaviors.

The consumers Customers arrive randomly at the firm with a known probability Γt at time t,

and purchase at most one unit/consumer of the chosen product (if any). Consumers may apply vari-

ous decision processes to make purchase decisions. Consistent with the marketing literature (see, e.g.,

Rangaswamy and Wu 2003, Gilbride and Allenby 2004, Gilbride and Allenby 2006 and Swait and Tulin

2007), we model the consumers’ choice behavior as a two-stage decision process, in which consumers

(i) choose a subset of products from the global set of all the available alternatives (i.e., consideration


stage), and then (ii) evaluate the products in the subset to select the product with the maximum gener-

ated utility (i.e., choice stage). The consumers’ utility gained by purchasing product j at time t is the

following:

ujt = µΩj − pjt, j ∈ N , (4.1)

where µ is a uniform random variable over [0, 1]. Uniform distribution has been commonly used in the

literature such as Bresnahan 1987 and Akcay et al. 2010 to capture heterogeneity among customers in

terms of relative importance between the quality and the price. In addition, we note that the linear

utility function given by (4.1) has been widely used in management science and economics literature

(see, e.g., Anderson et al. 1992, Bresnahan 1987, Roberts and Lilien 1993, Caplin and Nalebuff 1991,

Train 2003, Hensher and Greene 2003, Berry and Pakes 2007, Tirole 1988, Wauthy 1996, Bhargava and

Choudhary 2008, and Berry 1994). We assume that the no-purchase decision yields a zero utility.

Screening rules and consideration sets In order to form the consideration set (i.e., the first stage

of the consumers’ purchase process), consumers restrict the alternatives using screening rules. In this

paper, we consider the most generic screening rules which has been widely considered in marketing and

economics literature: quality-based and price-based screening rules (see, e.g., Arora et al. 2011, Gilbride

and Allenby 2004, Mehta et al. 2003 Kardes et al. 1993, Roberts and Lattin 1991). Let k indicate the

type of the consumers, where k = 1 and k = 2 represent the consumers who are quality-based and

price-based, respectively. Similar to Gilbride and Allenby (2004), we specify the products in the choice

set of the consumers with an indicator function as follows:

I(zkj , γ

k)

=

1 if the decision rule of the consumers type k is satisfied for the offered product j,

0 otherwise,

where zkj is the general argument of the indicator function I reflecting the decision rule that the consumers

type k apply to the j-th alternative, and γk is the consumers of type k’s predetermined level by which

they decided whether or not to consider the product in the choice set. If consumers are assumed to be

quality-based (i.e., k = 1), then the consideration set includes only the products above a predetermined

quality threshold, Ω. Hence, the indicator function I(z1j , γ

1) is

I(z1j

∆= Ωj , γ

1 ∆= Ω

)=

1 if Ωj ≥ Ω,

0 otherwise.

As the minimum acceptable quality is not deterministic from the firm’s perspective, let Ω be a random

variable with a cumulative distribution F (Ω) and density f(Ω). We can also take into account an upper


bound on the products’ quality above which products are not considered in the consumers’ consideration

set. However, for the expositional simplicity we use only the lower bound Ω for the quality threshold.

If price is applied by consumers to screen the products (i.e., k = 2), the products under a prespecified

threshold, p, are only qualified to be taken into consideration. Then the indicator function I(z2j , γ

2) will

be

I(z2j

∆= pjt, γ

2 ∆= p)

=

1 if pjt ≤ p,

0 otherwise.

As previously mentioned, the price (budget/income) threshold is not known (from the seller’s perspec-

tive). Hence, we let p be a random variable with a cumulative distribution G(p) and density g(p). The

consideration sets are defined as follows:

Cki =i ∈ N : I(zki , γk) = 1

,

where Cki is the i-th possible consideration set of consumers type k. Recall that, (i) products are sorted in

a descending order of quality, i.e., Ω1 > Ω2 > . . . > Ωn, and (ii) the products are vertically differentiated,

i.e., p1t > p2t > . . . > pnt if Ωs > Ωk ∀s < k ∈ N . Therefore, the consideration sets of quality-based and

priced-base consumers are

C1i = N/i, i+ 1, . . . , n = 1, 2, . . . , i ,

C2i = N/1, 2, . . . , i− 1 = i, i+ 1, . . . , n .

We also note that, for any choice set Cki , there is a chance of consumer’s no-purchase option, namely, the

consumers evaluate the products in their consideration sets and if they cannot find a desirable product

they leave without any purchase.

Probability of products’ choice Next, we calculate the consumers’ probability of purchase fol-

lowing the above-mentioned two-stage choice process. Let λk(j, Cki

)denote the purchase probability of

product j by a consumer type k whose consideration set is Cki

λk(j, Cki

)= αk(Cki ) βk(j|Cki ),

where αk(Cki ) is the probability that a consumer type k chooses the consideration set Cki (the first-stage

probability), and βk(j|Cki ) is the probability that a consumer buys product j given Cki (the second-

stage probability). In what follows, the first-stage probability of both the consumers of types 1 and 2 is


presented. If k = 1, then we have

α1(C1i ) = P

(C1i = 1, 2, . . . , i

)= P (Ω1 > Ω, . . . ,Ωi ≥ Ω,Ωi+1 < Ω, . . . ,Ωn < Ω)

= P (maxΩi+1, . . . ,Ωn < Ω < minΩ1, . . . ,Ωi)

= P (Ωi+1 < Ω ≤ Ωi) = F (Ωi)− F (Ωi+1).

(4.2)

If the consideration set is empty, then the corresponding probability would be α1(∅) = 1 − F (Ω1).

Similarly, in the case where k = 2, the choice set probability would be:

α2(C2i ) = P

(C2i = n, n− 1, . . . , i

)= P (pnt < p, . . . , pit ≤ p, pi−1t > p, . . . , p1t > p)

= P (maxpnt, . . . , pit < p < minpi−1t, . . . , p1t)

= P (pit ≤ p < pi−1t) = G(pi−1t)−G(pit) i = 2, . . . , n,

(4.3)

and for i = 1 (i.e., C21 = N = 1, 2, . . . , n), we will have α2(C2

1) = 1−G(p1t). In addition, the probability

that the consideration set is empty is α2(∅) = G(pnt). To compute the choice probability of a specific

product j, λk(j, Cki

), we need to obtain the consumers’ second-stage probability βk(j|Cki ) as follows:

βk(j|Cki ) = P

(j = argmax

l∈Cki ,n+1ult

). (4.4)

Invoking the utility function given in (4.1), we have

βk(j|Cki ) = P(ujt > ult, ∀l ∈ Cki

)= P

(µ(Ωj − Ωl)≥ pjt − plt,∀l ∈ Cki

),

(4.5)

We note that (Ωj − Ωl) < 0 ∀l ∈ Cki : j > l, and (Ωj − Ωl) > 0 ∀l ∈ Cki : j < l. We express Equation

(4.5) in the following form

βk(j|Cki ) = P(µkjt

(Cki ) ≤ µ ≤ µkjt(Cki ))

= µkjt(Cki )− µkjt

(Cki ),

where, (µ1jt

(C1i ), µ1

jt(C1i ))

=

(max

∀l∈Cki :j>l

pkt − pltΩk − Ωl

, min∀l∈Cki :j<l


). (4.6)

As can be seen from (4.6), in order to have a non-negative purchase probability (i.e., βk(j|Cki ) ≥ 0), we


should have

pjt − pj+1t

Ωj − Ωj+1≤ max∀l∈Cki :j>l


≤ min∀l∈Cki :j<l


≤ pj−1t − pjt

Ωj−1 − Ωj.

(4.7)

Thus, from (4.7) the selected prices should be set such away that

plt − pl+1t

Ωl − Ωl+1≥ pl+1t − pl+2t

Ωl+1 − Ωl+2≥ 0, ∀l ∈ Cki . (4.8)

That is, in order for the products in the consideration set to have all non-negative chance of purchase,

the difference in the prices of any two adjacent products to their quality differences is decreasing. From

(4.6) and considering (4.8), the purchase probabilities of products given a non-empty consideration set

Cki can be explicitly calculated as follows: (i) If the consumers are assumed to be quality-based (i.e.,

k = 1), and xl > 0 ∀l ∈ Cki then

(µ1jt

(C1i ), µ1

jt(C1i ))

=

(0, 0), if j /∈ C1i ;(

pjtΩj

, 1

), if j ∈ C1

i ∧ j = i = 1;(pjt − pj+1t

Ωj − Ωj+1, 1

), if j ∈ C1

i ∧ j = 1 ∧ i ≥ 2;(pjt − pj+1t

Ωj − Ωj+1,pj−1t − pjtΩj−1 − Ωj

), if j ∈ C1

i ∧ 2 ≤ j ≤ i− 1;(pjtΩj

,pj−1t − pjtΩj−1 − Ωj

), if j ∈ C1

i ∧ 2 ≤ j = i ≤ n;(0,pitΩi

), if j = n+ 1,

(4.9)

and (ii) in the case where consumers are budget-based (i.e., k = 2), and xl > 0 ∀l ∈ Cki then

(µ2jt

(C2i ), µ2

jt(C2i ))

=

(0, 0), if j /∈ C2i ;(

pjt − pj+1t

Ωj − Ωj+1, 1

), if j ∈ C2

i ∧ j = i < n;(pjt − pj+1t

Ωj − Ωj+1,pj−1t − pjtΩj−1 − Ωj

), if j ∈ C2

i ∧ i+ 1 ≤ j ≤ n− 1;(pjtΩj

,pj−1t − pjtΩj−1 − Ωj

), if j ∈ C2

i ∧ j = n ∧ i < n;(pjtΩj

, 1

), if j ∈ C2

i ∧ j = i = n;(0,pitΩi

), if j = n+ 1.

(4.10)


Clearly, if the choice set is empty then the consumers certainly leave the store without purchase (i.e.,

βk(n+1|∅) = 1). We note that, if xl = 0 for any l ∈ Cki , (i) the products are excluded from the available

products, and (ii) to find the purchase probabilities in (4.9) and (4.10), products (j + 1) and (j − 1) are

replaced with the adjacent product of product j with positive inventory.

4.3 The Firm Optimization Problem

The firm’s objective is to maximize its expected revenue given the inventory state x and remaining time

periods to the end of horizon, t. Let V kt (x, Cki ) be the marginal expected revenue of the firm if a consumer

chooses a specific non-empty consideration set Cki . The marginal expected revenue can be expressed as

follows:

V kt (x, Cki ) = maxpt

Γtα

k(Cki )

( ∑j∈Cki

βk(j|Cki ) (pjt + Vt−1(x− ej)) + βk(n+ 1|Cki )Vt−1(x)

)

+ (1− Γt)Vt−1(x)

, k = 1, 2.

(4.11)

A possible scenario is that the consumer cannot find any eligible product to take into consideration (i.e.,

the consideration set is empty). In this case, the firm’s marginal revenue would be constant and is given

by:

V kt (x,∅) =(Γt(α

k(∅)− 1) + 1)Vt−1(x), k = 1, 2. (4.12)

Equation (4.11) can be explained as follows. A consumer arrives at the store with probability Γt at

time t. The consumer chooses a consideration set Cki with probability αk(Cki ). Upon constituting the

consideration set, a product j ∈ Cki is purchased with probability βk(j|Cki ). The expected revenue

corresponding to a product j ∈ Cki is (βk(j|Cki )(pjt + Vt−1(x− ej)). Therefore, the firm’s total expected

revenue is the sum over the expected revenues generated by all the products in the choice set. Finally, a

consumer may not arrive at period t with probability (1− Γt),in which case, the firm does not sell any

product and only expects to gain V1,t−1(x). The following theorem states that the marginal expected

revenue function has a unique optimal solution in the case where consumers are assumed to be quality-

based type.

Theorem 8. The marginal expected revenue function has the following properties:

(i) Suppose that k = 1 (i.e., consumers form the choice sets based on products’ quality), then for any

probability distribution F of the minimum quality threshold Ω, the marginal expected revenue function

in (4.11) is concave in pt.


(ii) Suppose that k = 2 (i.e., consumers form the choice sets based on products’ price). The marginal

expected revenue function in (4.11) is log-concave in pt if αk(Cki ) is log-concave.

Next, we present the total expected revenue of the firm. Given marginal expected revenue in (4.11),

the total expected revenue function is obtained as follows:

V kt (x) = maxpt

Γt∑i

αk(Cki )

( ∑j∈Cki

βk(j|Cki )(pjt + V kt−1(x− ej)

)+ βk(n+ 1|Cki )V kt−1(x)

)

+ (1− Γt)Vkt−1(x) + Γtα

k(∅)V kt−1(x)

, k = 1, 2,

(4.13)

where the boundary conditions are:

V10(x) = 0, ∀ x,

V1t(0) = 0, ∀ t = T, · · · , 1.

Optimality equation in (4.13) can be explained similar to that of Equation (4.11). The only difference

is the inclusion of all the possible consideration sets. Hence, it is required to sum over the choice sets,

Cki ∀i, and to take into account the possibility of the consumers’ empty choice set to form the Equation

(4.13). The boundary conditions are self explanatory. We note that, as there can be inventory states

x : ∃ m ∈ N : xm = 0, therefore there would be∑n−1l=0

(nl

)types of optimality equations (4.13). This

implies that product of zero inventory is removed from the available products set and the new optimality

equation is constituted based on the available products with positive inventory.

Corollary 2. If consumers constitute the choice sets based on products’ quality, for any probability

distribution F of the minimum quality threshold Ω, the following holds:

(i) The total expected revenue function (4.13) is concave in pt.

(ii) The optimal price satisfies

∂V kt (x)

∂pjt= Γt

∑i

αk(Cki )∑l∈Cki

(∂βk(l|Cki )

∂pjt(plt + V kt−1(x− el))

+ βk(l|Cki ) +∂βk(n+ 1|Cki )

∂pjtV kt−1(x)

)= 0, j = 1, 2, · · · , n.

(4.14)

Corollary 2 guarantees the uniqueness of the optimal prices through the concavity of the optimality

equation in (4.13). The condition stated in part (ii) is the first-order condition that is sufficient for the

optimality of prices given concavity. To constitute the system of equations given in (4.14) for k = 1,


1 2 3 4 5 69.5

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

Remaining time

Op

tim

al p

rice

of

pro

du

ct

1

x = (1,1)

x = (1,2)

x = (2,1)

x = (1,0)

x = (2,0)

x = (2,2)

Figure 4.1: Optimal price of product 1 withvarious inventory levels for quality-based con-sumers where Ω ∼ U [0, 25]

1 2 3 4 5 6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Remaining time

Op

tim

al p

rice

of

pro

du

ct

2

x = (0,1)

x = (2,1)

x = (0,2)

x = (1,1)

x = (2,2)x = (1,2)

Figure 4.2: Optimal price of product 2 withvarious inventory levels for quality-based con-sumers where Ω ∼ U [0, 25]

we take the derivative of the purchase probabilities as follows. Thus, if i > 1 the following shows the

Jacobian matrix:

∂βk(Cki )

∂pt=

−1Ω1−Ω2

1Ω1−Ω2

0 0 0 · · · 0

1Ω1−Ω2

[−1

Ω1−Ω2− 1

Ω2−Ω3

]1

Ω2−Ω30 0 · · · 0

0 1Ω2−Ω3

[−1

Ω2−Ω3− 1

Ω3−Ω4

]1

Ω3−Ω40 · · · 0

......

.... . .

. . .. . .

...

0 0 · · · 0 0 1Ωi−1−Ωi

[−1

Ωi−1−Ωi− 1

Ωi

]

i×i

,

(4.15)

where each row of (4.15) represents

∂βk(l|Cki )

∂pt=

(∂βk(l|Cki )

∂p1t,∂βk(l|Cki )

∂p2t, · · · , ∂β

k(l|Cki )

∂pit

), l = 1, 2, · · · , i. (4.16)

If i = 1, then∂βk(l|Cki )∂pjt

= − 1Ω1

for j = 1, and zero otherwise. Finally, if l = n + 1, then∂βk(l|Cki )∂pjt

= 1Ωi

for j = i, and zero otherwise. We illustrate the optimal pricing procedure in the presence of consumers’

consideration sets through the following examples.

Example 8. Consider a firm offering two products 1 and 2 with qualities Ω1 = 20 and Ω2 = 8 over

a sales season of T = 6. In addition, suppose that the consumers are quality-based, and Ω follows a

uniform distribution over [0, 25]. Let also Γt = 0.8 for all t, and the initial inventory level be x = (2, 2).

The consumers’ consideration sets are C11 = 1, C1

2 = 1, 2, and the empty set ∅. In addition, given


1 2 3 4 5 60

2

4

6

8

10

12

14

16

18

Remaining time

Op

tim

al re

ve

nu

e V

t(x)

The optimal revenues of x=(2,2) and x =(2,1)nearly coincide.

x = (1,2)

x = (2,0)

x = (1,0)x = (1,1)

x = (0,2) x = (0,1)

Figure 4.3: Revenue with different inventory states for quality-based consumers where Ω ∼ U [0, 25]

(4.13) the optimality equations V kt (x) can be expressed as follows: (i) For x1 and x2 > 0,

V 1t (x) = max

pt

0.8

(8

25

)[(1− p1t − p2t

12

)(p1t + V 1

t−1(x− e1)) +

(p1t − p2t

12− p2t

8

)(p2t

+ V 1t−1(x− e2)) +

(p2t

8

)V 1t−1(x)

]+ 0.8

(12

25

)[(1− p1t

20

)(p1t + V 1

t−1(x− e1))

+(p1t

20

)V 1t−1(x)

]+ 0.2V 1

t−1(x) + 0.8

(5

25

)V 1t−1(x)

,

(4.17)

(ii) for x1 > 0 and x2 = 0,

V 1t (x) = max

p1t

0.8

(20

25

)[(1− p1t

20

)(p1t + V 1

t−1(x− e1)) +(p1t

20

)V 1t−1(x)

]

+ 0.2V 1t−1(x) + 0.8

(5

25

)V 1t−1(x)

,

(4.18)

and (iii) for x1 = 0 and x2 > 0,

V 1t (x) = max

p2t

0.8

(8

25

)[(1− p2t

8

)(p2t + V 1

t−1(x− e2)) +(p2t

8

)V 1t−1(x)

]

+ 0.2V 1t−1(x) + 0.8

(17

25

)V 1t−1(x)

.

(4.19)

Equations (4.17)-(4.19) can be optimized through Corollary 2. Figures 4.1 and 4.2 demonstrate the

optimal price of product 1 and 2 for different inventory states, respectively. As can be seen from the


1 2 3 4 5 67

8

9

10

11

12

13

Remaining time

Optim

al p

rice

of pro

duct

1

x = (1,1)

x = (1,2)

x = (2,2)

x = (2,1)

x = (2,0)

x = (1,0)

Figure 4.4: Optimal price of product 1 withvarious inventory levels for budget-based con-sumers where p ∼ U [0, 25]

1 2 3 4 5 6

3

3.5

4

4.5

5

5.5

6

Remaining time

Op

tima

l price

of

pro

du

ct 2

x = (0,1)

x = (0,2)

x = (1,1)

x = (2,1)

x = (1,2) x = (2,2)

Figure 4.5: Optimal price of product 2 withvarious inventory levels for budget-based con-sumers where p ∼ U [0, 25]

figures, the optimal prices of both the products are increasing in the remaining time. In addition, the

optimal prices are (weakly) decreasing as the products’ inventory level increases. Figure 4.3 shows the

optimal revenue of the firm for different inventory states over the planning horizon. The interesting

result is that, as consumers are quality-sensitive and prefer to consider high quality products, revenue

contribution of product 1 is much higher than that of product 2. For example, the revenue generated

for x = (2, 2) and x = (2, 1) (one less inventory of product 2) are almost the same, while the revenue

obtained for x = (1, 2) (one less inventory of product 1 compared to x = (2, 2)) leads to a substantial

reduction in revenue.

Example 9. Consider Example 8. Suppose that the consumers are budget-based, and p ∼ U [0, 25].

Following (4.13), the optimality equations are: (i) For x1 and x2 > 0,

V 2t (x) = max

pt

0.8(

1− p1t

25

)[(1− p1t − p2t

12

)(p1t + V 2

t−1(x− e1)) +

(p1t − p2t

12− p2t

8

)(p2t

+ V 2t−1(x− e2)) +

(p2t

8

)V 2t−1(x)

]+ 0.8

(p1t − p2t

25

)[(1− p2t

8

)(p2t + V 2

t−1(x− e2))

+(p2t

8

)V 2t−1(x)

]+ 0.2V 2

t−1(x) + 0.8(p2t

25

)V 2t−1(x)

,

(4.20)


1 2 3 4 5 60

2

4

6

8

10

12

14

16

Remaining time

Op

tim

al re

ve

nu

e V

t(x)

x = (2,0)

x = (2,2)

x = (2,1)

x = (1,2)x = (1,1)

x = (1,0) x = (0,1)x = (0,2)

Figure 4.6: Revenue with different inventory states for budget-based consumers where p ∼ U [0, 25]

(ii) for x1 > 0 and x2 = 0,

V 2t (x) = max

p1t

0.8(

1− p1t

25

)[(1− p1t

20

)(p1t + V 2

t−1(x− e1)) +(p1t

20

)V 2t−1(x)

]

+ 0.2V 2t−1(x) + 0.8

(p1t

25

)V 2t−1(x)

,

(4.21)

and (iii) for x1 = 0 and x2 > 0,

V 2t (x) = max

p2t

0.8(

1− p2t

25

)[(1− p2t

8

)(p2t + V 2

t−1(x− e2)) +(p2t

8

)V 2t−1(x)

]

+ 0.2V 2t−1(x) + 0.8

(p2t

8

)V 2t−1(x)

.

(4.22)

Figures 4.4 and 4.5 demonstrate the optimal price of product 1 and 2 for different inventory states,

respectively. A counter-intuitive result is that, unlike the quality-based model, when the aggregate

inventory level decreases it is possible for the optimal price of a product to decrease as well. For example,

right before period t = 2, the optimal price of product 1 for the inventory state x = (1, 0) is lower than

that of x = (1, 1). Moreover, Figure 4.6 shows the optimal revenue of the firm for different inventory

states over the planning horizon. Similar to the quality-based model, as consumers are price-sensitive

and willing to consider less expensive products, revenue contribution of product 2 is higher than that of

product 1.

Example 10. This example considers the optimal prices and revenues generated in the presence of


1 2 3 4 5 67.5

8

8.5

9

9.5

10

10.5

11

11.5

12

12.5

Remaining time

Op

tima

l price

of p

rod

uct

1

Quality−based consumers

Budget−based consumers

No consideration set

Figure 4.7: Optimal price of product 1 whereΩ, p ∼ U [0, 25], with that of no considerationset model

1 2 3 4 5 63

3.5

4

4.5

Remaining time

Op

tima

l price

of

pro

du

ct 2




Figure 4.8: Optimal price of product 2 whereΩ, p ∼ U [0, 25], with that of no considerationset model

consumers’ consideration sets and in the case where that is not taken into account. All the values are

the same as Examples 8 and 9. Figures 4.7, 4.8, and 4.9 compare the optimal price of product 1, product

2, and the optimal revenue when x = (2, 2), respectively. As can be seen from the figures, there is a

significant difference among the optimal prices and the resulting revenues which implies the necessity of

the assimilation of consideration sets concept to pricing models. The figures show that the optimal prices

and revenues of the budget-based model is less that that of quality-based model and the result without

inclusion of consideration set (this may differ as the parameter of the distribution of Ω and p changes).

In addition, Figure 4.10 reflects the revenue obtained from quality and budget-based consumers when

the upper-bound l in the distribution of Ω and p ∼ U [0, l] varies. Through Figure 4.10, we can find

the equivalent values of l for quality and budget-based models in terms of the revenue generation. For

example, l ≈ 27 is one of the equilibrium points at which both the models result in the same amount of

revenue.

4.4 Conclusion

In this chapter, we consider a firm offering vertically differentiated products over a finite sales season.

Consumers arrive at the firm randomly and apply a two-stage choice model to purchase a product (if any),

i.e., (i) constituting the consideration set by screening the products based on the desired criteria/features

(e.g., quality attributes, brands, price, etc); and (ii) choosing a product from the consideration set that

has the maximum utility level. On the other hand, the firm requires to solve its revenue maximization


1 2 3 4 5 60

5

10

15

20

25

Remaining time

Op

tima

l re

ven

ue

Vt(x

)




Figure 4.9: Revenue comparison of quality andbudget-based models with the no considerationset model where Ω, p ∼ U [0, 25], and x = (2, 2)

21 22 23 24 25 26 27 28 29 302.6

2.8

3

3.2

3.4

3.6

3.8

4

l

V1k (2

,2)

V12(2,2) of budget−based consumers

V11(2,2) of quality−based consumers

Figure 4.10: Revenue generated for quality andbudget-based models where the upper bound lchanges in Ω, p ∼ U [0, l] for x = (2, 2)

problem given the two-stage consumers’ choice behavior. As a result, the firm wishes to determine the

optimal price of each product at each time (given all the products’ inventory level). We show that the

optimal prices of products where consumers constitutes their consideration set through products’ quality

can be analytically obtained using the first order condition. The research in this chapter has limitations.

First, if consumers are assumed to consider the offered prices as the criterion for the consideration set

formation, then the optimization of the revenue function will be highly complex and deriving analytical

solution is complicated. Second, examining the problem when the consumers may constitute their

consideration set based on a mixture of products’ price and quality is also another interesting direction

for future work.

As the incorporation of consumers’ consideration sets into the pricing and revenue management

problems seems to be necessary, we hope the modeling approach in this chapter can serve as a basis for

many promising research directions beyond this work.

4.5 Proofs

Proof of Theorem 8

Proof of (i). The proof is by induction on i. Let i = 1 (i.e., only product 1 is qualified to be consid-

ered in the consumer’s choice set), then from (4.2) the corresponding probability of the consideration set

C11 = 1 would be α1(C1

1) = F (Ω1)−F (Ω2). In addition, from (4.9), the purchase probability of the the

products considered in the choice set would be β1(1|C11) = 1− p1t

Ω1and β1(n+ 1|C1

1) = p1tΩ1

. Considering


the alluded consideration and choice probabilities, the optimality Equation (4.11) will reduce to the

following:

Vt(x|C11) = max

pt

Γt (F (Ω1)− F (Ω2))

((1− p1t

Ω1

)(p1t + Vt−1(x− e1)) +

(p1t

Ω1

)Vt−1(x)

)+

(1− Γt)Vt−1(x)

, k = 1, 2.

The corresponding first-order derivative would be:

∂Vt(x, C11)

∂p1t= Γt (F (Ω1)− F (Ω2))

(1− p1t

Ω1− p1t + Vt−1(x− e1)

Ω1+Vt−1(x)

Ω1

).

Accordingly,

∂2Vt(x, C11)

∂p21t

= Γt (F (Ω1)− F (Ω2))

(−2

Ωj

)≤ 0

Note that since Ω1 > Ω2, therefore F (Ω1) > F (Ω2). As the second-order derivative is always non-

positive, then it is proven that the expected revenue function is concave if i = 1. Now, we prove that

the expected revenue function is concave for any i = k for 2 ≤ k ≤ n. The expected revenue function is

concave in pt(x), if the Hessian matrix is negative semi-definite (NSD).

∂Vt(x, C1i )

∂p1t= Γt (F (Ωi)− F (Ωi+1))

[(1− p1t − p2t

Ω1 − Ω2

)+

(1

Ω1 − Ω2

)((p2t − p1t)− (∆x2Vt−1(x)−∆x1Vt−1(x)))

].

For 2 ≤ j ≤ k − 1, we will have

∂Vt(x, C1i )

∂pjt= Γt (F (Ωi)− F (Ωi+1))

[(pj−1,t − pjtΩj−1 − Ωj

− pjt − pj+1,t

Ωj − Ωj+1

)+

(1

Ωj−1 − Ωj

)((pj−1,t − pjt)− (∆xj−1

Vt−1(x)−∆xjVt−1(x)))

+

(1

Ωj − Ωj+1

)((pj+1,t − pjt)− (∆xj+1

Vt−1(x)−∆xjVt−1(x)))

],

and finally, for j = i,

∂Vt(x, C1i )

∂pjt= Γt (F (Ωi)− F (Ωi+1))

[(pi−1,t − pitΩi−1 − Ωi

− pitΩi

)+

(1

Ωi−1 − Ωi

)((pi−1,t − pit)− (∆xi−1

Vt−1(x)−∆xiVt−1(x)))

−(

1

Ωi

)(pit −∆xiVt−1(x))

].


Now, according to the calculated first order derivative we can constitute the Hessian matrix as follows:

∂2Vt(x, C1i )

∂2pt= Γt(F (Ωi)−F (Ωi+1))

−2Ω1−Ω2

2Ω1−Ω2

0 0 0 · · · 0

2Ω1−Ω2

[−2

Ω1−Ω2+ −2

Ω2−Ω3

]2

Ω2−Ω30 0 · · · 0

0 2Ω2−Ω3

[−2

Ω2−Ω3+ −2

Ω3−Ω4

]2

Ω3−Ω40 · · · 0

......

.... . .

. . .. . .

...

0 0 · · · 0 0 2Ωi−1−Ωi

[−2

Ωi−1−Ωi+ −2

Ωi

]

i×i

(4.23)

The Hessian matrix (4.23) is a symmetric tridiagonal matrix which can be negative semi-definite under

if (i) The matrix is diagonally dominant, and (ii) Diagonal entries are all non-positive. Condition (i)

holds, as the magnitude of the main diagonal is greater than or equal to the sum of the off-diagonal

entries , and condition (ii) is trivial. Therefore, the Hessian matrix (4.23) is NSD and the expected

revenue function is concave in price.

Proof of (ii). Let

Π(x, Cki ) =∑j∈Cki

βk(j|Cki ) (pjt + Vt−1(x− ej)) + βk(n+ 1|Cki )Vt−1(x).

Following the proof of part (i), it is clear that the function Π(x, Cki ) is concave in pt(x). Given that

(1) any concave function is log-concave and (2) the product of two log-concave functions would also

be log-concave (Boyd and Vandenberghe 2004), it is proved that for k = 2, the function V kt (x, Cki ) =

Γtαk(Cki )Π(x, Cki ) + (1− Γt)Vt−1(x) is log-concave if the αk(Cki ) is log-concave.

Proof of Corollary 2

(i) In Theorem 8, we proved that for any specific choice set C1i , the expected revenue function is

concave. We know that the sum over a series of concave functions is concave, as well. Therefore, if If

k = 1, then the total revenue function (4.13) is concave in pt. (ii) Given part (i), the first order condition

would be the sufficient condition for the optimal solution. Therefore, the following holds:

∂V kt (x)

∂pjt= Γt

∑i

αk(Cki )∑l∈Cki

(∂βk(l|Cki )

∂pjt(plt + V kt−1(x− ej))

+ βk(l|Cki ) +∂βk(n+ 1|Cki )

∂pjtV kt−1(x)

)= 0, j = 1, 2, · · · , n.

Chapter 5

Conclusion

We consider a variety of pricing problems of a firm selling vertically differentiated products over a finite

horizon. In Chapter 2, we consider a consumer inspecting products sequentially to find a product to

purchase. At each stage of the search process, the consumer compares the incremental expected utility

with the loss of utility (or search cost) related to inspecting one additional product. Consequently, the

consumer may stop the search before inspecting all the products. The firm aims at maximizing the

expected revenue given the consumer’s search behavior. This arises the optimal sequencing and pricing

problem of the products, i.e., obtaining the best sequence of products’ presentation and the optimal

price of each product as a function of remaining time and inventory. We show that it is optimal for the

firm to display the products in the descending order of quality if the reservation utility is non-decreasing.

We also determine the closed-form solution for the optimal prices of products through which we derive

interesting managerial insights. We show that unlike the conventional pricing and revenue management

models, it is possible for the optimal price of a perishable product to increase over time. Moreover,

we extend the basic model to the case where the firm offers horizontally differentiated products, and

where consumers might choose random search sequences unknown to the firm. The research presented

in this chapter has limitations. First, if the sequencing is done by consumers and unknown to the firm,

the complexity of the optimal pricing problem becomes an issue for deriving analytical results. Second,

the stationary or increasing reservation utility is a critical assumption to reduce the complexity of the

optimization problem. Although this assumption holds true in several real search settings, it is unlikely

to be a good universal assumption for all search scenarios. In this paper, we retained it mostly for

analytical tractability. Third, we consider dynamic pricing of perishable products. Studying a firm

selling a set of non-perishable products would be an interesting direction. Finally, we mostly focused

on either vertically or horizontally differentiated products. Exploring the problem when the consumers

91

Chapter 5. Conclusion 92

may search a mixture of vertically and horizontally differentiated products is an interesting direction.

In Chapter 3 we consider a firm selling a line of substitutable products and is subject to a set of

sales milestone constraints that needs to be satisfied over the sales horizon. Due to stochastic nature

of demand, the firm specifies a maximum level of risk of not satisfying the constraints. The model

allows the firm to establish a balance between maximization of the expected total revenues and the risk

of not satisfying the constraints (i.e., losing a targeted market share). We formulate the problem as a

chance-constrained Markov decision process in which the firm wishes to obtain the optimal products’

prices simultaneously to maximize the expected profit, and achieve the sales constraints. We manage

to demonstrate that the KKT conditions are necessary and sufficient for the optimal price, and derive

the closed-form solution for the optimal price. This paper also explores a detailed analysis of the

structural properties of the model and the optimal pricing policy. Of particular interest is the following:

(i) Joint penetration-skimming pricing strategy is optimal, namely, firms which intend to implement

the penetration pricing policy may optimally end up the implementation of skimming pricing policy, if

there is sufficiently large periods to the milestone. In other words, there is a decoupling point before

which skimming pricing policy is optimal, and after which penetration pricing policy is optimal; (ii) Due

to the existence of the sales constraints, it is possible for firms to generate lower revenue from higher

inventory level. We also present a model incorporating decision makers’ degree of risk aversion. To

demonstrate how our model can be used in real cases, we implement our proposed pricing strategy for a

leading Canadian condominium developer (CCD). We compare the result generated by our model with

the pricing policy of CCD, and show a significant improvement in the firm’s profitability. This research

has also restrictions. Firs, incorporating different consumers’ choice behavior models may change the

optimal price properties. For example, customers’ sequential search can be integrated with the basic

pricing model. Second, considering a revenue constraint (i.e., a minimum revenue needs to be achieved

at different milestones) can also be interesting, as there are many cases in which firms sell a set of

substitutable products, and have both revenue and sales milestone constraints. Third, the problem can

be modeled as an infinite horizon MDP, as firms may have a line of non-perishable products with sales

milestone constraints.

In Chapter 4, we consider a firm offering perishable vertically differentiated products over a finite sales

season. Consumer applies a two-stage choice model to purchase a product (if any), i.e., (i) constituting

the consideration set by screening the products based on the desired criteria/features (e.g., quality

attributes, brands, price, etc); and (ii) choosing a product from the consideration set that has the

maximum utility level. On the other hand, the firm requires to solve its revenue maximization problem

given the two-stage consumers’ choice behavior. As a result, the firm wishes to determine the optimal

Chapter 5. Conclusion 93

price of each product at each time (given all the products’ inventory states). We show that the optimal

prices of products where customers constitutes their consideration set through products’ quality, can be

analytically obtained using the first order condition. This research has limitations. First, if consumers

are assumed to consider the offered prices as the criterion for the consideration set formation, then the

optimization of the revenue function will be highly complex and deriving analytical solution is very

complicated. Second, examining the problem when the consumers may constitute their consideration set

based on a mixture of products’ price and quality is also another interesting direction for future work.

We hope that the modeling approach in this research can be used as a basis for many promising

research directions beyond this work, and, in doing so, stimulate future research on the dynamic pricing

and revenue management in the presence of consumers’ sequential search, consumers’ consideration sets,

and sales milestone constraints in operations management and management science.

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