An economic model to compare the

profitability of pay-per-use and fixed-fee

licensing

Douwe Postmus a,∗, Jacob Wijngaard b, Hans Wortmann a

aDepartment of Business and ICT, University of Groningen, P.O. Box 800, 9700AV Groningen, The Netherlands

bDepartment of Operations, University of Groningen, P.O. Box 800, 9700 AVGroningen, The Netherlands

Abstract

This paper develops an economic model to compare the profitability of two strate-gies for the pricing of packaged software: fixed-fee and pay-per-use licensing. It isassumed that the market consists of a monopoly software vendor who is selling pack-aged software to customers who are homogeneous in marginal value of software usebut heterogeneous in level of use. In addition to obtaining the software package fromthe market, customers can develop the required software in-house. When in-housedevelopment costs are constant across customers, the results show that the softwarevendor prefers pay-per-use licensing over fixed-fee licensing if in-house developmentis relatively expensive, whereas fixed-fee licensing is optimal if the cost of in-housedevelopment drops below a certain threshold value. When the assumption of a con-stant in-house development cost is relaxed by letting it vary among customers, itstill holds that pay-per-use licensing is optimal if its average is relatively large.For low and medium values of the average cost of in-house development, however,fixed-fee licensing may no longer be optimal as the relative attractiveness of the twolicensing strategies now depends on how dispersed the in-house development costsof individual customers are.

Key words: Software pricing, Pay-per-use licensing, Fixed-fee licensing,Mathematical modeling

∗ Corresponding author. Tel.: +31 50 363 3982; fax: +31 50 363 2275Email addresses: d.postmus@rug.nl (Douwe Postmus), j.wijngaard@rug.nl

(Jacob Wijngaard), j.c.wortmann@rug.nl (Hans Wortmann).

Preprint submitted to Information and Software Technology 6 August 2008

1 Introduction

Packaged software is commercial software that is produced by a software ven-dor and sold on the market as a commodity item. Examples include Enter-prise Resource Planning (ERP) software, Customer Relationship Management(CRM) software, and Human Resource Management (HRM) software. Severalstrategies for the pricing of this type of software have been developed. Thispaper aims at comparing the profitability of two of them: pay-per-use andfixed-fee licensing.

When pay-per-use licensing is applied, pricing is based on the actual amountof use of the software, measured in units of use such as the number of users orthe number of transactions. Named-user licenses and floating licenses are well-known examples: named-user licenses impose customers to assign a separatelicense to each user, whereas floating licenses can be shared among a group ofusers but constrain the number of users that can work with the applicationconcurrently [1,12].

In contrast, when fixed-fee licensing is applied, customers obtain the rights touse a particular version of the software package by paying an amount that isindependent of product usage. The campus license is a typical example as itallows all faculty, supporting staff, and students of the participating collegeor university to use the software unlimitedly, while the sales price is usuallydetermined by the total number of faculty rather than the actual number ofusers.

To compare the profitability of pay-per-use and fixed-fee licensing, an eco-nomic model of a monopoly producer of packaged software is created. Althougha software vendor generally competes with other vendors to attract new cus-tomers, he has a monopoly position in the market of product upgrades andextensions, which can be explained as follows. Packaged software generally hasa modular structure, allowing customers to extend their systems over time byadding new modules [14]. Because of compatibility requirements, customershave large cost of switching from one software vendor to another: all existingmodules have to be replaced by comparable modules from the new softwarevendor. These switching costs result in lock-in and give a software vendor hismonopoly position over existing customers [2,10,15].

This paper contributes to the literature on software pricing and renting bytaking into account that customers can develop the required software in-house,rather than obtaining it from the market. In-house development of customizedextensions is a common solution for customers to tailor packaged software totheir specific requirements, without switching from the software vendor chosen.In this way, the monopoly position of the software vendor in the market of

2

product upgrades and extensions can be compared with a likely alternativechoice for the customer.

The remainder of the paper is organized as follows. First, related work onsoftware pricing and renting is discussed. Then, a formal presentation of ourmodel is provided. Next, the software vendor’s optimal licensing strategy isdetermined for the case when in-house development is equally expensive foreach customer. Subsequently, the assumption of a constant in-house develop-ment cost is relaxed by letting it vary among customers. Finally, the paperconcludes with a brief summary and suggestions for future research.

2 Related work

Software vendors can choose between two alternatives in delivering their prod-ucts to the market: selling and renting [5]. A software product is sold if thecustomer obtains the perpetual rights to use a particular version of the prod-uct. A separate maintenance agreement is usually required to receive upgradedversions. On the other hand, when the customer obtains the rights to use thesoftware product for a predefined period, such as one year, the product isrented. Upgrades are generally included, but the customer has to renew hiscontract if he wants to continue using the software after the expiry date. Inaddition, software vendors can either express their prices as a function of theamount of use of the software or apply a pricing strategy that is independentof product usage [16]. When combined, the two trade-offs yield the four genericstrategies shown in Table 1.

Table 1Four generic strategies for the pricing of packaged software.

Fixed-fee pricing Usage-based pricing

Selling Fixed-fee licensing Pay-per-use licensing

Renting Fixed-fee subscription Usage-based subscription

Based on these four generic strategies, related work can be divided into twocategories: (i) papers that compare fixed-fee licensing and subscription (firstcolumn in Table 1), and (ii) papers that compare fixed-fee and pay-per-uselicensing (first row in Table 1).

3

2.1 Papers that compare fixed-fee licensing and subscription

Choudhary et al. [3] develop a two-period model of a monopoly firm sellingand renting packaged software. In the first period, the firm sells and rents theinitial version of the software product; in the second period, the firm sells theupgraded version only. Customers who buy the upgraded version enjoy positivenetwork externalities, i.e. their valuation of the product depends positively onhow many other users there are [9], because of bug reports and requests foradditional features by first-period customers. The authors show that comparedto the case when the product is sold only, the introduction of the rental productin the first period leads to an increase in profit. They also find that as networkeffects become stronger, the firm reduces its prices in the first period in orderto expand the size of its network.

Zhang and Seidmann [19] study the optimal policy of a monopoly softwarevendor who provides packaged software to the market through licenses, sub-scriptions, and sales of upgrades. In their model, customers are heterogeneousin their quality preference but homogeneous in their sensitivity to network ex-ternality. It is shown that when there exists uncertainty with regard to productinnovations and upgrades, it is optimal to adopt a hybrid strategy rather thanpure selling or renting.

Choudhary [4] explores how the differences between fixed-fee licensing andsubscription affect a software vendor’s investment in product development. Inhis model, differences in the level of investment translate into differences insoftware quality, which in turn affects the software vendor’s profit. The resultsshow that compared to software licensing, the software vendor invests more inproduct development when software renting is applied, which results in highersoftware quality, larger profit, and higher social welfare.

2.2 Papers that compare fixed-fee and pay-per-use licensing

Gurnani and Karlapalem [7] use a monopoly pricing model to examine theoptimal pricing strategies for fixed-fee and pay-per-use licensing of packagedsoftware disseminated over the Internet. Customers are assumed to be ho-mogeneous in marginal value of software use but heterogeneous in level ofuse. In addition, the authors assume that customers incur inconvenience costswhen pay-per-use licensing is applied. The results show that compared to thecase when fixed-fee licensing is offered only, offering both types of licensingconcurrently increases the software vendor’s profit.

Jiang et al. [8] compare fixed-fee and pay-per-use licensing in a monopoly mar-ket where customers are heterogeneous along three dimensions: marginal value

4

of software use, level of use, and honesty type. Similar to [7], inconveniencecosts apply to customers that choose pay-per-use licensing. The authors showthat if the proportion of dishonest users in the user population is relativelylow, the software vendor will make higher profits by offering fixed-fee licens-ing. On the other hand, pay-per-use licensing is optimal in markets with arelatively high piracy rate.

This paper compares the profitability of fixed-fee and pay-per-use licensing ina monopoly market where customers are homogeneous in marginal value ofsoftware use but heterogeneous in level of use. Table 2 shows that our modeldiffers from the models presented in [7] and [8] in two aspects. First, customersare given the possibility to develop the required software in-house, rather thanpurchasing it from the software vendor. Empirical data show that in-housedevelopment can be a viable alternative, especially when organizations haveto pay large amounts of licensing fees if the software is obtained from themarket [17]. Second, customers do not incur inconvenience costs when pay-per-use licensing is applied. The next section provides a formal description ofour model.

Table 2An overview of the similarities and differences between our model and the modelsof Gurnani and Karlapalem [7] and Jiang et al. [8].

G & K J et al. Our model

In-house development no no yes

Customers incur inconvenience costs

when pay-per-use licensing is applied yes yes no

Software piracy no yes no

Customers heterogeneous in

marginal value of software use no yes no

Customers heterogeneous in

level of use yes yes yes

3 Model

The model considers a monopoly producer of packaged software. The marketconsists of N potential customers who are heterogeneous in level of use. Letθi ∈ [θL, θH ] denote the level of use of the ith customer, which is measuredin units of use such as the number of (concurrent) users or the number oftransactions. Throughout this paper, we assume that the value of N is suffi-

5

ciently large to approximate the distribution of levels of use in the customerpopulation by using a continuous distribution with density function fθ(θ) andcumulative distribution function Fθ(θ), where Fθ(θ) =

∫ θθL

fθ(θ)dθ denotes thefraction of customers with level of use less than or equal to θ. The softwarevendor knows how levels of use are distributed across customers (i.e. he knowsfθ(θ) and Fθ(θ)) but cannot directly observe the level of use of an individualcustomer. Because of this, the software vendor cannot apply first-degree pricediscrimination [18].

The value that customers derive from using the software can be comparedto the price that is charged by the software vendor through the use of so-called reservation prices [18]. Let ri(θi) = υθi denote the reservation price ofthe ith customer, which is defined as the maximum price that customer i iswilling to pay for the software: at any price lower than or equal to ri(θi), thevalue that the customer derives from using the software is sufficiently largeto make obtaining the software package economically feasible, whereas at anyprice greater than ri(θi), the customer is better of by spending his budget onsome of the other goods that are available to him. The parameter υ (referredto as the marginal value of software use) is assumed to be constant acrosscustomers.

In addition to buying a copy from the software vendor, customers can developthe required software in-house (or outsource it to an external organization atthe same cost). Let p(θ) denote the price that the software vendor chargesfor using the software, and let ci denote the cost of in-house development forthe ith customer. Compared to its initial development cost, the marginal costof reproducing a copy of the software package is negligible, i.e. the softwarevendor produces against zero unit cost.

Customer i obtains a copy from the software vendor if (i) he derives a nonneg-ative surplus from purchasing the software and (ii) the value of this surplusexceeds the net surplus from in-house development. Formally, these constraintscan be written as:

p(θi) ≤ υθi, (1)

p(θi) ≤ ci. (2)

The first constraint is the participation constraint: it says that the price thatthe software vendor charges for using the software cannot exceed the cus-tomer’s reservation price. The second constraint, the so-called incentive com-patibility constraint, says that this price must be less than the customer’sin-house development cost.

To compare the profitability of fixed-fee and pay-per-use licensing, two addi-

6

tional assumptions have to be made. First, we assume that every customerpays the same price for using the software when fixed-fee licensing is applied,i.e. p(θ) = pf . The constraints above can then be formulated as:

pf ≤ υθi, (3)

pf ≤ ci. (4)

Second, we assume that under pay-per-use licensing, payments to the softwarevendor depend linearly on the amount of use of the software, i.e. p(θ) = puθ.Constraints (1) and (2) can then be written as:

pu ≤ υ, (5)

puθi ≤ ci. (6)

4 Analysis for constant in-house development cost

We start our analysis by assuming that in-house development is equally ex-pensive for each customer, which implies that ci = c for all i ∈ {1, . . . , N}.Section 4.1 considers the case when both types of licensing are offered con-currently, whereas Section 4.2 considers the case when the software vendorexplicitly chooses between offering fixed-fee and pay-per-use licensing.

4.1 Offering both licensing variants concurrently

Customers who can choose between fixed-fee and pay-per-use licensing preferthe variant that gives them the highest net surplus. This causes customers toself-select into two different groups (occasional and frequent users) accordingto their levels of use and is referred to as second-degree price discrimination[18].

Offering both licensing variants concurrently allows the software vendor tosell his product at different prices to customers for which the participationconstraint is binding (occasional users) and customers for which the incentivecompatibility constraint is binding (frequent users), according to how muchthey are willing to pay for it. It follows from Equations (1) and (2) that theparticipation constraint is binding for all customers with level of use θ ≤cυ. Because of this, the software vendor can sell to each of these customers

at a price that equals his reservation price by setting pu = υ. The sameequations show that the incentive compatibility constraint is binding for all

7

customers with level of use θ ≥ cυ. The software vendor can therefore sell

to these customers at a price that equals their in-house development cost bysetting pf = c.

To conclude, when both licensing variants are offered concurrently, the soft-ware vendor can charge each customer his maximum willingness-to-pay bysetting pf = c and pu = υ, which results in a total profit of

πm = N∫ c/υθL

υθfθ(θ)dθ + N∫ θH

c/υ cfθ(θ)dθ

= N{∫ c/υθL

υθfθ(θ)dθ − c(1− Fθ(c/υ))}.

4.2 offering one licensing variant only

The previous section assumed that the software vendor offers his productunder two different licensing variants. Such an approach to providing packagedsoftware can generally be applied when the software vendor sells a new product(or a next generation of one of his current products) to his existing customerbase. In many situations, however, a software vendor must first compete withother software vendors to attract and lock in customers before he can exploithis monopoly position.

As an example, consider a two-period model in which a software vendor entersthe market with a new product in the first period and sells an extension ofthis product in the second period. In the first period, the software vendormust charge a low price to attract customers who know they are going to beexploited in the second period, where they have switching costs as a resultof their first-period purchase [10]. Competition will force the software vendorto break-even or even loose money in the first period, which explains whymany software vendors offer the basic version of their software products forfree [15]. In the second period, the software vendor sells an extension to hisexisting customer base. Now, the software vendor is locked into a specific typeof software license as legal and technical complexities restrict him to only offerthe licensing variant that is consistent with a customer’s first-period purchase.The software vendor’s optimal strategy is therefore to enter the market withthe licensing variant that maximizes his profits in the second period, wherehe can exploit his monopoly position. The analysis performed in this sectionsupports software vendors in making an explicit choice between offering fixed-fee and pay-per-use licensing.

8

4.2.1 Problem specification

When fixed-fee licensing is applied, it follows from (3) that every customerwith level of use θ ≥ pf

υgets a positive surplus from purchasing the software.

The profit maximizing problem of the software vendor under fixed-fee licensingcan therefore be written as

maxpf

πf (pf )

s.t. pf ≤ c,

(7)

where the function πf (pf ) =∫ θH

pfυ

pffθ(θ)dθ = pf (1 − Fθ(pf/υ)) returns the

software vendor’s average profit per customer for a given fixed-fee price pf

(multiplying πf (pf ) by N yields the software vendor’s total profit).

Similarly, when pay-per-use licensing is applied, it follows from (6) that everycustomer with level of use θ > c

puprefers to develop the required software

in-house as compared to obtaining it from the market. The profit maximizingproblem of the software vendor under pay-per-use licensing can therefore beformulated as

maxpu

πu(pu)

s.t. pu ≤ υ,(8)

where the function πu(pu) =∫ c

puθL

puθfθ(θ)dθ = pu

∫ cpu

θLθfθ(θ)dθ returns the

software vendor’s average profit per customer for a given pay-per-use price pu.

Let p∗f (c) and p∗u(c) be the optimal solutions to the problems (7) and (8),respectively. From the analysis in Section 4.1, we know that πu(p

∗u(c)) >

πf (p∗f (c)) for relatively large values of c, whereas πf (p

∗f (c)) > πu(p

∗u(c)) for rel-

atively small values of c. To explain why, first consider the case when c > υθH ,which implies that the participation constraint is binding for all customers.Because of this, the software vendor can charge each customer his maximumwillingness-to-pay by applying pay-per-use licensing (if the software vendorwere to apply fixed-fee licensing, he would only be able to sell to one typeof customer at this price: all customers with a reservation price less than pf

would not obtain the software package, whereas all customers with a reserva-tion price exceeding pf would derive a positive surplus from purchasing thesoftware). Next, consider the case when c < υθL, which implies that the incen-tive compatibility constraint is binding for all customers. Now, the softwarevendor can charge each customer his maximum willingness-to-pay by applyingfixed-fee licensing (if the software vendor were to apply pay-per-use licensing,

9

he would again only be able to sell to one type of customer at this price).Since pay-per-use licensing is optimal if the cost of in-house development ex-ceeds the reservation price of the customer with the highest level of use, andfixed-fee licensing is optimal if this cost is less than the reservation price of thecustomer with the lowest level of use, we conclude that the functions πf (p

∗f (c))

and πu(p∗u(c)) intersect at least once. The next two subsections show how to

determine the value c∗ ∈ [υθL, υθH ] of this intersect for the case when thelevels of use in the customer population are uniformly and beta distributed,respectively.

4.2.2 Uniform distribution

In this section, the software vendor’s profits under fixed-fee and pay-per-uselicensing are compared for the case when the levels of use in the customerpopulation are uniformly distributed. The density function of the uniformdistribution is

fθ(θ) =

1

θH−θLif θL ≤ θ ≤ θH

0 otherwise.(9)

In order to determine p∗f , we consider two mutually exclusive cases. The opti-mal solution for the software vendor is the maximum of these two cases. First,we consider the case when pf ≤ υθL, which implies that each customer getsa positive surplus from purchasing the software. Substituting (9) into (7) andadding the constraint pf ≤ υθL yields

maxpf

pf

s.t. pf ≤ c

pf ≤ υθL.

It is straightforward to see that the optimal solution is p∗f = c if c < υθL andp∗f = υθL if c ≥ υθL. In the second case, we assume that pf ≥ υθL. Now,depending on the fixed-fee price pf , there may be customers who do not geta positive surplus from purchasing the software. Substituting (9) into (7) andadding the constraint pf ≥ υθL yields

maxpf

pf θH− 1υ(pf )2

θH−θL

s.t. pf ≤ c

pf ≥ υθL.

10

The problem has no solution if c < υθL. The case when c ≥ υθL is different.Now, the optimal solution depends on the ratio of θL and θH . When 1

2θH ≤ θL,

p∗f = υθL. On the other hand, 12θH > θL implies that p∗f = c if c < 1

2υθH and

p∗f = 12υθH if c ≥ 1

2υθH . Taking the maximum of the two mutually exclusive

cases yields the results summarized in Proposition 1.

Proposition 1: The software vendor’s optimal price under fixed-fee licens-ing is

p∗f (c) =

c if c < υθL,

υθL if c ≥ υθL and 12θH ≤ θL,

c if υθL ≤ c < 12υθH and 1

2θH > θL,

12υθH if c ≥ 1

2υθH and 1

2θH > θL.

The corresponding average profit per customer is equal to

πf (p∗f (c)) =

c if c < υθL,

υθL if c ≥ υθL and 12θH ≤ θL,

θH−c/υθH−θL

c if υθL ≤ c < 12υθH and 1

2θH > θL,

(θH)2

4(θH−θL)υ if c ≥ 1

2υθH and 1

2θH > θL.

The optimal value of pu can be determined in a similar way. First, considerthe case when pu ≤ c

θH, which implies that all customers prefer to buy the

software from the market as compared to developing it in-house. Substituting(9) into (8) and adding the constraint pu ≤ c

θHyields

maxpu

θL+θH

2pu

s.t. pu ≤ υ

pu ≤ cθH

.

The reader can verify easily that the optimal solution is p∗u = cθH

if c ≤ υθH

and p∗u = υ if c > υθH . Next, consider the case when pu ≥ cθH

. Now, dependingon the pay-per-use price pu, there may be customers who prefer to developthe required software in-house as compared to obtaining it from the software

11

vendor. Substituting (9) into (8) and adding the constraint pu ≥ cθH

yields

maxpu

c2/pu−(θL)2pu

2(θH−θL)

s.t. pu ≤ υ

pu ≥ cθH

.

If c ≤ υθH , the optimal solution is p∗u = cθH

. The problem has no solutionwhen c > υθH . Proposition 2 summarizes the results that are obtained bycombining the two mutually exclusive cases.

Proposition 2: The software vendor’s optimal price under pay-per-use li-censing is

p∗u(c) =

c

θHif c ≤ υθH ,

υ if c > υθH .

The corresponding average profit per customer is equal to

πu(p∗u(c)) =

θH+θL

2θHc if c ≤ υθH ,

12(θH + θL)υ if c > υθH .

When the software vendor’s profits under fixed-fee and pay-per-use licensingare compared, it follows that pay-per-use licensing is more profitable thanfixed-fee licensing if in-house development is relatively expensive for customers,whereas fixed-fee licensing is optimal if the cost of in-house development dropsbelow a certain threshold value. Propositions 3 and 4 can be used to determinethis threshold value analytically for the case when 1

2θH ≤ θL and the case when

12θH > θL, respectively. For proofs of these propositions, the reader is referred

to Appendix A.

Case 1: 12θH ≤ θL

Proposition 3: The software vendor finds pay-per-use licensing more prof-itable than fixed-fee licensing if c ≥ 2υθL

(θL/θH)+1, whereas fixed-fee licensing is

optimal if c ≤ 2υθL

(θL/θH)+1.

Case 2: 12θH > θL

Proposition 4: The software vendor finds pay-per-use licensing more prof-

itable than fixed-fee licensing if c ≥ (θH)3

2((θH)2−(θL)2)υ, whereas fixed-fee licensing

12

is optimal if c ≤ (θH)3

2((θH)2−(θL)2)υ.

4.2.3 Beta distribution

This section generalizes the results from the previous section by consideringthe case when the levels of use in the customer population are beta distributed.The beta distribution is a continuous distribution defined on an interval witha minimum and maximum value and is parameterized by two parameters, de-noted by αθ and βθ. Depending on the value of these parameters, the betadensity function can take on different shapes, including the U-shape, the tri-angle shape, and the bell shape. For αθ = βθ = 1, the beta distribution reducesto the uniform distribution.

The density function of the beta distribution is

fθ(θ; αθ, βθ) =

C( θ−θL

θH−θL)αθ−1(1− θ−θL

θH−θL)βθ−1 if θL ≤ θ ≤ θH

0 otherwise,(10)

where C = 1θH−θL

Γ(αθ+βθ)Γ(αθ)Γ(βθ)

. Substituting (10) into (7) to obtain the softwarevendor’s profit maximizing problem under fixed-fee licensing yields

maxpf

pf (1− Fθ(pf

υ; αθ, βθ))

s.t. pf ≤ c,

(11)

where Fθ(θ; αθ, βθ) denotes the cumulative distribution function of the betadistribution. Similarly, by substituting (10) into (8), it follows (after rewriting)that the profit maximizing problem of the software vendor under pay-per-uselicensing can be written as

maxpu

pu{θLFθ(c

pu; αθ, βθ) +

(θH − θL)Γ(αθ+1)Γ(αθ+βθ)Γ(αθ+βθ+1)Γ(αθ)

Fθ(c

pu; αθ + 1; βθ)}

s.t. pu ≤ υ.

(12)

In the previous section, we have shown that when levels of use are uniformlydistributed across customers, the software vendor prefers pay-per-use licens-ing over fixed-fee licensing for relatively large values of c, whereas fixed-feelicensing is optimal if c drops below a certain threshold value c∗ ∈ [υθL, υθH ].The same is true for the more general case when levels of use are beta dis-tributed across customers. Although the properties of the beta distribution–

13

Fθ(θ; αθ, βθ) does not exist in closed form, in general [11]–restrict us fromdetermining the value c∗ for which the software vendor is indifferent betweenoffering fixed-fee and pay-per-use licensing analytically, we can approximateit numerically (within an error margin of δ) by applying the procedure out-lined in Figure 1. At each iteration, the algorithm requires us to determinethe function values πf (p

∗f (m)) and πu(p

∗u(m)) at the midpoint m = a+b

2of

the interval [a, b]. If πf (p∗f (m)) ≥ πu(p

∗u(m)), c∗ cannot be contained in the

subinterval [a, m), so this portion of the search interval is discarded. Similarly,πf (p

∗f (m)) ≤ πu(p

∗u(m)) implies that the subinterval (m, b] can be discarded

since it cannot contain c∗. The process of narrowing the search interval is con-tinued until c∗ has been isolated as accurately as required (i.e. within an errormargin of δ from its true value).

Fig. 1. A procedure to determine the value c∗ for which the software vendor isindifferent between offering fixed-fee and pay-per-use licensing.

Step 1: (initialization). Define an initial interval [a, b] such that a = υθL

and b = υθH .

repeat

Step 2: Compute the midpoint m = a+b2

.

Step 3: Determine the values of πf (p∗f (m)) and πu(p

∗u(m)) by solving the

software vendor’s profit maximizing problems under fixed-fee and

pay-per-use licensing, respectively, for the case when c = m.

Step 4: Narrow the search interval by applying one of the following rules:

a) If πf (p∗f (m)) ≥ πu(p

∗u(m)), update the left endpoint of the interval

[a, b] according to a := a+b2

.

b) If πf (p∗f (m)) ≤ πu(p

∗u(m)), update the right endpoint of the interval

[a, b] according to b := a+b2

.

until b−a2≤ δ

Step 5: Assign the value a+b2

to the threshold value c∗.

Table 3 summarizes the results of a numerical study that has been performedwith a program written in Matlab for the parameter values θL = 10, θH = 100,υ = 1000, and δ = 0.01. A graphical representation of the density functionscorresponding to the four different combinations of the shape parameters αθ

and βθ is given in Figure 2. To solve the problems (11) and (12), we reliedon Matlab’s implementation of the bound-constrained optimization algorithm,which is based on golden section search and parabolic interpolation [6,13].

14

Table 3Results of the numerical analysis.

Distribution Threshold value (c∗)

Numerical value Analytical value

Beta(1,1) 50,505 50,505 (Proposition 4)

Beta(2,2) 51,962 not available

Beta(3,6) 39,000 not available

Beta(4,2) 62,855 not available

Fig. 2. Density function of the beta distribution for four different combinations ofthe shape parameters αθ and βθ.

5 Analysis for variable in-house development cost

In the previous section, it was assumed that in-house development is equallyexpensive for each customer. In reality, however, this cost will differ signif-icantly, depending on the quality of a customer’s in-house solution. In thissection, we therefore relax the assumption of a constant in-house developmentcost by letting it vary among customers. Let fc(c) and Fc(c) denote the den-sity function and the cumulative distribution function of the distribution ofin-house development costs across customers, respectively. It is assumed thatthe distribution of the cost of in-house development is independent of the dis-tribution of the level of use, which implies that the density function of theirjoint distribution factors into the product of the density functions of the two

15

marginal distributions: f(c, θ) = fc(c)fθ(θ).

When fixed-fee licensing is applied, it follows from Equations (3) and (4) thatevery customer with (i) level of use θ ≥ pf

υand (ii) in-house development cost

c ≥ pf obtains the software package from the market. The profit maximiz-ing problem of the software vendor under fixed-fee licensing can therefore bewritten as

maxpf

pf [1− Fθ(pf

υ)][1− Fc(pf )]. (13)

Similarly, when pay-per-use licensing is applied, it follows from Equation (6)that every customer with in-house development cost c ≥ puθ prefers to obtain acopy from the software vendor as compared to developing the required softwarein-house. The profit maximizing problem of the software vendor under pay-per-use licensing can therefore be formulated as

maxpu

pu

∫ θHθL

θfθ(θ)[1− Fc(puθ)]dθ

s.t. pu ≤ υ.(14)

The first step in solving problems (13) and (14) is to specify how levels ofuse and in-house development costs are distributed across customers. In manyreal-life situations, we expect to find that the distribution of the cost of in-house development is skewed to the right (i.e. most of the distribution’s massis located at the left of its mean) as most customers will settle for a low-cost solution, while only a few require a high-quality implementation in termsof performance, scalability, etc. The gamma distribution, a continuous dis-tribution defined on the interval [0,∞) and parameterized through its shapeparameter αc and its scale parameter βc, provides a flexible set of densityfunctions that conform to such a shape. In the analysis below, it is thereforeassumed that in-house development costs are gamma distributed across cus-tomers. The levels of use in the customer population are assumed to be betadistributed, just as in the previous section.

Let µc = 1N

∑Ni=1 ci and σc =

√1N

∑Ni=1(ci − µc)2 be the mean and standard

deviation of the distribution of the cost of in-house development, respectively:the parameter µc reflects its average value in the customer population, whereasσc is a measure of how dispersed the in-house development costs of individualcustomers are. To fit the gamma distribution to these market characteristics,we must express the parameters αc and βc in terms of the population meanand standard deviation. This yields:

αc =µ2

c

σ2c

, (15)

16

βc =σ2

c

µc

. (16)

Based on Equations (15) and (16), the profitability of fixed-fee and pay-per-use licensing can be compared for any combination of µc and σc. Figure 3summarizes the results of a numerical study that has been performed in Matlab(again for the parameter values υ = 1000, θL = 10, and θH = 100). It showsthat the relative attractiveness of fixed-fee and pay-per-use licensing dependson the value of the population mean and standard deviation: fixed-fee licensingis optimal for all combinations of µc and σc in the black region of the figure,whereas pay-per-use licensing is optimal for all combinations of µc and σc inthe grey region. By distinguishing between “small”, “medium”, and “large”values of µc, we can draw the following conclusions:

a) For small values of µc, there exists a threshold value σc such that (i) fixed-fee licensing is optimal if σc ≤ σc and (ii) pay-per-use licensing is optimalif σc ≥ σc.

b) For medium values of µc, there exists an interval [σLc , σH

c ] such that (i)fixed-fee licensing is optimal if σc ∈ [σL

c , σHc ] and (ii) pay-per-use licensing

is optimal if σc /∈ [σLc , σH

c ].c) For large values of µc, pay-per-use licensing is more attractive than fixed-fee

licensing, independent of the value of σc.

For a constant in-house development cost, it holds that pay-per-use licensingis more profitable than fixed-fee licensing for relatively large values of µc = c,whereas fixed-fee licensing is optimal if µc drops below a certain thresholdvalue. The analysis performed in this section shows that when the assumptionof a constant in-house development cost is relaxed, pay-per-use licensing isstill optimal for relatively large values of µc (say ∀ µc ≥ µc). Depending onthe value of the standard deviation σc, it may happen, however, that fixed-feelicensing is no longer the preferred licensing strategy for small and mediumvalues of µc.

6 Conclusion

This paper compares the profitability of pay-per-use and fixed-fee licensingfor a monopoly software vendor who is selling packaged software to customerswho are homogeneous in marginal value of software use but heterogeneous inlevel of use. Next to obtaining a copy from the software vendor, customerscan develop the required software in-house. The situation when in-house de-velopment is not accounted for is equivalent to the case when the cost ofin-house development is infinitely large for all customers. Our results show

17

Fig. 3. Fixed-fee versus pay-per-use licensing for the case when the levels of use inthe customer population are (from left to right and top to bottom): (i) beta(1,1)distributed, (ii) beta(2,2) distributed, (iii) beta(3,6) distributed, and (iv) beta(4,2)distributed.

that the software vendor then prefers pay-per-use licensing over fixed-fee li-censing. The same results show, however, that fixed-fee licensing is optimal ifthe cost of in-house development drops below a certain threshold value. Whenthe assumption of a constant in-house development cost is relaxed by letting itvary among customers, it still holds that pay-per-use licensing is optimal if itsaverage is relatively large. For low and medium values of the average cost ofin-house development, however, fixed-fee licensing may no longer be optimalas the relative attractiveness of the two licensing strategies now depends onhow dispersed the in-house development costs of individual customers are.

Future research can extend our model in various ways. First, it has beenassumed that the software vendor charges a constant unit price when pay-per-use licensing is applied. In reality, however, unit prices often depend ona customer’s level of use, allowing the software vendor to price discriminatebetween different types of customer. It may therefore be interesting to explorehow nonlinear pricing affects the relative attractiveness of fixed-fee and pay-per-use licensing. Second, we have assumed that customers are homogeneousin marginal value of software use. Future research can generalize our resultsby letting this parameter vary among customers. Finally, it has been assumedthat the distribution of the cost of in-house development is independent of the

18

distribution of the level of use. A natural way to extend our model is thereforeto relax the assumption of a zero covariance.

References

[1] N. Bontis, H. Chung, The evolution of software pricing: from box licensesto application service provider models, Internet Research: Electronic NetworkApplications and Policy 10 (2000) 246-255.

[2] T.A. Burnham, J.K. Frels, V. Mahajan, Consumer switching costs: a typology,antecedents, and consequences, Journal of the Academy of Marketing Science31 (2003) 109-126.

[3] V. Choudhary, K. Tomak, A. Chaturvedi, Economic benefits of renting software,Journal of Organizational Computing and Electronic Commerce 8 (1998) 277-305.

[4] V. Choudhary, Comparison of software quality under perpetual licensing andsoftware as a service, Journal of Management Information Systems 24 (2007)141-165.

[5] M.A. Cusamo, The changing labyrinth of software pricing, Communications ofthe ACM 50 (2007) 19-22.

[6] G.E. Forsythe, M.A. Malcolm, C.B. Moler, Computer Methods forMathematical Computations, Prentice Hall, Englewood Cliffs, 1977.

[7] H. Gurnani, K. Karlapalem, Optimal pricing strategies for Internet-basedsoftware dissemination, Journal of the Operational Research Society 52 (2001)64-70.

[8] B.-J. Jiang, P.-y. Chen, T. Mukhopadhyay, Software pricing: pay-per-use versusperpetual licensing, Technical Report, Tepper School of Business, CarnegieMellon University, Pittsburgh, 2006.

[9] M.L. Katz, C. Shapiro, Network externalities, competition, and compatibility,The American Economic Review 75 (1985) 424-440.

[10] P. Klemperer, Competition when consumers have switching costs: an overviewwith applications to industrial organization, macroeconomics, and internationaltrade, Review of Economic Studies 62 (1995) 515-539.

[11] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 3rd ed., McGraw-Hill, Boston, 2000.

[12] M. Murtojarvi, J. Jarvinen, M. Johnsson, T. Leipala, O.S. Nevalainen,Determining the proper number and price of software licenses, IEEETransactions on Software Engineering 33 (2007) 305-315.

19

[13] J. Nocedal, S.J. Wright, Numerical Optimization, Springer-Verlag, New York,1999.

[14] K.A. Olson, P. Saetre, IT for niche companies: is an ERP system the solution?,Information Systems Journal 17 (2007) 37-58.

[15] C. Shapiro, H.R. Varian, Information Rules: A Strategic Guide to the NetworkEconomy, Harvard Business School Press, Boston, 1999.

[16] A. Sundararajan, Nonlinear pricing of information goods, Management Science50 (2004) 1660-1673.

[17] P. Thibodeau, New reasons to do it yourself, Computerworld 39 (2005) 38.

[18] H.R. Varian, Microeconomic Analysis, Third Edition, W. W. Norton &Company, New York, 1992.

[19] J. Zhang, A. Seidmann, The optimal software licensing policy under qualityuncertainty, Proceedings of the 5th international conference on Electroniccommerce, ACM Press, New York, NY, 2003, pp. 276-286.

Appendix A

Proof of Proposition 3:

First, consider the case when c < υθL. The profits of the software vendor equalc under fixed-fee licensing and θH+θL

2θHc under pay-per-use licensing. Because

θH+θL

2θHc < θH+θH

2θHc = c, the software vendor prefers fixed-fee over pay-per-use

licensing.

Next, consider the case when υθL ≤ c ≤ υθH . The profits of the software ven-dor equal υθL under fixed-fee licensing and θH+θL

2θHc under pay-per-use licensing.

Rewriting υθL ≥ θH+θL

2θHc gives c ≤ 2υθL

(θL/θH)+1. Because (i) 2υθL

(θL/θH)+1> υθL and

(ii) 2υθL

(θL/θH)+1< υθL+υθH

(θL/θH)+1= υθH{(θL/θH)+1}

(θL/θH)+1= υθH , we conclude that 2υθL

(θL/θH)+1

is within the interval [υθL, υθH ], as required.

Finally, consider the case when c > υθH . The profits of the software ven-dor equal υθL under fixed-fee licensing and 1

2(θH + θL)υ under pay-per-use

licensing. Because 12(θH +θL)υ > 1

2(θL +θL)υ = υθL, the software vendor finds

pay-per-use licensing more profitable than fixed-fee licensing, which completesthe proof. 2

20

Proof of Proposition 4:

First, consider the case when c < υθL. The profits of the software vendor equalc under fixed-fee licensing and θH+θL

2θHc under pay-per-use licensing. Because

θH+θL

2θHc < θH+θH

2θHc = c, the software vendor prefers fixed-fee over pay-per-use

licensing.

Second, consider the case when υθL ≤ c < 12υθH . The profits of the software

vendor equal θH−c/υθH−θL

c under fixed-fee licensing and θH+θL

2θHc under pay-per-use li-

censing. Because θH−c/υθH−θL

c >θH−( 1

2υθH)/υ

θH−θLc = θH

2(θH−θL)c ≥ θH−(θL)2/θH

2(θH−θL)c = θH+θL

2θHc,

the software vendor finds fixed-fee licensing more profitable than pay-per-uselicensing.

Third, consider the case when 12υθH ≤ c ≤ υθH . The profits of the software

vendor equal (θH)2

4(θH−θL)υ under fixed-fee licensing and θH+θL

2θHc under pay-per-use

licensing. Rewriting (θH)2

4(θH−θL)υ ≥ θH+θL

2θHc gives c ≤ (θH)3

2((θH)2−(θL)2)υ. Because (i)

(θH)3

2((θH)2−(θL)2)υ < (θH)3

2((θH)2− 14(θH)2)

υ < υθH and (ii) (θH)3

2((θH)2−(θL)2)υ > (θH)3

2(θH)2υ =

12υθH , we conclude that (θH)3

2((θH)2−(θL)2)υ is within the interval [1

2υθH , υθH ], as

required.

Finally, consider the case when c > υθH . The profits of the software vendor

equal (θH)2

4(θH−θL)υ under fixed-fee licensing and 1

2(θH + θL)υ under pay-per-use

licensing. Because 12(θH + θL)υ = (θH)2−(θL)2

2(θH−θL)υ >

(θH)2− 14(θH)2

2(θH−θL)υ > (θH)2

4(θH−θL)υ, the

software vendor prefers pay-per-use over fixed-fee licensing, which completesthe proof. 2

21