April 13, 1997

How to Measure Product Quality

under Monopolistic Product Market

Takanobu nakajima

How to Measure Product Quality under Monopolistic Product Market*

Takanobu nakajima

April 13, 1997

1 Introduction

The purpose of this paper is to present a simple method to measure product quality index based

on market information when a firm has a monopolistic power to a newly developed high-quality

product. The measurement of quality is not only an object of academic interest in inquiring an

unobservable variable, but a practical issue to a policy maker. When inflation is observed in

product market, it is crucial to judge what is causing the rise in price. A part of inflation rate

can be attributable to improvement in product quality which should be taken into account to

make an anti-inflation policy.

It is widely understood that "hedonic functional approach" is one of the most powerful method-

ologies to extract the effect of quality improvement from product price . Based on Rosen's

contribution to the theoretical background (Rosen[6]), it can be shown that a well-defined he-

donic function can give an unbiased estimate of pure inflation rate if a consumer's preference

independently varies and a product market is perfectly competitive.

Another method to measure quality is "Divisia quality index approach ." This approach is also

based on the neo-classical economic theory, where constant returns to scale (CRS) and perfect

competition are assumed . If this theoretical background is approved, product's (relative) price is equal to (ratio of) marginal utility. When the prices of two products in the same category

are different, the product of higher price must have higher quality which is appreciated by

consumer's rational behavior. We can calculate quality index as the ratio of Divisia quantity

index to the simple adding-up total.

*This paper is preliminary. tKeio University, Faculty of Business and Commerce. 2-15-45 Mita Minatoku, Tokyo 108, Japan Tel/Fax

+81-3-3453-4511 / +81-3-3798-7480, E-mail: nakajima@fbc.keio.ac.jp

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However, a newly developed high-quality product, in most cases, may hardly be under a

competitive market.' A computer which has the latest version of Intel's MPU is certainly more

expensive than one with a popular micro chip. The difference in prices may not only include a

value of quality appreciated in the product market but also an additional rent attributable to

a developer of a high-quality product. As economic growth geared by quantitative expansion

mostly ended, a firm has begun to spend more R&D investment for development of a new and

high-quality product seeking for monopoly rent. This paper is aiming at the measurement of

pure quality improvement when a new product is supplied by a monopolistic developer.

One of the important works in hedonic approach is to convert qualitative characteristics into

quantitative indicators. If a consumer appreciates speedy work in PC, we have to numerically

acquire PC's working speed which can be influenced by MPU's ability itself , various kinds of

memory, quality of hard disk, and so forth. All of these attributes should be taken into account

to get a precise hedonic function. On the other hand, a consumer may consider all these items

comprehensively and make a single quality index.2 This paper tries to get this consumer's quality

index on the basis of observable side information. If a monopolistic developer of a high-quality

product has the knowledge of a consumer's demand function, the profit maximizing price and

the amount of sales are reflected by the product's quality itself. The target is only this single

quality indicator.

Although a quality indicator is treated as an exogenous variable for simplicity, we can easily

elaborate the model to handle it endogenously. The model indicates the possibility of decreasing

marginal returns from raising product quality. If the increasing marginal cost for the develop-

ment is accepted, it can be shown that a firm may develop a new product the quality of which

'There are considerable number of research to target the relationship between product quality and monopolistic price. Spence[7] has pointed out the importance to identify a quality factor in the monopoly price in the context of finding an optimal regulative policy to a monopolistic output provider . Feenstra[2] has derived the condition of an exact hedonic functional form when a product market is not competitive. Berry, Levinsohn, and Pakes[1] can be one of the most sophisticated empirical researches in this field. Focusing on an automobile market, they construct the model composed of consumers' multiple choice among products differentiated by characteristics and auto providers' oligopolistic behavior, and estimate several parameters that show auto-market characteristics. 2Mussa and Rosen[4] present a theoretical model that treats quality characteristics as a single index measure. In their model, a consumer's utility function includes the quality index and a parameter which "measures intensity of a consumer's taste for quality." My paper is basically owing to this idea except that the M-R paper endogenizes quality index considering a firm's quality development cost in a static model. The endogenization of quality index might be necessary to derive a theoretical implication from the model, but makes the measurement of quality difficult in the two ways. Firstly, a firm's R&D behavior for raising product quality may require a model building from a dynamic point of view considering a firm's patent strategy, that will make a model too complicated for an empirical research. Secondly, the data related to a firm's R&D expenditure used only for quality improvement might not be available in Japan's current environment of R&D database .

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just equalizes the marginal returns to the marginal cost. It is widely known that the Grossman-

Helpman quality ladder model provides a powerful tool to handle product quality in a standard

dynamic economic modeL(Grossman and Helpman[3]) In their model, a quality indicator is in-cluded in the consumer's utility function as an inflater of quantity . In order to keep the balance

of budget, on the other hand, the product price is deflated by quality index . This framework is

certainly needed to build an elegant economic model , but may hardly seem real to consumers.

Suppose there are two kinds of computers on sale , one of which has twice higher quality and

price than the other. According to the G-H framework, a consumer who purchases a high-

quality computer consumes twice more than he/she who has an ordinary machine, although

the two prices deflated by quality are the same. If this is true, there is no one who purchases

an ordinary computer, because a high-quality machine has the same price to consumers . In

case of the purchase of durable consumer goods , like computers, furniture, cars, and so forth,

the G-H framework may not work properly. Instead we must separate' quality index from price

which should be considered as a factor to influence a consumer's cash flow. In the model of

this paper, although a consumer who purchases a high-quality product certainly consumes more

than an ordinary product, the high price affects his/her rest of money that would be used for

the purchase of other products.

The structure of the paper is as follows. In the next section we describe the basic framework

of the model. Section 3 illustrates how the model works and shows the design for an empirical

analysis. We also focus on the estimation bias when hedonic approach is applied to the artificial

data generated by monopolistic production circumstances . The final part of the paper is devoted

to the concluding remarks.

2 Model

2.1 Consumer's Choice

We assume that a consumer has the following utility function .

In U = u = v ln(a + X) + (1 - v) ln(M - E) (1)

where X stands for a quality-adjusted quantity of a product purchased by a consumer, M for income, and E for expenditure for the purchase of a product . The parameter v denotes a

consumer's relative preference of a product to other goods, and the parameter a shows a stock effect of a product.

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Suppose that there are two kinds of products for a consumer's choice: an ordinary product and

a high-quality product. An ordinary product is assumed to be under a perfectly competitive

market and its price is given as p. A high-quality product is monopolistically supplied by a

company and its price is p. It is supposed that a consumer has the following three alternatives

to decide an optimal choice.

(1) No purchase: X = 0, E = 0

(2) Purchase one unit of an ordinary product: X = 1, E = p

(3) Purchase one unit of a high-quality product: X = A, E = p

Here we are based on an assumption that a product quality for a consumer can be measured by

the increased quantity of consumption. In other words when a consumer purchases a product of

high quality, he feels as if he is consuming more quantity of an ordinary product . For example,

suppose that a new computer can work twice as faster than an ordinary computer . If a consumer

feels that a new computer deserves two ordinary ones , we define the quality of a new computer

is twice as much as that of an ordinary computer, and A = 2 .

We also assume that parameter v has a distribution among consumers. The density function

is defined as follows.

f(v)=(b+1)(1-v)b, 0v<1,-1<b<1 (2)

where b is a parameter showing the degree of a product's peculiarity . As b rises, less consumers

prefer the product.

A consumer chooses "purchasing a high-quality product" if and only if the following conditions

are satisfied:

v in(a + A) + (1 - v) ln(M - p) > v ln(a + 1) + (1 - v) ln(M - p). (3)

v ln(a + A) + (1 - v) ln(M - p) > v In a + (1 - v) In M. (4)

The inequalities (3) and (4) can be rewritten as follows.

In M-p v > M-p = G = 9(M, A, p, a, p) (5) ln~-a-- + In 1+ a M-p

M

v > la M-p M = H = h (M, A, p, a) (6) In a + In M-p

4

Under the feasible market situation where the price of a high-quality-product is higher than that

of an ordinary product, we have the following proposition .

Proposition I As a consumer's income level increases , more consumers prefer a product of

higher quality.

2.2 A Firm's Profit Maximization

In this subsection we consider the case where the quality change embodied in a new product

is given to a firm. For simplicity we assume that production technology is based on CRS , and

no variable cost is added for the production of a new product. In this case a firm's profit from

developing and producing a new product of quality A is defined as follows .

1 - f (p - c)(1 + b)(1 - v)bdv (7) Max(G,H)

where c stands for average cost for production.

The first order condition for maximization is derived as

In + a + ln(M - p) = ln(M - p) + (p - c)(b + 1), if G > H (8)

In + a + In M = In (M - p) + (p c) (b +_ 1) , if G < H (9) a M-p

We define the root of p in (8) as p1, and that in (9) as p2. If we define p which satisfies the

equality G = H as

ln(M - p) In ~a ° In 1 In M p- - M-exp I

n 1{a (10) a the profit maximizing price of a new product, p* is derived as follows (The proof is given in Appendix):

if p1 > p and p2 > p -~ p* = p1, (11)

if p1 < p < p2 - f p* = p, (12)

if p2 < p -~ p* = p2 • (13)

We can investigate how the profit maximizing price moves according to the changes in exogenous

variables. The results are shown as follows .3

* * * * *

da ->0, dp > 0, dp < 0, dp < 0, dp > 0 (14) dM db da do 3Because the cal culation is straightforward, here we leave it out except for describing that we should use the

inequality condition, M > p > c, and that only the effect of a on p is ambiguous .

5

Here we get the following proposition.

Proposition II When a firm which develops a high-quality-product has a monopolistic power in

the market, its price is determined not only by the quality itself but also by a consumer's income

level, product's peculiarity, a consumer's holding stock level, and average cost for production.

3 A Design for an Empirical Analysis

3.1 Estimation Procedure

In most cases an ordinary product and a high-quality product coexist at the same time in the

market. According to the study in the previous section , the optimal price of a high-quality

product must be p1. The purpose of this research is to know the value of an unobservable

quality index A. The share function for the estimation can be derived as follows.

In b+1 SN = 1 - a a-a---- M -~ no purchase (15) In a -~ In M_p

In 1 b+1 1n± a b+1 S _ o In 1±a -{- In M '] - In 1+a M -4 ordinary product (16) a _ M-p 1+a ~- In M_p

In ~ b+1 SH

In -4 high-quality product (17) 1+a M-p

As Sir + So + SH = 1 always holds, one of the three equations above is redundant . We can add

one equation derived from a necessary condition for a firm's profit maximization .

In i + a + ln(M - p) = ln(M - p) + (p - c) (b + 1) (18) ~a M p

Based on (15), (17), and (18), for example, we can estimate the theoretical values of b, a, and A.

3.2 Numerical Experiment

We investigate how the model works using artificial data. The numbers are set as given in Table

1. By solving (15), (17), and (18), we get the roots of b, a, and A in Table 2. Using the parameter estimates listed in Table 2, we investigate how the product market reacts according to the change

in product quality A.4 The figures (Figure 1 to 3) summarize the calculation results. Figure 4 We cannot deri

ve an explicit functional form of the profit function. The figures are based on numerical

calculations.

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Table 1: Artificial Data Set

M p p C SN

100,000 0.5

Sp

0.145

SH

6,000,000 100,000 120,000 0.355

Table 2: Estimation Result

b a A 13.02 2.54 1.16

1 shows the profit maximizing price increases when a firm develops higher quality of product .

The maximized profit also increases in Figure 2, but the marginal profit is declining in Figure

3. Although we treat quality index A exogenously in the model, the figures may partly show

how a firm determines A in the process of profit maximization. If a development cost for high

quality product is an increasing function of quality index, a firm's optimal choice is to develop

a product with quality index A which makes the marginal cost be equal to the marginal profit.

3.3 Sample Hedonic Estimation

In this subsection, we estimate a hedonic function using a sample variable set based on the

estimated parameter values in Table 2. Here we consider two time period, t and t + 1. From t to t + 1, we assume 1% growth of a consumer's income and 1% decrease in price of an ordinary

product and average (marginal) cost for production, although product quality is kept constant.

The hedonic estimates is expected to identify 20% of quality difference and 10% deflation rate.

The equation to be estimated is

In pi,t = +,Qit + ,32D, (19)

where D stands for a dummy variable with 0 for an ordinary product and 1 for a high quality

product, and t stands for a time variable with 0 at t and 1 at t + 1 period. The estimation

Table 3: Sample Variable Set

,

M p _p c- -- A t 6,000,000 100,000 124,776 100,000 1.2

t+1 6,600,000 90,000 117,338 90,000 1.2

7

2

80000

70000

60000

50000

40000w 0 a

30000

20000

10000

0

1

300000

280000

260000

240000

220000

m V n

200000

180000

160000

140000

120000

1.2

Figure 1: How p moves according to ,\

_,-I_ -~~ I --- , ---- Th

/home/taka/quality/resultllmdet.dar

1.4 1.6

Figure 2: ir

1.4 1 .6 1 .8

1.8 2 2.2 2.4 2.6 2.8 lambda

How moves according to A

-- --L °------ ~r IT.

/home/taka/quality/result/lmdet.dat

2 lam bda

8

2.2 2.4 2.6

3

2.8 3

4400

Figure 3: The increment in it

4200

4000

3800

3600

3400

0 a

C C

C) ro

E

3200

3000

2800

according to A

I

1.4 1.6

T

/home/taka/quality/result/Imdet.dat' -

1 .2

L-- -- 1

1.8 2 2.2 lam bda

- I ~

2.4 2.6 2.8 3

result is given in Table 4.5 The parameter X31 corresponds to deflation rate and ,Q2 to quality

difference of high-quality product from an ordinary product . The parameter estimates show

that underestimation of deflation rate and overestimation of quality index . The number in the

bottom of the table is the F-value when we set the null hypothesis that /31 = -0 .1 and ,Q2 = 0.2.

In spite of over- and under-estimation of parameters , the estimates are not significantly different

from the true values.

5The calculation is b ased on the generalized least square method adjusted by sales-share-weight of ordinary and high-quality product.

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Table 4: Hedonic

estimates

Function

t-value

10

R2

F-value

11.5041 653.64 -0 .08634 -3.9695

0.24262 11.060

0.99846

2.1303

4 Conclusion

We have presented a simple model to measure product quality index by using the market infor-

mation when a monopolistic provider starts to sell a high-quality product . As the theoretical

examination explains, when a high-quality product is supplied by a monopolist , its price includes the effects of not only marginal cost for production and product quality , but also consumer's income, consumer's preference variety , holding stock level. The result of sample hedonic es-

timation using artificially generated data shows the possibility of a quality parameter to be

overestimated.

The next step might be to apply the model to real observation . The model is based on

the assumption that qualitative factors of a product can be aggregated in consumer's mind to a

single quality index. Therefore, a product which has too many characteristics according to widely

varied consumer's preference such as an automobile is not suitable to the model . A computer

might be good target, almost all machines have Intel's MPU inside , and an ordinary product under perfect competition can hardly be found . One possibility is found in pharmaceutical

products. A newly developed medicine has in most cases higher quality than usual one and is

provided by a monopolistic company. When a new one appears, an old one loses monopolistic

power and is priced competitively. The price of high-quality medicine should certainly include

extra rent to cover R&D expenditure , but it might be useful to measure its real quality.

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Appendix

Proof of (11) to (13)

First we show that maximized profit based on (9) is greater than that on (8) at every value of

p. Substituting (8) for (7), we get

M p+a b+~ 7r G - (p - c) (p - c) (b -~ 1) In 1 + a (20)

In case of (9), we get

M p+a b+1 (p c)(b + 1) In a () It is obvious that 71H is greater than 7rG at every value of p . Now we have a following theorem.

Theorem Al The maximized profit under G < H is greater than that under G > H .

Next we show that valid region of p1 and p2 for the profit maximization . We rewrite g and h

function in (5) and (6) as follows.

G 1 1 ln(Mp)ln(Mp) X (22 1 +X' ln(A + a) - ln(1 + a) )

H 1- 1 Y _ lnM-ln(M-p) (23) _ 1+Y' ln(A + a) - In a

We can easily show that ap > ap holds at any value of p. Considering that G = H (X = Y) holds when p = p, it can be understood that G > H holds when p is greater than p

, and G < H

if p < p. This can lead to the following theorem.

Theorem A2 In the region of p which satisfies G > H , only p1 is feasible price for profit

maximization. In case of G < H, on the contrary , p2 is valid.

Finally we show that (7) is concave in the neighborhood of profit maximizing p. If we dif-

ferentiate (7) by second order considering the profit maximization conditions, (8) and (9), we

get

d~r2 = 1-GdG ~+a M-p

d2p - 1 + b In 1 + a + In M _-- + b < 0, for, G > H, (24) p p -

d~r2 1-HdH ~+a M

d2p = - 1 + b ~ In a -- + In M - + b < 0, for, G < H (25) p p

The calculation result shows that if (7) is continuous with respect with p, there is only one profit maximizing point which satisfies either (8) or (9). Now we have the following theorem.

Theorem A3 There exists only one profit maximizing p corresponding to each necessary con -

dition of profit maximization, (8) and (9).

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Considering the three theorems shown above, we can derive (11) to (13) directly. In case of

(11) where p1 exists in valid region but p2 does not, the profit maximizing price must be p1. On

the contrary, p2 is in valid region, p2 is only one profit maximizing price regardless the value

of p1, because from Theorem Al profit by p2 is always greater than that by p1 . This case

corresponds to (13). Theorem A3 can be used to derive (12), where both p1 and p2 drop in invalid region. A firm has no incentive to increase p from p, because the rise in p over p means leaving further from profit maximizing point, which exists in invalid region, and lowering profit. In the same way, the decrease in p under p is irrational, because it makes p further from p2.

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References

[1] Berry, S., J. Levinsohn, and A. Pakes, "Automobile Prices in Market Equilibrium," Econo-

metrica, 63 (July 1995), 841-890.

[2] Feenstra, R., "Exact Hedonic Price Indexes," The Review of Economics and Statistics, (1995), 634-653.

[3] Grossman, G. and E. Helpman, Innovation and Growth in the Global Economy, MIT Press (1991), Chapter 4, 84-111.

[4] Mussa, M. and S. Rosen, "Monopoly and Product Quality," Journal of Economic Theory,

18 (1978), 301-317.

[5] Shirotsuka, S., "The Effect of Quality Change on Price Index - Measurement of quality-

adjusted price index of PC by Hedonic Approach," (in Japanese) Kin-yu Kenkyu, 13 (4) (De- cember 1994), 61-96.

[6] Rosen, S., "Hedonic Prices and Implicit Markets: Product Differentiation in Pure Compe-

tition," Journal of Political Economy, Vol.82, No.1, (1974), 34-55.

[7] Spence, A. M., "Monopoly, Quality, and Regulation," The Bell Journal of Economics, 6 (2) (Autumn 1975), 417-429.

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