Quality competition: uniform vs. non-uniform consumer distribution Текст научной статьи по специальности «Математика»

Margarita A. Gladkova and Nikolay A. Zenkevich

St.Petersburg State University,

Graduate School of Management,

Volkhovsky Per. 3, St.Petersburg, 199004, Russia E-mail: rita.gladkova@gmail.com zenkevich@som.pu.ru

Abstract A two-stage game-theoretical model of duopoly and vertical product differentiation is examined. It is assumed, that there are two firms on some industrial market which produce homogeneous product differentiated by quality. The results of the research of a two-stage model of duopoly are presented. At the first stage of the model companies define quality level and at the second stage they compete in product price. It is supposed that consumers are uniformly distributed. This model was extended to the case when consumers are distributed non-uniformly. The research presents the comparative analysis of results in the case of uniform and non-uniform consumers’ distribution.

Keywords: vertical product differentiation, duopoly, multi-stage game, nonuniform consumer distribution, sub-game perfect equilibrium.

1. Introduction

Investigation of quality problems is in the focus of interest of the industrial organization. It is caused by the fact that quality is the most powerful instrument of the company management and it is required to provide the competitiveness of the company.

There are different approaches to quantitative quality estimation. In this paper we follow the idea of product quality estimation using the solution of corresponding game-theoretical model.

Models that are investigated in this paper are dedicated to the duopoly modeling under condition of quality competition and vertical product differentiation. Quality competition allows companies to obtain some competitive advantages. Making investments in technological innovations companies can manage and control its product quality range and gain the leadership position on the observed market. It is necessary to mention that making investments in technological innovations helps companies manage expenditures connected with the production of goods of certain quality, and it can give competitive advantage to the firm as well (costs leadership). The main aspect of costs leadership strategy lies in the lower net costs in comparison to the competing firms.

In this paper models of vertical product differentiation are suggested. The problem of optimal product quality estimation is solved for all feasible values of initial

* This work was supported by the Russian Foundation for Basic Research under grant No. 08-01-00301-a.

parameters of the models. The aim of the work is not only to build mathematical models of product quality assessment but also to solve numerical estimation examples of the product characteristics we are interested in.

There were created two game-theoretical models of company operation which is to produce goods of some quality under condition of competition on some industrial market. The first model is the model of product quality estimation in case of duopoly and uniform consumer distribution. Then in the second model we extended the first one to the case of non-uniform consumer distribution.

So, two two-stage game-theoretical models of duopoly and vertical product differentiation are examined. The basis for our models is the model described by Jean Tirol ( Tirol, 1988). It is supposed however that investigated market is uncovered.

It is assumed, that there are two firms on some industrial market which produce homogeneous product differentiated by quality. Also, quality range is supposed to be given in our investigations. The models consist of two stages: at the first stage companies define quality level and at the second stage they compete in product price. In both models we propose that each firm’s goal is the maximization of its profit while producing the goods of certain quality.

Using the backward induction the problem of optimal product quality estimation was solved.

To obtain the optimal product quality the sub-game perfect Nash equilibrium is found for the models described above. Methods of game theory and mathematical analysis are used to solve the problem. The suggested approach to the optimal product quality estimation for the described models of competition and vertical product differentiation can be used for recommendations on companies strategic planning.

Having the expert evaluation of market characteristics, the computation can be made using the models described in this paper. Thus, the analysis of quantitative results allow companies to formulate optimal production strategies.

2. Uniform Distribution

2.1. Problem Statement

This section represents the model of vertical differentiation - the differentiation by quality. When products are vertically differentiated all consumers agree about the most preferable set of product properties, so they agree that the higher is the product quality the better is the product. However, such natural ordering in the space of characteristics can be made only in case of equal product prices.

Two-stage game-theoretical model of duopoly under vertical product differentiation is investigated. It is assumed, that there are two firms on some industrial market which produce homogeneous product differentiated by quality. We will suppose that each consumer has unit demand and has different inclination to quality.

The utility function of the consumer with inclination to quality 0 when buying the product of quality s for price p is:

We call parameter 0 the ’’inclination to quality”, which indicates customer’s willingness to pay for quality. It is clear that a consumer with inclination to quality 0 (from now on we will simply call him/her ’consumer 0”) will consider the purchase

(1)

of the product of quality s for price p if and only if its utility from this purchase is non-negative. Note that the first item in equation (1) can be interpreted as maximum price that consumer 0 is ready to pay for the product of quality s (the worth of the product for the consumer 0).

As we have no information about the distribution of the parameter 0 we suppose that it is uniformly distributed with unit density over the interval [0_, 0], where

0 = 0 + 1, 0 > 0.0 and 0 are considered to be given. Otherwise we have to investigate separately the distribution of consumers’ willingness to pay for quality.

Firm i produces goods of quality si and we assume that the product will be sold at the price p > c(sj), where c(si) is firm’s cost function which express the production costs for the product of quality s.

Firm Vs profit of producing a product of quality Sj , where Sj <G [s, s] , will be defined as following function:

where pi - product price of the firm i, si - quality of the firm i’s product, P = (pi,p2) - a vector of product prices of the competing firms, s = (s1;s2) - a vector of product qualities, Di(p, s) - the demand function for the product of quality si , c(si) - production costs of firm i for the product of quality si .

The costs function is considered to be quadratic: c(si) = ks2, k > 0.

In this model we propose that each firm’s goal is the maximization of its profit function.

In such statement of the problem we have a two-stage model of duopoly, when:

— at the first stage companies define quality level;

— at the second stage they compete in product price.

2.2. Prices in Equilibrium

We assume that some regional industrial market is uncovered. In this case functions of the demand for the competing firms’ goods can be presented in the following way:

To find optimal product price for firms-competitors, we use the first order condition of extremum:

ffi(pi,Si) = pi(s)Di(p,s) - c(sj),i = 1, 2

(2)

(3)

( dlli = P2&1 - 2pis2 = n dpi siZis

I dil2 _ 6>Z\s - 2p2 + Pi

dp2 3s"

Then the firms reaction functions are:

pi = Ri{p2) = J^P2

d ( \ Pi + OAs P2 = R2 (pi) = 2-----

(4)

Hence, we receive equilibrium prices:

*/ \ OsiAs

Pi(s) = 4s2 — si

(5)

2 Os?, As S2

Equilibrium demand for firms’ products can be rewritten then such as:

P2(s) 4g2 _ Sl

d;<«) =

(6)

And firms’ profit functions of producing goods in equilibrium are defined as:

ni(s) = ni(p*s,s) = 0 sis2As -c(si)

(4s2 - si)

-2 (?)

272*(s) = n2(p*s,s) = 46> s2As -c(s2)

(4s2 - si)

2.3. Optimal quality choice in case of quadratic costs

Taking into consideration the results obtained in the subsection 2.2. we receive that firms profit functions of producing goods and selling them for optimal price look in the following way:

III (s) = 0 sis^As _ ^

(is2-sif K1>

-2 (8) tt* ( \ 4$ s^As 1 ^

n2 (s) = Ta------ “ C(S2)

(4s2 - si)

Using the extremum conditions we can define the optimal product quality. Let us recall as well that production costs function is supposed to be quadratic, i.e. c(si) = ks2, k > 0.

Then first order condition is:

<977* f)2 e2

^ (4S2 “ 7si) “ 2ksi = 0

si (4s2 - si)

±f_

s2 (4s2_si)

~g~ = tA ^ 82 ^3 (4s\ — 3sis2 + 2s2) — 2fcs2 = 0

Tablel. Initial data.

Os 6s s

350 420 70

300 367 67

320 388 68

290 355 65

350 410 60

To solve this system we can equate right parts of two equations. Then, we receive the following equation of the third degree:

4s2 - 23s2s1 + 12s2s1 - 84sf = 0. (9)

Let’s make the following substitution: s2 = ^s1. Then the solution of the cubic equation above is ^ = 5.2512 .

Then s2 = 5.2512s1. If we substitute this interdependence of product qualities in the extremum condition, we receive the following results:

1. If values of optimal product qualities from the system (12) belong to the interval si G [s;s] > then

s| = 0.0241^

s\ = 0.1266^ Hence, firms profits in the equilibrium are:

it

n\ = ni(s*) = 0.0125^

—4

27| = 27| (s*) = 0.0123^-

(10)

(11)

2. If s^ < s, s2 <G [s, s] , then sj* = s. From the first order condition we find the optimal product quality:

_2

^ = 49 32 , (4*1 - 3s2s + 2s2) - 2ks2 = 0. (12)

ds2 (4s2 — s)

Let’s make the following substitution: s2 = /xs. Then, we receive the following equation of the third degree in ^:

—64fcs3yU,3 + ^802s2 + 48fcs3^j /j2 — (q()2s2 + 12fcs3^j /j + 402s2 + fcs3 = 0. (13)

We obtain the solution of this cubic equation numerically.

We assume that we know maximum prices that consumers with the lowest and highest inclination to quality are ready to pay for products of the lowest possible quality s . As well, we suppose that the coefficient k = 0.02.

Table 1 presents the initial data for this case.

3.

Table2. Coefficients of equation and its solution.

M2

-439040 1740480 -1140720 712460 3,3304

-384977 1366244 -880317 544771,3 2,9339

-402473 1506207 -978728 608464,6 3,118

-351520 1271840 -822060 509592,5 2,9996

-276480 1552160 -1060440 676720 4,9376

Table3. Results.

sv m

70 233,128 30,82294 629,1544

67 196,5713 9,185541 388,6145

68 212,024 18,61086 486,4417

65 194,974 11,63705 393,1936

60 296,256 82,93263 1304,629

The coefficients at all powers of u in the equation, that we receive using the initial data given above, are presented in the Table 2 below.

So, in the next table (see Table 3) we present the problem solution, i.e. the optimal product quality for our initial data.

If ^ <G [s, s] , s2 > s , then s2* = s. The first order condition allows us to find the optimal product quality:

dm e s2 , x x

—- =---------------~ 4s - 7si - 2fcsi = 0 14

ds! (4s-Sl)3

Using the following substitution: si = /xs , we obtain the equation of degree four in yU,:

2fcsV - 24fcsV + 96fcs3y«2 - (jfs2 + 128fcs3) /x + 4tl2 = 0 (15)

This quadric equation is solved numerically. We assume that we know maximum prices that consumers with the lowest and highest inclination to quality are ready to pay for products of the highest quality - s .

Thus, we receive the highest possible quality that both firms can produce. Table

4 presents the initial data for our problem.

Table4. Initial data.

ffs 0s s

600 720 120

655 752 97

700 800 100

755 930 175

780 930 150

Table5. Coefficients of equation and its solution.

M4 M2

69120 -829440 3317760 -8052480 2073600 0,2896

36507 -438083 1752332 -6294971 2262016 0,3995

40000 -480000 1920000 -7040000 2560000 0,4038

214375 -2572500 10290000 -19774300 3459600 0,1935

135000 -1620000 6480000 -14694300 3459600 0,2642

Table6. Results.

si* s2* m

34,752 120 40,40304 603,6722

38,7515 97 77,85318 892,0378

40,38 100 86,52719 980,1692

33,8625 175 30,29724 487,8746

39,63 150 48,90488 765,9821

The coefficients at all powers of ^ in the equation, that we receive using the initial data given above, are presented in the Table 5 below.

So, Table 6 presents the problem solution, i.e. the optimal product quality for

our initial data.

4. If both values of optimal product qualities of the firms-competitors don’t belong to to the quality range interval, i.e. s^ < s, s| > s then

s* - s' ^

s2 = s

Thus we have the case of maximum differentiation. The profit functions in equilibrium are:

_2

27? (s) = nt(s*) = - fcs2

-2 S ” (1?)

n2 = n%(s*) = 46>. ^ (s ~ ^ - fcs2 2 2V 7 (4s -s)2

5. If sl,s% < s, then II* < 0 , which is infeasible.

The results of the modeling shows us that company’s decision on product quality strongly depends on the initial market conditions, which defines the model’s parameters.

3. Non-Uniform Consumer Distribution

In this section we consider a model for a vertically differentiated product when consumers are distributed non-uniformly over investigated market.

Assume that there are two firms i = 1,2 which produce products of quality

51 <G [s, s] at a cost independent of Sj, i.e. c(sj) = C = const. Suppose, that

52 > si.

As in the section 2 the utility function of the consumer with inclination to quality

0 when buying the product of quality s for price p is:

Ue (p)

(18)

0s — p, p < 0s

0, p > 0s

But here it is assumed that parameter 0 <G [0,1] which indicates the willingness to pay for quality is distributed over the population according to a continuous symmetric triangular density f (0).

Generally the density f (0) is defined as follows (see also Benassi et al., 2006):

f(0) =

40, for 0£ A= [0,

12

2, for0=-4(1-6»), for = 1]

Then distribution function can be presented as follows:

F (0)

202, for 6>gA=[0, h 12

20, for0=-

2

40 — 202, for 0 = 1]

Figure 1 represents the view of the density function f (0).

Figure1. The density function f (0)

First, let us introduce the following standard notations:

*i = ^; si

- the marginal consumer who is indifferent between purchasing the lower-quality product, the first firm’s product, and nothing at all.

q2 = P2 ~Pl S2 - Si '

-the marginal consumer who is indifferent between purchasing higher-quality product, the second firm’s product, and lower-quality product, the first firm’s product. Each firm’s demand can be written as follows:

f e2

Di(p,s) = Di(0i ,02)=/ f(0)d0 = F(02) — F(0i); (19)

Je 1

D2(p,s)= D2(02 )= / f (0)d0 = 1 — F (02). (20)

e2

where F(0) = /Qe f (z)dz is the distribution corresponding to the density f (z). Each firm’s profit of producing a product of quality Sj , where Sj <G [s, s] , is defined as follows:

i

n»(p, s) = pj(s)Dj(p, s) — C,i = 1, 2.

where pj - product price of the firm i, sj - quality of the firm i’s product, p = (pi,p2) - a vector of product prices of the competing firms, s = (si,s2) - a vector of product qualities, Dj(p, s) - the demand function for the product of quality sj , C - constant production costs.

The goal of each firm is its profit maximization, when at the first stage firms set the product quality and at the second stage they compete in product prices.

According to our proposition about the form of the density function, the explicit formulation of demand functions (19 and 20) differs depending on the value of the limits of integration. Therefore, demand functions view depends on the location of marginal consumers across the interval [0,1]. Thus, three co-locations of 0i,02 are possible:

1. 0i,02 € A;

2. 0i,02 € B;

3. 0i € A, 02 € B.

The proposition (Benassi et al., 2006) below allows us to cut the list of possible options to one case.

Proposition 1. Consider any concave symmetric density f (0) defined over [0,1], such that f (0) = f (1) = 0 and f (1/2) > 1. If (0j\ 0|), 0| > 0j* identifies the marginal consumers at the perfect Nash equilibrium in the two-stage game for vertical differentiated, products, then 0| is lower than the median of the distribution.

Proof. Consider the derivative of the second firm profit with respect to its product price and equal it to zero according to the first order condition:

= 1 - F(e2) - ~^—m) = 0.

d (p2) s2 — si

Thus,

1 — F (02) = b(02).

=_____el

S2 - ■

Notice some tendencies and properties of the equation above:

where b(02) = (a + d2) / (02), a= g2 ^ gi > 0.

1. 1 — F(02 ) is a strictly decreasing from 1 to 0.

2. b(02) is increasing till f (02) > 0 because b (02) = f (02) + (02 + a)f (02).

As b(02) is a continuous function such as b(1) = b(0) = 0, then the maximum of the function b(02) exists in some point 62.

When / (02) > 0 ^ which is true for sure for all 02 < ^, then b (02) > 0.

Thus, 02 > 7j.

3. 6 (^) = (a + £) / (^) > \ = i - F (X This is equivalent to: 6^) > 1 — F

As well b(0) = af (0) = 0 < 1 — F(0) = 1, which leads us to b(0) < 1 — F(0). Thus, 6>2 <

First two observations allow us to conclude that there is unique maximum 02 €

M2].

Now, we consider the interval 02 € [02,1] and show that there no extreme points there.

Let us introduce the function p(02) = 1 — F(02) — b(02).

— When 02 = 02, investigated function takes negative values p \ @2) < 0.

— p'(02) = —2f(02) — (a + 62)f'(02).

And p (#2) = — 2f (#2) — (a + ^2) f' (#2) = —f (^2) — b' (#2) < 0 (as

b' (02) =0).

— As density function f is decreasing in 02, 1 and convex, then f (02) is decreasing. Therefore, p (02) is increasing function.

— Taking into consideration that f (1) = 0 and f (1) < 0, we obtain that p (1) > 0. As p (02) is increasing function, then p = 0 in the unique point 02.

p(1) = 0 =^ p(0)2 < 0 when 02 < 02 < 1 =^ 1 — F(02) < b(02) for any

02 € 02, 1

Thus, we proved the proposition.

Therefore, to solve the problem only the first co-location of 0i,02 is possible. Thus, one case with explicit game formulation is considered.

Here, demand functions can be rewritten as follows:

Dl(p,s)= 2(^—— 2 ^

s2 — s1 s1

D2 (P, s) = 1 - 2

P2 ~ Pi S2 - Si

And profit functions of the investigated firms while producing goods of quality

ni (P,s) = Pi 2

22 P2 ~ Pi \ _ 2 / Pi

s2 - s1

s1

(21)

n2(p, s) = p2 ( 1 - 2 ( P2 _Pl 1 I - C.

(22)

The sub-game perfect Nash equilibrium is obtained using the backward induction.

Suppose first, that firms have chosen their product quality and compete by its prices knowing the quality choice of their competitor.

The first order condition is presented below:

dlh

dpi

n(P2-Pl\2 o (Pl\2 fi(P2-Pl) 4pi

dnL

dp2

(s2 - s1)2 s2

2 _ 1 ofP2~Pl\2 4p2(p2-pi)_n --1-2{W^) ~ (s,-si?

Thus, the solution of the system is:

(23)

Pi(s)

s1

z - 3

6 (3z - 1 )(z - 3) ’

(24)

p2(s)

s1

V& z(z — 3)

6 y/(3z-l)(z-3)'

(25)

where z

2+ i 1+ 3(52 Sl)/

Checking the second order condition, we see that both second derivatives are negative in equilibrium:

an 2

^(php*2)<o

dn 2

!|(p^)<0

(26)

Therefore, we showed that the solutions (24 and 25) are prices in equilibrium. Let us move to the second stage of the game - the choice of product quality.

2

s are:

0

l

By substituting equilibrium prices (24 and 25) into the profit functions (21 and 22) correspondingly, we get:

\/6 (3(s2 — s i)2 + s\(z — 3))

n{{s) = [ K 2 11 ^ U_3 ” - C- (27)

■y/ (3 z — 1 )(z — 3)

and

Si(z - 3) (3(s2 - Si)2 + Si(z - 3))

\/ (3^ — l)(z — 3)

It can be easily checked that the second firm’s profit is always increasing in its product quality, so we can confirm that the second firm chooses the maximum feasible product quality from the interval [s, s], which means

Substituting this result to the first order condition of the first firm and solving the equation in s1, we get that the first firm maximizes its profits by setting

s? = 0,4954s.

Therefore evaluated at the optimal quality the prices of the two firms products are:

* —

So = s.

pi = 0,24806s;

and

p*2 = 0, 061592s.

Generalizing the results of the investigation we should mention that optimal product quality of the firm which produce a higher-quality product is the highest possible product quality s. The optimal product quality of another firm varies depending on the market conditions.

4. Results of Comparison

So, the paper presents the results of investigation of two game-theoretical models, which are the models of quality competition for vertically differentiated products: when consumers are distributed uniformly and non-uniformly over some industrial market. In our research we focused on the investigation of uncovered market case which is a more realistic one. Moreover, in this case consumer distribution influences not only the demand shares but also the size of market demand.

For each proposed model the unique sub-game perfect Nash equilibrium is received.

We consider two critical cases when the distribution is uniform and when it is triangular. The research shows that in the second case the market coverage is higher.

Besides, high quality goods production brings the higher profit in all cases for all values of distribution parameters.

Therefore, our investigation has proven an idea that a more concentrated inclination to quality leads to the extended market coverage and increases the profit of the firm which produces higher-quality goods.

5. Conclusion

To summarize the results of the investigation it should be mentioned that the solution of game-theoretical models is obtained for all reasonable values of initial parameters for the uncovered market case. On the basis of the outcome of the research it is possible to evaluate quantitatively the following parameters in case of industrial competition for vertically differentiated product:

— product quality estimation for each competing firms;

— product price in competition;

— demand faced by competing firms in equilibrium;

— firms profit in equilibrium.

References

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