On the theory of monopolistic competition under ... - Springer Link

Journal of Economics

Vol. 54 (1991), No. 1, pp. 21-32

Zeitschrift

for

National6konomie

9 Springer-Verlag 1991 Printed in Austria

On the Theory of Monopolistic Competition Under Demand Uncertainty By

Yasunori Ishii, Yokohama, Japan* (Received January 30, 1990; revised version received April 19, 1991)

Establishing a model of a monopolistically competitive industry in which risk-averse Coumot finns act under demand uncertainty and in which the output of individual firms and the number of finns in the industry are both endogenously determined by free entry and exit, this paper attempts to investigate the effects of demand uncertainty on the market equilibrium of a monopolistically competitive industry. It is assumed, for calculus simplification, that the firms are identical in the sense that they have the same monopolistic power and the same production technology. The paper presents some interesting and useful comparative statics results which are contrary to those proposed in the existing papers.

I. Introduction Recently Appelbaum and Katz (1986) have established a model of industry equilibrium where the output of individual firms and the number of firms in the industry are both endogenously determined under demand uncertainty, and they have analyzed some comparative statics. Their striking contribution is, among others, to have demonstrated the following results: * This is a revised version of my paper which was firstly presented to the annual meeting of the Japanese Association of International Economics held in 1988 and then included partially in my book published in 1989. I am indebted to professors D. B6s, S. Fujino, M. Ohyama, M. Nishijima, to the members of the Public Economics Research Seminar in Bonn, and to two anonymous referees for their helpful discussions and useful suggestions. Any remaining errors, however, are my responsibility.

22

Y. Ishii:

(i) A spread-preserving change in demand leaves the output of individual firms unchanged. (ii) The effect of a mean-preserving change in demand uncertainty on the output of individual firms is ambiguous. (iii) Some comparative statics results such as the effects of changes in fixed costs, mean demand and demand uncertainty on the output of individual firms, on the number of firms in the industry, on industry output, and on the expected market price are signed independently of absolute and/or relative risk aversion of individual firms. The above results proposed by Appelbaum and Katz are contrary to those presented by Sandmo (1972), Ishii (1977), and others, and very interesting from a theoretical point of view. However, these were shown in a model assuming that the industry is perfectly competitive. In practice, the finite number of firms might be inconsistent with a perfectly competitive industry (see Fama and Laffer, 1972). The purpose of this paper is to present a model of monopolistically competitive industry under demand uncertainty in which the individual firms face a downward sloping demand curve characterized by a random shift parameter and in which the number of firms in the industry is endogenously determined by free entry and exit, and to examine how the above results (i)-(iii) shown by Appelbaum and Katz are modified in a monopolistically competitive industry.

2. The Basic Model and Industry Equilibrium Consider a monopolistically competitive industry of n Cournot firms which produce physically similar but economically differentiated commodities xi (i = 1 , . . . , n). It is here assumed, however, that all firms are identical and symmetrical in the sense that they have the same cost function, the same demand function, and the same monopolistic power. Then, the industry price-quantity relationship with additive uncertainty is given by P = f(x)

+ -yc,

(1)

where X = (2in__l xi) is total industry output, P is the price with the features of P ' = f ' ( X ) = O f / O X < 0 and P " = f " ( X ) = 0 2 f / O X 2 > 0, e is a random shift parameter such that E[e] = 0 and E[e 2] = 1, and "7 is a certain shift parameter such that 3' > 0. Here P ' < 0 and P " > 0 reflect the features of an ordinary inverse

On the Theory of Monopolistic Competition

23

demand curve which is downward sloping and convex to the origin. 1 The assumption of identical firms gives X = nx .

(2)

Moreover, it is assumed that while the number of firms in the monopolistically competitive industry is small enough so that every finn may affect the market price unlike in a perfectly competitive industry, it is large enough so that the finn can neglect its possible unsold output under demand uncertainty like in a perfectly competitive market. Thus, each finn's profit is formulated as 1] = P ( x ) z

- c(z)

- Co,

(3)

where Co is fixed costs and C ( x ) is the variable cost function with C(O) -= O, C ' ( x ) > 0, and C " ( x ) > 0 for x > 0. The finn acts so as to maximize the expected utility of its profits E[U(II)] where U(II) is a yon Neumann-Morgenstem utility function with U'(II) > 0 and U " ( H ) < 0 which implies that the finn is risk averse. Thus, the first-order optimal condition of a Cournot finn in a monopolistically competitive industry is given by

E[{P + P'x - C'(x)}V'(II)] = 0,

(4)

and the second-order condition is given by E[{2P' + P"x - C"(x)}g'(II) + {P + P'x - C'(x)}2U"(II)] < O . (5)

It should be noted that in (4) and (5) we assumed OP/Ox = p I and 0 2 p / O x 2 = P~, since in the Coumot industry each firm determines its output as if all other firms would not react to its own action. Since there exists a possibility that the first term of the left-hand side of (5) is positive and larger than the absolute value of the second term which is negative, the second-order condition is not always satisfied. Therefore, we assume

2 P ~ + P ' l x - C'' (x) < O f o r x > 0 ,

(6)

so that the second-order condition may hold even for small uncertainty of c [(6) is always true when P'~ is small enough or P ' - - 0 ] . Then, 1 With respect to the arguments of demand curves which are convex to the origin, see, for example, Denike and Parr (1970), Greenhut, Hwang, and Ohta (1975), and Ohta (1981).

24

Y. Ishii:

the second-order condition is always satisfied, and thus optimal output of the firm is uniquely obtained by solving (4), given the value of n. The number of firms in the monopolistically competitive industry is determined by free entry and exit. It is assumed that there is the reservation level ER of an individual firm's expected utility such that if ER is lower than the actual expected utility E[U(H)] of the existing firms, then some new firms enter the industry, and if E R is higher than E[U(II)], then some existing firms leave the industry. Thus, the industry equilibrium condition of a monopolistically competitive industry is given by C ( x ) - Co}] - ER = 0 .

E[U{P(X)x-

(7)

Since E[U(II)] is a strictly decreasing function of n, the number of firms in the industry is uniquely determined, given x, if E R is a reasonable value. The market equilibrium of a monopolistically competitive industry under demand uncertainty is defined by the pair (x, n) which satisfies (4) and (7) simultaneously. To examine the stability of the system, I make the usual assumptions about the response of individual firms and the industry when out of equilibrium. It is assumed that when E[U(II)] is larger (smaller) than ER, some firms enter (leave) the industry and that when E [ { P + P ' x - C ' ( x ) } U ' ( I I ) ] is positive (negative), the existing firms increase (decrease) their output. Thus the dynamics of the system is written as (i)

d x / d t = A1E[{P + P ' x - C ' ( x ) } U ' ( I I ) ] ,

(ii)

d n / d t = A 2 { E [ U { P ( X ) x - C ( x ) - Co}] - E R } ,

(8)

where A1 and A2 are positive adjustment parameters. The system is locally stable if trace [D] < 0 and I D I > 0 hold, where

[Dll D12]

[ D ] = [D21 Dll

D22

'

(2P' + P " x - C")E[U']

=

+ E [ ( P + P ' x - C ' ) 2 U ''] < 0 912

=

[from (5)],

x ( P f + P'Ix)E[U'] + x 2 P i E [ ( P d- P ' x - CI)U ''] ,

D21 = 0

[from (4)],

D22=x2P'E[U ']0).


25

Then, a simple calculation demonstrates that (i)

trace[D] = D l l q- D22

(x2p ' + 2P' + P " x - C")E[U'] + E[(P + P ' x -

C')ZU ''] < O, (9)

(ii)

I DI

= D11D22 x2P'E[U']{(2P ' + P " x - C")E[U'] + E[(P + P ' x Ct)2Utt]} > 0 -

hold locally in the neighborhood of the equilibrium point from P~ < 0, U ~ > 0, (5) and (6). Thus, the above arguments are summarized as the following:

Proposition 1: The equilibrium of a monopolistically competitive industry facing demand uncertainty is locally stable under some plausible assumptions adopted above. This result is independent of absolute and/or relative risk aversion of the finns. Proposition 1 implies that the comparative statics results investigated in the next section are meaningful at least in the neighborhood of the industry equilibrium point.

3. Comparative Statics As has been discussed in the previous section, the market equilibrium of a monopolistically competitive industry under demand uncertainty depends on all parameters included in (4) and (7). In this section I shall analyze the effects of changes in some main parameters such as fixed costs, mean demand, and demand uncertainty on the output of individual finns, on the number of finns in the industry, on industry output, and on the expected market price. The first investigation concerns the effects of a change in fixed costs on industry equilibrium under demand uncertainty.

Proposition 2: An increase in fixed costs reduces the number of finns in the industry, and vice versa, but the effects of a change in fixed

26

Y. Ishii:

costs on the output of individual firms, on industry output, and on the expected market price appear to be ambiguous in general. This result is independent of absolute and/or relative risk aversion of individual firms.

Proof." Totally differentiating both sides of (4) and (7) with respect to Co, one gets [Dll 0

D12] [Ox/OCo] = [ E [ ( P + P ' x - C ' ) U " ] ] [E[U'] 022 J L On/OCo J

.

(10)

Thus, (10) gives

Ox aGo

- x ( P ' + xP")E[U'] 2 [D[

,

On

OCo (2P' --Fx P " - C")E[U'] 2 + E[U']E[(P + P ' x - C ) 2 U '']

(11)

(12)

ID[ Next, substituting these results into the right-hand side of OX/OCo =

nOx/OCo + xOn/OCo, one obtains OX/OCo = {x(2P' + P " x - C" - n P ' - XP")E[U'] 2 + xE[U']E[(P + P'x - C')2U"]}/I D I (13) Finally, since the expected market price is defined by E[P] = f ( X ) , one can easily show

OE[P]/OCo = P' (OX/OCo) .

(14)

Thus, substituting [ D [ > 0, U' > 0, U" < 0, P ' < 0, P " > 0, and C " > 0 into the right-hand sides of (11)-(14), respectively, one gets the proposition. Q.E.D.

This result should be contrasted with the Appelbaum and Katz result that an increase in fixed costs reduces industry output and the number of


27

finns in the industry, but raises the output of individual firms remaining in the industry. When, as a special case, the inverse demand function is linear, i.e., p// = 0, one can definitely get Ox/OCo > 0 and On/OCo < 0 from (11) and (12) which are similar to the results shown by Appelbaum and Katz in the perfectly competitive industry model. However, the signs of OX/OCo and OE[P]/OCo are still ambiguous even in this case [see (13) and (14)].

Corollary l: When the inverse demand function of the industry is linear, an increase in fixed costs increases the output of individual firms and reduces the number of firms in the industry, and vice versa, but the effects of a change in fixed costs on industry output and on the expected market price are still ambiguous. This result is independent of absolute and/or relative risk aversion.

The next investigation concerns the effects of a spread-preserving change in random demand. Now, let us redefine the industry inverse demand function as P = f ( X ) + 7e + a by adding a certain shift parameter a to the right-hand side of (1). Then, a change in implies a spread-preserving change in the industry demand. Therefore, substituting this newly defined inverse demand function into P in (4) and (7) and differentiating totally both sides of the results with respect to a, one can derive the effects of a spread-preserving change in the industry demand on the output of individual firms and the number of firms in the industry:

[D011 D12 [ 02:/00~] D22 ] LOn/Oa J

E[(P + 1

L-xE[u']

c')u'] 1

J

(15) where the results are evaluated at c~ = 0. Thus, the next proposition is demonstrated:

Proposition 3: A spread-preserving increase in random demand raises the output of individual firms, the number of firms in the industry, and industry output, but reduces the expected market price and vice versa. This result is independent of absolute and/or relative risk aversion of individual firms.

28

Y. Ishii:

Proof: From (15), one can immediately obtain O_..f_x= x3 P " E[U'] 2 0OL IDI '

(16)

and

On Oa

(17)

- x E [ U q { ( 2 P ' + x P " - C")E[U'] + E [ ( P + P ' x - C')2U"]}

IDI Thus, taking account of (6), I D [ > 0, U' > 0, U " < 0, P ' < 0, p n > 0, and C " > 0, one can show that the right-hand sides of (16) and (17) are both positive. Furthermore, substituting these results into the right-hand side of OX/Oa = xOn/Oa + nOx/Oa, one immediately gets

OX/Oa > 0 .

(18)

Finally, since E[P] = f ( X ) + a holds, (16), (17), and OE[P]/O~ = P~OX/Oa + 1 combine to give

OE[P] _ n P ' P " x 3 E [ U ' ] 2 IDI

(19) '

which is negative. And the right-hand sides of (16)-(17) are all signed independently of absolute and/or relative risk aversion of individual firms. Q.E.D. The result about the individual firm's output should be contrasted with the Appelbaum and Katz result that a spread-preserving change in demand leaves the output of individual firms unchanged in a perfectly competitive industry. The difference between the result of Appelbaum and Katz and that in this paper stems from the fact that while any firm in a perfectly competitive industry does not affect the market price, individual firms in a monopolistically competitive industry do affect the market price. However, if the inverse demand function is linear, even in a monopolistically competitive industry model one can get the results which have the same signs as those in a perfectly competitive industry model:

Corollary 2: When the inverse demand function of the industry is


29

linear, a spread-preserving increase in demand increases the number of firms in the industry and industry output, but leaves the output of individual firms and the expected market price unchanged, and vice versa. This result is independent of absolute and/or relative risk aversion of individual firms.

It is obvious from (1) that a change in V is a mean-preserving change in demand uncertainty. Thus, I finally analyze the effects of a change in 7 on x, n, X, and E[P] in order to sign the effects of a mean-preserving change in demand uncertainty on the output of individual firms, on the number of firms in the industry, on industry output, and on the expected market price.

Proposition 4: A mean-preserving increase in demand uncertainty reduces the output of individual firms, the number of finns in the industry, and industry output, but raises the expected market price, and vice versa. This result is independent of absolute and/or relative risk aversion of individual firms.

Proof." Substituting (1) into (4) and (7) and differentiating totally the results with respect to V, we have [D0a

D12

=

On/O'7] -

-zE[~U']

+

P'x

-

C')g"]

]

(20)

Thus, from (20), ] D ] > 0, D l l < 0, and (A.1)-(A.3) in the Appendix, one obtains

Ox

0---~ =

(21)

x3{'TP"E[eU']E[U '] - P'E[U']E[(P + P'x - C')2U"]} and On 00'

-xE[eU']Dn IDI

< 0.

0 .

(24)

The signs of (21)-(24) are all determined independently of the firm's absolute and/or relative risk aversion. Q.E.D. Once again the result concerning the individual firm's output and the number of firms in the industry should be contrasted with Appelbaum and Katz's result that the effects of a mean-preserving increase in demand uncertainty on the output of individual firms and on the number of firms in the industry are both ambiguous in a perfectly competitive industry. The present paper demonstrates that those two effects are both judged definitely to be negative in a monopolistically competitive industry. Substituting P " = 0 into the right-hand side of (21), one gets

ox

0"/

+ P ' - c')

~ID}

u '']

(25)

Thus, the following corollary will be presented:

Corollary 3: Even if the inverse demand function of the industry is linear, a mean-preserving increase in demand uncertainty reduces the output of individual firms, and vice versa. 4. Conclusion

A model of a monopolistically competitive industry consisting of identical risk averse Cournot firms under demand uncertainty has been established to investigate the effects of changes in fixed costs, mean demand, and demand uncertainty on the output of individual firms, on the number of firms in the industry, on industry output, and on the expected market price. The paper has taken into consideration the fact that the firm faces a downward sloping demand curve because


31

of its monopolistic power and the number of firms in the industry is determined by free entry and exit. As a result, the paper has demonstrated that while the last one of the striking results (i)-(iii) of Appelbanm and Katz mentioned in Section 1 still remains valid, the other two, (i) and (ii), in general do not carry over to a monopolistically competitive industry. Furthermore, the paper has proved that the effects of a change in fixed costs are generally ambiguous except for its effect on the number of firms in the industry. It has also shown that some results in the present paper do not always equal the Appelbaum and Katz ones even if the inverse demand function of the industry is linear as a special case.

Appendix This appendix shows some calculations used in proving the propositions. A. Since c = ( P - E [ P ] ) / " / h o l d s equations:

~[~v']

E[P])v'j/.y

= E[(P

-

= E[(p

+ P'~

= {E[(P

-

from (1), one can get the following

+ P'x

(E[P]

- c' -

- ~[P]

- P'x + c')u'j/.y

c')s']

+ P'~ - c')E[u']}/~

= - ( E [ P ] + P'x - C')E[U']/',/

E[~(P

+ p'.

-

= E[(P

[from (4)].

(A.1)

c')u"] - E[pj)(p

+ P'.

-

c')u"]/~

= E [ ( P + P ' x - C ' - E[P] - P ' x + C ' )

• (p + p ' ~ - c')u"]/~/ = {E[(P

-

+

(E[P]

P'~ - c ' ) ~ u ''] + P'~

- c')E[(P

+ P'r - c')g"]}/',/. B.

(A.2)

(4) becomes E [ ( P + P ' x - C ' ) U ' ] = E [ P + P ' x - C']E[U'] +

32

Ishii: Monopolistic Competition cov (Te, U') = 0 where cov (Te, U') is a covariance between 7e and U/. Since OU'/Oe = "yxU" < 0 implies cov (Te, U~) < 0, we obtain E [ P + P~z - C'] > 0. Therefore, this result and (A.1) combine to give E[eU'] < 0 . (A.3)

References

Appelbaum, E., and Katz, E. (1986): "Measures of Risk Aversion and Comparative Statics of Industry Equilibrium." American Economic Review 76: 524-529. Baron, D. (1970): "Price Uncertainty, Utility, and Industry Equilibrium in Pure Competition." International Economic Review 11: 463--480. Denike, K. G., and Parr, J. B. (1970): "Production in Space, Spatial Competition, and Restricted Entry." Journal of Regional Science 10: 49-63. Fama, E. F., and Laffer, A. B. (1972): "The Number of Firms and Competition." American Economic Review 62: 670-674. Greenhut, M.L., Hwang, M., and Ohta, H. (1975): "Observations on the Shape and Relevance of the Spatial Demand Function." Econometrica 43: 669-682. Ishii, Y. (1977): "On the Theory of the Competitive Firm Under Price Uncertainty: Note." American Economic Review 67: 768-769. (1989): "Measures of Risk Aversion and Comparative Statics of Industry Equilibrium: Correction." American Economic Review 79: 285-286. -(1989): Competition, Monopoly, and International Trade Under Uncertainty. [In Japanese.] Tokyo: Toyokeizai New Press. Leland, H. E. (1972): "Theory of the Firm Facing Uncertain Demand." American Economic Review 62: 278-291. Ohta, H. (1981): "The Price Effects of Spatial Competition." Review of Economic Studies 68: 317-325. Sandmo, A. (1971): "On the Theory of the Competitive Firm under Price Uncertainty." American Economic Review 6 I: 65-73.

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Address of author: Prof. Dr. Yasunori Ishii, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236, Japan.