Imperfect competition and product differentiation

Mathematical

Social Sciences

25 (1993) 157-171

157

North-Holland

Imperfect competition differentiation

and product

Some further results Ben J. Heijdra OCFEB, Erasmus University,

Posthus 1738, 3000 DR Rotterdam,

Netherlands

Xiaokai Yang Department

of Economics,

Communicated

Monash University, Clayton, Melbourne,

Victoria 3168, Australia

by Y.-K. Ng

Received 1 June 1992

This paper

suggests

an alternative

solution

method more fully utilizes the structure reverses some old ones. The solution competition

method

method extends the applicability

by allowing for iso-elastic cost functions.

of the model of imperfect competition Key words: Imperfect

competition;

for a popular

model of imperfect

of the model, is simple to implement, The implications

are given throughout

international

competition.

yields some new predictions,

of the original

The and

model of monopolistic

of the method for recent applications

the paper.

trade; increasing

returns.

1. Introduction The standard model of imperfect competition is undoubtedly that developed by Dixit and Stiglitz (1977). Although this model was originally formulated in order to conduct a welfare theoretic analysis of the issue of quantity versus diversity under scale economies, the model has been successfully applied to such diverse fields as international trade (Krugman, 1980; Ethier, 1982; Grossman and Helpman, 1989), macroeconomics (Blanchard and Kiyotaki, 1987), the economics of patents and innovation (Judd, 1985), and growth theory (Grossman and Helpman, 1990; Romer, 1990). As the width of this range of applications indicates, the model is successful because it is flexible, analytically tractable, and yields intuitively sensible predictions. Correspondence to: B.J. Heijdra, OCFEB, lands.

Erasmus

0165-4896/93/$06.00

Science

0

1993-Elsevier

University,

Publishers

Postbus

1738. 3000 DR Rotterdam,

B.V. All rights

reserved

Nether-

158

B.J. Heudra, X. Yang / Imperfeci competition and product differentiation

The basic idea underlying the Dixit-Stiglitz model is a preference for diversity exhibited by the individual agents. The trade-off between the distortions arising from increasing returns to scale and this preference for diversity allows for the endogeneity of the number of consumer goods. In their solution method, Dixit and Stiglitz make a simplifying assumption about the number of goods that are produced. By assuming that the number of goods is large, they are able to derive a simple pricing rule in which the mark-up is constant. This particular solution concept was subsequently adopted by virtually all authors using their framework. In this paper we demonstrate that the a priori assumption of a large number of goods is unfortunate. On the one hand, the assumption is unnecessary; the model can still be solved even if it is abandoned. More importantly, the assumption may lead to misleading predictions, and may (spuriously) reduce the applicability of the Dixit-Stiglitz model. For this reason we suggest a more precise solution concept, one that makes fuI1 use of the symmetry assumption employed by Dixit and Stiglitz, but does not depend on the assumption of a large number of goods.’ The paper is organized as follows. In Section 2 the basic Dixit-Stiglitz model is developed and solved using the different solution concepts. We also discuss the impact of our solution method on the welfare theoretic results obtained by Dixit and Stiglitz. In Section 3 we discuss the Krugman (1980) model, an early and important application of the preference of diversity model. We show that some of the counterintuitive results obtained by Krugman disappear if our solution concept is used. In Section 4 we demonstrate an additional property of our solution method, namely its ability to accommodate iso-elastic cost functions. Hence, our solution method considerably extends the applicability of the original Dixit-Stiglitz model. Finally, in Section 5 some concluding remarks are given.

2. The preference

for diversity

model

Consider the following version maximize utility subject to a budget to be homethetic and separable in tiated commodities (Xi, i = 1, . . . , n).

t The following

features

differentiate

of the Dixit and Stiglitz model.* Consumers constraint. The utility function U( . ) is assumed the numeraire commodity x0 and the differenThe prices of the commodities are denoted by

this paper

from Yang and Heijdra

(1992). The condition

for the

existence of equilibrium is identified for the general model in this paper but is not in Yang and Heijdra; an Appendix with all the proofs of the propositions appears at the end of this paper. This includes a simple derivation of the two-stage budgeting solution that Dixit and Stiglitz (1977) themselves do not give. Also we give more numerical examples and intuition of our results than provided in Yang and Heijdra. Finally, we have shown that the productivity implications of the Krugman model which cannot be extracted by the Dixit-Stiglitz approach can be explored using our approach. 2 We follow the notation introduced by Dixit and Stiglitz (1977) with the exception that we do not set endowment income I equal to unity. The example discussed in the text refers to the constant elasticity case of section

I of their paper.

B. J. Heijdra,

pO( = 1) and pi, maximization

X.

Yang / Imperfect

respectively. problem

maxu

Income

competition

and product

differentiation

is equal to the endowment

income

159

(Z).3 The

is

= U(Xo,y)

S.t.

Z= 5

X0.x;

(1)

PjXi,

i=o

y E

i

X,”

[ i=l

4E i p;ve-“ae; I I

1

I’@;

i=l

1 rl’

(2)

1-e’

Agents are assumed to exhibit a preference for diversity so that O 1. The maximization problem can be solved in two stages owing to the separability assumption (see Green, 1964, ch. 3). The solutions for x0 and xi are as follows (see the Appendix for derivation): x0 =

y&WZ

11-s(q)lZ

Xi

=

0 -ny, pi

4

4

where s(q) is the share function whose form depends on the utility function U(. ). The elasticity of substitution between x0 and y is denoted by a(q), and the elasticity of the share function s(q) is denoted by e(q) = qs’(q)/s(q). Dixit and Stiglitz (1977, p. 298) derive the following useful expression for O(q): O(q)

= [1

-dq)lll

-s(dl<

The own elasticity of demand faced In the symmetric equilibrium calculated using (3) (see the Appendix): (pi/Xi)].

1.

(4)

by firm i is denoted by Ei[ =((aX,/ap,) with pi=pj Vi, j, this elasticity can

rl + O(4) - 1

&i=-~+ [

n

x

be

(5) 1.

Furthermore, the cross elasticity of demand (pj/xj)], is equal to the following:

for firm i, denoted

by cjj [ = (aXi/apj)

&ii = &j+ q.

x

(6)

The elasticity of demand (E;) as given in (5) differs from that calculated by Dixit and Stiglitz who ignore the term in square brackets altogether and find Ei= -q (p. 299). Their justification for this approach centres around the following two assumptions: (i) each producer i ignores the cross price elasticity of demand for a variety of goods (i.e. cjj = 0), and (ii) the influence of an individual price change on 3 The paper respects. As are zero. As requirement need not be

deals exclusively

with symmetric

equilibria,

ones in which

all firms are identical

in all

a result the marginal firm and all other active firms break even in equilibrium so that profits argued by Ng (1992). even if taking into account heterogeneous producers, the zero-profit is reasonable for modelling the effects of free entry/exit. For this reason distributed profits taken into account.

160

B. J. Heijdra, X. Yang / Imperfect competition and product differentiation

the general price index is ignored, i.e. 8 log q/a log pi= 0. Assumption (i) is ad hoc and is retained here since no generally accepted alternative is available. However, as we argue below, in any application of this model one must ascertain ex post that cij is indeed close to zero in equilibrium. Assumption (ii), however, is not innocuous. Indeed, it is straightforward to show that a log q/a log pi = l/n in symmetric equilibrium. Dixit and Stiglitz motivate (ii) on the grounds that the number of goods is extremely large. Whilst it is true that f310gq/810gpj-+0 as n+03, their claim may be criticized on the following grounds. First, the number of goods in their model is endogenously determined and is consequently a function of technology and endowment parameters. Hence we need some qualifications on these parameters to ensure that the number of goods is indeed large in equilibrium. We cannot simply assume that this condition is satisfied. Second, as we demonstrate below, the effects on the model’s solution of ignoring the price index in the calculation of the demand elasticity may not be trivial even if the number of goods is large, depending on the relative values of some key parameters. Third, there is no real need to make this restrictive assumption. The elasticity of demand as given in (5) incorporates the price index effect (in symmetric equilibrium) without significantly complicating the solution of the model as is shown below. Consider the Chamberlinian decentralized market solution, labelled market equilibrium by Dixit and Stiglitz. Each commodity is produced by one firm, all active firms exactly break even and the price equals a mark-up times marginal cost. Denoting the marginal cost by c and the fixed cost by a, the symmetric market equilibrium (SME) is characterized by the following conditions: (7)

(p-c)x=a, X=-

(8)

SW p*

q =pn’4-v)

(9)

’ 9

(10)

(114

Ei = -Yf,

&i=-~+

rl+wd-1 n

(lib)

Equation (7) is the (rewritten) marginal cost equals marginal revenue condition, (8) is the break-even condition, (9) is the demand for each produced differentiated commodity, and (10) is the definition of the price index in symmetric equilibrium. Dixit and Stiglitz close the model by (1 la), whilst we suggest using (1 lb) instead. Before investigating whether our solution method affects any of the qualitative conclusions reached by Dixit and Stiglitz, it must first be proved that a unique SME exists in our set-up. The proof is a generalization of that provided by Dixit and

B.J. Heijdra, X. Yang / Imperfect competition and product differentiation

Stiglitz

(1977, p. 300). Using

of firms (n,) can be expressed wp,

s(pn

n) =

(7)-(10)

the condition

yielding

the optimum

161

number

as follows:

141 pq)

)

+ ; [l + Ei@z’(’

-q), n)] = 0,

(12)

Pn

where it has been emphasized that Ei depends on both n and p. Provided Y(p, n) is a monotonic function of n for a given equilibrium price pe, the equilibrium is unique. Differentiating Y(p, n) yields the following expression:

(13) Dixit and Stiglitz suggest that the ‘natural assumption’ is that aY/an
rl+

1-

I? n

&q)

> l-O(q).

(14)

This condition can only be satisfied for sensible values for 12if [l - q - d(q)] < 0. This is the same condition derived by Dixit and Stiglitz (1977, p. 299) and will be adopted here as well. It is clear from (13), however, that this condition is not sufficient to render a Y/an < 0 since the second term in square brackets on the RHS of (13) may be negative as well. Using (4) the following expression for &9/dq can be obtained:

ae -=-,l-o(q)]~-[l-s(q)]~. a4

(19

Several interesting cases can be discussed. If U( . ) is Cobb-Douglas, then a(q) = 1, 0(q) = 0, and as/&7 = 0 SO that W/aq = 0. By (13) aY/an ~0 follows from the preference for diversity, q > 1. Hence, the equilibrium is unique. Another special case is where c’(. ) is CES with a(q)= CT.Then it can be shown that M/aq= -[l - o12[1 -s(q)]s(q)/q
162

B. J. Heijdra, X. Yang / Imperfect

competition and product differentiation

case both solution approaches yield unique closed-form solutions which can be analysed and compared easily. Proposition l(a) summarizes the SME solutions for the equilibrium price, output, and the number of products. For purposes of comparison, Proposition l(b) restates the results of Dixit and Stiglitz. Proposition 1. (a) The symmetric

equilibrium (SAL!?) solutions for n, xi =x, and pi =p using the model defined by (7)-( 10) and (11 b) are: market

x, =

e(vZ- 4 C[(YZ/Q)(l-@)+@I ;

r&(1-,)+,o. a

(16)

(b) The symmetric market equilibrium (SME) solutions for n, x;=x,and pi=p using the model defined by (7)-(10) and (1 la) (Dixit-Stiglitz approach) are p;=c;

x’

e Proof.

=

e

-f

ae

41 -e) ’

n: =

YZ(l-e) a

’

(17)

See the Appendix.

Comparing our solutions with those obtained using the Dixit-Stiglitz approximation reveals that the latter method leads to an underestimation of the number of firms (although by less than 1, since OpL, n,>n:, and x,
B.J. Heijdra, X. Yang / Imperfect competition and product differentiation

163

Under the CSO, the benevolent social planner chooses values for n, x,, and p; such that utility is maximized, profits of all active firms are non-negative, and the demand functions are satisfied. Owing to the symmetric nature of the model the CSO involves identical prices and output levels for all active firms and exactly zero profits. In the Cobb-Douglas case discussed here the symmetric CSO is easily calculated. Proposition 2(a) summarizes the SCSO solutions for the equilibrium price, output, and the number of products. Comparing the SCSO solutions (in (18)) with the SME solutions (in (16)) reveals that the market equilibrium does indeed lead to ‘a little too much’ diversification, i.e. n,>n,. By incorporating our solution method the surprising conclusion reached by Dixit and Stiglitz (1977, p. 301) regarding the equivalence of SME and SCSO no longer holds; it is simply an artifact of the approximate solution concept used by them. The symmetric unconstrained social optimum (SUSO), where lump-sum subsidies are allowed, can also be calculated. In the SUSO firms set prices equal to marginal cost (c), and receive a lump-sum subsidy from consumers to cover their fixed cost (a). Proposition 2(b) summarizes the SUSO solutions for the equilibrium price, output, and the number of products. Comparing SCSO and SUSO reveals that the quantity produced per active firm is identical, i.e. x, =x,,. As a comparison of SME and either SCSO or SUSO reveals, however, output per firm is lower in the market holds regardequilibrium than is socially optimal, i.e. x,n,. However, unlike Dixit and Stiglitz, no general conclusion can be drawn regarding the difference in the degree of diversification between SUSO and SME. As is reported in (20), there exists a critical value for e, denoted by ,Q, which determines the difference between n, and n,. Furthermore, 4 depends negatively on both fixed costs (a) and the share parameter (y). Proposition 2. (a) The symmetric

and pi=p

optimum (SCSO) using the model defined by (7)-(10) are:

PC=:; e (b) The symmetric and pi =p

constrained

x

ae

I-.

c

social

c(l -e) ’

unconstrained

solutions for n, x,=x,

n, = y’(l -Q). a

social optimum

(18)

(S(/SO)

solutions for n, x, =x,

are: P” =

c;

ae cc1-e)’

x, = ~

Yl(l -e)

” = a[@+ y(l

-e)]

’

(19)

164

B. J. Heijdra, X. Yang / Imperfect competition and product differentiation

(c) The degree of diversification in SME, SCSO, and SUSO compares as follows: n,> n,;

n,>n,;

eE[O,@)*n”>%_;

Q=@-n,=n,;

(20)

II* n,
ee(A

where @=@(a,y;Z) with @/8a
Proof.

1 -e

Eij=

’

+ (ae/rI)

e

(21)

(1 - e>W/a)U -e) + el .

See the Appendix.

3. International

trade and monopolistic

competition

An important early application of the Dixit-Stiglitz model to international trade issues is Krugman (1980) which we consider now. In this model there are L consumer-workers that supply 1 unit of labour each and receive wage w (=l). The maximization problem is essentially the same as (1) above, with s(q) = 1 and Z= L in (3). The production side of the model is as follows:

li = (Y+ BXi ;

L= i li. i=

(22)

I

Labour used in the production of Xi is li, total cost is li (since w = 1) SO that marginal cost is j?. Labour demand equals the given labour supply. Krugman (1980) himself uses the Dixit-Stiglitz approach to solve his model. He obtains the following equilibrium values for n, p, and x:

fl=P; e

a?

fi= LO-e)

z=p(l;

ff

(23)

.

Krugman’s model makes the sharp prediction that j and R are independent of the size of the labour force; that is, an opportunity for international trade (interpreted as an increase in L) has no effect on the wage price of goods and on the output level of each good. Krugman (1980, p. 953) expresses concern about this counter-intuitive feature of his model but claims that we need a more complicated model to sort out the effects of international trade on prices and output levels of each good. It is easily demonstrated that this claim loses its validity once our solution method is employed. The correct symmetric closed economy market equilibrium (SCEME) values of x, p, and n can be obtained by applying Proposition l(a), and setting y = 1, I= L, a = a, and c=/3: x, =

@(L-a) P[Wa)(l

-@)+@I ;

r&-Q)+@ o!

(24)

B.J. Heijdra,

In contrast

X.

to Krugman’s

l(a) do indeed of international

Yang / Imperfect

results

competition

and product

165

differentiation

(in (23)), x, and pe as obtained

from Proposition

depend on the size of the economy (and therefore on the opening trade) with the following comparative static signs:

up

(254

ax, -= aL

e2

~umu

>o.

-d + ~12

(25b)

The equilibrium price falls and the quantity produced per firm rises as a result of opening up trade. Using our solution method, no further complications to the model are needed in order to obtain plausible results. Of course, if, as Krugman assumes, L and n are large, then the comparative static predictions derived from Krugman’s solution in (23) are good proxies for the correct solutions given by our approach (25). As (24) furthermore shows, n, depends not only on L (the size of the economy), but also on Q and (r. Hence, n, may be small if (Yis large (the fixed labour cost in producing a good is close to the labour force size) or if Q sufficiently close to one (the substitution elasticity is extremely high) even if L is large. This has justified our claim that we need qualifications on parameters to ensure a large number of goods. Moreover, a large economy is not sufficient to ensure a large number of goods. Using (21) and making suitable substitutions, we can compute the equilibrium cross price elasticity of the Krugman model: &j) =

e (1 -&?)[@/a)(1

-e>+e*

(26)

E,~ may be non-trivial even for a large close to one. For example, assume that there are 1000 different varieties, i.e. ne= 1000, but that these varieties are very close substitutes, say Q = 0.9999. Using (24), this implies that L/a = 9.99 million; by any standards a large economy. For this combination of parameter values, however, it turns out that E;,= 10. Hence, a large value of n, is not by itself sufficient to ensure that the cross-elasticity is close to zero in equilibrium. The intuition behind this result is straightforward. The cross elasticity E,; can be written as @q/n, where q is the elasticity of substitution between the differentiated commodities (= l/(1 -Q)). From this it is easy to see that a large n does not ensure a small cij if q is also large. This statement attains even more relevance if it is assumed that the elasticity of substitution rises with the number of available varieties, i.e. q=q(n) with q’>O. Clearly, in such a case cij will only tend to zero as n + 03 if q(n) rises at a slower rate than n. This lends further support to our claim that simply assuming a large number of varieties may not be sufficient to ensure that cij=O. Finally, our approach also has implications for the comparative static effects of trade on factor productivity. Defining total labour productivity as rc = nx/L, (23) It is straightforward

to demonstrate

that

number of products (n,) if Q is sufficiently

166

B. J. Heijdra, X. Yang / Imperfect competition and product differentiation

implies 71 = e//3; factor productivity is constant in the Krugman model. This is a counter-intuitive result which is caused by the solution method adopted by Krugman. Using our approach, (24) implies n = ~(1 - a/L)/P so that &r/aL > 0; factor productivity increases with the opening up of international trade. Unlike Krugman’s approach, our approach distinguishes two distinct channels through which trade influences welfare: an increase in L increases the number of traded goods (&z,/aL > 0) and increases labour productivity (an/aL > 0).

4. Increasing

returns to scale and iso-elastic

In the original formulation by Dixit because of the existence of fixed cost. economies and imperfect competition terest to investigate whether the model cost functions. To illustrate that this is production functions for xi and x0 as xi = l//b;

p< 1;

cost functions

and Stiglitz increasing returns to scale exist Whilst this is a valid way of modelling scale (see, for example, Spence, 1976), it is of incan be generalized to include homogeneous indeed possible, we specify the homogeneous follows:

x0 = I,.

(27)

Here 1; denotes the quantity of labour employed to produce good Xj (j = 0, 1, . . . , n), and /? < 1 characterizes globally increasing returns to scale in producing good i. Each consumer is endowed with one unit of labour and the size of the labour force is L. Labour is assumed to be the numtraire and the nominal wage rate is w (=l). The demand side of the model is as above with c’( . ) Cobb-Douglas so that a(q) = y. Proposition 3 summarizes the results using our approach. Proposition 3. (a) The symmetric x;=x,

pi=p,

market

x0, pO, and

equilibrium

[ z;;_$)]‘“:

n e =p*@(l-P) Q-P (b) The market Proof.

exists if Q>P. The solutions for

x0=(1-y)L, 1

’

equilibrium

(28)

PO, = 1,

pe = [ y;y:;:l”-“‘“:

x,=

(SME)

TI are:

Ei=/ji

e-P ‘u = (1 -p)(l

(29)

-@I.

(30)

does not exist if Q
See the appendix.

In contrast, the solution obtained by using the Dixit-Stiglitz for this model except if Q =/3, but at that point the number

approach does not exist of firms (and the degree

B.J. Heijdra, X. Yang / Imperfect competition and product differentiation

of diversification) Stiglitz solution

167

is indeterminate. Although Q =p is not impossible, the Dixitmethod is unnecessarily restrictive in this case. Proposition 3(a)

shows that a well-defined market equilibrium does exist for values of Q in excess of p. The intuition behind this ‘knife-edge’ result is quite straightforward. The ratio of MC and AC is constant, i.e. MC=/MC. For the Dixit-Stiglitz method the solution must simultaneously satisfy MC =MR, p = AC, and MR = LIP. This can only be the case if P=Q. At that point there is nothing determining n. In our solution method this ‘knife-edge’ problem disappears because the elasticity of demand depends on n, and the solution satisfies MC = MR, and p = AC as before, but MR = [,g(n - l)/(n - e)]p. This can be solved for meaningful solutions if p< Q. This result is not specific to the simple production structure used in (27). In fact, for any multi-input homogeneous production function of degree l/A the cost function (i) is separable in output (y) and input prices (w), i.e. TC=@(w)y*, and (ii) the elasticity of scale k(y)/yk’(y) = l/1 is constant, where k(y) =y”. Suppose that A < 1, which implies that there are increasing returns to scale. Then the equilibrium number of goods given by our approach is n, = ~(1 - A)/(@ -A) if Q > A; n, = nmax (the number of goods cannot be greater than the number of firms which cannot be greater than some maximum number determined by the population size4) if @=A; and equilibrium does not exist if Q < A. According to the Dixit-Stiglitz approach, equilibrium does not exist if Q #A, and there is an infinite number of equilibria and n; can be any number if Q = A. Clearly, the Dixit-Stiglitz solution method breaks down in the iso-elastic case.’ The ‘razor-edge’ problem associated with the Dixit-Stiglitz approach is not trivial. Indeed, of the four production functions most commonly used by economists, i.e. Cobb-Douglas, CES, Leontief, and linear, only the last one is compatible with the Dixit-Stiglitz approach provided fixed costs exist. Furthermore, this same conclusion holds for the Ethier (1982) model as well. The Dixit-Stiglitz approach cannot generate a well-defined equilibrium if the production function for producer goods is Cobb-Douglas, CES, or Leontief, while our approach can.

5. Concluding The simple use of as the

remarks

main results of this paper are the following. First, we have shown that a solution method exists for the Dixit-Stiglitz model. This method makes full the structure of that model, and is applicable at the same level of generality original Dixit-Stiglitz method. Second, we have investigated the implications

4 In effect n, attains a corner labour, then the corner solution

solution. If we assume is nmax = L.

that the minimum

firm size is that of 1 unit of

5 The discussion is based on the assumption that all factors are flexible. To the extent that some factors are (quasi-)fixed in the short run, the short-run cost function will not be iso-elastic and the linear cost structure

assumed

in Sections

2 and 3 is more appropriate.

168

B. J. Heijdra, X. Yang / Imperfect competition and product differentiation

of this new solution method on the qualitative welfare theoretic conclusions derived from the Dixit-Stiglitz model itself and various models derived from it. It is shown that our solution method in some cases reverses the conclusions derived on the basis of the Dixit-Stiglitz solution method. Third, our solution method significantly widens the applicability of the Dixit-Stiglitz model. Specifically, whereas the Dixit-Stiglitz solution approach cannot accommodate iso-elastic decreasing cost functions, our method can. Hence, Cobb-Douglas, CES, and Leontief production technologies are not problematic in the Dixit-Stiglitz model, provided our solution method is employed. A final argument in favour of our approach is the fact that no additional complexity is introduced. The adjustment to the pricing rule is intuitively plausible and simple. On the basis of these arguments we suggest replacing the Dixit-Stiglitz solution method by ours.

Acknowledgements We thank Yew-Kwang

Appendix:

the following people for their helpful comments on an earlier Ng, Peter Dixon, Avinash Dixit, and Gene Grossman.

Derivation

draft:

of results

Equation (3) The first-order

au

conditions

au

-= ay

qaxg;

for the first stage optimization _ I

problem

are:

=x,+qy.

(AlI

Since U( . ) is homothetic (aU/ay)/(aU/axO) depends only on y/x,. Provided tT( . ) is concave, the first first-order condition defines y/x0 as a non-increasing function of q, say y/x,,=h(q). Using the budget constraint in (Al) this implies that x0 and q can be written as in (3), using a suitable definition of s(q):

s(q) =

qh(4

W)

1 +qW)’

and s(q)=y. If u(e) For example, if U( . ) =x0 leyyy, then h(q)=y/(l -y)(y/x,)-’ is CES with share parameter 1 - y (for x0) and y (for y) and substitution elasticity (T, then h(q) and s(q) are: -O;

In the second

(y/l s(q) =

stage of the maximization

1 +(y/l

problem

- y)Oq’ m0

(A3)

-y)Oq’pO’ y is maximized

by choice

of xi

B. J. Heijdra, X. Yang / Imperfect competition and product differentiation

subject

to the stage two budget

169

C pi Xi = s(q)Z. This yields the expression

restriction

for x,.

Equation (5) Using the demand

equation

for xi in (3) the elasticity

ei can be written

as follows:

aiogx, E;E

p=-q[ a log

l-==]

+ ‘;;&s(;)

+--s.

(A4)

Pi

the price index effect 8 log q/a log pi= l/n. Furthermore, in the form of (5). The cross elj can be calculated in a similar fashion.

In symmetric

equilibrium

B(q)=8 log s(q)/8 log q, so that (A4) can be rewritten elasticity

Proposition

I

(a) Using obtained:

(7)-(lo),

p=-

c

(1 lb), and imposing

n-e

(->

@ n-l These three equations (b) Using (7)-(lo), obtained: p2;

;

(p-c)x=

s(q) = y and B(q) =0, the following

a,

x=-.

are

YZ np

can be solved for p, n, and x as given in (16). (1 la), and imposing s(q) = y and B(q) = 0, the following

are

x=??

(p-c)x=a; e

nP .

These three equations

can be solved

for p, n, and x as given in (17).

Proposition 2 (a) Substituting

(3) into the utility

function

(1) yields the following:

u = YY(l -Y)‘-Y(pnl/(l-~))~YZ. Utility

is now only a function to the non-negativity

p subject

of n and p. It must be maximized by choice of n and condition for profits and the demand functions:

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