Scale economies and existence of sustainable monopoly prices

JOURNAL OF ECONOMIC THEORY 34, 149-163 (1984)

Scale Economies and Existence of Sustainable Monopoly Prices*

DANIEL F. SPULBER

Department of Economics. Universif.v of Southern California. Los Angeles, California 90089-0035

Received December 1, 1983: revised February 3, 1984

The size of the firm relative to market demand is crucial to a determination of whether there exist sustainable monopoly prices. In the one product case the size of the firm is its minimum efftcient scale. In the multiproduct case size is defined by a set of outputs at which cost complementarities are present. The analysis shows that when the size of the firm is sufficiently large, there exist anonymously equitable AumannShapley prices. Further, at these prices natural monopoly is sustainable against rival entry. The Aumann-Shapley price are also shown to be quantity sustainable in the sense of Brock and Scheinkman. Journal ofEconomic Literature Classification Numbers: 022. 61 1. 612. C 1984 Academic Press, Inc.

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The essential problem posed by the literature on natural monopoly’ is that while a monopoly may produce more efficiently than two or more firms there may not exist prices which deter rival entry. This paper addresses the crucial issue of existence of sustainable prices. It is argued that the principal requirement for entry deterring prices to exist is that the size of theflrm, as represented by its minimum efficient scale, be suflciently large relative to market demand. The literature on contestable markets has focused upon sufficient conditions for particular prices to be sustainable without verifying those conditions for actual cost and demand functions. In addition, the literature has imposed strong requirements upon firm technologies such as global increasing returns to scale and global cost complementarities for the multiproduct firm. These strong assumptions are neigher necessary nor sufficient for the existence of sustainable prices. This paper allows increasing

* Presented at the European meetings of the Econometric Society. Pisa. 1983. The support of the National Science Foundation under Grant 82-19 12 1 is gratefully acknowledged. I thank Michael Magill, Leonard Mirman and Wayne Shafer for very helpful discussions. I thank William Baumol for very helpful suggestions. Any errors are, of course, my responsibility,

I See Baumol. Panzar. and Willig [ 3 ] and Sharkey [ 191 for comprehensive overviews of the literature.

149 0022-053 l/84 $3.00

CopyrIght ‘Sl 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

150 DANIEL F. SPULBER

returns and cost complementarities to be absent for large outputs. It is shown that under certain conditions placed separately on cost and demand functions, the level of the firm’s minimum efficient scale is an easily verified sufficient condition for the existence of sustainable prices.

With the exception of Sharkey [ 18 1, little attention has been given to whether sufficient conditions for sustainability of monopoly prices are consistent or verifiable from cost data. Using an example with simple cost and demand functions, Sharkey shows that Ramsey optimal prices may not be sustainable. Sharkey presents existence results which depend upon conditions on the revenue vector that are difficult to verify for general nonseparable demand functions. Faulhaber [6] also presents a counterex- ample to the existence of sustainable Ramsey prices. Baumol, Bailey, and Willig (21 place conditions jointly on cost and revenue functions which guarantee existence of sustainable Ramsey prices and which are difficult to verify.’ Mirman, Tauman, and Zang [lo] present several sets of sufficient conditions for prices to be sustainable when these prices are obtained as imputations of a cost game; see also Faulhaber [6] and Panzar and Willig [ 141.

To formalize the notion of firm size, this paper generalizes Novshek [ 131, where the size of the firm is its minimum efficient scale (MES). Novshek [ 131 shows existence of Cournot-Nash equilibria when firms are small relative to market demand in a single output model. In this paper, the existence of sustainable monopoly prices depends on firms being large relative to market demand and is an opposite polar case to Novshek’s analysis. It is appealing that small MES results in existence of a competitive equilibrium while large MES results in existence of a monopoly equilibrium. This provides a natural economic explanation of the sustainability results.

In the multiproduct case, a frequent assumption in the literature on sustainable prices and on allocation of multiproduct costs is that global cost complementarities exist between outputs, 3 that is, an increase in the output of any product lowers the marginal cost of all other products. Like global decreasing returns to scale in the single product case, this is neither a necessary nor a sufficient condition for the existence of sustainable prices. Further, the assumption is not necessary for the core of the cost allocation game to be nonempty for a specific output level. Rather, it may be assumed that cost complementarities are present only for output levels that are not too large. The boundary of the set of outputs for which cost complementarities

’ Baumol, Bailey, and Willig / 2 ] require that the total cost function including entry costs be supported at the Ramsey output by a “pseudo-revenue hyperplane”; see also Baumol, Panzar, and Willig [3. Chap. 81.

3 See, for example, Panzar and Willig [ 141; Sharkey and Telser (201; and Mirman, Tauman, and Zang [lo]. However, in Baumol, Bailey, and Willig (21 trans ray convexity is assumed to hold only at the sustainable output point.

EXISTENCE OF SUSTAINABLE PRICES 151

hold provides a multiproduct analogue to minimum efficient scale. It is unreasonable to assume that cost complementarities may be obtained for all output levels no matter how large. One would expect that with all other outputs held constant returns to scale will eventually decrease for any output.

For the multiproduct monopoly, this paper shows that for MES large enough there exist market clearing Aumann-Shapley prices which are anonymously equitable; see Faulhaber and Levinson [7]. The Aumann-Shapley (AS) prices are shown to be sustainable against rival entry. AS prices are easily calculated based only upon the form of the cost function. AS prices are in the core of the cost sharing game given cost complementarities and have been examined by Billera and Heath [4]; Mirman and Tauman [9]; Mirman, Samet, and Tauman 181; and Mirman, Tauman, and Zang [ 10, 111. The latter provide a useful criterion for sustainability of prices in the core of the cost game.

Brock and Scheinkman [5] introduce the notion of quantity sustainability. A vector of zero profit monopoly output levels is quantity sustainable if the additional output supplied by an entrant will lower prices to the point where the entrant makes a loss. They show that when products are substitutes in demand, if a price-output pair is price sustainable it is also quantity sustainable. This paper shows that given a sufficiently large firm, the output quantities which clear the market at Aumann-Shapley prices are quantity sustainable.

2. ADDITIVELY SEPARABLE COST

Existence of sustainable prices must depend upon the characteristics of both market demand and firm costs. To build our intuition, consider the single product case. Global increasing returns to scale imply subadditivity of costs and yet are not sufficient for the existence of sustainable prices. In Fig. la,4 for example, where F(q) is the inverse demand and AC(q) is average cost at output q, the price p2 is clearly not sustainable. Any price below pz will cover the costs of supplying the market demand. While the price p, is locally sustainable, an entrant may still select prices below p2, draw the demand away from the established firm and earn positive profits. The problem in Fig. la is that at the last crossing of inverse demand and average costs, the inverse demand crosses average costs from below and is everywhere above average costs for larger outputs. To ensure that at the last crossing inverse demand crosses average costs from above, assume that

’ A figure similar to la is in Mirman, Tauman. and Zang (Ii I. A figure similar to 1 b appears in Baumol. Panzar, and Willig [ 14. p. 301.

DANIEL F. SPULBER

1 I

MES q q

b

FIG. 1. The prices p,, pz and p1 are not sustainable

average costs have a well-defined minimum and that market demand is positive and bounded at a price equal to minimum average costs. If decreasing returns to scale eventually set in, inverse demand will cross average costs from above at the intersection at which output is greatest. Although it is important for the monopoly cost function to exhibit increasing returns for small output levels, it is unreasonable to require that they hold globally. This would imply that existing technology places no limits on the size of an enterprise and that reductions in average costs can always be achieved by further expansion. In Fig. lb, p3 is not sustainable. It is possible for an entrant to undercut p3 and sell less than the total market demand, produce at lower average cost and earn positive profits. The source of the problem in Fig. lb is that the size of the firm is small relative to market demand. Existence of sustainable prices is obtained if the minimum efficient scale of the firm, MES. is sufficiently large relative to the size of the market, that is the output at which inverse demand equals minimum average costs 9*.

Let the firm’s cost function be given by C(q) = CJ’=, Cj(q,i), where q = (91 Y--*3 q,,) is the vector of outputs. Define ACj(.) = Cj(qj)/qj. TO

formalize the notion of the size of the firm we require the firm’s average cost to have a well-defined minimum efficient scale.

(Cl) The cost functions C’ are continuously differentiable for qj > 0, C”(Sj) > 0 and Cj(qj) > 0 for qj > 0, C’(0) > 0, for j = l,..., n.

(C2) There exists [&, ~j] c (0, co), $ > ij such that AC’(q,) = ACii, > 0 for qjE ]qji, tj] and ACJ(qj) > AC&, for qj& [cj, tj]. ACj(.) is nonincreasing for 0 < qj < Gj. Without loss of generality let qj = 1. Also, lim si,oACi(qj) < b < co, j = l,..., n.

Define MES(ACj) as the minimum efficient scale of the average cost


function. Thus, MES(ACj) = ej = 1. The cost function satisfying Cl and C2 may have a fixed cost component. Also, the assumption allows average costs with a flat bottom. Flat bottomed cost curves are discussed in Baumol, Panzar, and Willig [14].

Given the normalized cost function satisfying Cl and C2, we may rescale the size of the firm by adapting Novshek’s [ 131 definition of the size of a firm. 5

DEFINITION 1. An a-size firm corresponding to C is a firm with cost function C,(q) = aC(q/a).

Thus, C,(q) = xi”= I aC'(qj/a) = C~;-I C',(Sj>* Clearly, MES (ACd(qj)) = a, j = l,..., n. Also, the minimum average cost is not affected by CIY AChmin =ACj,,, =ACj(l).

Let D(p) represent the vector of demands (D’(p),...,D”(p)), where p = (p, ,...,p,) is the price vector for the n products. Let F(q) = (F’(q),..., F”(q)) be the vector of inverse demands and let N be the set of n products. Let pm and pe denote the vectors of prices set by the monopoly and entrant, respectively.

DEFINITION 2. Given C, F, and a E (0, co), the monopoly price vector pm is (a, C, F) sustainable if for every triple (M, qs,pR), M s N, satisfying

PZ,<PZ and q$ < D,&‘,,P&,), then &qZ, - C,(qL) < 0. It is also required that p”D(p”‘) - C,(D(pm)) > 0.

DEFINITION 3. Let s(a. C, F) represent the set of (a, C, F) sustainable monopoly prices.

The following assumptions on demand are used to examine the existence of sustainable prices in both the separable and nonseparable cost cases.

(F 1) Demands Dj(p) are continuously differentiable and downward sloping, D:(p) 3 aDj(p)/3pj < 0, j = l,..., n. Also, inverse demands are asymptotic to they axis, lirnqjA Fj(q) = CD given qi > 0, for all i #j.

(F2) Products are weak substitutes, D{(p) = aDj(p)/3pi > 0, i # j. Demands are bounded above, D’(p) < D’(pj) < co for all pj > 0 where D’(pj) is a downward-sloping continuous function of pj.

The assumption that inverse demands are asymptotic to the y axis is made to guarantee that small output levels may be profitably produced if average

5 Novshek 1 I3 1 employs the definition of an u-size firm to examine the existence and approximate competitiveness of a Cournot-Nash equilibrium when the size of firms is small relative to market demand. In our framework, the scale of the firm is increased relative to market demand to examine the existence of a sustainable monopoly equilibrium.

154 DANIEL F.SPULBER

costs are bounded. The assumption of weak substitutes permits demands for each good to be independent. The upper bound on market demand for any given price is due to consumers having budget constraints with fixed income levels. Define q: such that

D’(A &“) = qi* ) j = l,..., n. (1)

This is the size of the market for good j at the minimum average cost price. With additively separable costs, only average cost prices are sustainable,

since any other monopoly price can be undercut by entry in any single market. To show that average cost prices are sustainable, note that demand for each good j is shifted down if p& <pi since the products are substitutes; see Fig. 2. Thus, good j can only be supplied at a loss. This intuition is used to prove the following result.

PROPOSITION 1. Given C 1, C2, Fl, F2 and additively separable costs, there exists a* such that for all a E [a*, oo), S(a, C, F) is nonempty.

l?rooJ First, the existence of market clearing average cost prices is shown. Let AC,(q) = (ACL(q,),..., ACi(q,)), Q = [0, q:] x ... X [0, q,*] and A = [AC;,,, b] x a.- x [AC;,,, b] with qj* given by Eq. (1). Then AC,(q) is a continuous mapping from the compact set Q into the compact set A. The demand function D(p) is a continuous mapping from A to Q. Define the mapping T:A x Q-+,4 x Q by

T(P7 4) = AC,(q) x B(P).

Then the mapping r has a fixed point by Brouwer’s theorem, T(pm, qm) = (P”, q”)*

Now it is shown that for a sufficiently large, the equilibrium (pm, q*) occurs in the region of nondecreasing returns for each Cj(.). Let a,? = qT/&,

FIG. 2. Given p’ <p and weak substitutes, demand shifts down


j = l,..., n, where t is given in C2. Let a* = max{a:,..., a,*}. Then, for each a > a*, for all q such that q = D(p) and pj = AC’,(qj), j = l,..., n, the following holds,

qj<aj*Gj<a*tj, j = l,..., n.

Thus, given a > a*, qT/a < ti and AC;(x) is nonincreasing for x < qJY’, j = I,..., n.

Finally, it is shown that there exists a vector of sustainabk average cost prices. For a E [a*, 00) given, let pm be a vector of undominated average cost prices; i.e., there does not exist p’ <pm for pm, p’ in G(a), where G(a)= (pER::pj=ACj,(q),qj=D’(p)}. LetpR<pg,McN. Letpo’be the vector p excluding pi. Then for all j E M.

for all pj by F2. Thus, since average costs are nonincreasing and bounded below and pg < py, j E M,

for all qj < Dj(p&,~,“~) for all j E M. Thus p;qT < C’,(qP) for all j E M, M c N and so pm is sustainable. Q.E.D.

3. NONSEPARABLE COST FUNCTION

3.1. Aumann-Shapfey Cost Allocation

A necessarv condition for sustainability is that the price vector be subsidy free, that is, in the core of a cost allocation game; see Panzer and Willig [ 141. Accordingly, we focus upon a particular subsidy free price vector, the Aumann-Shapley allocation of costs examined by Billera and Heath [4]; Mirman and Tauman [9]; Samet and Tauman [ 151; Mirman, Samet, and Tauman [8]; and Mirman, Tauman, and Zang [ 10, 111. Mirman, Tauman, and Zang [ 10, 111 refer to the Aumann-Shapley pricing rule as an average cost pricing mechanism. The Aumann-Shapley average cost of product j for the cost function C(q, ,..., qn) is given by

Let AC(C, q) represent the vector of average costs (AC’(C, q),..., AC”(C, q)). Given additively separable costs, the Aumann-Shapley price equals the


average cost for goodj, Cj(qj)/qj. Clearly, average costs sum to total costs, CT= I A Cj(C, q) qj = C(q). Thus, average cost prices are zero profit prices.

The following assumptions on the general multiproduct cost function C(q) are useful.

(C3) The cost function C(q) is twice continuously differentiable, q E IR: and C(0) = 0. Let C,(q) = iC(q)/i3qj, j = l,..., n.

(C4) There is a comprehensive set6 KC E IR: such that for q E KC‘, Co(q) < 0 for all i, j. Also AC’(q) >FIC$,~, > 0 for all q, where AC;,, is independent of q, and lim,i,, A C’(q) < co for qi > 0, i #j.

Given condition C3, the average cost function AC’(C, q) is continuously differentiable in q. This assumption is made for convenience and may be relaxed. Note that condition C3 excludes lixed costs although these may be approximated.

Condition C4 guarantees that cost complementarities exist for small output levels. This is a much weaker requirement than global cost complementarities. Subadditivity of the cost function at an output q is implied if q E KC. Thus, if Cy= 1 xi = q, xi E IR:, then xi E KC and

C(q) < 2 C(x’). i=I

Therefore, C4 allows the cost function to not be subadditive for large output levels. The additional requirement of a lower bound on average costs essen- tially requires cost complementarities to be exhausted for large output levels. This restriction makes it possible to define a minimum efficient scale for each output. Let q”’ be the vector of outputs (ql,..., q,) excluding qj. Suppose that q”’ is an element of the projection of q on F!lP’. Then, it is possible to define a minimum efficient scale MES’(q”‘) as the output qj at which ACj(C, q”‘, qj) is minimized. Thus, C4 is the multiproduct analogue of a bounded region of increasing returns to scale. Note also, if q E KC, then q’ < MES’(q”‘) and &tCj(C, q)/aq, 5 0 for all j = l,..., ~1. This follows since Clj(q’) < 0 for all q’ < q and aACj(C, q)/8qi = s: tCij(fq) df < 0 for all i, j and q E KC. This is appealing since one would expect to eventually run into decreasing returns to scale for any output if all other outputs are constant. In addition, one expects to eventually run into decreasing returns if all outputs are increased. The boundary of KC may be seen as the multiproduct equivalent of minimum efficient scale.

The definition of an a-size firm is now extended to the multiproduct case.

6 KC is a comprehensive set if for any q E K’, K, s KC. where K, = {x E IR: : x ,< 9 ). The boundary of KC is strictly bounded away from zero for each goodj.


The a-size firm has the cost function C,(ql,..., qn) = aC(q,/a,..., q,/a) for a scalar a > 0. Thus, average costs are

AC'(C,,~)=J: Cj(tq/a)df=ACj(C,q/a) (3)

and for q E KC,

MES’ (4”‘) = a MES’(qo’/a). a (4)

Clearly ACj,,,, =ACii, by C4. Finally, q E KCa if q/a E KC. Consider now an example of a cost function satisfying C3 and C4,

C(q)= ($, QiYi)‘+ (, aiqi)D+” (5)

where 0 < p < 1, ai > 0. i = l,.... n. To verify whether C4 is satisfied, calculate C,,

Cij(q)=Qiaj [8(/-l) (\ln=,olYi)5p2+@+ l)~($,Uiqi)5p’]e (6)

Thus Cij(q)$$O as ~~=laiqiZG(l -/3)/(1 -t/3), and so KC= (qE IR:: C;=, aiqi < (1 -/?)/(l +/I)}. Next, calculate average costs,

ACj(C,q)=Qj [(cl ai4i)um’ + (sl ‘iqi)‘]’ (7)

Then aACj(C, q)/aqj$O as Cy=, c,qi$ (1 -p)/p. For JJi+jaiqi < (1 -/Q/p, AC’ has a minimum at

(1 --@//I? - x a;q; aj. i#cj Ii

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Thus.

AC’,&, qci), MES’) = aj( 1 - @-‘/jVr = A&,, (9)

where AC’(C, q) > ACii, for all j = l,..., n and for all q. For the a-size firm the average cost has the form

AC’(C,,q)= (aj/a’> [a( (loI

158 DANIEL F.SPULBER

where for q E KCu

a(1 -/3)/p - 1’ ajqi !T j Ii aj. (11)

An increase in the firm’s size a shifts the average cost curve upward and to the right.

3.2. Supportability and Anonymous Equity

It is now shown that for the size of the firm sufficiently large, there exist market clearing Aumann-Shapley average cost prices’ such that there are cost complementarities at the market output vector.

PROPOSITION 2. Given C3, C4, Fl and F2, there exists a* > 0 such that for each a~ [a*, co) there exist positive market clearing Aumann-Shapley prices and quantities (p”, qm) satisfying 4”’ = D(pm) and pj” = ACj(C, , qm), j = l,..., n, such that C,,(x) < 0 for all x < q”‘, x E F?: .

Proof: First, the existence of market clearing Aumann-Shapley prices is shown. Then, it is shown that for a* chosen appropriately, the market equilibrium outputs lie in the set KCu for a > a*. Let q,? = Dj(ACJAin) as in Eq. (1) and q* = (q;,..., q,*). Let Qj= [e,qF] and Q = Qi x ... x Q’ for some E > 0. Since ACj(C,, q) is continuous in q by C3, so is AC,(q) = (AC’(C,, q),..., AC”(C,, q)). Since Q is a compact set and q > E there exists a number B such that for each j and q E Q, ACii, < AC’(C,, q) <B. Consider the set A = [AC;,,, B] x ... x [AC;,,, B]. Then AC,(q) is a continuous map from Q to A. Given Fl and F2, D(p) is continuous and there exists an E > 0 such that D(p) is a mapping from A into Q. Define a mapping T:A x Q-A x Q by

P(P. q) = AC(C,, q) x D(P).

Since the mapping is continuous, r has a fixed point by Brouwer’s theorem, P(pm, q”) = (p”, q”). Thus, pm = AC(C,, qm) and 4”’ = D(p”).

Since Cajj(q) = (l/a) Cfj(q/a), C,, < 0 at q if C, < 0 at q/a. Choose a* such that q*ja E Kc for a > a*. Thus, q* E Kc* for a > a*. Since 4”’ < q*, C,,(x) ,< 0 for all x < q”, x E RI. Q.E.D.

Proposition 2 provides an existence result in the literature on cost sharing games. A price vector is subsidy free at q” if given pq” = C(q’), pwqw < C(q,), M g N; see Faulhaber [6]. Prices support the cost function C

’ Mirman and Tauman [9 1 prove that there exist demand compatible Aumann-Shapley cost sharing prices. Their analysis is more general in the sense that market demands are derived from individual consumer maximization. They do not consider the issues raised here regarding the effect of firm size.


at q”, if given pq” = C(q’), pq’ < C(q’), for q’ < q”; see Sharkey and Telser [ 201. Faulhaber and Levinson [7] define anonymously equitable prices as prices that are market clearing as well as supportable. Finally, ten Raa [23] assumes existence of supportable prices and shows that they are demand compatible.

In the literature on cost sharing games, emphasis has been placed upon global cost complementarities.s In particular, the Aumann-Shapley prices are shown to support the cost function for Cij(q) < 0 for all i, j, for all q; see Billera and Heath [4] and Mirman and Tauman [9]. It is sufficient, however, to assume that there are cost complementarities for small outputs with q E KC. Thus, given the conditions of Proposition 2, we immediately obtain the following.

COROLLARY. There exists a* > 0 such that for a E [a*, co), there exist anonymously equitable prices.

3.3. Sustainable Prices

Given existence of supportable Aumann-Shapley prices, the main sustainability result is now obtained. Mirman, Tauman, and Zang [ 10, Proposition 5 ] show that global cost complementarities, weak substitutes in demand, and inelastic demand for p <p are sufficient for a price p in the core of a cost game to be sustainable. The proof of the next result follows their proof, but weakens the cost requirement and shows existence of a specific sustainable price vector, the Aumann-Shapley prices.’

PROPOSITION 3. Given C3, C4, F2 and F3, there exists a* such that for all a E ]a*, a), there exist sustainable Aumann-Shapley prices pm, and thus S(a, C, F) is nonempty, if D$( p) pj/Di( p) > - 1 for each j E N and p <pm.

Proof Let a* be chosen as in Proposition 2. Choose a > a*. Let pm satisfy p.7 = ACj,(D(p”)), for all j E N, where AC;(q) = ACj(C,, q). The” entrant’s profit is given by rce =p&q:, - C,(qR), where qi, = D,,,(pi, ,pz\,,),

P’,<PL M c N. Let M, = (j E M: Dj(p&,p&) < D’(f)} and Mz = M\M, . By increasing costs,

C,VU P:,? P&,)) > C,(D.,,,(P;,, P;\+,)? D,,,~(P”)). (12) * Panzar and Willig I14 1 show that the core of a cost sharing game defined by subsidy free

prices is nonempty given C,,(q) < 0, i # j. for all q > 0. Sharkey and Telser [ 201 (see also Sharkey ( 19 1) show that Cjj(q) & 0 for all i, j is a sufficient condition for supportability at any output q. Mirman, Tauman. and Zang Ill ] show that for C,(q) < 0 for all i, j and q > 0 the sets of support and subsidy free prices coincide. Finally, Mirman and Tauman 191 show demand compatibility of Aumann-Shapley prices in a general equilibrium framework.

9 By weak substitutes in demand and by a result of Mirman, Tauman, and Zang [ 111 we need only consider the case where the entrant supplies ail of the demand in the market it enters. q;, = D(p:,, pyi,,,), M s N.


Since C,,(x) < 0 for x <O(@“) given a > a*, and Since C,(q) = CjcN AC’,(q) 4’ by definition,

+ L ACj,(D(p”‘)) D’(p”).

But, since py = AC’,(D( pm)), j E N, and p; <pE,

CS?~(P~>P&,N > 1 P;@CPR~ PE$,,) + x Pi”@. (14) jeM, .icMz

Thus,

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By weak substitutes in demand and p, <p)G,

ne < 1 [pj’Dj(p;,p$J -p;@(p”)]. jeM,

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By D{(p)pj/D’(p) > - 1, for each j E N, p <pm. ne < 0. Q.E.D.

Thus, without global restrictions on the cost function, subsidy free prices are sustainable against the entry of rival firms.

4. QUANTITY SUSTAINABILITY

Brock and Scheinkman [5] offer the concept of quantity sustainability as a means of verifying whether a vector of output levels set by a monopolist can deter entry in a subset or in all of the markets served by the monopolist. Brock and Scheinkman [5, p. 2] suggest that “Cournot’s notion is ‘dual’ to that of Bertand in that entrants anticipate that the monopolist’s quantities remain fixed but that prices adjust to absorb the extra output of the entrant. This motivates a ‘dual’ notion of sustainability,” Quantity sustainability thus involves entering firms making the conjecture of Cournot firms that rivals will hold their output constant.” The following definitions are based upon the Brock and Scheinkman [5] definition of quantitity sustainability.

“The quantity sustainable output is a multiproduct version of the entry deterring limit quantity discussed elsewhere by Bain [ 11, Sylos-Labini [22], Modigliani [ 121, and others. The belief on the part of entrants that an established firm will keep its output constant is referred to as the Sylos Postulate. See Spulber [21] for an evaluation of the Sylos and Excess Capacity hypotheses. Note that the quantity sustainability condition for an industry is related to the Cournot-Nash entry condition in Novshek’s model [ 131.

EXlSTENCEOFSUSTAlNABLE PRICES 161

DEFINITION 4. Given C, F and a E (0, co) the monopoly price vector pm and quantity vector qm = D(p”) are (a, C, F) quantity sustainable if for every (M, q;), ML N,

\\’ F’W + 4,) d, - C,(q;,) < 0 ,Zf

for all Q& > 0 for all M s N. It is also required that p”q” - C,(q”) > 0.

DEFINITION 5. Let Qs(a, C, F) represent the set of (a, C, F) quantity sustainable monopoly prices.

Brock and Scheinkman [ 5 ] have shown quantity sustainability to be a weaker requirement than price sustainability.

PROPOSITION (Brock and Scheinkman). Given downward-sloping demands and weak substitutes, then price sustainability impiies quantity sustainability.

Clearly by the proposition S(CY, C, F) g Qs(a, C, F). Thus, for the size of the firm a sufficiently large, the set of quantity sustainable prices wit1 be nonempty. Define Aumann-Shapley quantities as the market clearing quantities for Aumann-Shapley average cost prices,

qj = @(AC(q)), j = l,..., n.

Given a* as in Proposition 3, the following is obtained.

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PROPOSITION 4. Given the conditions of Proposition 3, there exists E* < a* such that for a E [E*, co), Aumann-Shapley quantities are quantity sustainable and thus QS(a, C, F) is nonempty.

5. CONCLUSION

The results in this paper add substance to the sustainability debate for two reasons. First, the demonstration of existence of sustainable prices shows that under certain market conditions entry deterring prices can indeed be found. The market conditions required for sustainable prices are easily verifiable from demand and cost data. In particular, the minimum efficient scale of the firm must be sufficiently large relative to market demand evaluated at minimum average cost. This criterion is meaningful in economic terms. Second, the results shift attention toward subsidy free prices and away from socially optimal pricing policies. Aumann-Shapley prices have the advantage of being easily calculable from market cost and output data. Furthermore, they satisfy the requirement of anonymous equity.


The analysis shows a trade-off between the strength of assumptions on cost and demand. When the multiproduct cost function is separable, weak substitutes in demand is sufficient for sustainability. When costs are not separable, a stronger condition such as inelastic demand is required. The question for future research on natural monopoly is to what extent costs are separable in markets with multiproduct firms and whether those markets have special demand structures of the type which yields sustainable prices.

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Scale economies and existence of sustainable monopoly prices