Monopolistic Competition and Optimum Product Diversity

By AVINASH K. DIXIT AND JOSEPH E. STIGLITZ*

The basic issue concerning production in potential commodity involves some fixed welfare economics is whether a market solu- set-up cost and has a constant marginal tion will yield the socially optimum kinds cost. Modeling the desirability of variety and quantities of commodities. It is well has been thought to be difficult, and several known that problems can arise for three indirect approaches have been adopted. broad reasons: distributive justice; external The Hotelling spatial model, Lancaster's effects; and scale economies. This paper is product characteristics approach, and the concerned with the last of these. mean-variance portfolio selection model The basic principle is easily stated.' A have all been put to use.3 These lead to re- commodity should be produced if the costs sults involving transport costs or correla- can be covered by the sum of revenues and tions among commodities or securities, and a properly defined measure of consumer's are hard to interpret in general terms. We surplus. The optimum amount is then therefore take a direct route, noting that the found by equating the demand price and the convexity of indifference surfaces of a con- marginal cost. Such an optimum can be ventional utility function defined over the realized in a market if perfectly discrim- quantities of all potential commodities al- inatory pricing is possible. Otherwise we ready embodies the desirability of variety. face conflicting problems. A competitive Thus, a consumer who is indifferent be- market fulfilling the marginal condition tween the quantities (1,0) and (0,1) of two would be unsustainable because total profits commodities prefers the mix (1/2,1/2) to would be negative. An element of monopoly either extreme. The advantage of this view would allow positive profits, but would is that the results involve the familiar own- violate the marginal condition.2 Thus we and cross-elasticities of demand functions, expect a market solution to be suboptimal. and are therefore easier to comprehend. However, a much more precise structure There is one case of particular interest on must be put on the problem if we are to which we concentrate. This is where poten- understand the nature of the bias involved. tial commodities in a group or sector or in- It is useful to think of the question as one dustry are good substitutes among them- of quantity versus diversity. With scale selves, but poor substitutes for the other economies, resources can be saved by pro- commodities in the economy. Then we are ducing fewer goods and larger quantities of led to examining the market solution in re- each. However, this leaves less variety, lation to an optimum, both as regards which entails some welfare loss. It is easy biases within the group, and between the and probably not too unrealistic to model group and the rest of the economy. We ex- scale economies by supposing that each pect the answer to depend on the intra- and intersector elasticities of substitution. To demonstrate the point as simply as possible, *Professors of economics, University of Warwick we shall aggregate the rest of the economy and Stanford University, respectively. Stiglitz's re- into one good labeled 0, chosen as the search was supported in part by NSF Grant SOC74- 22182 at the Institute for Mathematical Studies in the numeraire. The economy's endowment of it Social Sciences, Stanford. We are indebted to Michael is normalized at unity; it can be thought of Spence, to a referee, and the managing editor for com- as the time at the disposal of the consumers. ments and suggestions on earlier drafts. ISee also the exposition by Michael Spence. 2A simple exposition is given by Peter Diamond and 3See the articles by Harold Hotelling, Nicholas Daniel McFadden. Stern, Kelvin Lancaster, and Stiglitz.

297 298 THE AMERICAN ECONOMIC REVIEW JUNE 1977

The potential range of related products is varieties, or as diversification on the part labeled 1,2,3,.... Writing the amounts of of each consumer. the various commodities as x0 and x = (xl, X2, X3 ..., we assume a separable utility 1. Constant-ElasticityCase function with convex indifference surfaces: A. Demand Functions

(1) u = U(xO, V(x1,x2,X3..)) The utility function in this section is

In Sections I and II we simplify further (2) ( {x} I/P) by assuming that V is a symmetric function, and that all commodities in the group have For concavity, we need p < 1. Further, equal fixed and marginal costs. Then the since we want to allow a situation where actual labels given to commodities are im- several of the xi are zero, we need p > 0. We material, even though the total number n also assume U homothetic in its arguments. being produced is relevant. We can thus The budget constraint is label these commodities 1,2, ..., n, where n the potential products(n + 1), (n + 2), ... (3) xO + Pi= I are not being produced. This is a restrictive assumption, for in such problems we often where pi are prices of the goods being pro- have a natural asymmetry owing to grad- duced, and I is income in terms of the uated physical differences in commodities, numeraire, i.e., the endowment which has with a pair close together being better been set at I plus the profits of the firms mutual substitutes than a pair farther apart. distributed to the consumers, or minus the However, even the symmetric case yields lump sum deductions to cover the losses, as some interesting results. In Section III, we the case may be. consider some aspects of asymmetry. In this case, a two-stage budgeting pro- We also assume that all commodities cedure is valid.4 Thus we define dual quan- have unit income elasticities. This differs tity and price indices from a similar recent formulation by Michael Spence, who assumes U linear in xo, so that the industry is amenable to (4) y = {?, q= p partial equilibrium analysis. Our approach allows a better treatment of the intersectoral whereA = (I - p)/p, which is positive since substitution, but the other results are very O< p < 1. Then it can be shown5 that in the similar to those of Spence. first stage, We consider two special cases of (1). In s(q) Section I, V is given a CES form, but U is (S) y - I xo = I(1 - s(q)) allowed to be arbitrary. In Section II, U is q taken to be Cobb-Douglas, but V has a for a function s which depends on the form more general additive form. Thus the for- of U. Writing a(q) for the elasticity of sub- mer allows more general intersector rela- stitution between xo and y, we define 0(q) as tions, and the latter more general intra- the elasticity of the function s, i.e., qs'(q)/ sector substitution, highlighting different s(q). Then we find results. = - $1 - < 1 Income distribution problems are ne- (6) 0(q) 11 o(q)} s(q)} glected. Thus U can be regarded as repre- but 0(q) can be negative as a(q) can ex- senting Samuelsonian social indifference ceed 1. curves, or (assuming the appropriate aggre- gation conditions to be fulfilled) as a mul- tiple of a 4Sec p. 21 of John Green. representative consumer's utility. 5These details and several others are omitted to save Product diversity can then be interpreted space, but can be found in the working paper by the either as different consumers using different authors, cited in the references. VOL. 67 NO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 299

Turning to the second stage of the prob- downward. The conventional condition that lem, it is easy to show that for each i, the dd curve be more elastic is seen from (9) and (12) to be (7) = (13) + (q)>? where y is defined by (4). Consider the effect of a change in pi alone. This affects xi di- Finally, we observe that for i + j, rectly, and also through q; thence through y as well. Now from (4) we have the elasticity (14) xi Pi ]

(8)dlg =(q d logpi Pi Thus 1/(1 - p) is the elasticity of substitu- tion between any two products within the So long as the prices of the products in the group. group are not of different orders of mag- nitude, this is of the order (I/n). We shall B. Market Equilibrium assume that n is reasonably large, and ac- It can be shown that each commodity is cordingly neglect the effect of each Pi on q; produced by one firm. Each firm attempts thus the indirect effects on xi. This leaves us to maximize its profit, and entry occurs un- with the elasticity til the marginal firm can only just break () logx_ -I -(I + Oi) even. Thus our market equilibrium is the (9) =- .. familiar case of Chamberlinian monopolis- dlogpi (I -P) tic competition, where the question of In the Chamberlinian terminology, this is quantity versus diversity has often been the elasticity of the dd curve, i.e., the curve raised.6 Previous analyses have failed to relating the demand for each product type consider the desirability of variety in an ex- to its own price with all other prices held plicit form, and have neglected various constant. intra- and intersector interactions in de- In our large group case, we also see that mand. As a result, much vague presumption for i s j, the cross elasticity d log xi/d log p1 that such an equilibrium involves excessive is negligible. However, if all prices in the diversity has built up at the back of the group move together, the individually small minds of many economists. Our analysis effects add to a significant amount. This will challenge several of these ideas. corresponds to the Chamberlinian DD The profit-maximization condition for curve. Consider a symmetric situation each firm acting on its own is the familiar where xi = x and pi = p for all i from I equality of marginal revenue and marginal to n. We have cost. Writing c for the common marginal cost, and noting that the elasticity of de- xnI+ (10) Y= xn-IP= mand for each firm is (1 + ,B)/,B, we have q = pn - = pn t0-P)/P for each active firm: and then from (5) and (7),

pi (I_ d) c (11) X Is(q) pn Writing Pe for the common equilibrium The elasticity of this is easy to calculate; we price for each variety being produced, we find have (15) Pe = C(I + _ (12) Ig - - - d logp l (q)] p 6See Edwin Chamberlin, Nicholas Kaldor, and Then (6) shows that the DD curve slopes Robert Bishop. 300 THE AMERICAN ECONOMIC REVIEW JUNE 1977

The second condition for equilibrium is stant budget share for the monopolistically that firms enter until the next potential competitive sector. Note that in our para- entrant would make a loss. If n is large metric formulation, this implies a unit- enough so that I is a small increment, we elastic DD curve, (17) holds, and so equi- can assume that the marginal firm is exactly librium is unique. breaking even, i.e., (pn - c)xn = a, where xn Finally, using (7), (1 1), and (16), we can is obtained from the demand function and a calculate the equilibrium output for each is the fixed cost. With symmetry, this im- active firm: plies zero profit for all intramarginal firms as well. Then I = 1, and using (I I) and (15) (18) Xe we can write the condition so as to yield the number ne of active firms: We can also write an expression for the budget share of the group as a whole: S(Penf l)e a ( 16) - Pene f3c (1 9) Se = s (qe) where qe = Pen -0 Equilibrium is unique provided S(Penf-)/ Pen is a monotonic function of n. This re- These will be useful for subsequent com- lates to our carlier discussion about the two parisons. demand curves. From (11) we see that the behavior of s(pn -)/pn as n increases tells us how the demand curve DD for each firm C. Constrained Optimum shifts as the number of firms increases. It is The next task is to compare the equi- natural to assume that it shifts to the left, librium with a social optimum. With i.e., the function above decreases as n in- economies of scale, the first best or uncon- creases for each fixed p. The condition for strained (really constrained only by tech- this in elasticity form is easily seen to be nology and resource availability) optimum requires pricing below average cost, and (17) 1 + f3(q) > 0 therefore lump sum transfers to firms to This is exactly the same as (13), the condi- cover losses. The conceptual and practical tion for the dd curve to be more elastic than difficulties of doing so are clearly formid- the DD curve, and we shall assume that it able. It would therefore appear that a more holds. appropriate notion of optimality is a con- The condition can be violated if c(q) is strained one, where each firm must have sufficiently higher than one. In this case, an nonnegative profits. This may be achieved increase in n lowers q, and shifts demand by regulation, or by excise or franchise towards the monopolistic sector to such an taxes or subsidies. The important restriction extent that the demand curve for each firm is that lump sum subsidies are not available. shifts to the right. However, this is rather We begin with such a constrained opti- implausible. mum. The aim is to choose n, Pi, and xi so Conventional Chamberlinian analysis as- as to maximize utility, satisfying the de- sumes a fixed demand curve for the group mand functions and keeping the profit for as a whole. This amounts to assuming that each firm nonnegative. The problem is n *x is independent of n, i.e., that s(pn -I) is somewhat simplified by the result that all independent of n. This will be so if , = 0, or active firms should have the same output if' (q) = I for all q. The former is equiv- levels and prices, and should make exactly alent to assuming that p = 1, when all zero profit. We omit the proof. Then we can products in the group are perfect substi- set I = 1, and use (5) to express utility as a tutes, i.e., diversity is not valued at all. That function of q alone. This is of course a de- would be contrary to the intent of the whole creasing function. Thus the problem of analysis. Thus, implicitly, conventional maximizing u becomes that of minimizing analysis assumes o(q) = 1. This gives a con- q, i.e., VOL. 67 NO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 301

min pn (25) -(a + cx)Uo + (1 + /3)xn4LJy = 0 n,p From the first stage of the budgeting prob- subject to lem, we know that q = U,,/UO. Using (24) and (10), we find the price charged by each (20) (p - c) s(-pn a active firm in the unconstrained optimum, pn PU,equal to marginal cost To solve this, we calculate the logarithmic (26) Pu = c marginal rate of substitution along a level curve of the objective, the similar rate of This, of course, is no surprise. Also from transformation along the constraint, and the first-order conditions, we have equate the two. This yields the condition a C (27) xu= + 0(q) (21) p - c _ Finally, with (26), each active firm covers its 1 + /O(q) - variable cost exactly. The lump sum trans- The second-order condition can be shown fers to firms then equal an, and therefore to hold, and (21) simplifies to yield the price =I 1 - an, and for each commodity produced in the con- pn strained optimum, pc, as ( an) ws( pn = (22) pC c(l + d) The number of firms nu is then defined by Comparing (15) and (22), we see that the s(cn-) a/d two solutions have the same price. Since (28) = l they face the same break-even constraint, nu 1I- anu they have the same number of firms as well, We can now compare these magnitudes and the values for all other variables can be with the corresponding ones in the equilib- calculated from these two. Thus we have a rium or the constrained optimum. The most rather surprising case where the monopo- remarkable result is that the output of each listic competition equilibrium is identical active firm is the same in the two situations. with the optimum constrained by the lack The fact that in a Chamberlinian equilib- of lump sum subsidies. Chamberlin once rium each firm operates to the left of the suggested that such an equilibrium was "a point of minimum average cost has been sort of ideal"; our analysis shows when and conventionally described by saying that in what sense this can be true. there is excess capacity. However, when variety is desirable, i.e., when the different D. Unconstrained Optintunt products are not perfect substitutes, it is not These solutions can in turn be compared in general optimum to push the output of to the unconstrained or first best optimum. each firm to the point where all economies Considerations of convexity again establish of scale are exhausted.7 We have shown in that all active firms should produce the one case that is not an extreme one, that the same output. Thus we are to choose n firms first best optimum does not exploit econo- each producing output x in order to maxi- mies of scale beyond the extent achieved in mize the equilibrium. We can then easily con- ceive of cases where the equilibrium exploits (23) u = U(1 - n(a + cx),xn'+') economies of scale too far from the point of where we have used the economy's resource view of social optimality. Thus our results balance condition and (10). The first-order undermine the validity of the folklore of ex- conditions are cess capacity, from the point of view of the

(24) -ncUo + n'l+U Y = 0 7Scc David Starrctt. 302 THE AMERICAN ECONOMIC REVIEW JUNE 1977

y Using (29) we can easily compare the budget shares. In the notation we have been using, we find s,, e s, as 0(q) e 0, i.e., as (1/anq)/q r(q) e 1 providinig these hold over the en- tire relevant range of q. It is not possible to have a general result A concerning the relative magnitudes of x0 in the two situations; an inspection of Figure I shows this. However, we have a sufficient /q condition:

Xou = (1 - anu)(l - su) < 1 - su < 1 - SC = xocif r(q) > 1

1-anu 1 0 In this case the equilibrium or the con- strained optimum use more of the nu- FIGURE I meraire resource than the unconstrained optimum. On the other hand, if c(q) = 0 we unconstrained optimum as well as the con- have L-shaped isoquants, and in Figure 1, strained one. points A and B coincide giving the opposite A direct comparison of the numbers of conclusion. firms from (16) and (28) would be difficult, In this section we have seen that with a but an indirect argument turns out to be constant intrasector elasticity of substitu- simple. It is clear that the unconstrained tion, the market equilibrium coincides with optimum has higher utility than the con- the constrained optimum. We have also straincd optimum. Also, the level of lump shown that the unconstrained optimum has sum income in it is less than that in the lat- a greater number of firms, each of the same ter. It must therefore be the case that size. Finally, the resource allocation be- tween the sectors is shown to depend on the (29) qu < qc = qe intersector elasticity of substitution. This Further, the difference must be large elasticity also governs conditions for enough that the budget constraint for xo uniqueness of equilibrium and the second- and the quantity index y in the uncon- order conditions for an optimum. strained case must lie outside that in the Henceforth we will achieve some analytic constrained case in the relevant region, as simplicity by making a particular assump- shown in Figure 1. Let C be the constrained tion about intersector substitution. In re- optimum, A the unconstrained optimum, turn, we will allow a more general form of and let B be the point where the line joining intrasector substitution. the origin to C meets the indifference curve in the unconstrained case. By homotheticity II. VariableElasticity Case the indifference curve at B is parallel to that The utility function is now at C, so each of the moves from C to B and from B to A increases the value of y. Since (31) u = x0 -YjZv(xi)}Y the value of x is the same in the two optima, we must have with v increasing and concave, 0 < y < 1. This is somewhat like assuming a unit inter- = (30) nu > nc ne sector elasticity of substitution. However, Thus the unconstrained optimum actually this is not rigorous since the group utility allows more variety than the constrained V(x) = Ziv(xi) is not homothetic and there- optimum and the equilibrium; this is fore two-stage budgeting is not applicable. another point contradicting the folklore on It can be shown that the elasticity of the excessive diversity. dd curve in the large group case is VOL. 67 NO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 303

We wish to choose n and x to maximize u, (32) d logx1 _ _ v'(xi) foranyi a logpi XiV"(Xi) subject to (34) and the break-even condition px = a + cx. Substituting, we can express u This differs from the case of Section I in as a function of x alone: being a function of xi. To highlight the sim- ilarities and the differences, we define O(x) by (39) u =y(I - y) -()a+ cx- 1 + (x) v'(x) : The first-order condition defines xc: )(x) xv " (x)

Next, setting = x and pi = p for i = 1, (40) cx- - -= 1 W(xi)xcp(x) xi a + cxc 1 + 3(xc) 'yp(xc) 2, .. ., n, we can write the DD curve and the demand for the numeraire as Comparing this with (37) and using the second-order condition, it can be shown (34) x = I w(x), XO - 1[1 - w(x)] that provided p'(x) is one-signed for all x,

where (41) xc Q Xe according as p'(x) 5 0

(35) w(X) = yp (x) With zero pure profit in each case, the [,yp(x) ? (I - Y) points (Xe, Pe) and (xc, pc) lie on the same xv ' (x) declining average cost curve, and therefore p (x) =v(x) (42) Pc f? Pe according as xc > Xe We assume that 0 < p(x) < 1, and therefore have 0 < w(x) < 1. Next we note that the dd curve is tangent to Now consider the Chamberlinian equilib- the average cost curve at (Xe, Pe) and the rium. The profit-maximization condition DD curve is steeper. Consider the case for each active firm yields the common XC > Xe. Now the point (xc, pC)must lie on a equilibrium price Pe in terms of the common DD curve further to the right than (Xe, Pe), equilibrium output xe as and therefore must correspond to a smaller number of firms. The opposite happens if (36) Pe = c[D + /3(Xe)] XC < xe. Thus, Note the analogy with (15). Substituting > (43) nc neaccording as xc Xe (36) in the zero pure profit condition, we ? have xe defined by Finally, (41) shows that in both cases that arise there, p(xc) < Pp(Xe). Then w(xc) < CXe _ 1 (37) W(Xe), and from (34), a + cxe I + A(Xe) > Finally, the number of firms can be calcu- (44) XOc XOe lated using the DD curve and the break- A smaller degree of intersectoral substitu- even condition, as tion could have reversed the result, as in Section I. (38) ne - W(Xe) An intuitive reason for these results can be given as follows. With our large group For uniqueness of equilibrium we once assumptions, the revenue of each firm is again use the conditions that the dd curve is proportional to xv'(x). However, the con- more elastic than the DD curve, and that tribution of its output to group utility is entry shifts the DD curve to the left. How- v(x). The ratio of the two is p(x). Therefore, ever, these conditions are rather involved if p'(x) > 0, then at the margin each firm and opaque, so we omit them. finds it more profitable to expand than what Let us turn to the constrained optimum. would be socially desirable, so Xe > Xc. 304 THE AMERICAN ECONOMIC REVIEW JUNE 1977

Given the break-even constraint, this leads This is in each case transitive with (41), and to there being fewer firms. therefore yields similar output comparisons Note that the relevant magnitude is the between the equilibrium and the uncon- elasticity of utility, and not the elasticity of strained optimum. demand. The two are related, since The price in the unconstrained optimum is of course the lowest of the three. As to (x) 1 (4) x P' l_ - the of we note (45) p(x) 1+ (3(x) p(x)px number firms,

- (x8) __ __ Thus, if p(x) is constant over an interval, so C a + cx a + cx is /3(x) and we have 1/(1 + 3) = p, which is the case of Section I. However, if p(x) and therefore we have a one-way compari- varies, we cannot infer a relation between son: the signs of p'(x) and d'(x). Thus the varia- (51) < > tion in the elasticity of demand is not in Ifxu xC,thennu nc general the relevant consideration. How- Similarly for the equilibrium. These leave ever, for important families of utility func- open the possibility that the unconstrained tions there is a relationship. For example, optimum has both bigger and more firms. for v(x) = (k + mx)j, with m > 0 and 0 < That is not unreasonable; after all the un- j < 1, we find that -xv"/v' and xv'/v are constrained optimum uses resources more positively related. Now we would normally efficiently. expect that as the number of commodities produced increases, the elasticity of substi- tution between any pair of them should in- III. AsymmetricCases crease. In the symmetric equilibrium, this is The discussion so far imposed symmetry just the inverse of the elasticity of marginal within the group. Thus the number of varie- utility. Then a higher x would correspond ties being produced was relevant, but any to a lower n, and therefore a lower elasticity group of n was just as good as any other of substitution, higher -xv"/v' and higher group of n. The next important modifica- xv'/v. Thus we are led to expect that p'(x) > tion is to remove this restriction. It is easy 0, i.e., that the equilibrium involves fewer to see how interrelations within the group and bigger firms than the constrained opti- of commodities can lead to biases. Thus, if mum. Once again the common view con- no sugar is being produced, the demand for cerning excess capacity and excessive di- coffee may be so low as to make its produc- versity in monopolistic competition is called tion unprofitable when there are set-up into question. costs. However, this is open to the objection The unconstrained optimum problem is that with complementary commodities, to choose n and x to maximize there is an incentive for one entrant to pro- duce both. However, problems exist even (46) u = - n(a +cx)]--- [nv(x)]i[l when all the commodities are substitutes. It is easy to show that the solution has We illustrate this by considering an industry which will produce commodities from one (47) pu= c of two groups, and examine whether the (48) c u = P(xu) choice of the wrong group is possible.8 a +~ cxi, Suppose there are two sets of commodi- ly ties beside the numeraire, the two being per- (49)(49) nu = ~~~a+ cxi, fect substitutes for each other and each hav- ing a constant elasticity subutility function. Then we can use the second-order condition Further, we assume a constant budget share to show that 8For an alternative approach using partial equilib- (50) xu S x, according as p'(x) e 0 rium methods, see Spence. VOL. 67 NO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 305 for the numeraire. Thus the utility function only if is (55) q2 < sc- (52) s - a, Now consider the optimum. Both the ob- u n + P2]I/P2}s [PI jective and the constraint are such as to lead the optimum to the production of com- We assume that each firm in group i has a modities from only one group. Thus, sup- fixed cost ai and a constant marginal cost ci. pose ni commodities from group i are being Consider two types of equilibria, only produced at levels xi each, and offered at one commodity group being produced in prices pi. The utility level is given by each. These are given by + (56) u = x -Sfxlln+Ol X2nf+$2 Is (53a) x = a,1x2=O and the resource availability constraint is c, O, = c(l + 31) (57) xo + n1(al + clxl) + n2(a2 + C2X2) = a,(l + 131) Given the values of the other variables, the level curves of u in (nl, n2) space are con- cave to the origin, while the constraint is q, = p1n,' I= c,(l + f3)l+/(5-) linear. We must therefore have a corner u 1 I -s = ss(l _) Iq -s optimum. (As for the break-even con- straint, unless the two qi = pini-,i are equal, (53b) -2 = a2 x,5 = 0 the demand for commodities in one group C2d2' is zero, and there is no possibility of avoid- P2 = C2(1 + /2) ing a loss there.) Note that we have structured our ex- ample so that if the correct group is chosen, a2(1 + /32) the equilibrium will not introduce any further biases in relation to the constrained = p2n22 = c2(1 + 2) 42 optimum. Therefore, to find the constrained optimum, we only have to look at the U2 = s(l - ) 2 values of ui in (53a) and (53b) and see which Equation (53a) is a Nash equilibrium if is the greater. In other words, we have to and only if it does not pay a firm to produce see which 4i is the smaller, and choose the a commodity of the second group. The de- situation (which may or may not be a Nash mand for such a commodity is equilibrium) defined in (53a) and (53b) cor- responding to it. [ 0 for P2 >q1 Figure 2 is drawn to depict the possible X2 S1P2 for P2 < equilibria and optima. Given all the rele- vant parameters, we calculate (41, 12) from Hence we require (53a) and (53b). Then (54) and (55) tell us whether either or both of the situations are possible equilibria, while a simple compari- max(P2 - C2)X2 = 5(I - 4) < a2 son of the magnitudes of q1 and 42 tells us which is the constrained optimum. In the or figure, the nonnegative quadrant is split into regions in each of which we have one (54) S< C2 combination of equilibria and optima. We s - a2 only have to locate the point (41, 72) in this Similarly, (53b) is a Nash equilibrium if and space to know the result for the given 306 THE AMERICAN ECONOMIC REVIEW JUNE 1977

equilibria, and it is therefore possible that C E the high elasticity group is produced in equi- librium when the low elasticity one should I eqm No eqm F I opt I opt No eqm have been. If the difference in elasticities is large enough, the point moves into region s 11I opt C, where (53b) is no longer a Nash equilib- s-aa G rium. But, owing to the existence of a fixed A 11eqm cost, a significant difference in elasticities is i11 eqm I opt necessary before entry from group 1 com- I opt modities threatens to destroy the "wrong" D equilibrium. Similar remarks apply to re- II eqm gions B and D. 11 opt Next, begin with symmetry once again, /B and consider a higher cl or This in- 111 eqm a,. II opt creases q1 and moves the point into region B, making it optimal to produce the low- sc2/(s-a2) 1 cost group alone while leaving both (53a) and (53b) as possible equilibria, until the FIGURE2. SOLUTIONSLABELED I REFERTO EQUATION(53a); SOLUTIONSLABELEI) 11 difference in costs is large enough to take REFERTO EQUATION(53b) the point to region D. The change also moves the boundary between A and C up- ward, opening up a larger region G, but parameter values. Moreover, we can com- that is not of significance here. pare the location of the points correspond- If both q1 and q2 are large, each group is ing to different parameter values and thus threatened by profitable entry from the do some comparative statics. other, and no Nash equilibrium exists, as in To understand the results, we must ex- regions E and F. However, the criterion of amine how qi depends on the relevant constrained optimality remains as before. parameters. It is easy to see that each is an Thus we have a case where it may be neces- increasing function of ai and ci. We also sary to prohibit entry in order to sustain the find constrained optimum. If we combine a case where c1 > c2 (or > a2) and /3, > 02, i.e., where commodi- (58) daIog0 = -log ni a, ties in group 2 are more elastic and have lower costs, we face a still worse possibility. and we expect this to be large and negative. For the point (4q, 42) may then lie in region Further, we see from (9) that a higher j3i G, where only (53b) is a possible equilib- corresponds to a lower own-price elasticity rium and only (53a) is constrained opti- of demand for each commodity in that mum, i.e., the market can produce only a group. Thus qi is an increasing function of low cost, high demand elasticity group of this elasticity. commodities when a high cost, low demand Consider initially a symmetric situation, elasticity group should have been produced. with scl/(s - a,) = SC2/(S - a2), I, = /2 Very roughly, the point is that although (the region G vanishes then), and suppose commodities in inelastic demand have the the point (q-I 42) is on the boundary be- potential for earning revenues in excess of tween regions A and B. Now consider a variable costs, they also have significant change in one parameter, say, a higher own- consumers' surpluses associated with them. elasticity for commodities in group 2. This Thus it is not immediately obvious whether raises q2, moving the point into region A, the market will be biased in favor of them and it becomes optimal to produce com- or against them as compared with an opti- modities from group 1 alone. However, mum. Here we find the latter, and inde- both (53a) and (53b) are possible Nash pendent findings of Michael Spence in other VOL. 67 NO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 307 contexts confirm this. Similar remarks D apply to differences in marginal costs. In the interpretation of the model with heterogenous consumers and social indif- ference curves, inelastically demanded com- modities will be the ones which are inten- sively desired by a few consumers. Thus we have an "economic" reason why the market will lead to a bias against opera relative to football matches, and a justification for subsidization of the former and a tax on the ACA latter, provided the distribution of income MCA is optimum. Even when cross elasticities are zero, D A ACB there may be an incorrect choice of com- McB modities to be produced (relative either to an unconstrained or constrained optimum) output as Figure 3 illustrates. Figure 3 illustrates a case where commodity A has a more FIGURE 4 elastic demand curve than commodity B; A is produced in monopolistically competitive sumer surplus from A (at its equilibrium equilibrium, while B is not. But clearly, it level of output) is less than that from B is socially desirable to produce B, since ig- (i.e., the cross hatched area exceeds the noring consumer's surplus it is just mar- striped area), then constrained Pareto opti- ginal. Thus, the commodities that are not mality entails restricting the production of produced but ought to be are those with in- the commodity with the more elastic elastic demands. Indeed, if, as in the usual demand. analysis of monopolistic competition, elimi- A similar analysis applies to commodities nating one firm shifts the demand curve for with the same demand curves but different the other firms to the right (i.e., increases cost structures. Commodity A is assumed to the demand for other firms), if the con- have the lower fixed cost but the higher marginal cost. Thus, the average cost curves cross but once, as in Figure 4. Commodity A is produced in monopolistically com- petitive equilibrium, commodity B is not (although it is just at the margin of being B produced). But again, observe that B should be produced, since there is a large con- sumer's surplus; indeed, since were it to be A produced, B would produce at a much higher level than A, there is a much larger consumer's surplus. Thus if the government were to forbid the production of A, B AC would be viable, and social welfare would B AC: increase. In the comparison between constrained Mc Pareto optimality and the monopolistically MRA competitive equilibrium, we have observed that in the former, we replace some low fixed cost-high marginal cost commodities output with high fixed cost-low marginal cost com- FIGURE 3 modities, and we replace some commodities 308 THE AMERICAN ECONOMIC REVIEW JUNE 1977 with elastic demands with commodities with presented here, in conjunction with other inelastic demands. studies cited, offer some useful and new insights. IV. ConcludingRemarks We have constructed in this paper some models to study various aspects of the rela- REFERENCES tionship between market and optimal re- R. L. Bishop, "Monopolistic Competition source allocation in the presence of some and Welfare Economics," in Robert nonconvexities. The following general con- Kuenne, ed., Monopolistic Competition clusions seem worth pointing out. Theory, New York 1967. The monopoly power, which is a neces- E. 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