Monopolistic Competition and Optimum Product Diversity

B>'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.^ 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 fuHilling the marginal condition tween the quantities (1,0) and (0,1) of two wouldbe 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.^ 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 ofthe bias involved. tial commodities in a group or sector or in- It is useful to think ofthe 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 eaeh 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 pan by NSF Grant SOC74- numeraire. The economy's endowment of it 22182 at the Institute for Mathematical Studies in the is normalized at unity; it can be thought of Sociai Sciences, Stanford. We are indebted to Michael 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, 'Sec also the exposition by Michael Spence. A simple exposition is given by Peter Diamond and •'Sec the articles by Harold Hoiclling, Nicholas Daniel McFadden. Stern, Kelvin Lancaster, and Stiglilz.

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 Vo and .V = (x,. Xi, Xi ..., we assume a separable utility [. Constant-Elasticity Case function with convex indifference surfaces: A. Demand Functions 0) U - The utility function in this section is

In Sections I and II we simplify further (2) u = by assuming that K is a symmetric function, and that all commodities in the group have For concavity, we need p < \. Further, equal fixed and marginal costs. Then the since we want to allow a situation where actual labels given tu commodities are im- several of the Xj 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 the potential products (« + 1), (« + 2). ... (3) 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 / 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 1 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.'' Thus we define dual quan- have unit income elasticities. This differs tity and price indices from a similar recent formulation by Michael Spencc, who assumes U linear in .-\/ti Xo, so that the industry is amenable to (4) >•= q = partial equilibrium analysis. Our approach allows a better treatment ofthe interscctoral where /!i = (I - p)/p, which is positive since substitution, but the other results are very 0 < p < 1. Then it can be shown^ that in the similar to those of Spence. first stage. We consider two special cases of (1). In Section I, V is given a CES form, but U is (5) y - = /{I allowed to be arbitrary. In Section II. U is 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 (T{q) for the elasticity of sub- mer allows more general intersector rela- stitution between XQ and V, we define ffiq) as tions, and the latter more general intra- the elasticity ofthe function .v. i.e.. qs'{q)l sector substitution, highlighting different siq). Then we find results. Income distribution problems are ne- (6) \\ glected. Thus U can be regarded as repre- but can be negative as o{q) can ex- senting Samuelsonian social indifference ceed 1. curves, or (assuming the appropriate aggregation conditions to be fulfilled) as a mul- tiple of a representative consumer's utility. ScL- p. 21 of John Green. Product diversity can then be interpreted These details and several others arc omitted to save 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 DtXIT AND STIGLITZ: PRODUCT DIVERSITY 299

Turning to the second stage ofthe prob- downward. The conventional condition that lem, it is easy to show that for each /, the dd curve be more elastic is seen from (9) and (12) to be (7) X, - y (13) h, + Hq) > 0 where v is defined by (4). Consider the effect of a change in /?, alone. This affects x, di- Finally, we observe that for / # j. rectly, and also through q: thence through v 1/(1- as well. Now from (4) we have the elasticity (14)

Thus 1/(1 - p) is the elasticity of substitu- ^ log Pi \Pil 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 magnitude, this is of the order (l/«). 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 p, on q; produced by one firm. Each firm attempts thus the indirect effects on x,. This leaves us to maximize its profit, and entry occurs un- with the elasticity til the marginal firm can only just break even. Thus our market equilibrium is the dlogx, -1 -(1 + (i) (9) familiar case of Chamberlinian monopolis- 0 log Pi (1 - 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.*" 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 / ^ j. the cross elasticity c* log xjb log p, 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 x, = x and p, = p for all / from 1 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- (10) y = xi = xn mand for each firm is (1 -I- 0)/0, we have q = pn~'^ = p for each active firm: and then from (5) and (7), = c 1 + (11) X = pn Writing p^ for the common equilibrium The elasticity of this is easy to calculate; we price for each variety being produced, we find have

f* log X (15) p, = f(l -h /3) = -^ (12) = -[I - P ^Scc Edwin Chamberlin, Nicholas Kaldor. and Then (6) shows that the DD curve slopes Robtrl 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., (/)„ - c)x„ = a, where x„ Finally, using (7), (11), 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 / = 1, and using (11) and (15) (18) X. = we can write the condition so as to yield the (ic number/7j. of active firms: We can also write an expression for the i\t.\ •''iPe^i'^) fl budget share ofthe group as a whole: (16) = _ where q^ = p^n;^ Equilibrium is unique provided sip^n''^)/ p^n is a monotonic function of n. This re- These will be useful for subsequent com- lates to our earlier 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. Consirained 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 1 + mq) > 0 requires pricing below average cost, and 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 ddcurvt to be more elastic than difficulties of doing so are clearly formid- the/)/) 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 a(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 lo 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 «. /?,, and x, so Conventional Chamberlinian analysis as- as lo 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 oi'n, i.e., that sipn -'^) is somewhat simplified by the result that all independent of«. This will be so if ,t^ = 0, or active firms should have the same output \f

mm pn (25) -{a + (1 + = 0 n,p From the firsl stage ofthe budgeting prob- subject to lem, we know that q = UjL'o. Using (24) and (10), we find the price charged by each (20) active firm in the unconstrained optimum, = a /?„, equal to marginal cost To solve this, we calculate the logarithmic (26) p. = c marginal rate of substitution along a level eurve of the objective, the similar rate of This, of course, is no surprise. Also from transformation along ihe constraint, and the first-order conditions, we have equate the two. This yields the condition (27) X., =

p - c + (21) Finally, wiih (26), each active firm covers its I + 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 1 = \ - an, and for eaeh commodity produced in the constrained optimum, p,. as ,v = (I -an) pn (22) /?, = r(l + (i) The number of firms «„ is then defined by Comparing (15) and (22), we see that the alii Iwo solutions have the same price. Since (28) 1 - an. they face the same break-even constraint, 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 Ihe 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. Unconsl rained Optimum 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 besi optimum. each firm lo the point where all economies Considerations of convexity again establish of scale are exhausted.' We have shown in that al! 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 v 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) - n[a economies of scale too far from the point of view of social optimality. Thus our results where we have used the economy's resource undermine the validity ofthe folklore of ex- balance condition and (10). The firsl-order cess capacity, from the poinl of view ofthe conditions are

(24) + 0 Djvid Siarrctl. 302 THE AMERICAN ECONOMIC REVIEW JUNE 1977

Using (29) we can easily compare the budget shares. In the notation we have been using, we find s^ § .v^ as d{q) ^ 0, i.e., as a{q) $ 1 providing these hold over the en- tire relevant range of q. It is not possible to have a general result concerning the relative magnitudes of .VQ in the two situations; an inspection of Figure 1 shows this. However, we have a sufficient condition:

(1 - anj{\ < \ - < I -

In this case the equilibrium or the constrained optimum use more of the nu- Kltil RH 1 meraire resource than the unconstrained optimum. On the other hand, if a(^) = 0 we unconstrained optimum as well as the con- have L-shaped isoquants, and in Figure I, strained one. points A and B coincide giving the opposite A direct comparison of the numbers of eonclusion. 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 strained 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- (29) tween the sectors is shown to depend on the < Qc = intersector elasticity of substitution. This Further, the difference must be large elasticity also governs conditions for enough that the budget constraint for XQ 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 I. 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. Variable Elasticity Case the indifference curve at B is parallel to that at C, so each ofthe moves from C to fi and The utility function is now from B io A increases the value of >". Since the value of x is the same in the two optima, we must have with V increasing and concave. 0 < y < 1. (30) «, > n, = n. This is somewhat like assuming a unit intersector elasticity of substitution. However, Thus the unconstrained optimum actually this is not rigorous since the group utility allows more variety than the constrained ^U) ^ Z-K-v,) 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. i/f/curve in the large group case is VOL. 67 NO. 3 DIXIT AND STIGUTZ: PRODUCT DIVERSITY 303

d log X, We wish to choose n and x to maximize u, (32) - for any / subject to (34) and the break-even condition d log PI px = a + ex. Substituting, we can express u This differs from the case of Section I in as a function of x alone: being a iunclion of x,. To highlight the sim- ilarities and the differences, we define ii{x) a + ex by (39) u = ypix) + (1 - 7) 1 + (33) The first-order condition defines x/.

Next, setting x, = x and /^^ = /? for / = 1. (40) ex, 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) . = /[I - U>{X)] that provided p'{x) is one-signed for all x,

where (41) X, ^ X, according as p'(x) $ 0 With zero pure profit in each case, the (35) [yp(x) - 7)1 points (x^, p^) and (x,, p,) He on the same xv'{x) declining average cost curve, and therefore (42) p, ^ Pt according as x^ J x. We assume that 0 < p(x) < I, and therefore haveO < aj(x) < I. Next we note that the dd curve is tangent to Now consider the Chamberlinian equilib- the average cost curve at (x^, pt) and the rium. The profit-maximization condition DD curve is steeper. Consider the case for each active firm yields the common X, > X,. Now the point (x^, Pc) must lie on a equilibrium price/*, in terms ofthe common DD curve further to the right than (x,, pj, equilibrium output x, as and therefore must correspond to a smaller number of firms. The opposite happens if (36) p, = c[\ + 0{Xt)\ x^ < X,. Thus, Note the analogy with (15). Substituting (43) n, $ «s according as X, ^ x. (36) in the zero pure profit condition, we have x^ defined by Finally, (41) shows that in both cases that arise there, p(x^) < p(x^). Then u;(xj < I (37) ex., ijj(x^), and from (34), a + 1 + (44) xo, > Xo, Finally, the number of firms can be calculated 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) t An intuitive reason for these results can a + ex. 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 ofthe 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 x, > x^. 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 p'W 1 (45) the number of firms, we note

Thus, if p{x) is constant over an interval, so is/i(A) and we have 1/(1 -\- ^) = p, which is a + ex, a + cx^ the case of Section I. However, if p{x) and therefore we have a one-way compari- varies, we eannot infer a relation between son: the signs o{p'(x) and /3'(A). Thus the varia- tion in the elasticity of demand is not in (51) < x^, then n^ > 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)\ with m > 0 and 0 < That is not unreasonable; after ali 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 substitution between any pair of them should in- III. Asymmetric Cases crease. In the symmetric equilibrium, this is just the inverse of the elasticity of marginal The discussion so far imposed symmetry utility. Then a higher x would correspond within the group. Thus the number of varie- to a lower n, and therefore a lower elasticity ties being produced was relevant, but any of substitution, higher -xv"/v' and higher group of n was just as good as any other xv'/v. Thus we are led to expect that p'{x) > group of n. The next important modifica- 0, i.e., that the equilibrium involves fewer tion is to remove this restriction. It is easy and bigger firms than the constrained opti- to see how interrelations within the group mum. Once again the common view con- of commodities can lead to biases. Thus, if cerning excess capacity and excessive di- no sugar is being produced, the demand for versity in monopolistic competition is called coffee may be so low as to make its produc- into question. tion unprofitable when there are set-up 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 produce both. However, problems exist even (46) u = [rtv(j:)]''[l - n(a + cx)]^-^ when alt 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 choice ofthe wrong group is possible.** (48) ex.. Suppose there are two sets of commodities beside the numeraire, the two being per- (49) n.. = a 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 *For an alternative approach using partial equilib- (50) jff according as P'(A:) ^ 0 rium methods, see Spence. VOL. 67 /VO. 3 DIXIT AND STIGLITZ: PRODUCT DIVERSITY 305 for the numeraire. Thus the utility function only if is (55) sc. (52) s - n Now consider the optimum. Both the ob- u = jective and the constraint are such as to lead L'2- I 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 fl, and a constant marginal cost c,. pose n, commodities from group i are being Consider two types of equilibria, only produced at levels x, each, and offered at one commodity group being produced in prices/»,. The utility level is given by each. These are given by (56) u = xMxiwI^'^i

(53a) X, = = 0 and the resource availability constraint is (57)

Given the values of the other variables, the level curves of u in («], /I2) space are concave to the origin, while the constraint is linear. We must therefore have a corner optimum. (As for the break-even con- u, = straint, unless the two 17, = p/ir*^' are equal, the demand for commodities in one group (53b) = 0 is zero, and there is no possibility of avoid- ing a loss there.) Note that we have structured our ex- n-, = ample so that if the correct group is chosen, the equilibrium will not introduce any further biases in relation to the constrained optimum. Therefore, to find the constrained optimum, we only have to look at the values of u, 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 q, 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) corresponding to it. 0 for P2 > Figure 2 is drawn to depict the possible X2 = equilibria and optima. Given all the rele- ^IPi for P2 < vant parameters, we calculate (q[, Q2) 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 = s\\ - —] < 02 son of the magnitudes of ^, and ^2 te'ls us which is the constrained optimum. In the figure, the nonnegative quadrant is split or into regions in each of which we have one SC2 (54) ^1 combination of equilibria and optima. We only have to locate the point (^i, ^2) '" this space to know the result for the given Similarly, (53b) is a Nash equilibrium if and 306 THE AMERICAN ECONOMIC REVIEW JUNE 1977

equilibria, and it is therefore possible that the high elasticity group is produced in equilibrium when the low elasticity one should have been. If the difference in elasticities is large enough, the point moves into region C, where (53b) is no longer a Nash equilibrium. But, owing to the existence of a fixed cost, a significant difference in elasticities is necessary before entry from group I commodities threatens to destroy the "wrong" equilibrium. Similar remarks apply to regions Sand D. Next, begin with symmetry once again, and consider a higher r, or a^. This increases^, and moves the point into region B, making it optimal to produce the low- sCjAs-a^) cost group alone while leaving both (53a) and (53b) as possible equilibria, until the FIGURE 2. SOLUTIONS LABLLED I TO EQUATION (53a): SOLUTIONS difference in costs is large enough to take REKER TO RgUATlON {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 ofthe points correspond- If both ^1 and ^2 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 £ and F. However, the criterion of amine how q^ 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 a, and c,- We also sary to prohibit entry in order to sustain the find constrained optimum. If we combine a case where c, > C2 (or (58) a, > Qj) and &\ > ^2. ie-, where commodities 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 (^1, cfi) ma> then lie in region Further, we see from (9) that a higher {3, 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 ^, 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 SCJ(S - fl|) ^ SC2/is - (32), fj^ = 1^2 Very roughly, the point is that although (the region G vanishes then), and suppose commodities in inelastic demand have the the point (^|, ^.) 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 ^2. 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 us 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 apply lo differences in marginal costs. in the interpretation of the model with heterogenous consumers and social indil- ference curves, inelastically demanded commodities 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 latter, provided the distribution of income is optimum. Even when cross elasticities are zero, there may be an incorrect choice of commodities 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 FiGURt 4 elastic demand curve than commodity fi; 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 arc nol 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 wiih 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 competitive equilibrium, commodity B is not (although il is just at the margin of being produced). But again, observe that B should be produced, since there is a large consumer's surplus; indeed, since were it lo be 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 /I, B would be viable, and social welfare would increase. In the comparison between constrained Pareto optimality and the monopolistically 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- FiGLRf- 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. Concluding Remarks 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 Compelliion clusions seem worth pointing out. Theory. New York 1967. The monopoly power, which is a neces- E. Chamberlin, "Product Heterogeneity and sary ingredient of markets with noncon- Public Policy." Amer. Econ. Rev. Proc. vexities, is usually considered to distort May 1950. "^0, 85-92. resources away from the sector concerned. P. A. Diamond and D. L. McFadden, "Some However, in our analysis monopoly power Uses of the Expenditure Function In enables firms to pay fixed costs, and entry Public Finance," J. Publ. Econ., Feb. cannot be prevented, so the relationship be- 1974. ,^2. 1-23. tween monopoly power and the direction of A. K. Dixit and J. E. Stiglitz. "Monopolistic market distortion is no longer obvious. Competition and Optimum Product Di- In the central case of a constant elasticity versity."" econ. res. pap. no. 64. Univ. utility function, the market solution was Warwick. England 1975. constrained Parelo optimal, regardless of H. A. John Green. Aggregation in Economic the value of that elasticity (and thus the Analysis. Princeton 1964. implied elasticity of the demand functions). H. Hotelling. "Stability in Competition." With variable elasticities, the bias could go Econ.J,.UdX. 1929, J9. 41-57. either way, and the direction of the bias de- N. Kaldor,*'Market Imperfeetion and Excess pended not on how the elasticity of demand Capacity." Economica. Feb. 1934, 2, changed, but on how the elasticity of utility 33-50. changed. We suggested that there was some K. Lancaster, "Socially Optimal Product Dif- presumption that the market solution ferentiation," Amer. Econ. Rev.. Sept. would be characterized by loo few firms in 1975,65,567 85. the monopolistically competitive sector. A. M. Spence, "Product Selection, Fixed With asymmetric demand and cost condi- Cosis. and Monopolistic Competition," tions we also observed a bias against com- Rev. Econ. Stud.. June \976,43. 217-35. modities with inelastic demands and high D. A. Siarrett. "Principles of Optimal Loca- costs. tion in a Large Homogeneous Area." The general principle behind these results J. Econ. Theory. Dec. 1974, 9. 418-48. is that a market solution considers profit at N. H. Stern, "The Optimal Size of Market the appropriate margin, while a social opti- Areas." J. Econ. Theory. Apr. 1972, 4. mum lakes into account the consumer's sur- 159-73. plus. However, applications of this principle J. E. Sttglirz, "Monopolistic Competition in come lo depend on details of cost and de- the Capital Market." tech. rep. no. 161, mand functions. We hope that the cases IMSS, Stanford Univ.. Feb. 1975.