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 aggre- gation 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 mag- nitude, 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.