Dixit-Stiglitz1977垄断竞争模型

0521819911C04 June 20, 2003 18:10 Char Count= 0 4 Monopolistic competition and optimum product diversity (February 1975) Avinash K. Dixit and Joseph E. Stiglitz 4.1 Introduction This chapter began as an attempt to formalise a simple general equilib- rium version of the Chamberlinian monopolistic competition model, in order to see whether there is any validity in the common assertion that mo- nopolistic competition leads to too much product diversification. Cham- berlin (1950, p. 89) himself was careful not to make any such assertion; as a matter of fact, he saw the central issue very clearly, and expressed it succinctly as follows: ‘It is true that the same total resources . . . may be made to yield more units of product by being concentrated on fewer firms. The issue might be put as efficiency versus diversity’. Kaldor (1935, p. 50), too, saw that excess capacity was not the same as excessive diversity. He said that if economies of scale were exploited to a greater extent, ‘the public would be offered finally larger amounts of number of commodi- ties; and it is impossible to tell how far people prefer quantity to diversity and vice versa’. What this really means is that a model must be specified in greater detail to determine conditions under which the one or the other might be expected. The one example which has been worked out in de- tail, the Hotelling spatial location model, has led many economists to the presumption that there is in fact excessive diversity associated with mo- nopolistic competition.1 The results of our analysis throw considerable doubt on this presumption. Further, it soon becomes apparent that the issue of diversity is only part of the more general question of a comparison between the market equilibrium and the optimum allocation: not only may the numbers pro- duced be incorrect, but the choice of which commodities to produce, The research for this chapter was initiated while Dixit was at Balliol College, Oxford, and Stiglitz was Visiting Fellow at St Catherine’s College, Oxford. Stiglitz’s research was supported in part by National Science Foundation Grant SOC74-22182 at the Institute for Mathematical Studies in the Social Sciences, Stanford University. The authors are indebted to Michael Spence for comments and suggestions on an earlier draft. 1 Hotelling (1929). However the article by Stern (1972) casts doubt on that presumption even in the context of location. 89 0521819911C04 June 20, 2003 18:10 Char Count= 0 90 Avinash K. Dixit and Joseph E. Stiglitz and how much of each to produce, may differ. There are a number of effects at work. Whether a commodity is produced depends on revenues relative to total costs. Social profitability depends, on the other hand, on a number of factors. In deciding whether to produce a commodity, the government would look not only at the profitability of the project, but also at the consumer surplus (the profitability it could attain if it were acting as a completely discriminating monopolist), and the effect on other industries and sectors (on the consumer surplus, profitability and viability). The effects on other sectors result both from substitution and income effects. The whole problem hinges crucially on the existence of economies of scale. In their absence, it would be possible to produce infinitesimal amounts of every conceivable product that might be desired, without any additional resource cost. Private and social profitability would co- incide given the other conventional assumptions, and the repercussions on other sectors would become purely pecuniary externalities. With non- convexities, however, we shall see that all these considerations are altered. Moreover, given economies of scale in the relevant range of output, market realisation of the ‘unconstrained’ or first-best optimum, i.e. one subject to constraints of resource availability and technology alone, re- quires pricing below average cost, with lump-sum transfers to firms to cover losses. The conceptual and practical difficulties of doing so are clearly formidable. It would therefore appear that perhaps a more appro- priate notion of optimality is a constrained one, where each firm must operate without making a loss. The government may pursue conventional regulatory policies, or combinations of excise and franchise taxes and sub- sidies, but the important restriction is that lump-sum subsidies are not possible. The permissible output and price configurations in such an optimum reflect the same constraints as the ones in the Chamberlinian equilibrium. The two solutions can still differ because of differences implicit in the objective functions. Consider first the manner in which the desirability of variety can enter into the model. Some such notion is already implicit in the convexity of indifference surfaces of a conventional utility function defined over quan- tities of all the varieties that might exist. Thus, a person who might be indifferent between the combinations of quantities (1, 0) and (0, 1) of two product types would prefer the combination ( 12 , 1 2 ) to either extreme. If this is the only relevant consideration, we shall show that in one central case the Chamberlinian equilibrium and the constrained optimum coin- cide. In the same case, we shall also show that the first-best optimum has firms of the same size as in the other two solutions, and a greater number 0521819911C04 June 20, 2003 18:10 Char Count= 0 Monopolistic competition and optimum product diversity (1975) 91 of such firms. These results undermine much of the conventional wisdom concerning excess capacity as well as excessive diversity. However, it is conceivable that the range of products available is by itself an argument of the utility function, over and above what is taken into account through the amounts actually consumed. This may reflect the desirability of accommodating a sudden future change of tastes, or of retaining one’s identity by consuming products different from those consumed by one’s neighbours, or some such consideration. Variety then takes on some aspects of a public good, and this raises the usual problems for the optimal provision of such goods in a market system. Even if variety is not a public good, its private and social desirability can still differ on account of the failure to appropriate consumers’ sur- plus as noted above. In the large group case, it so happens that if the elasticity of demand is constant and the same for all products, the con- sumers’ surplus is proportional to the revenue, with the same factor of proportionality for all goods. The difference in the objectives of firms and of welfare maximisation then does not matter. Otherwise, we expect the equilibrium outcome to be biased against those varieties for which the ratio of consumers’ surplus to revenue is large. However, this simple principle does not yield much direct insight. A change in the output of a commodity, or the introduction of a new commodity, affect the de- mands for all other goods. With possible changes in the levels as well as the elasticities of all demands, the consumers’ surpluses and revenues can change in complicated ways. Therefore the answers to the questions of the equilibrium and the optimum levels of output, including possibly considerations of the viability of these commodities, involve a very large range of possibilities. We need an explicit model with a detailed formu- lation of demand, in order to isolate and analyse the various questions. The rest of the chapter attempts to provide such analyses. In section 4.2 we discuss the problems of modelling the demand for variety, and set up the model of the special case mentioned above. In section 4.3, this case is analysed in detail. Sections 4.4–4.6 consider the various gener- alisations mentioned above. In each case, we compare various features of the Chamberlinian equilibrium with those of the two types of optima, with particular regard to (1) the number and mix of products, (2) their prices and quantities and (3) the total resource allocation for this group of products. We focus on the allocation problems that are of interest here, and ne- glect two other issues. The first is that of income distribution. We assume utility to be a function of market aggregate quantities. This is justified if the consumers have identical tastes, and either identical incomes or linear Engel curves; alternatively we can assume that lump-sum redistributions 0521819911C04 June 20, 2003 18:10 Char Count= 0 92 Avinash K. Dixit and Joseph E. Stiglitz take place to maximise an individualistic social welfare function, thus yielding Samuelsonian social indifference curves. Also, we assume that the consumers’ preferences are exogenous, thus excluding considerations of advertising and its welfare implications. We feel that prevailing think- ing has overstressed this aspect at the expense of some basic allocative issues, and that the qualitative effects of adding these considerations to our model should in any case be fairly evident. Our model differs from the spatial location model in one important respect. There, each consumer purchases only one of the products in the industry. Increasing product differentiation leads to the consumer being able to purchase a commodity closer to his liking, i.e. to go to a store closer to his residence. Our model includes such considerations in its interpretation with heterogeneous consumers and social indifference curves. But in addition, it can allow each consumer to enjoy product diversity directly. There are numerous examples where this formulation is clearly more appropriate than one modelled on location. The ability to diversity a portfolio by spreading one’s wealth over a large number of assets was one of the instances that provided the original motivation for formulating this kind of model, and is discussed in more detail elsewhere (Stiglitz, 1973). Clothes suited to different climatic conditions, or flavours of ice-cream, are other examples of this type. 4.2 The demand for variety Consider a potentially infinite range of related products,2 numbered 1,2, . . . n, . . . A competitive sector labelled 0 aggregates the rest of the economy. Good 0 is chosen as the nume´raire and the amount of the econ- omy’s endowment of it is normalised at unity; this can be thought of as the time at the disposal of the consumers. If the amounts of the commodities consumed are x0 and x = (x1, x2, ..xn, . . .) we define a utility function u = U(x0, x1, x2, . . ., xn, . . .). (4.1) This function, assumed to have convex indifference surfaces, considers variety as a private good in the sense defined before. If a sub-set S of commodities is actually being produced, i.e. xi > 0 for i ∈ S and xi = 0 for i /∈ S, then the public good case can be modelled by allowing u to 2 An earlier version of this chapter considered the aesthetically more pleasing case of a continuum of products. However, it was discovered that [the] technical difficulties of that case led to unnecessary confusion. 0521819911C04 June 20, 2003 18:10 Char Count= 0 Monopolistic competition and optimum product diversity (1975) 93 depend explicitly on S, i.e. u = U(x0, x1, x2, . . . , xn, . . . ; S ). (4.2) We shall take up this case in section 4.4. It is clear that at this level of generality, nothing specific or interesting could be said. We proceed to impose some structure on U in order to iso- late issues for sharper focus. First, we assume that the group of products in question is separable from the aggregated sector, i.e. u = U(x0, V(x1, x2, . . . , xn, . . .)). (4.3) For most of this chapter, we assume that V is a symmetric function. This, combined with an assumption about the symmetry of costs, removes the issue of the product mix. The number of products is still a relevant con- sideration, but given this number n, it does not matter what labels they bear. Then we may as well label them 1, 2, . . .n, and potential products (n + 1), (n + 2), . . . are not being produced. This is a restrictive assump- tion, for in such problems we often have a natural sense of order along a spectrum, and two products closer together on this spectrum are better substitutes than two products farther apart. This makes V asymmetric, and the actual labels of products available become important. This is naturally recognised in the spatial context, but the Chamberlin tradition where the nature of the products in the group is left unspecified has im- plicitly assumed symmetry. We shall follow this tradition, but in section 4.6 we shall return to the question of the product mix.3 The next simplification is to consider an additively separable form for the function V(x), i.e. u = U ( x0, ∑ i v(xi ) ) . (4.4) We take up this case in section 4.5. In section 4.4, we consider an even more special form where V(x) has a constant elasticity of substitution, i.e. u = U  x0, [∑ i xρi ]1/ρ . (4.5) For concavity, we need ρ < 1. Further, since we wish to allow a situation where several of the xi are zero, we need ρ > 0. 3 Spence (1974) focuses on this issue in greater detail. 0521819911C04 June 20, 2003 18:10 Char Count= 0 94 Avinash K. Dixit and Joseph E. Stiglitz Finally, we assume that U is homogeneous of degree one in x0 and V(x). Then, with unit income elasticities, we can study substitution between the sectors without the added complication of unequal income effects. In the remainder of this section we shall derive the demand functions for the special case (4.5), and comment on their properties. Suppose products 1, 2, . . .n are being produced, and write the budget constraint as x0 + n∑ i=1 pi xi = I, (4.6) where I is income in terms of the nume´raire, i.e. the endowment which has been normalised at 1, plus the profits of firms distributed to con- sumers, or minus the lump-sum transfers to firms, as the case may be. We omit the details of utility maximisation. The interesting feature is that a two-stage budgeting procedure is applicable.4 Thus we can define a quantity index y = V(x), and a price index q = Q(p) such that (x0, y) maximise U(x0, y) subject to x0 + q y = I , and then x maximises V(x) subject to ∑ i pi xi = q y. Moreover, with the quantity index of a constant elasticity form, so is the price index. Thus, when y = [ n∑ i=1 xρi ]1/ρ (4.7) we have q = [ n∑ i=1 p−1/βi ]−β , (4.8) where β = (1 − ρ)/ρ. From the conditions imposed on ρ, we know that β is positive. Now consider the first stage of budgeting. Since U is homogeneous of degree one, x0 and y are each proportional to I , and the budget shares are functions of q alone. Let s (q ) be the budget share of y, i.e. y = Is (q ) q . (4.9) The ratio x0/y is a function of q alone, and its elasticity is defined as the intersectoral elasticity of substitution, which we shall write as σ (q ). The behaviour of budget shares depends on the relation between σ (q ) and 1 4 See, e.g., Green (1964, p. 21). 0521819911C04 June 20, 2003 18:10 Char Count= 0 Monopolistic competition and optimum product diversity (1975) 95 in the standard manner; thus we have the elasticity θ(q ) = q s ′(q ) s (q ) = [1 − σ (q )] [1 − s (q )] . (4.10) We see at once that θ(q ) < 1. (4.11) Turning to the second stage of the problem, it is easy to show that for each i , xi = y ( q pi )1/(1−ρ) , (4.12) where y is defined by (4.9). Consider the effect of a change in pi alone. This affects xi directly, and also through q and thence through y as well. Now from (4.8) we have the elasticity ∂ log q ∂ log pi = ( q pi )1/β . (4.13) So long as the prices of the producers in the group are not of different orders of magnitude, this is of the order (1/n). We shall assume that n is reasonably large, and accordingly neglect the effect of each pi on q and thus the indirect effects on xi . This leaves us with the elasticity ∂ log xi ∂ log pi = − ( 1 1 − ρ ) = 1 + β β . (4.14) In the Chamberlinian terminology, this is the elasticity of the dd curve, i.e. the curve relating the demand for each product type to its own price with all other prices held constant. In our large group case, we also see that for i �= j , the cross-elasticity ∂ log xi/∂ log p j is negligible. However, if all prices in the group move together, the individually small effects add to a significant amount. This corresponds to the Chamberlinian DD curve. Consider a symmetric situation where xi = x and pi = p for all i from 1 to n. We have y = xn1/ρ = xn1+β (4.15) q = pn−β = xn−(1−ρ)/ρ (4.16) and then, from (4.8) and (4.12) x = Is (q ) pn . (4.17) 0521819911C04 June 20, 2003 18:10 Char Count= 0 96 Avinash K. Dixit and Joseph E. Stiglitz The elasticity of this is easy to calculate; we find ∂ log x ∂ log p = − [1 − θ(q )] . (4.18) Then (4.11) shows that the DD curve slopes downward. The conventional condition that the dd-curve be more elastic is seen from (4.14) and (4.18) to be 1 β + θ(q ) > 0. (4.19) Finally we observe that; for i �= j , xi x j = ( p j pi )1/(1−ρ) . (4.20) Thus 1/(1 − ρ) is the elasticity of substitution between any two products within the group. This calls for some comment. A constant intra-sectoral elasticity of substitution has some undesirable features in a model of product diversity. Some problems of assuming symmetry were pointed out earlier. For a spectrum of characteristics, we would expect the elas- ticity to depend on the distance between i and j . In addition, the total number of products being produced may be thought to influence the elasticities. If the total conceivable range of variation is finite, then prod- ucts have to crowd closer together as their number increases, and thus the elasticity of substitution should on the whole increase and tend to infinity in the limit. However, it is often the case that the total range is very large, and most practicable product ranges can only hope to cover a negligible fraction of it. This is particularly true if there are several relevant charac- teristics, and therefore several dimensions to the spectrum. Since this is a very likely situation, we think it interesting to have a model where there is an infinity of conceivable products but only a finite number are ever produced, and the elasticities of substitution are all bounded above, thus always leaving some monopoly power in existence. With fresh apologies for symmetry, the assumption of constancy then offers some simplicity and an interesting result. In section 4.5, we shall relax constancy to some extent. As regards production, we assume for most of the chapter that each firm has the same fixed cost, a, and a constant marginal cost, c, also equal for all firms. All our results remain valid if the variable cost of production is allowed to depend on output, but the algebra is considerably more complicated. In section 4.6 we consider a case where different firms have different values of a and c, and in the concluding remarks we mention some other problems. 0521819911C04 June 20, 2003 18:10 Char Count= 0 Monopolistic competition and optimum product diversity (1975) 97 4.3 The constant elasticity case 4.3.1 Market equilibrium In this section we study the consequences of the utility function (4.5) and the associated demand functions derived in section 4.2. Let us be- gin with the Chamberlinian group equilibrium. The profit-maximisation condition for each firm is the familiar equality of marginal revenue and marginal cost. With a constant elasticity of demand and constant marginal cost for each firm, this becomes pi [ 1 − 1 1/(1 − ρ) ] = c, for i = 1, 2, .., n. Write pe for the common equilibrium price for each variety being pro- duced. Then we have pe = c ρ = c(1 + β). (4.21) The second condition of equilibrium is that firms enter until the next potential entrant would make a loss, i.e. n is defined by (pn − c)xn ≥ a (pn+1 − c)xn+1 < a } . We shall assume that n is large enough t