Monopolistic Competition in a Large Economy with Differentiated Commodities: A Correction

In Hart (1979), a model of monopolistic competition in a large economy with differentiated commodities was developed. In this model, firms had a choice whether to set up or not. One feature of the model was that free entry of firms was not assumed. Barriers to entry were captured by assuming that there was a large (generally, infinite) set of potential firms F. Corresponding to each f ∊ F, there was a firm (called "firm f") with a production set Y(f). Each firm had a set-up cost associated with it. Only very weak conditions were placed on the set F and the production set mapping Y(·), so that in particular the case where different firms could produce a commodity on different terms was allowed for. The economy was made large by replicating the consumer sector, keeping the production sector, i.e. the set of potential firms F, fixed. The number of operating firms in equilibrium generally increased, however, since in view of the set-up costs there was "room" for more firms in a large economy. Unfortunately, it turns out that this procedure, while correct, does not capture quite what was intended. In particular, while in the resulting monopolistically competitive equlibrium, some firms will earn supernormal profits, it can be shown that, for any η > 0, the per capita number of firms earning profits in excess of η tends to zero as the size of the consumer sector tends to infinity (see Corollary 6 in the Appendix to Hart (1979)). In other words, in per capita terms, almost all firms earn approximately zero profits in a large economy. Thus while barriers to entry may be significant in absolute terms, in per capita terms they are negligible. The way round this difficulty is to drop the assumption that the set of potential firms is fixed. Instead substitute the assumption that the set of potential firms in the economy rE, where the consumer sector is replicated r times, is given bywhere F is as before. That is, one replicates the set of potential firms at the same time as the consumer sector. Then the theorems of Hart (1979) continue to hold. Corollary 6 in the Appendix must be modified as follows. Corollary 6′. There exists h > 0 such thatfor all f ∊ F. Corollary 6′ is proved below. Otherwise the proofs of Theorem 1 and Proposition 2 are unchanged (one no longer sets h = 1 after Corollary 6). As an example, F might consist of one firm with an efficient technology for producing some commodity and one firm with an inefficient technology. Then in the economy rE, there will be r potential firms with the efficient technology and r potential firms with the inefficient technology. It is easy to construct cases where both types of firms operate in the monopolistically competitive equilibrium in rE and the efficient firms earn supernormal profits which are bounded away from zero as r → ∞. Thus barriers to entry which are significant in per capita terms are now allowed for. A justification for replicating F along with the consumer sector can be given. In the above example, the efficient firms may owe their superior technology to the fact that they are situated on good land, say, of which there is a scarcity (thus the supernormal profits are just rents on the land). When one replicates the economy, it is natural to replicate the scarce land and hence the number of firms which are situated on it, so as to keep everything constant except for scale. Note finally that it may be possible to generalize the analysis to the case where the set of potential firms in the economy rE is given by rF, where 1F, 2F are exogenously specified sets and rF is not necessarily the r-fold union of some set F. We have not investigated this, however. Proof of Corollary 6′. Suppose not. Then for each h > 0, we can find f ∊ F with . By Lemma 5 (2), rπf > h for all r ≧ some r*. But in rE there are r firms identical to firm f and so each of these firms makes profit in excess of h in the monopolistically competitive equilibrium when r ≧ r*. Hence total per capita profits of all firms exceed h in equilibrium when r ≧ r*. It follows that, letting h → ∞, we can find a subsequence of the economies rE such that total per capita profits tend to infinity along the subsequence. However, applying Corollary 4 and an argument similar to that in (A.30)–(A.32), we see that ʃArp(a)drY1(a) is bounded. Hence so are per capita profits, ʃArp(a)drY1(a) + rY0. Contradiction. ||