Does welfare always fall when concentration rises?

Suppose that industry X has a lower measure of market concentration than industry Y. Let HHIj be the concentration index in industry j, with j = X,Y, and assume that HHIX < HHIY. Is welfare always higher in market X? The answer is no. To see this, suppose that the distribution of marginal costs in market X is c1,c2,...,cn, and that the cost structure in market Y is a mean-preserving spread of the cost structure in market X, so that the cost structure in market Y is c1 + h1,c2 + h2,...,cn + hn, with ^ h = 0. Then c is unchanged, and by equations (10.30) and (10.33), we know that QX = QY, PX = PY, and CSX = CSY. The concentration index in market Y is:

where we have used the assumption that E”=i ht = 0. Since

П = HHIY , we have:

?

This immediately gives us the following set of results. Suppose that the cost structure in industry Y is a mean-preserving spread of the cost structure in market X. Then:

  • 1. If 2^”=iCh + E”-ih? > 0, then HHIY > HHIX & nY > ПX & WY > WX.
  • 2. If 2E ”=1 ctht is sufficiently negative so that 2^”=1 ctht + E”=1 h2 < 0, then HHIY < HHIX & nY < ПX & WY < WX.

Suppose that the mean-preserving spread of costs is independent of the original cost structure, so that Eblcfa = 0. If markets X and Y are identical in all respects except for their cost structures, and if the cost structure in market Y is an independent mean-preserving spread of the cost structure in market X, then case 1 above applies, and both market concentration and welfare will be higher in market Y than they are in market X.

This is exactly the opposite result than the one we would obtain if we relied on naive intuition about the relationship between market concentration and welfare. Intuitively, the mean-preserving spread of costs keeps the average marginal cost (and therefore equilibrium price) the same, but results in a cost structure whose distribution has a larger number of lower-cost firms. The market share (and profits) of these

low-cost firms increases by more than the decrease in the market share and profits of high-cost firms, and so both the market concentration measure and aggregate welfare must also both rise.

A higher concentration index is therefore neither a necessary nor a sufficient condition for aggregate welfare to be lower in an industry, even with the same number of firms and the same market demand.

 
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