# Does welfare always fall when concentration rises?

Suppose that industry *X* has a lower measure of market concentration than industry *Y*. Let *HHI** _{j}* be the concentration index in industry

*j*, with

*j*

*=*

*X,Y*, and assume that

*HHI*

_{X}*<*

*HHI*. Is welfare always higher in market

_{Y}*X*?

*The answer is no*. To see this, suppose that the distribution of marginal costs in market

*X*is

*c*c

_{1},_{2},...,

*c*and that the cost structure in market

_{n},*Y*is a

*mean-preserving spread*of the cost structure in market X, so that the cost structure in market

*Y*is

*c*

_{1}*+*

*h*+ h

_{1},c_{2}_{2},...,

*c*

_{n}*+*

*h*, with ^

_{n}*h*

*=*0. Then

*c*is unchanged, and by equations (10.30) and (10.33), we know that

*Q*

_{X}*=*

*Q*,

_{Y}*P*

_{X}*=*

*P*, and

_{Y}*CS*

_{X}*=*

*CS*. The concentration index in market

_{Y}*Y*is:

where we have used the assumption that *E”=i **h** _{t} =* 0. Since

П = *HHI** _{Y}* , 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^”=i
*Ch*+ E”-i^{h}*? >*0, then*HHI*_{Y}>*HHI*n_{X}&_{Y}> П_{X}&*W*_{Y}>*W*._{X} - 2. If 2E ”=
_{1}*c*_{t}*h*is sufficiently negative so that 2^”=_{t}_{1}*c*_{t}*h*+ E”=_{t}_{1}h^{2}< 0, then*HHI*_{Y}<*HHI*n_{X}&_{Y}< П_{X}&*W*_{Y}<*W*._{X}

Suppose that the mean-preserving spread of costs is independent of the original cost structure, so that E_{bl}*cfa **=* 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.