By Schmalensee R., Willig R.

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The following propositions make this standard practice precise. 25 Proposition 15 Assume that the average cost function C(y)/y has a unique minimum at yM, is strictly decreasing for 0 < y < yM, and is strictly increasing for y > ym. Then the cost-minimizing number of firms for the production of the industry output y l is exactly yl/yM if that number is an integer. In this case, CI(y I) = (yI/yM)C(yM) _ y l . AC M. If yI/yM is not an integer, then the cost-minimizing number of firms is either the integer just smaller or the integer just larger than yI/yM.

Panzar 22 Proposition 6 A twice-differentiable multiproduct cost function that exhibits weak cost complementarities over N up to output level y exhibits economies of scope at y with respect to all partitions of N. Proof Since any partition of N can be obtained by a sequence of binary partitions, it suffices to demonstrate the result for the partition T, T, where N 4: T = ~ . Rearrange terms so that the condition to be demonstrated is [C(y~+y~)--C(YT)]- [C(y~)-C(O)] <0. The first term in brackets can be rewritten as fr~i~ ~[Ci( YT + Xr~)dxi] and the second bracketed term as fr~,i~~[Ci(x~)dxi], where F is any smooth monotonic arc from 0 to yT~.

Since the primary focus of this analysis is on long-run industry structure, the first definition limits attention to industry situations in which supply equals demand and the second requires that each firm earns non-negative profits from its market activities. One must go further than this, of course. For there are situations in which there may exist feasible industry configurations containing one, four, or a hundred firms. 4 illustrates such a situation, if one imagines that the average cost curve rises, but only imperceptibly, beyond M.