By Schmalensee R., Willig R.
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To see this, first note that, when there are rising marginal costs, any industry output y is divided in positive portions most cheaply among k different firms if each firm produces the same amount, y / k . Then (minimized) total industry costs for a k firm industry are k C ( y / k ) = k F + a y 2 / k > F + ay 2 for all y < Ys. The foregoing discussion has revealed that the relationship between economies of scale and natural monopoly is nontrivial, even in the single product case. In the multiproduct world things are even more complicated.
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~.
It is important to note, however, 35Most of this section is drawn from the discussion in Baumol, Panzar and Willig (1982, pp. 446-448). 36See Blackorby, Primont and Russell (1977) or Denny and Fuss (1977) for discussions of econometric aggregationtests. Ch. 1: Determinants of Firm and Industry Structure 43 that it is not generally valid to extrapolate those measures to output mixes outside the sample. As the discussion of the previous section indicates, knowledge of the entire M-locus is generally required when calculating bounds on the number of firms in an efficient industry configuration for any output mix.