Externalities

Hal Varian
January 1992

Efficient allocations with externalities

Firms 1 and 2 have cost functions

c1[y1_] := y1^2
c2[y1_,y2_] := y2^2 + y1 y2
Profit1[y1_] := p1 y1 - c1[y1]
Profit2[y1_,y2_] := p2 y2 - c2[y1,y2]

Given prices p1 and p2, the firms operated where

Solve[{D[Profit1[y1],y1]==0,
       D[Profit2[y1,y2],y2]==0},{y1,y2}][[1]]

The efficient solution involves maximizing the sum of the profits.  Hence we want

Solve[{D[Profit1[y1]+Profit2[y1,y2],y1]==0,
      D[Profit1[y1]+Profit2[y1,y2],y2]==0},
      {y1,y2}][[1]]

Note that the efficient solution involves firm 1 producing less output.