Welfare

Hal Varian
January 1992

Utility possibilities frontier

Let's calculate the utility possibility frontier for two Cobb-Douglas consumers.  We'll maximize a weighted sum of the two utility functions, and then vary the weighting parameter to see how it sweeps out the utility possibilities frontier.

W= a(Log[x1] + Log[x2]) + 
  (1-a)*((1/3)*Log[10-x1]+ (2/3)*Log[10-x2])
solution=Solve[{D[W,x1]==0, D[W,x2]==0},{x1,x2}][[1]]

u1[a_] := Evaluate[Log[x1] + Log[x2]]/.solution
u2[a_] := Evaluate[(1/3)*Log[10-x1]+ (2/3)*Log[10-x2]]/.solution
u1[a]
u2[a]

ParametricPlot[{u1[a],u2[a]},{a,.1,.95}]