Consumers' Surplus

Hal Varian
January 1992

Compensating and Equivalent Variation

A consumer has a demand function of the form x=0.1 m - p, where m is income and p is price.  The consumer has income of 100 and faces a price of 2.  If the price increases to 4, what is the compensating and equivalent variation?

The first thing we have to do is to compute the expenditure function associated with this demand function.  We know from Chapter 8 that there is an analytic expression for e[p,u], but we'll use Mathematica's numerical differential equation solver here just to be different:

solution1=NDSolve[{e'[p]==.10*e[p] - p, e[2]==100},e,{p,1,5}]
Plot[Evaluate[e[p]/.solution1],{p,1,5}]

The compensating variation will be the amount of money the consumer needs at the new price, p=4, to restore the original utility level, minus the original income.  This will be

CV= (e[4]/.solution1) - 100

The equivalent variation will be the amount of money that the consumer needs at the old price to make him as well off as he would be at the new price.  We have to solve the integrability equation again with a new boundary condition.

solution2=NDSolve[{e'[p]==.10*e[p] - p, e[4]==100},e,{p,1,5}]
EV= 100-(e[2]/.solution2)

Finally, we'll calculate the consumer's surplus for the move from p=4 to p=2:

CS=Integrate[10-p,p]
Change =  (CS/.{p->4})-(CS/.{p->2})

What relationship has to hold between CV, EV, and CS?