Question

In the consumption/leisure model, let the consumer’s utility function be U(C,l)=C.25+l.25. Suppose Pc=$1 and w1=$10 determine the optimal amount of C and L and the equation for the labor supply.

Answer 1

Solution:

With the utility function of U(C, L) = C^0.25 + L^0.25 where C is consumption level in units and L is leisure hours.

Then marginal utility of consumption, MUc = d(U, C)/dC

MUc = 0.25*C^0.25-1 + 0 = 0.25*C^-0.75

And marginal utility of leisure, MUl = 0.25*L^0.25-1 + 0 = 0.25*L^-0.75

Assuming total number of available hours as the standard value of 24 hours and non labor income as 0, the required budget line becomes:

Pc*C + w*L = w*N ; where N is the number of labor hours. So, it must be L + N = 24, with total available hours of 24.

Then, budget line becomes: 1*C + 10*L =.10*(24 - L)

C + 10L = 240 - 10L

C + 20L = 240

Optimal level occurs where slope of indifference curve equals the ratio of prices. So, MUl/MUc = w/Pc

0.25*L^-0.75/0.25*C^-0.75 = 10/1

(C/L)^0.75 = 10

Or C = 10^4/3*L

Substituting this in the budget line, we get:

10^4/3L + 20*L = 240

Solving this we get optimal value of L, and the corresponding optimal value of C.