Question

Phil’s utility for consumption and leisure can be expressed as:U(C, L) = C^1/3L

a. Find the expression for Phil’s marginal rate of substitution between leisure and consumption, as a function of C and L.

b. What does Phil’s utility function tell us about his relative preference between consumption and leisure?

c. When L = 100 and C = 500, how much would Phil be willing to pay for an additional hour of leisure? How about when L=50 and C = 500?

Phil has 128 hours available to be allocated between work and leisure every week (he spends the rest of the time sleeping). He faces a market wage of $12 per hour. Assume Phil owns a rental property and receives $640 in rental income each week from his tenants.

d. Compute Phil’s reservation wage.

e. In this solution, what is Phil’s optimal labor supply choice? How many hours does he choose to enjoy as leisure? What is his optimal consumption level? What is his level of utility at this optimal point?

All of Phil’s tenants move out and his weekly rental income drops to $0.

f. What is Phil’s reservation wage now?

g. What is Phil’s optimal amount of consumption and leisure now? How many hours does he choose to work? What level of utility does he attain?

h. Suppose that the government starts a welfare policy that pays $G to non-workers (and $0 to workers). At what value G will Phil find it worthwhile to opt out of the labor force in order to go on welfare?

i. If Phil drops out of the labor market to go on welfare, his consumption level will be given by the $G you found in part (h) above. This is lower than the level of consumption he enjoyed in the absence of the welfare program, which you found in part (g) above. Therefore, the welfare program induces Phil to reduce his consumption. In other words, Phil chooses to stay on welfare, even though going back to work would clearly increase his income and hence his consumption). How is this possible?

Answer 1

We are given U(C,L)= C^1/3 L

a) marginal rate of subsitution is given by Marginal utility of L/ marginal rate of C

marginal utility of C == 1/3 C^-2/3L  

marginal utility of L    = C^1/3

marginal rate of subsitution is given by = C^1/3/1/3 C^-2/3L = C/(1/3 L)

with increase in the Leisure hours and decrease in consumption the MRS increases and vice versa when the MRS decreases.

b) with every 1 unit increase in consumption his utility increases by less amoint than by 1 init increase in leisure . so we get that he prefers more leisure .

c) MRS when  L = 100 and C = 500 is 500*3/ 100 = 15

when L=50 and C = 500 is 500*3/50 = 30

d) the reservation wage is given by the maximum leisure ie not worku=ing at all . so since he receives a $ 640 of rent per week , his reservation wage is given by total rent received / total hours in aweek = $640/128= $5

so reservation wage is $5.