Woody lives in Houston, and works at a college restaurant where the minimum wage

Question

Woody lives in Houston, and works at a college restaurant where the minimum wage per hour, w, is paid. The current minimum wage rate per hour is $8. Woody allocates his daily hours either for working or for leisure. That is, for the total 24 hours he has a day, he can choose to work (denoted as H), or do other things (denoted as L). If he works, all the money he earns, w x H, is spent on his consumption. Saving is not poosible. Both consumption and leisure are goods for him.

Woody's utility function is
U(L,C)= Squre root of CL2
MUC=L/(2*Square root of C)
MUL=Sqaure root of C

a) Derive Woody's MRS function. What is the MRS at (L,C) = (20,32)? State the meaning of MRS in terms of leisure and consumption.

b) Derive Woody's budget constraint equation. (Hint: C=w x H, and the total hours a day is 24)

c) What is Woody's optimal (L*,C*) and utility level U*? How many hours does Woody work each day? Clearly state the utility maximization condition.  

Explanation / Answer

Woody earns a minimum wage rate per hour of $8. Woody allocates the total 24 hours in working (denoted as H), or doing other things (denoted as L). All income is spent on his consumption.

Woody's utility function is U(L,C)= (CL2) ½ or U = C½L, which gives marginal utility from consumption as

MUC = L/(2C½)
MUL= C½

a) MRS function is the ratio of marginal utility of consumption and marginal utility of leisure. This gives MRS = L/(2C½)*C½ or simply MRS = L/2C.

MRS at (L,C) = (20,32) will be equal to 20/64 or 0.3125. MRS shows the rate of substitution between leisure and consumption.

b) Woody's budget constraint equation is given by C = (24 – L)*8 or C = 192 – 8L. The budget equation becomes C + 8L = 192.

c) Woody's optimal (L*,C*) are determined when MRS = slope of the budget equation

L/2C = 1/8 which gives C = 4L.

C + 8L = 192

4L + 8L = 192

L= 192/12 = 16

C = 64

Hence the utility is U = 64½16 = 128. Woody work each day for 8 hours and earn $64. The utility is maximized when MRS = slope of budget equation.

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