Question

Charlie likes buying shoes and going out to sing. His utility function for pairs of shoes, S, and the number of times he goes singing per month, T, is U(S,T)=2S^2T^2 . It costs Charlie $50 to buy a new pair of shoes or to spend an evening out singing. Assume that he has $500 to spend on shoes and singing.

a. What is the equation for her budget line? Draw the budget constraint with the T on the vertical axis, and label the slope and intercepts.

b. Draw Charlies’s indifference curves.

c. What is Charlie’s MRS? Explain.

d. Write out his constrained optimization problem.

e. Solve mathematically for his optimal bundle with the Lagrange multiplier method.

f. Solve mathematically for his optimal bundle with a different method.

g. Suppose that the price of shoes increases to $75. What is the new optimal bundle?

h. Are shoes normal goods and why?

Answer 1

a) It is given to us that the total money Charlie has to spend is $500.

Cost of a new pair of shoes in $50 . Cost for going out to sing once is $50.

Thus, BUDGET CONSTRAINT is :

500 = 50*S + 50*T

Intercepts : When S = 0, T = 500/50 = 10

  When T = 0, S =500/50 = 10

Slope: Applying total differentiation to the constraint,

0 = 50 dS + 50 dT

or, dT/dS = -1

b) MRS : - (MU_T/MU_S) = - (4TS²/4ST²) = -(S/T)

Second order derivative: - [ T (dS/dT) - S(dT/dS) / T² ] <0 So, IC is downward sloping convex.

  

c) U = 2S²T²

Marginal Utility of S = dU/dS = 4ST²

Marginal Utility of T = dU/dT = 4TS²

MRS : - (MU_T/MU_S) = - (4TS²/4ST²) = -(S/T)

This explains that to consume one pair of shoes , Charlie has to give up singing for (S/T) evenings.

d) Constrained Optimization problem :

MAXIMIZE U = 2S²T²

Subject to : 500 = 50*S + 50*T