Question 1 (50 marks + 5 bonus marks): Skip has the following utility function: U(x; y) = x(y1),

where x and y are quantities of two consumption goods whose prices are px and py respectively. Skip

has a budget of B; B > py.

Therefore, Skip's maximization problem is to maximize U(x; y) = x(y 1) subject to the budget

constraint g(x; y) = B pxx pyy = 0:

(1) Write down the Lagrangian function as the Lagrangian multiplier. (5 marks)

(2)Find the expressions for the demand functions x = x(px; py;B) and y = y(px; py;B) from the rst

order conditions. (5 marks)

(3) Verify that Skip is at a maximum by checking the second-order conditions. (5 marks)

(4) Find an expression for the indirect utility function U = U(px; py; B) by substituting x and y into

the utility function. (5 marks)

(5) By rearranging the indirect utility function, derive an expression for the expenditure function B =

B(px; py; U). Interpret this expression. Find @B=@px, @B=@py, and @B=@U. (10 marks)

Skip's maximization problem could be recast as the following minimization problem:

min B(x; y) = pxx + pyy s:t: g(x; y) = U x(y + 1) = 0:

(6) Write down the Lagrangian for this problem. (5 marks)

(7) Find the values of x and y that solve this minimization problem and the expenditure function. Are

the optimal values, x and y, equal to the partial derivatives of the expenditure function, @B=@px and

@B=@py respectively. Explain these results using the Envelope Theorem. (15 marks)

(8) What's the relationship between the values of Lagrangian multiplier in the utility-maximization

problem and the expenditure-minimization? Explain this relationship using their economic meanings.

(Hint: Use the relationship between B and U.) (Bonus: 5 marks.)

Question 1 (50 marks + 5 bonus marks): Skip has the following utility function: U(x, y) = x(y-1), where x and y are quantities of two consumption goods whose prices are Px and Py respectively. Skip has a budget of B, B > Py. Therefore, Skip's maximization problem is to maximize U(x, y) X(y 1) subject to the budget constraint g(x, y) = B PrX Pyy = 0. (1) Write down the Lagrangian function with l as the Lagrangian multiplier. (5 marks) (2)Find the expressions for the demand functions r* = X (Px, Py, B) and y* = y(Px, Py, B) from the first order conditions. (5 marks) (3) Verify that Skip is at a maximum by checking the second order conditions. (5 marks) (4) Find an expressions for the indirect utility function U* = U(Pc, Py, B) by substituting x* and y* into the utility function. (5 marks) (5) By rearranging the indirect utility function, derive an expression for the expenditure function B* B(Px, Py, U*). Interpret this expression. Find 8B*/Opx, 8B* /apy, and B*/2U*. (10 marks) Skip's maximization problem could be recast as the following minimization problem: min B(x, y) = PzX+ Pyy sit. g(x, y) = U* x(y + 1) = 0. (6) Write down the Lagrangian for this problem. (5 marks) (7) Find the values of x and y that solve this minimization problem and the expenditure function. Are the optimal values, x* and y*, equal to the partial derivatives of the expenditure function, B*/px and B* /apy respectively. Explain these results using the Envelope Theorem. (15 marks) (8) What's the relationship between the values of Lagrangian multiplier in the utility-maximization problem and the expenditure-minimization? Explain this relationship using their economic meanings. (Hint: Use the relationship between B* and U*.) (Bonus: 5 marks.)