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 P.x2 Pyy = 0. (1) Write down the Lagrangian function with l as the Lagrangian multiplier. (5 marks) (2)Find the expressions for the demand functions x* = = X(Px, Py, B) and y* = y(Pr, 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(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(P.x, Py, U*). Interpret this expression. Find 8B*/@px, B* /py, and 8B*/@U*. (10 marks) Skip's maximization problem could be recast as the following minimization problem: min B(x, y) = P.zX+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, aB*/@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.)