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 pz and P, respectively. Skip has a budget of B, B > Py: Therefore, Skip's maximization problem is to maximize U(r, y) = x(y - 1) subject to the budget constraint g(x,y) = B - P.2 - Pyy = 0. (1) Write down the Lagrangian function with , as the Lagrangian multiplier. (5 marks) (2) Find the expressions for the demand functions 2* = 1(Pu: 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(P2, 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., Py, U*). Interpret this expression. Find 8B* /Op., a B* /apy, and aB* /au. (10 marks) Skip's maximization problem could be recast as the following minimization problem: min B(x, y) = P_X +Pyys.t. g(x, y) = U' x(y+1) = 0. (6) Write down the Lagrangian for this problem. (5 marks) (7) Find the values of u 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* /apa and aB*/ap, 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.)