(a) Write the Lagrangian function? (b) Find the optimal levels of purchase of x*

(a) Write the Lagrangian function?
(b) Find the optimal levels of purchase of x* and y*?
(c) Is the second-order sufficient condition for maximum satised?
Question 1 A rm has the following total-cost and demand functions: C = Q3 ?? Q2 + Q + Q = ?? P (b) Write out total-revenue function R in terms of Q. (c) Formulate the total-prot function in terms of Q. (d) Find the prot-maximizing level of output Q. (e) What is the maximum prot? (f) Check that it actually is maximum. Question 2 Consider the following macroeconomic model: C = C + (Y ?? T) T = T + tY I = I ?? R G = G X = X?? Y L = Y ?? R M = M In this model Y is national income C is consumption T is taxes I is invest- ment R is the interest rate G is government expenditure X are net exports L is money demand and M is money supply. All barred variables are exogenous. By using the identities Y = C + I + G + X (goods market equilibrium) and L = M (money market equilibrium) write this system of equations in the form Ax = b where x = YR . Solve for the equilibrium levels of Y and R. (a) What is the impact of increased lump-sum taxation (higher T) on Y ? (b) What is the impact of increased variable tax rate (higher t) on Y ? (c) What happens when government expenditures decline? (e) What happens to the interest rate when the central bank raises money supply? Question 3 A two-product rm faces the following demand and cost functions: Q1 = 10 ?? P1 + P2 Q2 = 10 + 2P1 ?? P2 C = Q21+ Q22+ 20 (a) Find output levels that satisfy the rst-order condition for maximum prot. (b) Check the second order sucient condition. (c) What is the maximal prot? Question 4 1
Given U = ln x + (1 ?? ) ln y and prices are px and py while the income is equal to I = 1. (a) Write the Lagrangian function. (b) Find the optimal levels of purchase of xand y. (c) Is the second-order sucient condition for maximum satised? 2
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