1. A firm produces @ = J() units of a commodity using _ units of labour. Assume f,(L) > 0 and fir(L) 0. b.Let f(L) = VL. Find an explicit expression for * as a function of p and w.Also find the partial derivatives of * with respect to p and w and verify the signs obtained are as expected. (10 marks) 2. Find the degree of homogeneity of the following functions: a. f(x,y) = 25 + x392 b.x(p, r) = Ap_1.52.08 c.f(x, y ) = 12 ty2 d.F(K, L) = A(aK-P + bL-P)-1/p (5 marks) 3. a. A firm produces two different kinds A and B of a commodity. The daily cost of producing a units of A and y units of B is C(x,y) = 0.04x2 + 0.0lay + 0.0ly2 + 4x + 29 + 500 A sells for $15 per unit and B sells for $9 per unit. Find the daily production levels of x and y that maximize profit per day b.A firm's production function is given by F(K, L) = K 2 LI . Each unit of K costs $1.2 and each unit of _ costs $ 0.6. Each unit of output sells at $12. Find the optimal level of K and _ that maximize profit. (10 marks) 4 Complete the following implications: a. z = (224 + 2)5 - az = b.F(K, L) = (Ka + La)1/0 _, KF* + LF) = c.g(t, w) = 3t + wt2 - awat d.g (t 1, t2, t3) = (th + t; + +3)2 _. 94(t1, t2, t3) = (5 marks) 5 Find the partial elasticities of z w.r.t. x and y a. z = x3y-4 b.z = In(x2 + y2) c.z = erty (Hint: The partial elasticity of z w.r.t. ~ is given by El z = z z (5 marks) 6 a. Prove that if z = (ax + by)2, then xzz + yzy = 2z b.Let z = , In(x2 + y2). Show that ozz + 92z dy ? = 0 (5 marks) 7 You are given the following Cobb-Douglas utility function: u(x, y) = Axay and the budget constraint px + qy = m. (A, a, b, p, q, m > 0) a. Solve the following maximization problem using the Lagrange multiplier method: max Axayb s.t . px + qy = m b. Give an interpretation of the Lagrange multiplier A. c.Show that the demand functions are homogeneous of degree zero. (15 marks)