1. A firm's cost minimization problem is given by min w.x subject to x e V(y) where x is the vector of factor inputs, w is the price vector of the factors, and V(y) is the input requirement set for a given output y. The optimised value function of the problem is then the cost function c(w, y). (a) Prove that the cost function is concave in w. Illustrate this graphically. (b) State the Shephard's Lemma. Using this lemma and the concavity of the cost function, what predictions can you make about the properties of the price derivatives matrix of the factor input demands 2. (a) A Leontif technology production function is f(1, 12) = min[ax1, ax2] where r, and r2 are the two inputs and a, b > 0 are parameters. Define the corresponding input requirement set and isoquant. Is this technology monotonic? (b) A competitive firm produces a single output y according to the production function y = 40z - 2" where z is the single input. The price of y is denoted by p and the price of z is denoted by w. It is necessary that z 2 0. (a) What is the first-order condition for profit maximization if > > 0? Check the second-order condition. (b) For what values of w and p will the profit maximizing z be zero? (c) For what values of w and p will the profit maximising z be 20? d) What are the output supply and input demand functions? (e) What is the profit function (i.e. the value function of the profit maximization program)? (f) What is the derivative of the profit function with respect to w? 3. Consider a perfectly competitive market. Each firm has a cost function given by c(q) = 8+292 where q is the firm's output. Any firm which stops production bears no cost. The inverse demand function is Q = 20 - p.(a) Suppose the number of firms is fixed at 4. Find the equilibrium price and quantity. (b) Now, suppose there is free entry and exit in the market. New entrants will have the same cost function as the incumbents. What is the equilibrium price, quantity and number of firms in the long run? (c) The government imposes a sales tax of 2 per unit. Find the effect on market price, quantity and number of firms in the long-run. 4. Consider the CES production function with two inputs: f (L, K) = [aLP + (1 - a)Ke]# . (a) What is the returns to scale for this production function? (b) Define the elasticity of substitution as follows O = d In (K/L) dIn (fL/ fk) Calculate o. (c) Show that for special values of the parameter p, the CES production function re- duces to Cobb-Douglas, linear and Leontief functions. (d) Derive the cost function