(a) Solve for the firm's short-run cost function. [1 marks] (b) Solve for the firm's long-run cost function. [4 marks] (c) Draw a graph of the firm's short-run and long-run average cost curves. (1 marks] Question 5 Show that if a production function exhibit constant returns to scale, then the cost function can be written as c(w,y) = y . c(w, 1) Explain. [8 marks] Question 6 Consider the cost function in the case of two inputs, c(w1, w2, y). (a) Show that ac(w1, (2, y) dy where A is the Lagrange multiplier from the cost-minimization problem. [5 marks] [Hint: You will need to use the first order conditions from the cost-minimization problem along with the derivative of the firm's constraint at its optimal choice with respect to y.] (b) Hence, show that at the firm's optimal input choices, the clasticity of scale, e(x) = af(tx) _ AC(@ 5 marks can be re-written as e = MC() . In doing this assume all of the firm's costs are variable (i.c., no fixed set-up costs).] (c) What does this imply about the relationship between returns to scale in production and economics of scale in costs? [5 marks]