Problem 1 The point of this problem is to derive some properties of cost functions and how they relate to the returns to scale of the production function. Use the general Cobb Douglas function a. C. d. q=f(K,L)=KL" What restriction would you have to place on a and b for this function to be constant returns to scale? b. Suppose the condition you found in (a) is met. Show that q=MPK.K+MPL.L. Continue to assume that the condition you found in (a) holds. Show that if MPL>APL then MPK must be negative. What does this imply about what range of K and L production must take place in? Can a firm ever produce at a point where APL is increasing? Show that the result you found in (b) holds in the case of any function that has constant returns to scale (hint: start with the definition of constant returns to scale: g(tL,tk)=tg(L,K), and differentiate both sides with respect to t). Assume the restriction you found in (a) holds. What is the marginal product of labor? The marginal product of capital? Are they decreasing? f. Find contingent demand for labor and capital as functions of w,r and q, in the case where the condition you found in (a) holds. What happens to contingent demand for labor when q is multiplied by a constant m? What happens to contingent demand for capital when q is multiplied by a constant m? Can you provide intuition for this result? g. What is the cost function associated with this production function when the restriction you found in (a) is met? Show that to increase production by one unit, the additional quantities of labor and capital needed do not depend on q. i. What are the average and marginal cost functions associated with this production function? How do they depend on q? How can you provide intuition for that? How is this related to your answer to (f)? e. h