4. (2 marks for each part) Consider a price-taking, profit-maximizing firm in a one-output, one-input world. Write the production function as y = h() and let p denote the price of output and w denote the input price. Assume the production function is strictly concave and differentiable and both prices are positive. (i) Describe how one would obtain the input demand function z(w,p) and output supply y(w,p), and show these functions must be homogeneous of degree zero in (w,p). (ii) Prove the profit function, II(w,p), is convex in (w,p). (iii) Set w equal to some positive real number, wo. As precisely as you can, sketch a graph of II(wo,p) against p. (iv) One implication of the convexity of II is that the main diagonal of its Hessian matrix is nonnegative. Use this fact and the envelope theorem to sketch a proof that yz(w,p) > 0. 4. (2 marks for each part) Consider a price-taking, profit-maximizing firm in a one-output, one-input world. Write the production function as y = h() and let p denote the price of output and w denote the input price. Assume the production function is strictly concave and differentiable and both prices are positive. (i) Describe how one would obtain the input demand function z(w,p) and output supply y(w,p), and show these functions must be homogeneous of degree zero in (w,p). (ii) Prove the profit function, II(w,p), is convex in (w,p). (iii) Set w equal to some positive real number, wo. As precisely as you can, sketch a graph of II(wo,p) against p. (iv) One implication of the convexity of II is that the main diagonal of its Hessian matrix is nonnegative. Use this fact and the envelope theorem to sketch a proof that yz(w,p) > 0