4.1. Prove rigorously that profit maximization implies cost minimization. 4.2. Use the Kuhn-Tucker theorem to derive conditions for cost minimiza- tion that are valid even if the optimal solution involves a boundary solution. 4.3. A firm has two plants with cost functions c (y) = y/2 and c2(y2) = 32. What is the cost function for the firm? 4.4. A firm has two plants. One plant produces output according to the production function 21x2. The other plant has a production function xx. What is the cost function for this technology? 4.5. Suppose that the firm has two possible activities to produce output. Activity a uses a units of good 1 and a2 units of good 2 to produce 1 unit of output. Activity buses b units of good 1 and b units of good 2 to produce 1 unit of output. Factors can only be used in these fixed proportions. If the factor prices are (w, w2), what are the demands for the two factors? What is the cost function for this technology? For what factor prices is the cost function not differentiable? = 4.6. A firm has two plants with cost functions c(31) = 4 and c(32) = 22. What is its cost of producing an output y? 4.7. The following table shows two observations on factor demand x1, x2, factor prices, w, w2, and output, y for a firm. Is the behavior depicted in this table consistent with cost-minimizing behavior? Obs 1 W2 x1 x2 A 100 2 1 10 20 B 110 1 2 14 10 Suppose the second factor in a Cobb-Douglas technology is restricted to operate at a level k. Then the cost-minimizing problem is min 121 + 2k such that y=xk-a Solving the constraint for x as a function of y and k gives x1 = (yk-1) Exercises 3.1. A competitive profit-maximizing firm has a profit function (w, W2)= 1(1)+(w). The price of output is normalized to be 1. (a) What do we know about the first and second derivatives of the functions (w)? (b) If (w, w) is the factor demand function for factor i, what is the sign of Oz,/Ow;? (c) Let (1, 2) be the production function that generated the profit function of this form. What can we say about the form of this production function? (Hint: look at the first-order conditions.) 3.2. Consider the technology described by y = 0 for r1 and y = Inz for >1. Calculate the profit function for this technology. 3.3. Given the production function f(1,2) = a In + a ln 22, calcu late the profit-maximizing demand and supply functions, and the profit function. For simplicity assume an interior solution. Assume that a > 0. 3.4. Given the production function f(x1, x2)=2, calculate the profit- maximizing demand and supply functions, and the profit function. Assume a>0. What restrictions must a and a2 satisfy? 3.5. Given the production function f(x1,2) = min{1,2}, calculate the profit-maximizing demand and supply functions, and the profit function. What restriction must a satisfy?