Questions 5-7

5. (30 pts) Use the following production function: 9 = L''K's, where L is labor, K is capital, to derive a. contingent demand functions for capital and labor and the cost function. Be sure to show your work. SOC. Discuss whether the FOC and the SOC are satisfied and what it means to satisfy the FOC and the b . Do your contingent demand functions for capital and labor portray the expected relationships the demand for the input and input prices? Explain using calculus. C. Does your cost function portray the expected relationship between cost and production? Between cost and input prices? Explain using calculus. d. Use your cost function to derive the marginal cost and average variable cost functions. e. Do these cost functions have the degree of homogeneity with respect to input prices that you expected? Explain. f. Use your cost function to derive a supply function. Does the supply function have the expected relationship between supply and output and input prices? Explain. 6. (12 pts) The handmade snuffbox industry is composed of 100 identical firms, each having short-run total cost given by, SRC(Q) = 0.502 + 100 +5, where 2 is the output of snuffboxes per day for a representative firm. a. Derive the short-run supply function for the representative firm. b. Derive the short-run supply function for the market. c. Solve for market equilibrium price/quantity if the market demand for total snuffboxes is, QD = 1, 100-50P . d. What is the representative firm's profits? What do you predict will happen in the long run? Explain. 7. (10 pts) Suppose there are only three goods ( X,, X, and X, ) in an economy and that the excess demand functions for X, and X, are given by: ED, = -3 P2/ P, + 2 P, / P, -1 and ED, = 4 P2 / P, - 2 P, / P, - 2. a. Show that these functions are homogeneous of degree zero in P,, P, and Py . b. Use Walras' law to show that if ED, = ED, = 0, ED, also must be 0. c. Solve this system of equations for the equilibrium relative prices P, / P, and P, / P, . What is the equilibrium value for P, / P2