Explanation

Problem 18 Consider again Worked-Out Problem 8.6 but assume that a unit of capital costs $1,000 per week and that Hannah and Sam are initially remodeling 200 square feet per week. What are their short-run and long-run cost functions if capital is fixed in the short run, but variable in the long run? Worked-Out Problem 8.6 Consider again Hannah and Sam's remodeling business described in Worked-Out Problems 8.2 and 8.3. They have a Cobb-Douglas production function Q = F(L,K) = 1010505, and face a wage rate of $1,000 per week, and a capital price of $250 per week per unit. Suppose that they are initially remodeling 100 square feet per week using the least-cost input combination for producing that output level. What are their short-run and long-run cost functions if their capital is fixed in the short run, but variable in the long run? The solution to Worked-Out Problem 8.2 tells us that they initially use 20 units of capital. So, if their capital is fixed at 20 units, to remodel Q square feet Hannah and Sam need the amount of labor _ that solves the formula 10V ZV 20 = Q which means that L = (Q2/2,000). So, their short-run cost function is C!(Q) = (250)(20) + (1,000) 02 2,000 or equivalently, CSR(Q) = 5,000 + In contrast, the solution to Worked-Out Problem 8.3 tells us that Hannah and Sam's long-run cost function is CLR(@) = 1000 Observe, though, that their short-run and long-run costs are equal when Q = 100 [since Cox (100) = Cox( 100) = 10,000], just as our discussion indicated they must be