[Solved] (Consider again Worked-Out Problem 8.6 (page 271) but assume that Hannah and Sam are initially remodeling 200 square feet per week. What are their short-run and long-run cost functions if

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(Consider again Worked-Out Problem 8.6 (page 271) but assume 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
The Problem Consider again Hannah and Sam’s remodeling business described in Worked-Out Problems 8.2 and 8.3 (pages 254 and 256). They have a Cobb-Douglas production function Q = F ( L , K ) = 10 L 0.5 K 0.5 , 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 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 L that solves the formula
which means that L = ( Q 2 /2,000). So, their short-run cost function is
Or equivalently,
In contrast, the solution to Worked-Out Problem 8.3 tells us that Hannah and Sam’s long-run cost function is
C LR ( Q ) = 100 Q
Observe, though, that their short-run and long-run costs are equal when Q = 100
, just as our discussion indicated they must be.
Worked-Out Problem 8.2
Worked-Out Problem 8.3


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