Production Function problem set solutions PDF

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Production Function problem set solutions.
Derive short-run average and marginal costs....

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Long Exercise Solutions Ramon manages a branch office of a large financial services firm. He uses computers (capital, K) and people (labour, L) to produce consulting advice Q, according to the production function: Q=K * L. Employing people costs the wage rate w = 1, while renting computers costs the rental rate r. Suppose computers cost twice what people do, i.e., r=2w=2. For now, the number of computers in the branch is fixed at K = k 0 . A. How much labour does Ramon employ if he needs to produce output Q? Show that short-run total cost is SRTC(Q) = 2 k 0 + Q/ k 0 . Answer Given the assumption that capital is constant at K=k0, we have that output is Q=K*L= k 0*L. From this, for a given level of output Q, we need to employ L=Q/ k 0 . Short-run total cost: notice that SRTC = wL + rK = 1*(Q/ k 0 )+2* k 0 = 2 k 0 + Q/ k 0 . B. Derive short-run average and marginal costs. How do short-run average and marginal cost vary with the output? Draw a diagram. Answer The short-run (SR) average cost is ATC = SRTC / Q = (2 k 0 + Q/ k 0 ) / Q = 2 k 0 / Q + 1/ k 0 . The marginal cost is then MC = 1/ k 0 . Notice that SR average cost declines with output but SR marginal cost is constant. Here is a diagram illustrating the average and marginal cost curves.

C. Assume that Ramon decides to optimally choose capital. Derive the long-run total cost function, average cost function, and marginal cost function. Answer The long-run total cost is given by the minimum cost to provide a specific level of output, by optimally choosing both inputs. In our case, we need to choose k0 to minimise the short-run cost for each level of output. To minimise the SRTC, we take the derivative with respect to k 0 and equalise it to 0: ∂SRT C ∂k0

Which implies k 0 =

√

Q 2

=0 ⇒ 2−

Q (k 0)

2

= 0.

. We can substitute this into the SRTC to get the

2 √2 Q + Q √2 = 2√2Q . The long-run average cost is then LRTC/Q = Q √Q and the long-run marginal cost is LRMC = Q2 .

LRT C = 2

√

Q 2

√...