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Question: 4. (Short Run vs Long Run Input Demands) (25 points)A firm uses three factors of production: cap...

4. (Short Run vs Long Run Input Demands) (25 points)

A firm uses three factors of production: capital (K), raw
materials (M), and labor (L). The rental rate of capital is r, the
per unit cost of raw material is m, and the wage rate is w. The
production function is

Q = KML

(a) Suppose that capital is fixed in the short-run at a value
K0. Find the short-run input demand functions Ms (Q,K0) and Ls
(Q,K0) and the short-run cost function Cs (Q,K0).

(b) Use Ms (Q,K0) and Ls (Q,K0) determined in (a) to solve the
following problem minK0 rK0 +mM(Q,K0)+wL(Q,K0)

Let K0 (Q) be the solution to this problem. K0 (Q) is the amount
of capital that minimizes costs, for given output target.

(c) In the long-run, capital can be adjusted. Find the long-run
input demand functions

Kl (Q), Ml (Q) and Ll (Q) and the long-run cost function Cl
(Q).

(d) Confirm that for any Q

Kl (Q) = K0 (Q)

Ml (Q) = Ms (Q, K0(Q))

Ll (Q) = Ls (Q, K0(Q))

Cl (Q) = Cs (Q,K0 (Q))

where K0 (Q) is the function identified in (b).

Show transcribed image text 4. (Short Run vs Long Run Input Demands) (25 points) A firm uses three factors of production: capital (K), raw materials (M), and labor (L) The rental rate of capital is r, the per unit cost of raw material is m, and the wage rate is w. The production function is KML (a) Suppose that capital is fixed in the short-run at a value Ko. Find the short-run input demand functions M5 (Q, Ko) and L5 (Q, Ko) and the short-run cost function Cs (Q, Ko) (b) Use Ms (Q, Ko and La (Q, Ko) determined in (a) to solve the following problem min r Ko mM (Q, Ko) w (2, Ko) Ko Let Ko (Q) be the solution to this problem. Ko (Q) is the amount of capital that minimizes costs, for given output target. (c) In the long-run, capital can be adjusted. Find the long-run input demand functions Kl (Q), M (Q) and Li (Q) and the long-run cost function Cl (Q) (d) Confirm that for any Q Ki (Q) Ko (CR) Cl (Q) CS (Q, Ko (Q))

4. (Short Run vs Long Run Input Demands) (25 points) A firm uses three factors of production: capital (K), raw materials (M), and labor (L) The rental rate of capital is r, the per unit cost of raw material is m, and the wage rate is w. The production function is KML (a) Suppose that capital is fixed in the short-run at a value Ko. Find the short-run input demand functions M5 (Q, Ko) and L5 (Q, Ko) and the short-run cost function Cs (Q, Ko) (b) Use Ms (Q, Ko and La (Q, Ko) determined in (a) to solve the following problem min r Ko mM (Q, Ko) w (2, Ko) Ko Let Ko (Q) be the solution to this problem. Ko (Q) is the amount of capital that minimizes costs, for given output target. (c) In the long-run, capital can be adjusted. Find the long-run input demand functions Kl (Q), M (Q) and Li (Q) and the long-run cost function Cl (Q) (d) Confirm that for any Q Ki (Q) Ko (CR) Cl (Q) CS (Q, Ko (Q))