A firm which hires K units of capital at a price of r per unit, and L units of l
ID: 1103567 • Letter: A
Question
A firm which hires K units of capital at a price of r per unit, and L units of labor at a wage of w per unit produces an output of Q= root (K) + root (L).
a. Find the optimal choices K* and L* that minimizes the cost of rK+wL of producing the target output Q, where r, w, and Q are positive constants.
b. Compute the partial derivatives of the minium cost function C* (r, w, Q) = rK* + wL* wrt r, w, and Q.
A firm which hires K units of capital at a price of r per unit and L units of labor at a wage of w per unit produces an output of Q = VK + VL Find the optimal choices K* of capital and L* of labor that minimizes the cost rK+ wL of producing the target output Q, where r, w, and Q are positive constant. Compute the partial derivatives of the minimum cost function C*(r, w, Q) = rK* + wL* w.r.t· r, w, and Q, and give economic interpretations. a) b)Explanation / Answer
a)
The long run cost minimization problem can be solved using Lagrangian method.
The cost structure of the firm is given by
C = rK + wL
Setting Lagrangian would imply:
Minimize C = rK + wL – (K0.5 + L0.5)
Finding the partial derivatives and setting them equal to zero gives
w – 0.5L-0.5 = 0
r – 0.5K-0.5 = 0
The first two equations give
K/L = w/r
K = L(w/r)
Substitute this value in the production function
Q = K0.5 + L0.5
Q = (L(w/r))0.5 + L0.5
L* = Q2r/(w0.5 + r0.5)2
K* = L(w/r) = Q2w/(w0.5 + r0.5)2
These are the optimal choices for K and L
b) Cost function C = Q2rw/(w0.5 + r0.5)2 + Q2rw/(w0.5 + r0.5)2
C = 2Q2rw/(w0.5 + r0.5)2
dC/dQ = 2Qrw/(w0.5 + r0.5)2
dC/dr = 2Q2w [1/(w0.5 + r0.5)2 - r0.5/(w0.5 + r0.5)3]
dC/dw = 2Q2w [1/(w0.5 + r0.5)2 - w0.5/(w0.5 + r0.5)3]
For increase in quantity, marginal cost dC/dQ is increasing.
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