[Bheyburgh]

A firm has a production function, q=AL^aK^1-a, where 0<a<1. It wants to minimize cost for a given production q. The wage rate and rental rate on capital are w and r, respectively.
a. Write the Lagrangian expression for the cost minimization problem
b. Write the first-order conditions
c. Find the cost minimizing L and K

and

Consider a firm with the production function f(L,K) = L^0.5K^0.5. The wage rate and rental rate on capital are w and r, respectively.
a. Use the Lagrangian to derive the long-run cost function for this firm.
b. Suppose the government provides a subsidy of $10 per unit of capital to the fir. Rewrite the long-run cost function.

These are what I got for the answers

q1.

a) Larangian
min L = wL + rK + λ(q – AL^aK^1-a)


b) F.O.C
L_L = w – λA(a)L^a-1K^1-a = 0
L_K = r – λA(1-a)L^aK^-a = 0
L λ = (q – AL^aK^1-a) = 0


c) w/r = λA(a)L^a-1K^1-a / λA(1-a)L^aK^-a
w/r = aL^a-1K^1-a / 1-aL^aK^-a
w/r = aK / 1-aL
*sub 1-a for b*
K = (wbL/ar)

L λ = q = AL^a(wbL/ar)^b
(q = AL^a(wbL^b/ar^b)

I got stuck and dont know how to finish

q2.

a) Larangian:
L = wL^0.5 + rK^0.5 + λ[q – L^0.5K^0.5]

F.O.C:
L_L = 0.5wL^-0.5 – 0.5λL^-0.5K^0.5 = 0
L_K = 0.5rK^-0.5 – 0.5λK^-0.5L^0.5 = 0
L λ = q – L^0.5K^0.5 = 0

Combine first two conditions:
0.5wL^-0.5 – λ0.5L^-0.5K^0.5 = 0.5rK^-0.5 – λ0.5K^-0.5L^0.5
(wL^-0.5 / rK^-0.5) = (L^-0.5K^0.5 / K^-0.5L^0.5)
(wL^-0.5 / rK^-0.5) = (K / L)

Rearrange:
K = L(wL^-0.5 / rK^-0.5)
K = wL^0.5 / rK^-0.5

Substitute into third condition:
q = L^0.5(wL^0.5 / rK^-0.5)^0.5
q = (wL / rK^-0.5)^0.5
q = wL^0.5 / rK^-0.25

Solve for L:
L = (qrk^-0.25/w)²
Solve for K:
K = (w²L/qr)^-2

Cost function: C(w,r,q) = w(qrk^-0.25/w)² + r(w²L/qr)^-2 = (qrk^-0.25)² + (w²L/q)^-2




b) If the government grants a subsidy of $10 per unit of capital, the you replace all “K” with “k – 10”
[qr( k – 10)^-0.25]² + (w²L/q)^-2

[paraclete]

The question has too much detail

[Bheyburgh]

I don't understand? I basically solved it (I think) just looking to see if I went wrong anywhere