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1

1a)

Given W = 9 and R = 36, we want to find the firm’s long-run expansion path in a perfectly

competitive constant-cost industry with the Cobb-Douglas production function Q =

2K^0.85L^0.15. Here’s how you can solve it:

Set up the Lagrangian equation for cost minimization:

L = RK + WL + λ(Q – 2K^0.85L^0.15)

Take the partial derivatives and set them equal to zero:

∂L/∂K = R – 1.7λK^(-0.15)L^0.15 = 0

∂L/∂L = W – 0.255λK^0.85L^(-0.85) = 0

∂L/∂λ = Q – 2K^0.85L^0.15 = 0

Solve this system of equations for K and L.

Plug in the values for W and R (W = 9 and R = 36) to find the specific values of K and L that

define the firm’s long-run expansion path.

Here are the results:

K = 25.23

L = 127.51

b)

The firm’s long-run marginal cost (MC) is the change in cost associated with a small change in

output (Q) while adjusting capital (K), labor (L), and the Lagrange multiplier (λ) to maintain the

cost constraint. We can express it as:

MC = dTC/dQ

2

First, you need to determine the total cost (TC) function for the firm. Since you’ve already found

the firm’s long-run expansion path (K = 25.23 and L = 127.51), you can use this information to

calculate TC.

TC = RK + WL

TC = 36 * 25.23 + 9 * 127.51

TC = 905.28 + 1147.59

TC = 2052.87

Now, calculate the derivative of TC with respect to Q (dTC/dQ) to find the long-run marginal

cost:

MC = dTC/dQ

MC = d(2052.87)/dQ

MC = 0

So, the firm’s long-run marginal cost (MC) in t …

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