Economics 2

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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
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
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
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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