This question has you determine the effect of a tax on labor on the lo

This question has you determine the effect of a tax on labor on the long-run cost function. Consider a firm with the production function f(L,K) = LK. The wage rate and rental rate on capital are w and r, respectively. a.

Using the Lagrangian, derive the long-run cost function for this firm. b. Suppose the government taxes labor at by an amount t per unit of labor. Rewrite the long-run cost function including the tax. Hint: the effective wage rate is now w + t. c. Compute the marginal effect of the tax on the long-run cost function. To do so, compute the partial derivative of the cost function with respect to t. Does an increase in the tax increase the cost linearly?

ANSWER

a. The Lagrangian is
L = wL + rK + λ[q – LK]
The first-order conditions are
LL = w – λaK = 0
LK = r – λbL = 0
Lλ = q – LK
Combining the first two conditions:
w/r = K/L
Rearranging yields
K = wL/r
Substituting into the third condition:
q = wL2/r
Solving for L:
L = (qr/w)1/2
Solving for K:
K = (qw/r)1/2
The cost function is
C(w,r,q) = w(qr/w)1/2 + r(qw/r)1/2 = 2(qwr)1/2
b. Replace “w” with “w + t” in the cost function above.
C(w + t, r, q) = 2[q(w + t)r]1/2
c. The derivative is:
Ct = (qr)1/2(w + t)-1/2
The tax does not increase the costs linearly in q.