Author Question: This question has you determine the effect of a tax on labor on the long-run cost function. Consider ... (Read 38 times)

codyclark · **on:** Jul 1, 2018

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?

Question 2

Social Security contributions are a tax on labor.

a. True b. False

tuwy · **Reply #1 on:** Jul 1, 2018

Answer to Question 1

a. The Lagrangian is
L = wL + rK + q LK
The first-order conditions are
LL = w aK = 0
LK = r bL = 0
L = 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) = 2q(w + t)r1/2
c. The derivative is:
Ct = (qr)1/2(w + t)-1/2
The tax does not increase the costs linearly in q.

Answer to Question 2

a