Question

13.1 Homework. Unanswered A firm's production function is x = k0.220.6. We have p=100,w=18 and r=18. What is long-run firm output when the government imposes a tax of 11.1% on the market price that the firm has to pay (An an intuition exercise, compare this to similar exercises in last week's homework). Enter a number only, round to two decimal places. If the answer is o enter 999. If the answer is a range (any range) e.g. (0,2] enter 111. Numeric Answer:

Answer 1

Solution:

Firm's production function: X = k^0.2*l^0.6

p = 100, r = 18, w = 18

With a tax on market price, p, that firm has to pay, the price that firm receives reduces by factor of this tax. Thus, price for firm is p = 100*(1 - 11.1%) = $88.9

Funding long run output using the optimality condition (or profit maximization or cost minimization) condition of martial rate of technical substitution (MRTS) = w/r

MRTS = marginal product of labor/marginal product of capital

MPL = dX/dl

MPL = 0.6*k^0.2*l^0.6-1 = 0.6*k^0.2l^-0.4

And MPK = dX/dk

MPK = 0.2*k^0.2-1*l^0.6 = 0.2*l^0.6k^-0.8

So, MRTS = (0.6*k^0.2*l^-0.4)/(0.2*l^0.6*k^-0.8)

MRTS = 3*k/l

Then, with MRTS = w/r

3k/l = 18/18

So, 3k/l = 1 or optimality condition is: 3k = l

Then, output function gives:

X = k^0.2*(3*k)^0.6 = 3^0.6*k^0.8

Or k = (X^1.25/3^0.75)

And so, l = 3*(X^1.25/3^0.75) = 3^0.25*X^1.25

So, total cost = w*l + r*k

TC = 18*3^0.25*X^1.25 + 18*X^1.25/3^0.75

TC = X^1.25*72/3^0.75 = 31.586*X^1.25

So, marginal cost = dTC/dX = 1.25*31.586*X^1.25-1 = 39.482*X^0.25

Optimal output level is where marginal cost = price

So, 39.482*X^0.25 = 88.9

X = (88.9/39.482)^1/0.25 = 25.7 approx

Long run firm output is 26 units approximately.