Solution:
Firm's production function: X = k0.2*l0.6
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*k0.2*l0.6-1 = 0.6*k0.2l-0.4
And MPK = dX/dk
MPK = 0.2*k0.2-1*l0.6 = 0.2*l0.6k-0.8
So, MRTS = (0.6*k0.2*l-0.4)/(0.2*l0.6*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 = k0.2*(3*k)0.6 = 30.6*k0.8
Or k = (X1.25/30.75)
And so, l = 3*(X1.25/30.75) = 30.25*X1.25
So, total cost = w*l + r*k
TC = 18*30.25*X1.25 + 18*X1.25/30.75
TC = X1.25*72/30.75 = 31.586*X1.25
So, marginal cost = dTC/dX = 1.25*31.586*X1.25-1 = 39.482*X0.25
Optimal output level is where marginal cost = price
So, 39.482*X0.25 = 88.9
X = (88.9/39.482)1/0.25 = 25.7 approx
Long run firm output is 26 units approximately.
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