The Economic Planning Department at International Chemicals, Inc. has used regre

Question

The Economic Planning Department at International Chemicals, Inc. has used regression analysis to estimate the firm's production function as ln Q = 3 + 0.25 ln K + 0.75 ln L

where "ln" denotes the natural logarithm of the variable.

a. Convert this production back to its original (i.e., multiplicative of Cobb-Douglas) form. (HINT: Find the antilogarithm of both sides of the equation.)

b. If the capital stock is fixed at 16, the price of labor is $200 per unit, and the price of the firm's only product, sulfuric acid, is $10 per unit, determine the rate of labor input that will maximize the firm's profit.

Explanation / Answer

The firm's production function is ln Q = 3+ 0.25 ln K + 0.75 ln L

Taking logarithm of either side, we get

LogQ = Log3 + logK0.25 + logL0.75 = log(3 * K0.25 * L0.75 )

a) i.e., Q = 3 x K0.25 x L0.75 which represents Cobb Douglas production function

b) Given physical Capital (K) = 16 , price of labour or w = $200, price of sulphuric acid (r) = $10

Then Quantity, Q = 3 X (16)1/4 x L3/4 = 6 x L3/4

i.e., L = (Q/6)4/3

The Total Cost function of the firm,

TC = wL +rK = $200L + $16 X10 = 200L +160 = 200 (Q/6 )4/3 + 160

Thus marginal cost is,

d(TC)/dQ = 200 x 4/3 (Q/6)1/3

Therefore, if a firm has to maximize its profit then

   Marginal Cost = Marginal Revenue = Average Cost = Total Cost/Quantity

i.e., 200 X 4/3(Q /6)1/3 = (200L + 160)/Q

i.e., 200 x 4/3 x Q x (Q/6)1/3 = 200L +160

i.e., 400 x 4 x (Q/6) X (Q/6)1/3 = 200L + 160

i.e., 1600 x (Q/6)4/3 = 200L +160

i.e., 1600L-200L = 160

i.e., 1400L = 160

i.e., L = 160/1400 = 4/35

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