Question #63042

CARICOM Products Limited production function is lnQ = 0.63 + 0.43lnK + 0.56lnL. Given that price of labour (L) is $20 and the price of capital (K) is $33

(i) What is the optimal mix?

(ii) What is the firm’s output elasticity and returns to scale? Explain.

(i) What is the optimal mix?

(ii) What is the firm’s output elasticity and returns to scale? Explain.

Expert's answer

lnQ = 0.63 + 0.43lnK + 0.56lnL, PL = $20, PK = $33.

(i) If Q = e^0.63 x K^0.43 x L^0.56. This is a Cobb-Douglas Production Function.

The optimal mix is in the point, for which MPK/PK = MPL/PL, so:

MPK = Q'(K) =e^0.63*0.43/K^0.57

MPL = Q'(L) = e^0.63*0.56/L^0.44

0.43/(e^0.63*0.43/K^0.57) = 0.56/(e^0.63*0.56/L^0.44)

e^0.63*0.43/K^0.57*0.56 = 0.43*e^0.63*0.56/L^0.44

K^0.57 = L^0.44

K = L^0.77

ii) Output elasticity of capital is 0.43, while output elasticity of labor is 0.56.

0.43 + 0.56 = 0.99 which is less than one. So, the firm is experiencing decreasing returns to scale.

(i) If Q = e^0.63 x K^0.43 x L^0.56. This is a Cobb-Douglas Production Function.

The optimal mix is in the point, for which MPK/PK = MPL/PL, so:

MPK = Q'(K) =e^0.63*0.43/K^0.57

MPL = Q'(L) = e^0.63*0.56/L^0.44

0.43/(e^0.63*0.43/K^0.57) = 0.56/(e^0.63*0.56/L^0.44)

e^0.63*0.43/K^0.57*0.56 = 0.43*e^0.63*0.56/L^0.44

K^0.57 = L^0.44

K = L^0.77

ii) Output elasticity of capital is 0.43, while output elasticity of labor is 0.56.

0.43 + 0.56 = 0.99 which is less than one. So, the firm is experiencing decreasing returns to scale.

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