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.
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.