Question #50824

Use the production function: Q=10K^0.5L^0.6, suppose that the wage rate is $28, the price of capital also is $28, and the firm currently is producing 30.3 units of output per period using four units of capital and two units of labor. Is this an efficient resource combination? Explain. What would be a more efficient (not necessarily the best) combination? Why? (Hint: Compare the marginal products of capital and labor at the initial input combination.)

Expert's answer

Q=10K^0.5L^0.6, w = $28, k = $28, Q = 30.3 units, K = 4 units, L = 2 units.

MPL = Q'(L) = 6K^0.5/L^0.4

MPk = Q'(K) = 5L^0.6/K^0.5

Marginal rate of technical substitution MRTS = MPL/MPk =

(6K^0.5/L^0.4)/( 5L^0.6/K^0.5) = 1.2*4/2 = 2.4, so we need 2.4 units of K

for every unit of L.

So, K = 4 units and L = 2 units is not an efficient resource

combination. A more efficient combination will be 4.8 units of K and 2

units of L.

MPL = Q'(L) = 6K^0.5/L^0.4

MPk = Q'(K) = 5L^0.6/K^0.5

Marginal rate of technical substitution MRTS = MPL/MPk =

(6K^0.5/L^0.4)/( 5L^0.6/K^0.5) = 1.2*4/2 = 2.4, so we need 2.4 units of K

for every unit of L.

So, K = 4 units and L = 2 units is not an efficient resource

combination. A more efficient combination will be 4.8 units of K and 2

units of L.

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