Question #57137

a) Derive the expansion path for a firm with the production function Q=10K^0.6 L^0.4 and input prices r = 10 and w = 5.
b) The firm is currently using fifty units of capital and forty units of labor. Is this an efficient input combination?

Expert's answer

a) Derive the expansion path for a firm with the production function Q=10K^0.6 L^0.4 and input prices r = 10 and w = 5.

Expansion path is a curve in a graph with quantities of two inputs, typically capital and labor, plotted on the axes. in our case the expansion path is the same as the production function Q = 10K^0.6 L^0.4.

b) The firm is currently using fifty units of capital and forty units of labor. This input combination is efficient, if the total output Q is maximized, so Q = 10K^0.6 L^0.4 = max, Q' = 0, 6L^0.4/K^0.4 = 0, 4K^0.6/L^0.6.

Expansion path is a curve in a graph with quantities of two inputs, typically capital and labor, plotted on the axes. in our case the expansion path is the same as the production function Q = 10K^0.6 L^0.4.

b) The firm is currently using fifty units of capital and forty units of labor. This input combination is efficient, if the total output Q is maximized, so Q = 10K^0.6 L^0.4 = max, Q' = 0, 6L^0.4/K^0.4 = 0, 4K^0.6/L^0.6.

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