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Answer to Question #48041 in Microeconomics for Ieva

Question #48041
Suppose that the production function for iPods is Q = 20K0.5L0.5. The marginal product of labour is 10(K/L)0.5, and the marginal product of capital is 10(L/K)0.5.
Suppose that labour can be hired for $6, and capital can be hired for $9. When the firm is producing 49 units at lowest cost, what will the firm’s marginal rate of technical substitution be?

Expert's answer
Q = 20K^0.5*L^0.5, MPL = 10(K/L)^0.5, MPK = 10(L/K)^0.5.
PL = $6, PC = $9,  Q = 49 units at lowest cost.
In microeconomic theory, the Marginal Rate of Technical Substitution (MRTS) is the amount by which the quantity of one input has to be reduced when one extra unit of another input is used, so that output remains constant.
The firm’s marginal rate of technical substitution will be:
MRTS = MPL/MPK = (10(K/L)^0.5) / ((10(L/K)^0.5) = (K/L)^0.5 / (L/K)^0.5 = (K/L)^0.5*(K/L)^0.5 = K/L

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