Answer to Question #245847 in Microeconomics for Mendie

Question #245847
Coosider the production function characterizing garri plant as:

Q= 20L + 60k - L^2 - K^2

If total outlay for both capital, K =N50, and a Labour, L= N20, is N4,600. Find:

a. The optimal quantity of K* and L* to be employed.

b. Calculate the maximum possible output of garri that the plant can produce.
1
Expert's answer
2021-10-03T18:18:20-0400

The problem is to maximize the function "Q = 12K^{0.4}L^{0.4}" subject to the budget constraint

"40K + 5L = 800"

The theory of the firm tells us that a firm is optimally allocating a fixed budget if the last £1 spent on each input adds the same amount to output, i.e. marginal product over price should be equal for all inputs. This optimization condition can be written as

"\\frac{MP_K}{P_K} = \\frac{MP_L}{P_L}"

The marginal products can be determined by partial differentiation:

"MP_K = \\frac{dQ}{dK} = 4.8K^{-0.6}L^{0.4} \\\\\n\nMP_L = \\frac{dQ}{dL} = 4.8K^{0.4}L^{-0.6} \\\\\n\n4.8K^{-0.6}L^{0.4} = 4.8K^{0.4}L^{-0.6}"

Dividing both sides by 4.8 and multiplying by 40 gives:

"K^{-0.6}L^{0.4}=8K^{0.4}L^{-0.6}"

Multiplying both sides by "K^{0.6}L^{0.6}" gives:

"L = 8K \\\\\n\n40K + 5 \\times 8K = 800 \\\\\n\n40K + 40K = 800 \\\\\n\n80K=800 \\\\\n\nK = 10 \\\\\n\nL = 8 \\times 10 = 80"



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