A firm has a production function of Q=4K^2L^2 (4K squared multiplied by L square

Question

A firm has a production function of Q=4K^2L^2 (4K squared multiplied by L squared). Wage rates (W)=$12/hour and Rental Rate (R)=$20/hour.
1a.) In the short run K=5. What is the total cost of producing 400 units?
1b.) In the long run, what is the minimum cost to produce 400 units? A firm has a production function of Q=4K^2L^2 (4K squared multiplied by L squared). Wage rates (W)=$12/hour and Rental Rate (R)=$20/hour.
1a.) In the short run K=5. What is the total cost of producing 400 units?
1b.) In the long run, what is the minimum cost to produce 400 units?
1a.) In the short run K=5. What is the total cost of producing 400 units?
1b.) In the long run, what is the minimum cost to produce 400 units?

Explanation / Answer

Using the tangency condition, initially K/L= w/r

K/L = 12/20 = 0.6

K= 0.6*L

1.a) At k=5, L= 5/0.6 = 8.33

Cost of producing 400 units, 400 = 4*5^2 *8.33^2 = 6941.67

1. b) Long run,

dq/dK = 8*K*L^2 , dq/dL= 8*K^2*L

Equating the above to 0

d2Q/ dk*dl = 16*K*L >0 hence minimum

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