Question

In: Economics

Suppose a competitive firm’s production function is q = L 0.5K0.5 and the wage rate is...

Suppose a competitive firm’s production function is q = L 0.5K0.5 and the wage rate is w = $5 and the rental rate is r = $0.8

(a) If K is fixed at 25 units in the short-run, what is the short-run production function?

(b) What is the equation that determines how much labor the firm should hire to produce some quantity, q?

(c) Use your answer to (b) to derive the short-run total cost curve.

(d) What is the firm’s marginal cost curve? What is the firm’s average total cost curve?

(e) If the price of the firm’s product is $6, how many units should the firm choose to sell?

(f) What are its short-run profits?

Thanks for your help!

Expert Solution

q = L^0.5K^0.5

w = $5

r = $0.8

K = 25 in short run

a) Putting the value of K = 25 in the production function given above,

q = L^0.5 x (25)^0.5

q = L^0.5 x 25

q = 5L^0.5 is the answer.

b) The equation that determines how much labor the firm should hire is the total cost function of the firm.

Total cost = (wage x units of labor) + (rent x units of capital)

Total cost = wL + rK = 5L + 0.8K

c) The short run total cost curve can be found by putting the value of K = 25 in the total cost function derived in part b.

Short run total cost curve = 5L + 0.8K

Short run total cost curve = 5L + 0.8(25)

Short run total cost curve = 5L + 20 is the answer.

d) Marginal cost is the addition to the total cost when an additional unit of labor is employed. It is found out by differentiating the total cost function with respect to L. So, marginal cost = 5 is the answer.

Average total cost is the per unit total cost which is found out by dividing the the total cost function by labor.

So, Average total cost curve = (5L + 20) / L = 5 + 20/L is the answer.

Rahul Sunny answered 1 year ago

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