Question

Question 4

A firm has the short-run production function as follows:

Q = L + 15L² – 0.5L³,

where Q = total products per period and L = number of workers employed per period.

4.1) Derive the marginal product of labor (MP_L). At what number of workers (L) does the law of diminishing returns begin?

• MP_L = f(L) = __________________________________
• Law of diminishing return begins when L =   ___________ workers.

4.2) Derive the average product of labor (AP_L). Find the number of workers (L) that maximizes the average product of labor.

• AP_L = f(L) = __________________________________
• When AP_L is at maximum point L =   ___________ workers.

4.3)  Determine the boundaries (ranges of number of workers) for the three stages of production process. (Hint: you might use this    to solve x in a quadratic equation ax² + bx + c = 0)

• Stage I:    _____ < L £  _____
• Stage II:   _____ < L £ _____
• Stage III: L > __________

4.4) Suppose that the price of a product is $2 per unit. How many workers will the firm hire to maximize its profit if the wage rate is $100 per period?

(Hint: you might use this    to solve x in a quadratic equation ax² + bx + c = 0)

• The firm should hire ___________ workers to get maximum profit.

Answer 1

4.1) MP_L = dQ/dL ( Marginal Product of Labour is the change in output that results from employing an added unit of labor )

dQ/dL = d(L + 15L² - 0.5L³) / dL (first order differentiation)

MP_L = f(L) = 1 + 30L - 1.5L²

Law of diminishing marginal return begins when MP_L = 0

Therefore, law of diminishing marginal returns begin at f(L) = 1 + 30L - 1.5L² = 0

By solving the quadratic equation we get L = 20 workers (approximately)

4.2) AP_L = Q/L = (L + 15L² - 0.5L³ )/ L

AP_L = f(L) = 1 + 15L - 0.5L²

AP_L reaches maximum when AP_L = MP_L

1 + 15L - 0.5L² = 1 + 30L - 1.5L²

L² - 15L = 0

L(L-15) = 0

Therefore, L=15

When AP_L is at maximum point L = 15 workers

4.3) The three stages of the production will be

(1) When MP_L is increasing at an increasing rate ( MP_L > AP_L )

(2) When MP_L is increasing at a decreasing rate ( MP_L < AP_L )

(3) When MP_L is becomes zero and then negative with further increase in input ( MP_L < = 0)

Stage 1:

MP_L > AP_L

1 + 30L - 1.5L² > 1 + 15L - 0.5L²

15L > L²

L² - 15L < 0

L(L-15) < 0

Therefore, 0 < L < 15

Stage 2:

MP_L < AP_L && MP_L > 0

1 + 30L - 1.5L² < 1 + 15L - 0.5L²

15L < L²

L² - 15 > 0

L(L-15) > 0

Therefore, L (-infinity,0) (15, infinity)

but as we know L cannot be negative and MP_L needs to be positive thus we will only take 15< L< 20

Stage 3:

MP_L < = 0

1 + 30L - 1.5L² <= 0

Solvind this equation we will get L >= 20

Stage I: 0 < L < 15

Stage II: 15 < L < 20

Stage III: L > 20

4.4) Profit maximization happens when MC (Marginal Cost) is minimum.

and from the graph we can that MC is minimum when MP_L is at max that is when dQ/dL = 0

dQ/dL = d(L + 15L² - 0.5L³) / dL = 0

1 + 30L - 1.5L² = 0

By solving the quadratic equation we get L = 20 workers (approximately)

The firm should hire 20 workers to get maximum profit.