Question

production Managers for the Pottsville Fishing Corporation estimate their production function to be TP=30L+13.5L2-L3 where TP is the number of fish processed, L is labour, the variable input. Illustrate your answer using diagrams.

1. What are the equations for MP and AP

2. At what level of input do diminishing marginal returns begin?

3. At what level of input does average product peak?

4. What is the maximum output that can be produced?

5. If the price of fish was $10 what wage rate would you be willing to pay to 8th Labour?

Answer 1

1. AP = TP/L = (30L/L) + (13.5L2/L) - (L3/L) = 30 + 13.5L - L²
MP =

2.

So, 27 - 6L = 0
So, 6L = 27
So, L = 27/6 = 4.5

So, diminishing returns begin after L = 4.5

3.

So, 2L = 13.5
So, L = 13.5/2 = 6.75

So, AP is maximum when L = 6.75

4) TP is maximum when MP = 0
So, 30 + 27L - 3L² = 0
So, -3(L² - 9L - 10) = 0
So, L² - 9L - 10 = 0
So, L² - 10L + L -10 = 0
So, L (L - 10) + 1(L - 10) = 0
So, (L+1) (L-10) = 0
So, L+1 = 0 or L-10 = 0
So, L = -1 or L = 10
As L can't be negative so TP is maximum at L = 10
Maximum TP = 30L+13.5L2-L3 = 30(10)+13.5(10)2-(10)3 = 300 + 1350 - 1000 = 650

5) L = 8
MP (at L = 8) = 30 + 27L - 3L² = 30 + 27(8) - 3(8)² = 30 + 216 - 192 = 54

Wage rate for 8th unit of labor = Price*MP = 10*(54) = $540