Question

Suppose that you have a production function Y(L)=540L-L^3. Assume that the production of this good occurs in a perfectly competitive market and can receive a price of $2/unit. Additionally, assume that the cost of labor is a function of labor and is C(L)=9L^2 + 48. Given this information, what is this firm’s total revenue, total cost and total profit functions? What would be the optimal number of people (L) for this producer to hire given this information? If the project were to operate until breakeven, how many people would be hired?

Answer 1

Y = 540L - L³

C(L) = 9L² + 48

(a)

(i) Total revenue (TR) = Price x Y = 2 x (540L - L³) = 1,080L - 2L³

(ii) Total cost (TC) = C(L) = 9L² + 48

(iii) Profit (Z) = TR - TC = 1,080L - 2L³ - 9L² - 48

(b) Profit is maximized when dZ/dL = 0

1,080 - 6L² - 18L = 0

6L² + 18L - 1,080 = 0

L² + 3L - 180 = 0

L² + 15L - 12L - 180 = 0

L(L + 15) - 12(L + 15) = 0

(L + 15) (L - 12) = 0

L = 12 (Since L = - 15 is inadmissible as L >= 0)

(c) In breakeven, TR = TC

1,080L - 2L³ = 9L² + 48

2L³ + 9L² - 1,080L + 48 = 0

Solving this polynomial function using online solver,

L = 21 [Rejecting the negative root of L = -26 and the third root of 0.04 (being too small)].