Question

A firm produces its product using only labor. Its production function is Q​ = 20Lminus−Upper L squaredL2​, where Q is the number of units of output produced and L is the number of labor hours used. The firm purchases labor in a competitive labor market at the going wage rate of w​ = ​$1010 per hour. The firm sells its output in a competitive market at the market price of P​ = ​$44. To maximize​ profit, the firm should use_________ nothing hours of labor and produce_____ nothing units of output. ​(Enter your responses rounded to two decimal​ places.)

Answer 1

Answer:

The firm should use 10 hours of labor (L) and produce 10 units of output (Q).

Explanation:

In economics,  Total revenue (TR) is the multiplication of the number of units of output produced (i.e. Q) and the market price (i.e. P). Since  Q = 20L - L^2 and P = $44, therefore,

TR = Q*P  

    =  (20L - L^2)44

TR =  880 - 44L^2 ........................................... (1)

Total cost (TC) is the multiplication of the number of units of output produced (i.e. Q) and the wage rate (i.e. w). Since  Q = 20L - L^2 and w = $1010, therefore,

TC = Q*w

     =  (20L - L^2)1010

TC = 20200 - 1010L^2 ....................................... (2)

In economics,  is Profit (Pr) TR minus TC, therefore:

Pr = TR - TC

Pr =  (880 - 44L^2) - (20200 - 1010L^2)

    =  880 - 44L^2 - 20200 + 1010L^2  

    =  1010L^2 - 44L^2 - 20200 + 880

Pr = 966L^2 - 19320 ....................................... (3)

In economics, profit is maximized when equation (3) is differentiated with respect to L and equated to zero as follows:

dPr/dL = 2(966)L - 19320 = 0

1932L = 1932 0

L = 19320 /1932

L = 10

To get Q, we substitute L = 1 into Q = 2L - L^2 as follows:

Q = 2(10) - (10)^2

   = 20 - 10

Q = 10

Since L = 10 and Q = 10 when the profit is maximized, the firm should use 10 hours of labor (L) and produce 10 units of output (Q).