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.)

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).

answer; balance sheets;