Question

1. For the following total revenue and total cost functions of a firm:
   TR = 802.5Q – 10Q²
   TC = (2/3)Q³ -30Q² + 672Q +4000
   

1. Determine the level of output at which the firm maximizes total profit
2. Calculate the profit
   (20 points)

• Reference problem on pp. 104-106 and graphs on p. 77.

1. Solution writes a total profit equation (TR equation minus the TC equation and simplify) and takes its derivative to get a marginal profit equation.
2. If you set the marginal profit or derivative equation equal to zero you find the points where the rate of change is zero; i.e., the peak or trough (see diagram of profit function in text).
3. To solve it you will probably want to use the Quadratic Formula. If you are unfamiliar with the Quadratic see http://www.purplemath.com/modules/quadform.htm
4. The quadratic formula gives two values of Q for which the marginal profit is zero. In the text example, they take the second derivative (the derivative of the marginal profit function) and plug in each quantity. If the 2^nd derivative is negative you have reached a peak, if negative, a trough. You can take the 2^nd derivative if you prefer but for the functions you will encounter in this course, the answer is ALWAYS THE LARGER ONE. Note that on the total profit graph (p. 77) there is both a peak and a trough. The peak occurs at the larger quantity.

Answer 1

Solution:

We first find the profit function

Profit = total revenue - total cost

Given the Total revenue (TR) and total cost (TC) functions, we have profit function as follows:

Profit, P = (802.5*Q - 10*Q²) - ((2/3)*Q³ - 30*Q² + 672*Q + 4000)

P = -(2/3)*Q³ + 20*Q² + 130.5*Q - 4000

(a) To find the level of output which maximizes the total profit, we will solve the first order condition (FOC): = 0

So, the marginal profit, = 3*(-2/3)*Q^3-1 + 2*20*Q^2-1 + 130.5*Q^1-1 + 0

= -2*Q² + 40*Q + 130.5

So, using FOC, we have, -2*Q² + 40*Q + 130.5 = 0

Solving this quadratic equation, we get: Q = [-(40) (+/-) [(40)² - 4(-2)(130.5)]^1/2]/(2*(-2))

Q = [-40 (+/-) 51.42]/(-4)

So, Q = (- 40 + 51.42)/(-4) = -2.855 OR Q = (- 40 - 51.42)/(-4) = 22.855

Since, Q is quantity, it cannot be negative, so rejecting the negative value, we have final answer as Q = 22.855 units

(b) Profit at Q = 22.855, by substituting in the profit function, we have

P = -(2/3)*Q³ + 20*Q² + 130.5*Q - 4000

P = -(2/3)*(22.855)³ + 20*(22.855)² + 130.5*(22.855) - 4000

P = -7958.89 + 10447.02 + 2982.58 - 4000

Profit = $1,470.71 (approx)