Q = 39,000 – 500P

AVC = 30 + 0.005Q
Q is quantity demanded and produced, and P is price of the product. Total fixed cost is $50,000. Assume these functions are based on monthly demand and production.
A. What are the inverse demand and the MR revenue equations?

B. What is the profit maximizing output of this firm?

C.What is the price at the profit maximizing output?

d. What is the firm’s total profit?

A. To find the inverse demand equation, we need to solve the original demand equation for P (price). Let's start with the given demand equation:

Q = 39,000 – 500P

Rearranging the equation by isolating P, we get:

500P = 39,000 - Q

Divide both sides by 500:

P = (39,000 - Q) / 500

So, the inverse demand equation is:

P = 78 - 0.002Q

To find the marginal revenue (MR) equation, we differentiate the inverse demand equation with respect to Q. Since the inverse demand equation is linear, the slope remains constant, hence the derivative is a constant. The derivative of a constant is zero. Therefore, the MR equation is:

MR = 78 - 0.002Q

B. To find the profit-maximizing output, we need to set marginal cost (MC) equal to marginal revenue (MR) and solve for Q. However, we are given the average variable cost (AVC) equation. To find MC, we need to find the derivative of AVC with respect to Q.

AVC = 30 + 0.005Q

Differentiating AVC with respect to Q, we get:

MC = d(AVC)/dQ = 0.005

Set MC equal to MR:

0.005 = 78 - 0.002Q

Rearrange the equation to solve for Q:

0.002Q = 78 - 0.005

0.002Q = 77.995

Q = 77,995 / 0.002

Q = 38,997.5

Since the firm cannot produce fractional quantities, the profit-maximizing output is 38,997 units (rounded down to the nearest whole number).

C. To find the price at the profit-maximizing output, substitute the found Q value back into the inverse demand equation:

P = 78 - 0.002Q

P = 78 - 0.002 * 38,997

P = 78 - 77.994

P = 0.006

So, the price at the profit-maximizing output is 0.006 (rounded to three decimal places).

D. To find the firm's total profit, we need to calculate the total cost and total revenue.

Total Fixed Cost (TFC) = $50,000 (given)

Total Variable Cost (TVC) = AVC * Q

TVC = (30 + 0.005Q) * Q

TVC = 30Q + 0.005Q^2

Total Cost (TC) = TFC + TVC

TC = $50,000 + (30Q + 0.005Q^2)

Total Revenue (TR) = P * Q

TR = (0.006) * Q

Total Profit (TP) = TR - TC

TP = (0.006 * Q) - (50,000 + 30Q + 0.005Q^2)

Substitute the value of Q (38,997) into the profit equation and calculate to find the firm's total profit.