Profit Maximization for a Firm in a Competitive Market

Understanding the principles of profit maximization in a competitive market is fundamental for any business. This involves determining the optimal level of output and pricing strategies to achieve maximum profit. In this article, we delve into the mathematics behind finding the profit-maximizing level of output and associated price, using an example where the demand equation and total cost function are given.

Given Data and Equations

The problem at hand involves a firm where the price P and the quantity Q are related through the demand equation, and the total cost TC is given as a function of Q.

Step 1: Determine the Revenue Function

The relationship between the price and quantity is given by the demand equation:

P 60 - 0.25Q

Total Revenue (TR) is calculated as:

TR P * Q (60 - 0.25Q) * Q 60Q - 0.25Q^2

Step 2: Calculate the Marginal Revenue (MR)

To maximize profit, we need to find the point where Marginal Revenue (MR) equals Marginal Cost (MC). The Marginal Revenue is the derivative of the Total Revenue with respect to Q:

MR d(TR)/dQ d(60Q - 0.25Q^2)/dQ 60 - 0.5Q

Step 3: Determine the Marginal Cost (MC)

The total cost is given by:

TC Q^2 - 40Q 50

The Marginal Cost is the derivative of the Total Cost with respect to Q:

MC d(TC)/dQ 2Q - 40

Step 4: Set Marginal Revenue Equal to Marginal Cost

To find the profit-maximizing quantity, we set MR equal to MC:

MR MC

60 - 0.5Q 2Q - 40

Rearranging the equation:

60 40 2Q 0.5Q

100 2.5Q

Q 100 / 2.5 40

Step 5: Calculate the Profit-Maximizing Price

Substitute the profit-maximizing quantity back into the price equation:

P 60 - 0.25Q 60 - 0.25 * 40 60 - 10 50

Summary

The profit-maximizing level of output is 40 units, and the corresponding price is $50.

Alternative Approach

We can also approach this problem using an alternative set of given data:

For the data where Q 60 - 0.25P, the revenue function can be derived as follows:

The demand equation is rearranged as:

P 240 - 4Q

Total Revenue (TR) is:

TR P * Q (240 - 4Q) * Q 240Q - 4Q^2

Total Cost (TC) is:

TC 1000 - 40Q Q^2

Profit (PQ) is the difference between total revenue and total cost:

PQ TR - C 240Q - 4Q^2 - (1000 - 40Q Q^2) 280Q - 5Q^2 - 1000

For the second approach, the profit function is:

PQ -1.25Q^2 200Q - 500

Deriving the Marginal Revenue ( MR ) and setting it to the Marginal Cost (MC), we get:

MR -2.5Q 200 dPQ/dQ

MC 2Q - 40

Setting MR MC:

-2.5Q 200 2Q - 40

240 4.5Q

Q 240 / 4.5 53.33

P 240 - 4 * 53.33 240 - 213.33 26.67

Second Alternative Approach

For a simpler example, the original problem was given as:

Q 60 - 0.25P

P 60 - Q / 0.25

P 240 - 4Q

Total Revenue (TR) is:

TR PQ (240 - 4Q) * Q 240Q - 4Q^2

Total Cost (TC) is:

TC Q^2 - 40Q 50

Profit (PQ) is:

PQ TR - C 240Q - 4Q^2 - (Q^2 - 40Q 50) -5Q^2 280Q - 50

R'Q -10Q 280 0

Q 280 / 10 28

P 240 - 4 * 28 240 - 112 128

Conclusion

This detailed analysis shows how to find the profit-maximizing level of output and associated prices by setting marginal revenue equal to marginal cost. In the first example, the profit-maximizing level of output is 40 units, and the price is $50. In the second example, the output is adjusted to 28 units, with a price of $128. These examples highlight the importance of maintaining balance between total revenue and total cost to achieve the ultimate goal of profit maximization in a competitive market.