Monopolist Profit Maximization: A Case Study Using Revenue and Cost Functions

In this article, we will explore the profit maximization for a monopolist using marginal revenue (MR) and marginal cost (MC) functions. By applying the principles of calculus and algebra, we will derive the profit-maximizing output, the equilibrium price, and the minimum profit. This analysis is essential for understanding how monopolistic firms determine their optimal production levels and prices to maximize profit.

An Overview of Marginal Revenue and Marginal Cost Functions

The given monopolist has the following functions:

MR 30 - 2Q, where Q is the output.

MC 3Q^2 - 30Q 10, where Q is the output.

To determine the profit-maximizing output, we need to equate the marginal revenue (MR) and the marginal cost (MC) and solve for Q.

Determining the Profit-Maximizing Output

Given these equations:

MR 30 - 2Q

MC 3Q^2 - 30Q 10

We set MR equal to MC:

30 - 2Q 3Q^2 - 30Q 10

Simplifying the equation:

3Q^2 - 28Q 20 0

To solve this quadratic equation, we can use the quadratic formula:

Q frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 3, b -28, and c 20.

Substituting the values into the quadratic formula:

Q frac{28 pm sqrt{(-28)^2 - 4 cdot 3 cdot 20}}{2 cdot 3}

Q frac{28 pm sqrt{784 - 240}}{6}

Q frac{28 pm sqrt{544}}{6}

Q frac{28 pm 23.32}{6}

There are two possible solutions for Q:

Q frac{28 23.32}{6} approx 8.22

Q frac{28 - 23.32}{6} approx 0.78

Since the monopolist's profit-maximizing output cannot be negative, we discard the smaller solution.

Calculating the Equilibrium Price

To determine the equilibrium price, we substitute the value of Q into the demand function:

P 30 - 2Q

Using Q 8.22:

P 30 - 2 cdot 8.22

P ≈ 13.56

Therefore, the equilibrium price is approximately 13.56.

Calculating the Minimum Profit

To calculate the minimum profit, we need to subtract the total cost (TC) from the total revenue (TR).

First, we calculate the total revenue (TR):

TR P cdot Q

TR 13.56 cdot 8.22 ≈ 111.55

Next, we calculate the total cost (TC):

TC MC cdot Q

TC (3Q^2 - 30Q 10) cdot Q

TC 3Q^3 - 30Q^2 10Q

Using Q 8.22:

TC 3 cdot 8.22^3 - 30 cdot 8.22^2 10 cdot 8.22

TC ≈ 215.70

Now, we calculate the profit:

Profit TR - TC

Profit 111.55 - 215.70 ≈ -104.15

Therefore, the minimum profit is approximately -104.15, indicating a loss.

Conclusion

In summary, the profit-maximizing output is approximately 8.22 units, the equilibrium price is approximately 13.56, and the minimum profit is approximately -104.15. This analysis demonstrates the complexities and challenges faced by monopolists in determining their optimal production levels and prices.

This study is pertinent for understanding the economic behavior of monopolistic firms, which are characterized by a single seller with significant market power. By mastering these concepts, business leaders and economists can better guide decision-making processes in monopolistic markets.

Keywords: monopolist equilibrium, profit maximization, marginal revenue, marginal cost