Maximizing Total Revenue in a Competitive Market

Introduction

The concept of revenue maximization is crucial in economics and business strategy. In a competitive market, understanding how to optimize revenue by setting the right price and quantity is fundamental. This article delves into a specific scenario where the demand function is given by the equation P 1000 – 2Q_d, and the task is to determine the maximum level of total revenue.

The Demand Function and Revenue Maximization

The equation P 1000 – 2Q_d represents the demand function, where P is the price and Q_d is the quantity demanded. Assuming that the term “d” is a typo and should be dropped, we proceed with the assumption that the function is correctly interpreted as P 1000 – 2Q.

Total revenue (TR) is calculated as the product of price and quantity:

TR P × Q (1000 – 2Q) × Q -2Q^2 1000Q.

To find the maximum level of total revenue, we need to determine the point where the derivative of the revenue function with respect to Q equals zero. This point represents the marginal revenue (MR), which is the derivative of total revenue.

Calculating Marginal Revenue and Maximizing Revenue

The total revenue function is:

TR -2Q^2 1000Q.

The derivative of this function is:

TR' d(TR)/dQ -4Q 1000.

Setting the derivative equal to zero to find the maximum point:

-4Q 1000 0.

Solving for Q:

Q 250.

Thus, the quantity that maximizes revenue is 250. Substituting Q 250 into the original demand function:

P 1000 - 2(250) 1000 - 500 500.

So, the price at the maximum revenue point is 500. The maximum revenue can be calculated as:

TR 500 × 250 125000.

Conclusion

In this scenario, it is clear that the maximum level of total revenue is achieved when the quantity is 250, and the corresponding price is 500, resulting in a maximum revenue of 125,000 units. This analysis helps businesses understand how to optimize pricing and sales strategies to maximize their revenue.

Related Keywords

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