Question

Suppose that the demand curve for a good is given by \(D(p)=100 / p\) What price will maximize revenue?

Short answer

Revenue is constant at 100, so any price will yield a maximum revenue.

Step-by-step solution

Understand the Revenue Function

Revenue is calculated as price multiplied by quantity. The demand function is given as \(D(p) = \frac{100}{p}\). Therefore, Quantity \(Q = \frac{100}{p}\). The revenue function \(R(p)\) can be written as:\[ R(p) = p \times Q = p \times \frac{100}{p} = 100. \]So, the revenue function simplifies to \( R(p) = 100 \).

Analyze the Revenue Function

Since the revenue function \(R(p) = 100\) is constant, it does not depend on the price \(p\) at all. This indicates that the revenue remains constant, irrespective of the price set.

Conclusion on Maximum Revenue

As the revenue remains constant at 100 across all prices, any price will technically maximize revenue, given that the total always equals 100. There is no specific price that maximizes revenue more than another.

Key concepts

Demand Curve

The demand curve is a fundamental concept in economics that relates the price of a good or service to the quantity demanded by consumers. In our exercise, the demand curve is expressed as:\[ D(p) = \frac{100}{p} \]This equation shows an inverse relationship between price \( p \) and quantity demanded.

• As the price \( p \) increases, the quantity demanded \( D(p) \) decreases.
• Conversely, as the price \( p \) decreases, the quantity demanded \( D(p) \) increases.

This particular form of the demand curve is hyperbolic, indicating that demand changes in an inverse proportion to price changes. Understanding the nature of the demand curve helps in predicting how changes in price can affect overall sales volume, which is crucial for making pricing decisions aimed at revenue maximization.

Revenue Function

The revenue function tells us how much money a firm will make based on different prices and quantities. Revenue is calculated by multiplying the price per unit \( p \) by the quantity of units sold \( Q \). For the demand curve \( D(p) = \frac{100}{p} \), the quantity \( Q \) is also \( \frac{100}{p} \). Therefore, our revenue function \( R(p) \) is:\[ R(p) = p \times Q = p \times \frac{100}{p} = 100 \]

• The expression shows the revenue is constant at 100.
• This means that regardless of price changes, the total revenue remains the same.

Such a result is quite unusual because, generally, revenue varies with price. Here, however, no matter the changes in price \( p \), the product's total revenue remains unaffected. Understanding the revenue function is key as it provides the direct relationship between pricing strategies and economic outcomes.

Economics Problem Solving

Economics problem solving involves analyzing different variables and understanding how they interact in order to make informed decisions. In the context of the given exercise, the main objective was to determine the price that maximizes revenue. However, the constant revenue function implies that:

• Any price will yield the same total revenue, making this a unique economic problem.
• In standard scenarios, revenue varies with price, and identifying an optimal price is essential.

Solving economics problems like this requires a solid understanding of fundamental concepts such as demand curves and revenue functions. Identifying unusual patterns, as seen in this problem, can also be crucial. When revenue stays constant, strategic decisions can focus more on other factors, such as cost minimization or market positioning, rather than price adjustments. This approach illustrates the necessity and application of economic theory in practical decision-making.