Question

True or false? If the demand function is \(x_{1}=-p_{1},\) then the inverse demand function is \(x=-1 / p_{1}\)

Short answer

False. The inverse function should be \(p_1 = -x_1\).

Step-by-step solution

Understand the Demand Function

The given demand function is \(x_1 = -p_1\). This equation shows that the quantity demanded \(x_{1}\) is a linear function of the price \(p_{1}\), with a slope of -1.

Define the Inverse Demand Function

The inverse demand function expresses the price \(p_{1}\) as a function of the quantity \(x_1\). For the demand function \(x_1 = -p_1\), we need to solve this for \(p_1\) in terms of \(x_1\).

Solve for Inverse Price Function

Starting from \(x_1 = -p_1\), we need to rearrange to find \(p_1\). Flip the equation so that \(p_1 = -x_1\). This is the inverse demand function.

Evaluate the Given Inverse Function

The problem states the inverse demand function as \(x = -1 / p_1\). Compare this with our derived inverse demand function \(p_1 = -x_1\). They are not the same as the problem suggests the inverse as a reciprocal which does not match.

Key concepts

Demand Function

In microeconomics, understanding the demand function is crucial. It's a mathematical representation of the relationship between the price of a product and the quantity demanded. The demand function is generally expressed as a formula showing how the quantity demanded (\(x_1\)) changes in response to variations in price (\(p_1\)). For instance, in the given exercise, the demand function is \(x_1 = -p_1\). This means every time the price changes, the demand responds in a linear fashion.
The negative sign indicates an inverse relationship—when price goes up, the quantity demanded goes down, which is an intuitive rule in economics. This specific formula shows a direct, proportional change because it has a constant rate of change (slope of -1). This behavior is typical in competitive markets, where price changes significantly influence consumer behavior.

Price-Quantity Relationship

The price-quantity relationship represents how demand responds to changes in price. It provides insights into consumer purchasing patterns and market dynamics. Usually, demand decreases as price increases — a core principle known as the law of demand. The given problem illustrates this with a simple linear model: \(x_1 = -p_1\). Here's how this relationship can be understood:

• The slope of the demand curve here is -1, which means for every 1 unit increase in price, the quantity demanded decreases by 1 unit.
• This proportional change highlights a predictable reaction, making it easy for businesses to predict sales volume changes based on price adjustments.
• Understanding such relationships helps in strategic pricing and maximizing revenue.

In practical terms, a steeper slope would imply that consumers are more sensitive to price changes, whereas a flatter slope would suggest less sensitivity.

Microeconomic Concepts

Microeconomic concepts help explain the behaviors of individuals and firms in making decisions about the allocation of scarce resources. Central to these is the analysis of supply and demand and how they balance in the market.
In this context, the inverse demand function is an important concept. It describes how prices must adjust to various quantities being demanded. Solving the demand function for price, as demonstrated, gives us the inverse demand: \(p_1 = -x_1\). This shows the price necessary for a given demand quantity.

• Using inverse demand functions helps firms determine optimal pricing strategies to fit desired sales volumes.
• It aids in understanding consumer preference shifts and adjusting marketing tactics accordingly.
• Recognizing these relationships can drive better competitive positioning and profitability.

Understanding and applying these microeconomic principles are vital in predicting market trends and effectively navigating economic landscapes.