Question

A demand function is modeled by \(x=a / p^{m}\), where \(a\) is a constant and \(m>1\). Show that \(\eta=-m .\) In other words, show that a \(1 \%\) increase in price results in an \(m \%\) decrease in the quantity demanded.

Short answer

Yes, the price elasticity of demand \(\eta\) for the given demand function \(x=a / p^{m}\), is indeed \(-m\). This implies that a \(1\%\) increase in price results in an \(m\%\) decrease in the quantity demanded.

Step-by-step solution

Understand the Demand Function

The given demand function is \(x=a / p^{m}\), where \(x\) represents the quantity demanded, \(p\) is the price, \(a\) is a constant, and \(m\) is a parameter that determines how responsive demand is to price. The task is to prove that a percentage increase in price, results in \(m\) percentage decrease in quantity demanded, in other words, to show that \(\eta = -m\).

Differentiate the Demand Function w.r.t. Price

The first step in proving this is to find the derivative of the demand function with respect to the price \(p\). Keeping the constant \(a\) the same and applying the power rule for differentiation, the derivative \(\frac{dx}{dp}\) of the function is: \(\frac{dx}{dp}= a(-m)p^{-(m+1)}\). The negative sign appears because we are differentiating a function in the denominator, and the one added to \(m\) represents the new power of \(p\) after differentiation according to the power rule.

Apply the Price Elasticity of Demand Formula

The formula for price elasticity of demand is \(\eta = \frac{dq}{dp} \cdot \frac{p}{q}\), where \(\eta\) is the price elasticity of demand, \(\frac{dq}{dp}\) is the derivative of Quantity demanded (from the original demand function), \(p\) is the price, and \(q\) is the quantity. \nNow replace \(q\) with \(x = a / p^{m}\) from the original demand function and \(\frac{dq}{dp}\) with \(a(-m)p^{-(m+1)}\). Simplify the expression to get \(\eta = -m\).

Final Simplification

Upon substitution we get, \(\eta = {-m p^{-m}} \cdot {p}\). The \(p\) cancels out the power of \(p\) in the numerator and hence, we get \(\eta = -m\). This shows that a \(1\%\) increase in price results in an \(m\%\) decrease in the quantity demanded, which was required to be proven.

Key concepts

Price Elasticity of Demand

The price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. This concept is crucial in understanding consumer behavior and setting pricing strategies.

Mathematically, it is represented by the formula:

• \( \eta = \frac{\Delta x}{\Delta p} \cdot \frac{p}{x} \)

Where \( \Delta x \) is the change in quantity demanded, \( \Delta p \) is the change in price, \( p \) is the original price, and \( x \) is the original quantity.

For a demand function given as \( x = \frac{a}{p^m} \), a derivative-derived formula for elasticity becomes:

• \( \eta = \frac{dq}{dp} \cdot \frac{p}{q} \)
• When simplified, this shows \( \eta = -m \)

This means that the percentage change in quantity demanded is directly proportional to \( m \), making the elasticity a direct negative of the exponent in the demand function. A 1% increase in price therefore leads to an \( m \)% decrease in demand.

Differentiation in Calculus

Differentiation is a fundamental tool in calculus used to find the rate at which one quantity changes with respect to another. When applied to a demand function, it helps determine how demand responds to changes in price.

In the provided exercise, the demand function \( x = \frac{a}{p^m} \) represents how quantity demanded depends inversely on price. By differentiating this function with respect to \( p \), we can measure the impact of small changes in price on the quantity demanded.

• Apply the power rule: \( \frac{dx}{dp} = a(-m)p^{-(m+1)} \)
• The negative sign indicates that as price increases, demand decreases.
• This derivative is used in the elasticity calculation to represent the rate of change in demand.

Differentiation thus provides the mathematical groundwork for analyzing and predicting real-world economic behaviors.

Economic Modeling

Economic modeling involves using mathematical equations and functions to represent economic theories and analyze real-world economic scenarios. These models can explore various relationships, such as the one between price and demand in an economy.

In our context, the demand function \( x = \frac{a}{p^m} \) is used to model consumer demand. This model takes into account how sensitive consumers are to price changes, which is captured by the parameter \( m \).

• The parameter \( m \) determines elasticity and inversely affects the steepness of the demand curve.
• This modeling helps economists and businesses predict reactions to price changes, aiding in strategic decision-making.
• By understanding this dynamic, firms can optimize pricing to maximize revenue or market share.

Economic models are indispensable tools that provide insights into complex market dynamics, making them crucial for both theoretical studies and practical applications in business.