Question

Consider the following demand function: \(Q^{D}=25 / \mathrm{P}^{2}\). Show that the point elasticity of demand for that function is always equal to \(-2\).

Short answer

The point elasticity of demand is always \(-2\).

Step-by-step solution

Understand the Demand Function

We are given the demand function \(Q^{D}=\frac{25}{P^2}\). This describes how quantity demanded changes with price.

Recall the Formula for Point Elasticity of Demand

The point elasticity of demand is given by the formula \(E = \frac{dQ}{dP} \times \frac{P}{Q}\), where \(\frac{dQ}{dP}\) is the derivative of \(Q\) with respect to \(P\).

Find the Derivative of the Demand Function

Differentiate \(Q^{D}\) with respect to \(P\). We have \(Q^{D} = 25 \cdot P^{-2}\). The derivative is \(\frac{dQ}{dP} = -50 \times P^{-3} = -\frac{50}{P^3}\).

Substitute into the Elasticity Formula

Substitute \(\frac{dQ}{dP} = -\frac{50}{P^3}\), \(P\), and \(Q = \frac{25}{P^2}\) into the elasticity formula: \[E = \left(-\frac{50}{P^3}\right) \times \frac{P}{\frac{25}{P^2}}.\]

Simplify the Elasticity Expression

Simplify the expression: \[E = -\frac{50}{P^3} \times \frac{P^3}{25}.\]Cancel the \(P^3\) terms and simplify to \[E = -2.\]

Conclusion on Constant Elasticity

No matter the value of \(P\), the elasticity of demand for this function is always \(-2\). This confirms that the point elasticity of the demand function given is indeed always \(-2\).

Key concepts

Understanding the Demand Function

A demand function, like the one given in the problem, is a mathematical expression that describes the relationship between the quantity demanded of a good and its price. In our example, the demand function is defined as \[ Q^D = \frac{25}{P^2} \]This specific form of the function tells us that the quantity demanded, \(Q^D\), is inversely related to the square of the price, \(P\).

• As the price \(P\) increases, \(Q^D\) decreases.
• The function is hyperbolic, meaning it forms a curve that bends toward the horizontal axis as \(P\) increases.

Understanding the demand function is crucial because it allows economists to predict and analyze how consumers will behave in response to price changes. In simple terms, a higher price typically leads to less demand, while a lower price encourages more demand. To solve this exercise, we need to understand how changes in price affect the quantity demanded, which leads us to the concept of elasticity.

The Role of Calculus in Economics

Calculus is a powerful tool in economics, enabling us to analyze rates of change. This is particularly relevant when dealing with demand functions and elasticity. In this exercise, we use calculus to determine how sensitive the quantity demanded is to a change in price. To find this sensitivity, or point elasticity of demand, we need to calculate the derivative of the demand function with respect to price. This derivative represents the instantaneous rate of change of the quantity demanded as price changes. For \(Q^D = \frac{25}{P^2}\), the derivative with respect to \(P\) is \[ \frac{dQ}{dP} = -\frac{50}{P^3} \]The derivative tells us that as \(P\) changes, the quantity demanded changes at a rate of \(-\frac{50}{P^3}\). This negative sign indicates that \(Q^D\) decreases as \(P\) increases, which aligns with the inverse relationship described by the demand function.

Using the Elasticity Formula

The elasticity formula helps economists measure how responsive the quantity demanded is to changes in price. The point elasticity of demand is given by:\[E = \frac{dQ}{dP} \times \frac{P}{Q}\]To use this formula, we first identify the components:

• The derivative \(\frac{dQ}{dP}\) is \(-\frac{50}{P^3}\),
• The price \(P\) is our current price level,
• And \( Q = \frac{25}{P^2} \) is our calculated quantity demanded.

Substituting these values into the elasticity formula, we simplify: \[ E = \left(-\frac{50}{P^3}\right) \times \frac{P}{\frac{25}{P^2}} \]The \( P^3 \) terms in the numerator and denominator cancel each other out:\[ E = -\frac{50}{25} = -2 \]This result means that the elasticity is \(-2\), showing that for every 1% increase in price, the quantity demanded decreases by 2%. This constant elasticity indicates that consumers are somewhat sensitive to price changes, doubling their rate of quantity change compared to price changes.