Problem 20 - Applied Calculus

Question

Show that a demand equation \(q=k / p^{r},\) where \(r\) is a positive constant, gives constant elasticity \(E=r\)

Step-by-Step Solution

Answer

The demand equation gives constant elasticity \( E = r \).

Step 1: Understanding the Demand Equation

The demand equation given is \( q = \frac{k}{p^r} \), where \( q \) is the demand, \( k \) is a constant, \( p \) is the price, and \( r \) is a positive constant.

Step 2: Recall the Elasticity Formula

The elasticity of demand formula is \( E = \frac{dq/dp}{q/p} \). This measures the percentage change in quantity demanded resulting from a one percent change in price.

Step 3: Differentiate the Demand Equation

Differentiate \( q = \frac{k}{p^r} \) with respect to \( p \). Use the power rule for differentiation: \( dq/dp = -rk \cdot p^{-(r+1)} \).

Step 4: Find \( q/p \) in the Elasticity Formula

In the formula \( q/p \), simply use the given \( q = \frac{k}{p^r} \) as: \( \frac{q}{p} = \frac{k}{p^{r+1}} \).

Step 5: Substitute into the Elasticity Formula

Substitute \( dq/dp = -rk \cdot p^{-(r+1)} \) and \( q/p = \frac{k}{p^{r+1}} \) into \( E = \frac{dq/dp}{q/p} \). This gives:\[E = \frac{-rk \cdot p^{-(r+1)}}{k/p^{r+1}}\].Simplify the expression by canceling \( k \) and rearranging the exponents.

Step 6: Simplify to Find the Elasticity

The expression simplifies to:\[E = \frac{-rk \cdot p^{-(r+1)}}{k \cdot p^{-(r+1)}} = -r\].Note the negative sign indicates the direction of change (as price increases, demand decreases), thus elasticity \( E = r \).

Step 7: Conclude the Elasticity is Constant

Since the elasticity \( E = r \) is independent of \( p \) and other variables, the elasticity is constant and equal to \( r \).

Key Concepts

Differentiation in Calculus

Differentiation is a key concept in calculus that focuses on how functions change over time. When we talk about differentiation, we're often concerned with finding the derivative of a function. The derivative essentially measures how a function changes as its input changes.
An important aspect of differentiation is the power rule, which is a straightforward formula for finding the derivative of polynomial functions. For example, if you have a function of the form \( f(x) = x^n \), the derivative of this function using the power rule is \( f'(x) = nx^{n-1} \).
In the context of our original exercise, differentiation is used to analyze the change in demand relative to the change in price. By applying the power rule to the demand equation, we can find out how sensitive demand is to price changes. This derivative is crucial to determining the elasticity of demand, which in turn tells us how much quantity demanded changes when the price changes by a certain percentage.

Demand Equation

A demand equation is an equation that represents the relationship between quantity demanded and price within a market. It reflects how much of a product consumers want to buy at different price levels. In mathematics, this relationship can be expressed as a formula like \( q = \frac{k}{p^r} \), where:

• \( q \) represents the quantity demanded.
• \( p \) is the price of the good or service.
• \( k \) is a constant that represents other non-price factors affecting demand.
• \( r \) is a positive constant showing the influence of price on demand.

A demand equation like this is particularly useful for analyzing price elasticity because it captures the inverse relationship between price and demand. As price increases, demand generally decreases, and vice versa. This aspect is explicitly captured by the exponent \( r \) in the equation.
Understanding this equation helps economists and businesses determine pricing strategies that maximize revenue while considering consumers' responsiveness to price changes.

Elasticity of Demand Formula

The elasticity of demand is a crucial economic concept that measures how quantity demanded reacts to changes in price. The formula for price elasticity of demand is expressed as:
\[ E = \frac{dq/dp}{q/p} \]
Where:

• \( \frac{dq}{dp} \) is the derivative of demand with respect to price, indicating the rate of change in demand for an infinitesimal change in price.
• \( q/p \) is the ratio of quantity to price, essentially showing how much demand corresponds to a certain price level.

Elasticity can be greater than 1 (elastic demand), less than 1 (inelastic demand), or equal to 1 (unitary elastic demand). Elasticity offers insights into consumer behavior, helping businesses and policymakers understand and predict how changes in pricing will affect overall demand.
In the specific case of our original exercise, the demand elasticity is shown to be a constant \( r \). The negative sign indicates the inverse relationship; as price increases, the quantity demanded decreases. Simplifying the elasticity formula in the exercise confirms this by showing how the influence of price through the \( r \) exponent results in constant elasticity, encapsulated simply as \( E = r \).