Problem 40 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Suppose that the demand equation for a certain commodity is \(q=60-0.1 p\) (for \(0 \leq p \leq 600\) ). a. Express the elasticity of demand as a function of \(p\). b. Calculate the elasticity of demand when the price is \(p=200\). Interpret your answer. c. At what price is the elasticity of demand equal to 1 ?

Step-by-Step Solution

Answer

a) \( E_d(p) = -0.1 \times \left( \frac{p}{60 - 0.1p} \right) \) b) \( E_d(200) = -0.5 \), demand is inelastic at \( p = 200 \). c) Elasticity is 1 at \( p = 300 \).

Step 1: Understanding the Demand Equation

We are given the demand equation: \[ q = 60 - 0.1p \] where \( q \) is the quantity demanded and \( p \) is the price.

Step 2: Deriving the Elasticity Formula

Elasticity of demand (\( E_d \)) is given by:\[ E_d = \left( \frac{dq}{dp} \right) \times \left( \frac{p}{q} \right) \]First, find the derivative \( \frac{dq}{dp} \) from the demand equation \( q = 60 - 0.1p \): \[ \frac{dq}{dp} = -0.1 \].

Step 3: Expressing Elasticity in Terms of p

Substitute \( \frac{dq}{dp} = -0.1 \), \( q = 60 - 0.1p \), and \( p \) into the elasticity formula: \[ E_d(p) = (-0.1) \times \left( \frac{p}{60 - 0.1p} \right) \].

Step 4: Calculating Elasticity at p = 200

Substitute \( p = 200 \) into the elasticity function: \[ E_d(200) = (-0.1) \times \left( \frac{200}{60 - 0.1 \times 200} \right) = (-0.1) \times \left( \frac{200}{40} \right) = -0.1 \times 5 = -0.5 \].

Step 5: Interpreting the Elasticity

An elasticity of \(-0.5\) means that the demand is inelastic at \( p = 200 \), because the magnitude of elasticity is less than 1. This implies that a 1% increase in price leads to a less than 1% decrease in the quantity demanded.

Step 6: Finding Price Where Elasticity is 1

Set the absolute value of elasticity equal to 1 and solve for \( p \): \[ 1 = 0.1 \times \frac{p}{60 - 0.1p} \], Solve for \( p \): \[ 60 - 0.1p = 0.1p \], \[ 60 = 0.2p \], \[ p = 300 \].

Step 7: Conclusion

From the previous steps, the demand elasticity function, elasticity at a specific price, and the price for unit elasticity are computed.

Key Concepts

demand equation

The demand equation shows the relationship between the price of a commodity and the quantity demanded. In the given exercise, the demand equation is represented as \( q = 60 - 0.1p \). This means the quantity demanded (\( q \)) decreases by 0.1 units for every 1 unit increase in price (\( p \)). The demand equation helps us understand how changes in price impact the purchase behavior of consumers. For example, when the price is 0, the quantity demanded is 60 units.

price elasticity

Price elasticity of demand (\( E_d \)) measures the responsiveness of the quantity demanded to changes in price. It is calculated as \( E_d = \frac{dq}{dp} \times \frac{p}{q} \). This formula shows how much demand changes with a given change in price. If the price elasticity is greater than 1, demand is elastic. If it is less than 1, demand is inelastic. For instance, in the exercise, when the price is 200, the calculated elasticity is -0.5, which classifies the demand as inelastic.

inelastic demand

Inelastic demand occurs when the absolute value of price elasticity of demand is less than 1 (\( |E_d| < 1 \)). This means that a change in price results in a less than proportional change in the quantity demanded. In simple terms, consumers are not very sensitive to changes in price. For example, with an elasticity of -0.5 calculated at a price of 200, it means a 1% increase in price results in only a 0.5% decrease in the quantity demanded. Products considered necessities often exhibit inelastic demand.

unit elasticity

Unit elasticity occurs when the absolute value of price elasticity of demand is exactly 1 (\( |E_d| = 1 \)). This indicates that the percentage change in quantity demanded is exactly equal to the percentage change in price. In the exercise, we determine when the elasticity equals one by solving \( 1 = 0.1 \times \frac{p}{60 - 0.1p} \). The calculation results in a price of 300. At this price, the demand changes proportionally with price changes, meaning consumers are equally responsive to price increases and decreases.

derivative

The derivative gives us the rate at which one variable changes with respect to another. In the context of the demand equation, the derivative \( \frac{dq}{dp} \) represents the rate of change of quantity demanded with respect to price. For the provided demand equation \( q = 60 - 0.1p \), the derivative is \( \frac{dq}{dp} = -0.1 \). This negative sign indicates that an increase in price leads to a decrease in quantity demanded. This derivative is crucial for calculating the price elasticity of demand.