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Chapter 4: Problem 14

Find the elasticity \(E\) at the given points and determine whether demand is inelastic, elastic, or unit elastic. \(x=20-4 p\), a. \(p=1\) b. \(p=2.5\) c. \(p=4\)

Short Answer

Expert verified
At \(p=1\), demand is inelastic; at \(p=2.5\), demand is unit elastic; at \(p=4\), demand is elastic.

Step by step solution

01

Understand the demand equation

The demand equation is given as \(x = 20 - 4p\). In this context, \(x\) represents the quantity demanded, and \(p\) represents the price. Our goal is to find the elasticity of demand at specific prices \(p = 1\), \(p = 2.5\), and \(p = 4\).
02

Find the derivative of the demand equation

The elasticity formula involves the derivative of the demand function. Differentiate the demand function \(x = 20 - 4p\) with respect to \(p\):\[ \frac{dx}{dp} = -4. \]
03

Formula for elasticity of demand

The formula for elasticity of demand \(E\) is given by:\[ E = \frac{p}{x} \cdot \frac{dx}{dp}. \] We will use this formula to calculate the elasticity at each specified price.
04

Calculate elasticity at \(p = 1\)

First, find the quantity \(x\) when \(p = 1\): \[ x = 20 - 4(1) = 16. \] Now apply the elasticity formula: \[ E = \frac{1}{16} \cdot (-4) = -\frac{4}{16} = -0.25. \] Since \(|E| < 1\), the demand is inelastic at \(p = 1\).
05

Calculate elasticity at \(p = 2.5\)

First, find the quantity \(x\) when \(p = 2.5\): \[ x = 20 - 4(2.5) = 10. \] Now apply the elasticity formula: \[ E = \frac{2.5}{10} \cdot (-4) = -1. \] Since \(|E| = 1\), the demand is unit elastic at \(p = 2.5\).
06

Calculate elasticity at \(p = 4\)

First, find the quantity \(x\) when \(p = 4\): \[ x = 20 - 4(4) = 4. \] Now apply the elasticity formula: \[ E = \frac{4}{4} \cdot (-4) = -4. \] Since \(|E| > 1\), the demand is elastic at \(p = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elastic Demand
When we talk about elastic demand, we mean situations where the quantity demanded is highly sensitive to price changes.
This typically occurs when there are many substitutes available or when the product is not a necessity. In the step-by-step solution, we see this concept in action at price point \( p = 4 \). As the calculated elasticity \( E \) is \(-4\), this is further than \(-1\) which indicates a responsive or elastic demand.
Some quick facts about elastic demand:
  • Even a small price increase can lead to a large decrease in quantity demanded.
  • Absolute value of elasticity is greater than 1.
  • Reflects a highly competitive market where consumers can easily switch to substitutes.
Understanding this behavior is crucial for businesses as it affects pricing strategies drastically. For instance, lowering prices can significantly boost sales.
Inelastic Demand
Inelastic demand, on the other hand, reflects a much less responsive relationship between price changes and quantity demanded.
Products in this category are usually necessities or have few substitutes available. From the exercise, when \( p = 1 \), the elasticity \( E \) is \(-0.25\). This shows the demand is inelastic as \( |E| < 1 \).
Key points about inelastic demand:
  • Price changes do not significantly affect the quantity demanded.
  • Absolute value of elasticity is less than 1.
  • Typically includes essential goods and services.
In such cases, businesses might not experience a drastic fall in sales despite price increases.
However, it does suggest that price cuts may not significantly increase sales volume either.
Unit Elastic Demand
Unit elastic demand is an interesting and balanced scenario where the percentage change in quantity demanded equals the percentage change in price. Thus, total revenue remains constant as price changes. In our problem, at \( p = 2.5 \), elasticity \( E \) is calculated to be \(-1\), indicating unit elasticity. A few points about unit elastic demand:
  • The demand curve is perfectly balanced in terms of responsiveness.
  • Absolute value of elasticity equals 1.
  • Slightly harder to find in real-life cases, but important for theoretical understanding.
Understanding this equilibrium can be exceptionally useful for businesses looking to stabilize revenue during pricing adjustments.
In such scenarios, revenue maximization strategies would differ greatly compared to other elasticity types.

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