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Chapter 4: Problem 27

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=4000 e^{-0.01 p}, \quad p=200 $$

Short Answer

Expert verified
The demand is elastic with \(E(200) = -2\).

Step by step solution

01

Write the formula for elasticity of demand

The elasticity of demand with respect to price, \(E(p)\), is given by the formula: \[E(p) = \frac{D'(p) \times p}{D(p)}\]where \(D'(p)\) is the derivative of the demand function.
02

Differentiate the demand function

Given the demand function \(D(p) = 4000 e^{-0.01p}\), we need to find \(D'(p)\). Using the chain rule, the derivative is:\[D'(p) = \frac{d}{dp} [4000 e^{-0.01p}] = 4000 \cdot (-0.01) \cdot e^{-0.01p} = -40 e^{-0.01p}\]
03

Substitute the derivative and demand function into the elasticity formula

Plug the derivative \(D'(p) = -40 e^{-0.01p}\), \(D(p) = 4000 e^{-0.01p}\), and \(p = 200\) into the elasticity formula:\[E(200) = \frac{-40 e^{-0.01 \times 200} \times 200}{4000 e^{-0.01 \times 200}}\]Simplify the expression to:\[E(200) = \frac{-8000 e^{-2}}{4000 e^{-2}} = -2\]
04

Interpret the elasticity value

The elasticity \(E(200) = -2\) indicates that the demand is elastic. In economic terms, if the absolute value of elasticity is greater than 1, the demand is considered elastic.

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Key Concepts

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

Understanding the Demand Function
A demand function represents the relationship between the price of a product and the quantity demanded by consumers. It's typically expressed as a mathematical equation. For instance, in the given exercise, the demand function is \(D(p) = 4000 e^{-0.01p}\). This tells us that the quantity demanded decreases exponentially as the price \(p\) increases.
  • The function involves an exponential term \(e^{-0.01p}\), highlighting how demands diminish with rising prices.
  • An essential aspect of understanding demand functions is how they capture consumer behavior in response to price changes.
To grasp how changes in price affect quantity demanded, we'll need to explore derivatives, elasticity, and more.
Derivatives in Calculus for Economic Analysis
Derivatives play a crucial role in economic analysis, especially when it comes to understanding changes in demand with respect to price. The derivative of a function provides the rate at which the function's value changes with respect to a change in its input. In the context of demand functions:
  • The derivative \(D'(p)\) indicates how the quantity demanded changes as the price changes.
  • Using calculus, we differentiate the given demand function \(D(p) = 4000 e^{-0.01p}\) and find \(D'(p) = -40 e^{-0.01p}\).
This derivative informs us that as the price \(p\) increases, the quantity demanded decreases at a rate proportionate to \(-40 e^{-0.01p}\). Differentiation provides valuable insights into consumer reactions to price changes.
Exploring Economic Elasticity
Economic elasticity measures how responsive the quantity demanded is to changes in price. The elasticity of demand, specifically, helps us understand whether a demand curve is more flexible or stiff:
  • If demand is elastic, consumers significantly change their quantity demanded when prices change (elasticity absolute value is greater than 1).
  • If demand is inelastic, the changes in price don't greatly affect the quantity demanded (elasticity absolute value is less than 1).
  • If the demand is unit-elastic, the percentage change in quantity demanded is equal to the percentage change in price (elasticity absolute value is 1).
For the given exercise, the elasticity calculated at \(p = 200\) was \(E(200) = -2\). This negative value indicates a decrease in demand as price rises, which is typical. The absolute value (2) suggests the demand is elastic: consumers would substantially reduce demand if prices were raised.
Applying the Chain Rule in Calculus
The chain rule is a fundamental tool in calculus used to differentiate composite functions. In this exercise, it's used to find the derivative of the demand function, which includes an exponential component:
  • The demand function \(D(p) = 4000 e^{-0.01p}\) contains two parts: a constant (4000) and an exponential term \(e^{-0.01p}\).
  • To differentiate, the chain rule helps us "unlink" these parts, allowing us to take the derivative of the outer function along with the derivative of the inner function.
With the chain rule, the derivative calculated is \(D'(p) = -40 e^{-0.01p}\), leading to insights on how demand shifts with price changes. The chain rule proves essential in unraveling changes across these interconnected components.

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Most popular questions from this chapter

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=300-p^{2}, \quad p=10 $$

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