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Chapter 5: Problem 16

For each demand function, find \(E(p)\) and determine if demand is elastic or inelastic (or neither) at the indicated price. $$ q=\left(77 / p^{2}\right)+3, p=1 $$

Short Answer

Expert verified
The price elasticity of demand is \( -1.925 \) and the demand is elastic at the price \( p = 1 \).

Step by step solution

01

Find the derivative of the demand function with respect to price

The demand function is given as \( q = \left(\frac{77}{p^2}\right) + 3 \). To find the price elasticity of demand, first find the derivative of the demand function with respect to price, \( p \). The derivative \( \frac{dq}{dp} \) can be found using the power rule and chain rule of differentiation.
02

Apply the power rule

For the term \( \left(\frac{77}{p^2}\right) \), rewrites as \( 77 p^{-2} \). Its derivative with respect to \( p \) is \( -154 p^{-3} \). The term 3 is a constant, so its derivative is 0. Hence, \( \frac{dq}{dp} = -154 p^{-3} \).
03

Use the formula for price elasticity of demand

The formula for price elasticity of demand is \( E(p) = \left( \frac{p}{q} \right) \cdot \left( \frac{dq}{dp} \right) \). We already have \( \frac{dq}{dp} = -154 p^{-3} \). Now, substitute it along with \( q \) and \( p \) into the elasticity formula.
04

Substitute \( p = 1 \)

Given that \( p = 1 \), first find \( q \). Substituting \( p = 1 \) into the demand function, \( q = \left( \frac{77}{1^2} \right) + 3 = 77 + 3 = 80 \).
05

Calculate \( E(1) \)

Substitute \( p = 1 \), \( q = 80 \), and \( \frac{dq}{dp} = -154 \) into the elasticity formula: \[ E(1) = \left( \frac{1}{80} \right) \cdot (-154) = -\frac{154}{80} = -1.925 \]
06

Determine elasticity type

Price elasticity of demand \( E(p) = -1.925 \). Since \( |E(p)| > 1 \), the demand is elastic at the price \( p = 1 \).

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Key Concepts

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

Differentiation
Differentiation is a mathematical process where we find the rate at which a function changes. It is essential when working with demand functions, as it helps to determine how the quantity demanded changes with a change in price.
In this exercise, we differentiate the demand function with respect to price to find the elasticity. For a function like this: \( q = \frac{77}{p^2} + 3 \), we apply rules of differentiation.
Remember that for a term like \( \frac{77}{p^2} \), you can rewrite it as \( 77p^{-2} \). Differentiating this term involves using the power rule.
Demand Function
A demand function describes the relationship between the quantity of a good demanded and its price. It is typically written as \( q = f(p) \), where \( q \) is quantity and \( p \) is price.
Understanding the demand function is crucial in economics because it helps businesses set prices optimally. In our exercise, the function given is \( q = \frac{77}{p^2} + 3 \). This function indicates that as the price \( p \) changes, the quantity demanded \( q \) also changes.
Elasticity Calculation
Price elasticity of demand measures how responsive the quantity demanded is to a change in price. It is calculated using this formula: \[ E(p) = \frac{p}{q} \times \frac{dq}{dp} \].
Let's break it down:
  • Find \( \frac{dq}{dp} \), the derivative of the demand function with respect to price.
  • Substitute the price \( p \) and quantity \( q \) values into the formula.
  • The result tells whether the demand is elastic (\( |E(p)| > 1 \)), inelastic (\( |E(p)| < 1 \)), or unitary elastic (\( |E(p)| = 1 \)).
  • In our example: \( E(p) = -1.925 \), the demand is elastic since its absolute value is greater than 1.
Power Rule
The power rule is a fundamental tool in differentiation. It states that if you have a function of the form \( f(x) = x^n \), its derivative is \( f'(x) = nx^{n-1} \).
When using the power rule on our demand function term \( 77p^{-2} \), we get:
  • \( f(p) = 77p^{-2} \)
  • Differentiating: \( f'(p) = -154p^{-3} \)
Thus, the power rule simplifies finding the rate of change of terms in the demand function with respect to price.

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Most popular questions from this chapter

Currently, 1800 people ride a certain commuter train each day and pay $$\$ 4$$ for a ticket. The number of people \(q\) willing to ride the train at price \(p\) is \(q=600(5-\sqrt{p})\). The railroad would like to increase its revenue. (a) Is demand elastic or inelastic at \(p=4 ?\) (b) Should the price of a ticket be raised or lowered?

The size of a certain insect population is given by \(P(t)=300 e^{.01 t}\), where \(t\) is measured in days. (a) How many insects were present initially? (b) Give a differential equation satisfied by \(P(t)\). (c) At what time will the population double? (d) At what time will the population equal \(1200 ?\)

After \(t\) hours there are \(P(t)\) cells present in a culture, where \(P(t)=5000 e^{.2 t}\). (a) How many cells were present initially? (b) Give a differential equation satisfied by \(P(t)\). (c) When will the population double? (d) When will 20,000 cells be present?

A movie theater has a seating capacity of 3000 people. The number of people attending a show at price \(p\) dollars per ticket is \(q=(18,000 / p)-1500\). Currently, the price is $$\$ 6$$ per ticket. (a) Is demand elastic or inelastic at \(p=6 ?\) (b) If the price is lowered, will revenue increase or decrease?

How much money must you invest now at \(4.5 \%\) interest compounded continuously to have $$\$ 10,000$$ at the end of 5 years?
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