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Chapter 4: Problem 18

Find the elasticity \(E\) at the given points and determine whether demand is inelastic, elastic, or unit elastic. \(x=10+\frac{1}{p}\) a. \(p=0.1\) b. \(p=1\)

Short Answer

Expert verified
Demand is inelastic at both \(p=0.1\) and \(p=1\).

Step by step solution

01

Calculate the demand function

The demand function is given as: \(x = 10 + \frac{1}{p}\). It shows the quantity demanded as a function of the price \(p\).
02

Find the derivative of x with respect to p

Differentiate the demand function \(x = 10 + \frac{1}{p}\) with respect to \(p\). This gives \(\frac{dx}{dp} = -\frac{1}{p^2}\).
03

Use the formula for price elasticity of demand

Price elasticity of demand is given by the formula: \(E = \left( \frac{p}{x} \right) \cdot \frac{dx}{dp}\). We will use this formula to calculate the elasticity at the given prices.
04

Calculate elasticity at p=0.1

Substitute \(p = 0.1\) into the demand function: \(x = 10 + \frac{1}{0.1} = 10 + 10 = 20\). Now calculate the elasticity: \[E = \left( \frac{0.1}{20} \right) \cdot \left( -\frac{1}{(0.1)^2}\right) = \left( \frac{0.1}{20} \right) \cdot (-100) = -0.5\]Thus, the elasticity at \(p=0.1\) is \(-0.5\).
05

Determine elasticity type at p=0.1

Since \(|E| = 0.5 < 1\), the demand is inelastic at \(p=0.1\).
06

Calculate elasticity at p=1

Substitute \(p = 1\) into the demand function: \(x = 10 + \frac{1}{1} = 10 + 1 = 11\). Now calculate the elasticity:\[E = \left( \frac{1}{11} \right) \cdot \left( -\frac{1}{1^2} \right) = \frac{1}{11} \cdot (-1) = -\frac{1}{11} \approx -0.09\]So, the elasticity at \(p=1\) is approximately \(-0.09\).
07

Determine elasticity type at p=1

Since \(|E| = 0.09 < 1\), the demand is inelastic at \(p=1\).

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Key Concepts

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

Demand Function
A demand function is a mathematical expression that shows the relationship between the quantity of a good demanded and its price. In our exercise, the demand function is given by: \[ x = 10 + \frac{1}{p} \]This formula tells us how much of a product will be demanded at different prices. Here, \(x\) represents the quantity demanded, and \(p\) is the price per unit.
  • The term \(10\) could represent a baseline demand that exists regardless of price.
  • The term \(\frac{1}{p}\) shows that as the price decreases, the quantity demanded increases, demonstrating the law of demand.
Understanding a demand function is crucial as it helps businesses determine how changing prices can affect sales. It provides valuable insight into consumer behavior and informs pricing strategies.
Differentiation
Differentiation is a process used in calculus to find the rate at which a function is changing at any given point. In economics, we use differentiation to understand how a small change in one variable, like price, will affect another, like quantity demanded.Given the demand function in our problem:\[ x = 10 + \frac{1}{p} \]We differentiate this function with respect to \(p\) to find \(\frac{dx}{dp}\). The differentiation gives us:\[ \frac{dx}{dp} = -\frac{1}{p^2} \]
  • This derivative \(-\frac{1}{p^2}\) shows the rate of change of the quantity demanded concerning the price.
  • It indicates how responsive the demand for the product is when the price changes slightly.
Differentiation is key to calculating the price elasticity of demand, which helps to analyze consumer responsiveness to price changes.
Inelastic Demand
Inelastic demand describes a situation where a change in the price of a product results in a less-than-proportional change in the quantity demanded. In other words, consumers are not very responsive to price changes with inelastic demand.To determine if demand is inelastic at specific price points, we use the formula for price elasticity of demand:\[E = \left( \frac{p}{x} \right) \cdot \frac{dx}{dp}\]Two cases are considered in our exercise:- **At \(p = 0.1\):** The elasticity \(E = -0.5\). Since \(|E| = 0.5 < 1\), demand is inelastic.- **At \(p = 1\):** The elasticity \(E \approx -0.09\). Since \(|E| = 0.09 < 1\), demand remains inelastic.In these examples, the absolute values of elasticity are less than 1. This means that changes in price lead to smaller changes in the quantity demanded. Understanding inelastic demand is essential for businesses as it helps them predict how price adjustments will affect total revenue.

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Most popular questions from this chapter

Biological Control Lysyk \(^{44}\) (refer to previous exercise) also constructed a mathematical model given approximately by the equation \(r(t)=0.0002 t(t-14) \sqrt{36-t}\) where \(t\) is temperature in degrees Celsius and \(r\) is the intrinsic rate of increase (females per females per day). Find \(r^{\prime}(t)\)

Biology The free water vapor diffusion coefficient \(D\) in soft woods \(^{21}\) is given by $$ D(T)=\frac{167.2}{P}\left(\frac{T}{273}\right)^{1.75} $$ where \(T\) is the temperature and \(P\) is the constant atmospheric pressure measured in appropriate units. Find the rate of change of \(D\) with respect to \(T\).

Medicine Suppose that the concentration \(c\) of a drug in the blood \(t\) hours after it is taken orally is given by $$ c=\frac{3 t}{2 t^{2}+5} $$ What is the rate of change of \(c\) with respect to \(t ?\)

Transportation Costs Koo and coworkers \(^{89}\) used production cost data from the U.S. Department of Agriculture to estimate transportation costs for durum wheat. They found that the rail transportation cost \(R\) in dollars per ton from producing regions to durum mills was approximated by the equation \(R(D)=0.0143 D^{0.67743},\) where \(D\) is the distance in miles. Graph. Find values for \(R^{\prime}(D)\) at \(D=100,200\), \(400,\) and \(800 .\) Interpret what these numbers mean. What is happening? What does this mean in terms of economies of scale?

Economies of Scale in Food Retailing Using data from the files of the National Commission on Food Retail- ing concerning the operating costs of thousands of stores, Smith \(^{24}\) showed that the sales expense as a percent of sales \(S\) was approximated by the equation \(S(x)=0.4781 x^{2}-\) \(5.4311 x+16.5795,\) where \(x\) is in sales per square foot and \(x \leq 7\) a. Find \(S^{\prime}(x)\) for any \(x\). b. Find \(S^{\prime}(4), S^{\prime}(5), S^{\prime}(6),\) and \(S^{\prime}(7)\). Interpret what is happening. c. Graph the cost function on a screen with dimensions [0,9.4] by \([0,12] .\) Also graph the tangent lines at the points where \(x\) is \(4,5,6,\) and 7 . Observe how the slope of the tangent line is changing, and relate this to the observations made above concerning the rates of change. d. Use the available operations on your computer or graphing calculator to the find where the function attains a minimum.
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