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Chapter 9: Problem 51

For a demand function \(D(p),\) the elasticity of demand (see page 294 ) is defined as \(E=\frac{-p D^{\prime}}{D} .\) Find demand functions \(D(p)\) that have constant elasticity by solving the differential equation \(\frac{-p D^{\prime}}{D}=k,\) where \(k\) is a constant.

Short Answer

Expert verified
The demand functions with constant elasticity are of the form \(D(p) = A p^{-k}\).

Step by step solution

01

Rewrite the Differential Equation

Given the equation \(-\frac{p D'}{D} = k\). First, multiply both sides by \(-1\) to get \(\frac{p D'}{D} = -k\).
02

Rearrange Terms

Rearrange the equation to separate variables. You get \(D' = \frac{-k D}{p}\).
03

Multiply Through by \(p\)

Multiply every term by \(p\) to facilitate integration: \(p D' = -k D\).
04

Separate Variables

Write the equation in separable form: \(\frac{D'}{D} = \frac{-k}{p}\).
05

Integrate Both Sides

Integrate both sides of the equation: \(\int \frac{1}{D} \, dD = \int \frac{-k}{p} \, dp\). This gives \(\ln |D| = -k \ln |p| + C\), where \(C\) is the integration constant.
06

Solve for \(D(p)\)

Exponentiate both sides to solve for \(D\): \(D = e^C p^{-k}\). Rename \(e^C\) as a new constant, say \(A\), to get the general solution \(D(p) = A p^{-k}\).

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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 describes the relationship between the price of a good and the quantity demanded by consumers. In mathematical terms, it is often expressed as a function \(D(p)\), where \(D\) represents demand and \(p\) represents price. Understanding this function is crucial for businesses as it helps predict how changes in price affect demand.
  • It provides insights into consumer behavior and helps optimize pricing strategies to maximize revenue and profit.
  • The function is typically downward sloping, indicating that as price decreases, demand increases.
  • The precise form of a demand function can vary depending on market conditions and consumer preferences.
Grasping the demand function helps in illustrating how sensitive demand is to price changes, which is analyzed through the elasticity of demand.
Differential Equation
Differential equations are mathematical equations that involve derivatives, reflecting how a function changes over time or space. In the context of elasticity of demand, the particular differential equation here is \(- \frac{p D'}{D} = k\).
This equation represents how changes in price \(p\) and the demand function \(D(p)\) interact with the elasticity \(E\).
  • The given differential equation models the condition for demand functions that maintain constant elasticity.
  • Rewriting and solving this equation allows us to find demand functions that exhibit this trait.
Solving such equations usually involves steps like multiplying through by constants, separating variables, and integrating to find the solution.
Constant Elasticity
Constant elasticity in a demand function implies that the elasticity of demand is the same for any price level. This means the percentage change in quantity demanded is proportional to the percentage change in price.
  • The constant elasticity of demand equation is given by \(E = \frac{-p D'}{D} = k\).
  • This concept is key when examining goods whose demand is relatively stable irrespective of price changes.
  • For constant elasticity, the demand equation simplifies to \(D(p) = A p^{-k}\), where \(k\) is a constant elastic factor.
Such functions are especially useful in economic modeling as they simplify analysis while still capturing essential behaviors of real-world demand.
Integration Constant
When solving differential equations, like the one for constant elasticity, an integration constant \(C\) emerges from the process of integration. This constant represents the initial conditions of the system and is vital for determining the precise form of the demand function.
  • During integration of the separated variables, \(\ln |D| = -k \ln |p| + C\) arises, where \(C\) is the integration constant.
  • Meglecting this constant can lead to incorrect modeling of the demand function.
  • By exponentiating to solve for \(D(p)\), the term \(e^C\) is recognized as another constant \(A\), representing the demand at a particular price level.
Understanding the integration constant is essential for fully determining how a chosen model fits real-world data.

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Most popular questions from this chapter

Determine whether each differential equation is separable. (Do not solve it, just find whether it's separable.) $$ y^{\prime}=\ln (x y) $$

A 500-gallon tank is filled with water containing 0.2 ounce of impurities per gallon. Each hour, 200 gallons of water (containing 0.01 ounce of impurities per gallon) is added and mixed into the tank, while an equal volume of water is removed. a. Write a differential equation and initial condition that describe the total amount \(y(t)\) of impurities in the tank after \(t\) hours. b. Solve this differential equation and initial condition. c. Use your solution to find when the impurities will reach 0.05 ounce per gallon, at which time the water may be used for drinking. d. Use your solution to find the "long-run" amount of impurities in the tank.

Let \(y(t)\) be the value of a commercial building (in millions of dollars) after \(t\) years. a. Write a differential equation that says that the rate of growth of the value of the building is equal to two times the one-half power of its present value. b. Write an initial condition that says that at time zero the value of the building is 9 million dollars. c. Solve the differential equation and initial condition. d. Use your solution to find the value of the building at time \(t=5\) years.

A Ponzi scheme is an investment fraud that promises high returns but in which funds, instead of being invested, are merely used to pay returns to the investors and profits to the fund manager. To avoid running out of money, new investors must be brought in at an increasing rate to provide funds to pay off existing investors. Eventually the scheme must collapse in debt when not enough new investors can be found. Suppose that each of 10 investors deposits \(\$ 100,000\) into a fund and is promised a \(20 \%\) annual return. However, the \(\$ 1,000,000\) collected is used to pay each investor the required \(\$ 20,000\) return, with the remaining \(\$ 800,000\) kept by the fund manager. Let \(y(t)\) be the total number of investors needed after \(t\) years so that incoming funds will be enough to pay the existing investors plus the manager's \(\$ 800,000 .\) Representing dollar amounts in thousands, we have $$ \begin{array}{l} \left(\begin{array}{c} \text { Annual } \\ \text { inflow } \end{array}\right)=100 \cdot \frac{d y}{d t} \\ \left(\begin{array}{l} \text { Annual } \\ \text { outflow } \end{array}\right)=20 \cdot y+800 \end{array} $$ For inflow to equal outflow, we must have the differential equation and initial condition $$ \left\\{\begin{array}{l} 100 \cdot \frac{d y}{d t}=20 y+800 \\ y(0)=10 \end{array}\right. $$ a. Solve this differential equation and initial condition. b. Use your solution to find how many investors would be needed after 10 years, after 20 years, after 30 years, and after 50 years.

A Ponzi scheme is an investment fraud that promises high returns but in which funds, instead of being invested, are merely used to pay returns to the investors and profits to the fund manager. To avoid running out of money, new investors must be brought in at an increasing rate to provide funds to pay off existing investors. Eventually the scheme must collapse in debt when not enough new investors can be found. Suppose that each of 20 investors deposits \(\$ 100,000\) into a fund and is promised a \(25 \%\) annual return. However, the \(\$ 2,000,000\) collected is used to pay each investor the required \(\$ 25,000\) return, with the remaining \(\$ 1,500,000\) kept by the fund manager. Let \(y(t)\) be the total number of investors needed after \(t\) years so that incoming funds will be enough to pay the existing investors plus the manager's \(\$ 1,500,000\). Representing dollar amounts in thousands, we have $$ \begin{array}{l} \left(\begin{array}{c} \text { Annual } \\ \text { inflow } \end{array}\right)=100 \cdot \frac{d y}{d t} \\ \left(\begin{array}{l} \text { Annual } \\ \text { outflow } \end{array}\right)=25 \cdot y+1500 \end{array} $$ For inflow to equal outflow, we must have the differential equation and initial condition $$ \left\\{\begin{array}{l} 100 \cdot \frac{d y}{d t}=25 y+1500 \\ y(0)=20 \end{array}\right. $$ a. Solve this differential equation and initial condition. b. Use your solution to find how many investors would be needed after 10 years, after 20 years, after 30 years, and after 50 years.
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