Suppose that the market equilibrium price for printers is P*=100 and quantity traded of printers is Q=250. At the equilibrium point, the own price elasticity of demand is: Eqx,px = -4. a) Find the demand equation. b) What is the price that maximizes revenues? What are total revenues at that price?
Suppose that the market equilibrium price for printers is P*=100 and quantity traded of printers is Q=250. At the equilibrium point, the own price elasticity of demand is: Eqx,px = -4. a) Find the demand equation. b) What is the price that maximizes revenues? What are total revenues at that price?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
CHECK OUT IMAGE & PLEASE SHOW WORK
![### Equilibrium Price and Quantity Analysis
**Problem Statement:**
Suppose that the market equilibrium price for printers is \( P^* = 100 \) and the quantity traded of printers is \( Q = 250 \). At the equilibrium point, the own price elasticity of demand is:
\[ E_{Qx,Px} = -4 \]
**Tasks:**
a) Find the demand equation.
b) What is the price that maximizes revenues? What are total revenues at that price?
### Detailed Explanation:
**Elasticity and Demand Equation:**
The own price elasticity of demand ( \( E_{Q,P} \) ) is given by:
\[ E_{Q,P} = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} \]
Given:
- \( P = 100 \)
- \( Q = 250 \)
- \( E_{Qx,Px} = -4 \)
To find the demand equation, which is generally of the form \( Q = a - bP \), follow these steps:
1. Use the elasticity formula:
\[ -4 = \frac{\partial Q}{\partial P} \cdot \frac{100}{250} \]
2. Rearrange to find \( \frac{\partial Q}{\partial P} \):
\[ -4 = \frac{\partial Q}{\partial P} \cdot 0.4 \]
\[ \frac{\partial Q}{\partial P} = \frac{-4}{0.4} = -10 \]
So, \( -b = -10 \), thus \( b = 10 \).
3. Use the equilibrium condition \( Q = a - bP \):
\[ 250 = a - 10 \cdot 100 \]
\[ 250 = a - 1000 \]
\[ a = 1250 \]
Thus, the demand equation is:
\[ Q = 1250 - 10P \]
**Revenue Maximization:**
Revenue (\( R \)) is given by:
\[ R = P \cdot Q \]
To maximize revenue, we use the relationship between revenue and elasticity:
- Revenue is maximized when the price elasticity of demand equals -1.
Given the initial elasticity and the linear demand equation, compute:
\[ E_{Q,P} = -1 = \frac{\partial Q}{\partial P} \cdot \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F300cbca4-f6ea-4709-a050-de562527f506%2F932a2883-feb1-4bbb-ab15-3b68350b5ba8%2Fi280wo9_processed.png&w=3840&q=75)
Transcribed Image Text:### Equilibrium Price and Quantity Analysis
**Problem Statement:**
Suppose that the market equilibrium price for printers is \( P^* = 100 \) and the quantity traded of printers is \( Q = 250 \). At the equilibrium point, the own price elasticity of demand is:
\[ E_{Qx,Px} = -4 \]
**Tasks:**
a) Find the demand equation.
b) What is the price that maximizes revenues? What are total revenues at that price?
### Detailed Explanation:
**Elasticity and Demand Equation:**
The own price elasticity of demand ( \( E_{Q,P} \) ) is given by:
\[ E_{Q,P} = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} \]
Given:
- \( P = 100 \)
- \( Q = 250 \)
- \( E_{Qx,Px} = -4 \)
To find the demand equation, which is generally of the form \( Q = a - bP \), follow these steps:
1. Use the elasticity formula:
\[ -4 = \frac{\partial Q}{\partial P} \cdot \frac{100}{250} \]
2. Rearrange to find \( \frac{\partial Q}{\partial P} \):
\[ -4 = \frac{\partial Q}{\partial P} \cdot 0.4 \]
\[ \frac{\partial Q}{\partial P} = \frac{-4}{0.4} = -10 \]
So, \( -b = -10 \), thus \( b = 10 \).
3. Use the equilibrium condition \( Q = a - bP \):
\[ 250 = a - 10 \cdot 100 \]
\[ 250 = a - 1000 \]
\[ a = 1250 \]
Thus, the demand equation is:
\[ Q = 1250 - 10P \]
**Revenue Maximization:**
Revenue (\( R \)) is given by:
\[ R = P \cdot Q \]
To maximize revenue, we use the relationship between revenue and elasticity:
- Revenue is maximized when the price elasticity of demand equals -1.
Given the initial elasticity and the linear demand equation, compute:
\[ E_{Q,P} = -1 = \frac{\partial Q}{\partial P} \cdot \frac{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education