Problem 23 - Calculus Concepts

Question

Wooden Chairs The demand for wooden chairs can be modeled as $$ D(p)=-0.01 p+5.55 \text { million chairs } $$ where \(p\) is the price (in dollars) of a chair. a. According to the model, at what price will consumers no longer purchase chairs? Is this price guaranteed to be the highest price any consumer will pay for a wooden chair? Explain. b. What quantity of wooden chairs will consumers purchase when the market price is \(\$ 99.95 ?\) c. Calculate the amount that consumers are willing and able to spend to purchase 3 million wooden chairs. d. Calculate the consumers' surplus when consumers purchase 3 million wooden chairs.

Step-by-Step Solution

Answer

a. Consumers stop buying at $555, not the guaranteed max. b. 4.55 million chairs are bought at $99.95. c. Spend $765 million for 3 million chairs. d. Consumer surplus is 4.425 million dollars.

Step 1: Analyzing the Demand Function

The demand function given is \( D(p) = -0.01p + 5.55 \), where \( D(p) \) is the demand in millions of chairs and \( p \) is the price in dollars. This linear model shows the relationship between price and demand. As the price \( p \) increases, \( D(p) \) decreases.

Step 2: Finding the Zero Demand Price (Part a)

To find the price where no chairs are purchased, set \( D(p) = 0 \) and solve for \( p \):\[ 0 = -0.01p + 5.55 \] \[ 0.01p = 5.55 \] \[ p = \frac{5.55}{0.01} = 555 \]. Therefore, consumers will no longer purchase chairs at a price of \$555. This is the theoretical maximum price according to this model; however, in real life, it is not guaranteed to be the highest price any consumer might pay due to factors like individual valuation and market dynamics.

Step 3: Calculating Demand at a Specific Price (Part b)

Calculate \( D(p) \) when \( p = 99.95 \):\[ D(99.95) = -0.01(99.95) + 5.55 \] \[ D(99.95) = -0.9995 + 5.55 \] \[ D(99.95) = 4.5505 \]. Thus, consumers will purchase approximately 4.55 million chairs when the price is \$99.95.

Step 4: Solving for Consumer Expenditure (Part c)

To find the price at which consumers will buy 3 million chairs, set \( D(p) = 3 \):\[ 3 = -0.01p + 5.55 \] \[ 0.01p = 5.55 - 3 \] \[ p = \frac{2.55}{0.01} = 255 \]. So, consumers are willing to pay \$255 each for 3 million chairs, leading to a total expenditure of \( 3 \times 255 = 765 \) million dollars.

Step 5: Calculating Consumer Surplus (Part d)

Consumer surplus is the area above the price line and below the demand curve up to the quantity purchased. It is given by the integral from 0 to 3 million of \( D(p) \) minus the total expenditure. The function must be integrated:\[ \int_0^{3} (5.55 - 0.01p) \, dp \]First, find the total area under the demand curve up to 3 million:\[ = (5.55 \times 3) - 0.5(0.01 \times 3 \times 255) \]Calculate consumer surplus:\[ = 8.25 - 0.5(7.65) = 8.25 - 3.825 = 4.425 \] million dollars.

Key Concepts

Consumer Surplus

Consumer surplus represents the difference between what consumers are willing to pay for a good or service and what they actually pay. Imagine going to a market where the price of a wooden chair is set at $255. If you were willing to pay up to $300 for that chair, the $45 difference is your consumer surplus. This surplus indicates the additional satisfaction or economic benefit you receive.
In economic terms, consumer surplus can be visualized as the area under the demand curve and above the price level up to the quantity consumed. It's a measure of the consumer's gain from participating in the market. In our original exercise, when consumers purchase 3 million chairs, the consumer surplus was calculated to be $4.425 million.

• Consumer surplus highlights the efficiency and welfare aspects of markets.
• It can also indicate the strength of demand and how much more consumers value a product than its market price.
• In this model, despite the price being $255, consumers derive additional value adding up to a noticeable economic gain.

Linear Model

A linear model is a mathematical representation that describes a relationship between two variables with a straight line. It is defined by the equation of a line, typically written as:\[ y = mx + b \]where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope, and \( b \) is the y-intercept.
In the context of our demand function \( D(p) = -0.01p + 5.55 \), we can see it's a linear equation where:

• The independent variable \( p \) is the price of the wooden chair.
• The dependent variable \( D(p) \) is the demand for chairs in millions.
• The slope \(-0.01\) indicates that for every dollar increase in price, the demand decreases by 0.01 million chairs.
• The y-intercept \(5.55\) represents the quantity demanded when the price is zero, though practically, this scenario is not feasible.

Linear models are easy to work with because the relationship they describe is straightforward and predictable. These models are powerful tools in economic modeling and decision-making, providing a fundamental understanding of how price adjustments impact demand.

Price-Demand Relationship

The price-demand relationship is a fundamental concept in economics showing how the quantity demanded of a good changes as its price changes. Generally, demand curves slope downwards, indicating that as prices decrease, demand increases and vice versa.
For the wooden chairs, the demand function \( D(p) = -0.01p + 5.55 \) illustrates this relationship clearly:

• When prices are low, consumers buy more chairs—this is a direct portrayal of higher affordability attracting more buyers.
• Conversely, as prices increase, fewer chairs are bought—illustrating price sensitivity and budget constraints affecting buying decisions.
• The negative slope (-0.01) quantifies this sensitivity by showing exactly how much demand shrinks with each dollar increase in price.

Understanding this relationship helps businesses and economists predict consumer behavior and make informed pricing decisions. It allows for anticipating changes in market behavior based on price adjustments, assisting in optimizing sales strategies and maximizing revenue.

Economic Modeling

Economic modeling involves constructing abstract representations of economic processes to predict, analyze, and explain economic phenomena. These models can be simplistic or complex, but their essence is to provide clarity on how different variables interact.
In this exercise, the demand function for wooden chairs \( D(p) = -0.01p + 5.55 \) is a simplified economic model capturing how demand fluctuates with price.

• Such models offer insights into consumer behavior, market trends, and price strategies.
• They are valuable tools for forecasting future market conditions, assessing the impact of economic policies, and strategizing business operations.
• Despite their limitations in addressing every real-world intricacy, economic models like this linear demand function are crucial for understanding broad market dynamics.

By modeling these relationships, economists gain the ability to simulate different scenarios, like predicting what happens when prices change, allowing businesses to optimize their production and pricing strategies more effectively.