Problem 24 - Calculus Concepts

Question

The demand for ceiling fans can be modeled as \(D(p)=25.92\left(0.996^{p}\right)\) thousand ceiling fans where \(p\) is the price (in dollars) of a ceiling fan. a. According to the model, is there a price above which consumers will no longer purchase fans? If so, what is it? b. Calculate the amount that consumers are willing and able to spend to purchase 18 thousand ceiling fans. c. How many fans will consumers purchase when the market price is \(\$ 100\) d. Calculate the consumers' surplus when the market price is \(\$ 100\).

Step-by-Step Solution

Answer

a. No finite price; b. $57.13; c. 17,364 fans; d. $144.48 in surplus.

Step 1: Understanding the No Purchase Condition

In part (a), we want to determine if there is a price above which no ceiling fans are purchased. This occurs when the demand, \(D(p)\), equals zero. Analyzing the function, \(D(p) = 25.92 \times 0.996^p\), we see that \(0.996^p\) approaches zero as \(p\) approaches infinity but never truly reaches zero. Therefore, according to the model, there is no finite price at which consumers stop purchasing entirely.

Step 2: Calculating Price for 18,000 Ceiling Fans

To solve part (b), set the demand function equal to 18 thousand fans: \(25.92 \times 0.996^p = 18\). Divide both sides by 25.92 to get \(0.996^p = \frac{18}{25.92}\). Taking the natural logarithm of both sides gives \(p = \frac{\ln(\frac{18}{25.92})}{\ln(0.996)}\). Solving this equation yields \(p \approx 57.13\). So the price consumers are willing to pay for 18 thousand fans is approximately \$57.13.

Step 3: Calculating Demand at $100 Price Point

For part (c), substitute \(p = 100\) into the demand function: \(D(100) = 25.92 \times 0.996^{100}\). Calculating \(0.996^{100} \approx 0.670\), hence \(D(100) \approx 25.92 \times 0.670 = 17.364\). Therefore, at a \$100 price point, consumers will purchase approximately 17.364 thousand ceiling fans, or 17,364 fans.

Step 4: Finding Consumer Surplus at $100

For part (d), the consumer surplus is the area under the demand curve above the price line \(p = 100\). It is calculated as the integral from 0 to 100 of \(D(p)\) minus \(100 \times 17.364\), the cost when \(p = 100\). Compute \(\int_{0}^{100} 25.92 \times 0.996^p \, dp\), which evaluates to approximately \( \frac{25.92}{\ln(0.996)} \times (0.996^100 - 1) \). After calculating, the consumer surplus is approximately \$144.48.

Key Concepts

Consumer Surplus

Consumer surplus is a key economics concept that measures the benefit to consumers from purchasing goods at a price lower than what they would be willing to pay. Essentially, it represents the difference between what consumers are prepared to pay for a good or service and the market price they actually pay. It can be visualized as the area between the demand curve and the price line in a market graph.
For instance, if consumers are willing to pay $150 for a ceiling fan, but the market price is only $100, then the consumer surplus is $50 per fan. To find the consumer surplus, we typically compute the area under the demand curve from zero to the quantity purchased, subtracting the total money spent at market price. This is often done using integral calculus when the demand function is complex.
The concept of consumer surplus is crucial for economic policy and business strategies as it highlights consumer valuations and potential welfare gains from different market interventions.

Market Price

Market price represents the monetary value assigned to a good or service in a marketplace, where sellers and buyers agree upon an exchange rate. It is the price that clears the market by balancing supply and demand, and all transactions occur at this price.
The market price can fluctuate based on several factors, such as changes in consumer preferences, costs of production, or the introduction of new competitors in the market. When calculating demand or analyzing market dynamics, market price serves as a fundamental benchmark.
In our example, when the market price of the ceiling fans is set at $100, we can determine other important metrics such as how many fans consumers will purchase or the associated consumer surplus. Hence, knowing the market price allows us to understand consumer behavior and overall market efficiency.

Natural Logarithm

The natural logarithm, represented as \( ext{ln}(x)\), is a mathematical function that helps us solve equations involving exponential relationships. It is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. Natural logarithms are crucial in analyzing economic models, especially where growth or decay is involved, such as computing compound interest or decay rates.
In the context of demand functions like \(D(p)=25.92(0.996^{p})\), natural logarithms can be used to simplify and solve equations for price or quantity demanded. For example, when solving for the price point at which demand equals a specific quantity, we manipulate the exponential equation into a logarithmic form to find the unknown variable. This is because natural logarithms are the inverse operation of exponentiation.
Understanding how to work with natural logarithms can be powerful, especially when predicting future trends or solving complex economic functions. It's an essential tool for economists and financial analysts alike.