Problem 17 - Calculus Concepts

Question

Determine whether there is a specific market price above which demand is zero or price per unit is unbounded. Write the maximum possible market price, using dollars per unit as the units of measure for input. \(D(p)=35-7 \ln p\) units

Step-by-Step Solution

Answer

Demand becomes zero at approximately $148.41 per unit.

Step 1: Understanding the demand function

The given demand function is \( D(p) = 35 - 7 \ln p \), where \( p \) is the market price. Our task is to find if there is a market price \( \bar{p} \) such that the demand becomes zero.

Step 2: Set demand to zero

To find the market price where demand is zero, we set the demand function equal to zero: \( 35 - 7 \ln p = 0 \).

Step 3: Solve for the market price

Rearrange the equation from Step 2 to find \( \ln p \):\[ 7 \ln p = 35 \]Divide both sides by 7:\[ \ln p = 5 \].

Step 4: Exponentiate to find \( p \)

To solve for \( p \), take the exponential of both sides of the equation from Step 3:\[ p = e^5 \].

Step 5: Calculate the maximum possible price

The market price \( \bar{p} \), which makes the demand zero, is calculated as:\[ p = e^5 \ \approx 148.41 \].Therefore, demand becomes zero when the price is approximately \$148.41 per unit.

Key Concepts

Market Price

The market price is a critical component when analyzing demand functions. It refers to the current price at which a good or service is sold in the marketplace. Understanding the relationship between market price and demand is vital for businesses, as it can determine their revenue potential. Market price can fluctuate due to various factors, including supply and demand, consumer preferences, and market competition.

In the given exercise, we are tasked with identifying a specific price point where the demand for a product drops to zero. This involves analyzing the demand function provided, which is dependent on the market price. The market price is key here, as it directly influences how much consumers are willing to purchase.

Zero Demand

Zero demand occurs when consumers have no interest in buying a product at a given price. In economic models, zero demand signifies a point at which the market no longer supports buying behavior for a particular price.

In the context of the exercise, the demand function is represented as: \[ D(p) = 35 - 7 \ln p \]To find the point of zero demand, we set this function equal to zero.
This leads us to the equation: \[ 35 - 7 \ln p = 0 \] By solving this equation, we can identify the market price where demand is non-existent.

Exponential Function

An exponential function is a mathematical expression where a variable appears in the exponent. This kind of function is very useful when modeling growth or decay processes, like population growth or radioactive decay, but it also plays a crucial role in economics and finance.

In our exercise, the concept of the exponential function comes in while solving for the market price where demand is zero. After finding \( \ln p = 5 \), we need to exponentiate both sides to solve for \( p \): \[ p = e^5 \].This is where the exponential function transforms a logarithmic equation back to its original nonlinear form, providing the numerical market price solution.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse operation of taking an exponential function with base \( e \), where \( e \approx 2.71828 \). The natural logarithm is particularly significant in scientific and financial contexts because it simplifies the manipulation of exponential relationships.

In the provided demand function \( D(p) = 35 - 7 \ln p \), the natural logarithm of the price \( ( \ln p ) \) is what helps us relate the market price to the demand. To find the point where demand becomes zero, the equation \( \ln p = 5 \) is solved using the natural logarithm properties, highlighting how this concept is essential for transforming between different types of mathematical expressions.