Problem 15 - 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. \(D(p)=50-2 p\) units

Step-by-Step Solution

Answer

The maximum possible market price is $25 per unit.

Step 1: Understand the Problem

We need to find a market price at which the demand for units becomes zero according to the demand function \( D(p) = 50 - 2p \). This involves solving for \( p \) such that \( D(p) = 0 \).

Step 2: Set Demand to Zero

Set the demand equation \( D(p) = 50 - 2p \) equal to zero to find the price \( p \) at which demand is zero. \[ 50 - 2p = 0 \]

Step 3: Solve for Price

Solve the equation \( 50 - 2p = 0 \) for \( p \). Add \( 2p \) to both sides: \[ 50 = 2p \] Divide by 2 to isolate \( p \): \[ p = 25 \]

Step 4: Interpret the Result

The solution \( p = 25 \) means that when the price is \(25 per unit, the demand becomes zero. This suggests that the highest price at which demand is zero (and thus unbounded) is \)25 per unit.

Key Concepts

Understanding Market Price in Demand Functions

In economics, the market price is the amount consumers are willing to pay for a product or service at a given time. It is a fundamental concept in determining how much of a product will be sold.
The market price is influenced by various factors such as competition, demand, and supply levels.
It helps businesses and consumers make decisions about production and consumption.

• **Influence on Demand**: As the market price rises, the demand typically decreases because fewer consumers are willing to pay the higher price.
• **Market Equilibrium**: The market price plays a crucial role in achieving market equilibrium, where the quantity supplied equals the quantity demanded.
• **Dynamic Nature**: Market prices can change based on shifts in consumer preferences and external economic conditions.

In the context of a demand function, the market price can directly impact the quantity demanded, as shown in the example demand function \(D(p) = 50 - 2p\). Here, changes in \(p\) (the price), determine how many units are demanded.

Exploring the Concept of Zero Demand

Zero demand refers to a situation where no consumers are willing to buy a product at a particular price point. It is critical in understanding price elasticity and helps identify the limits of pricing strategies.
Zero demand does not necessarily mean the product is undesirable but rather that at a specific price, demand falls to zero.

• **Price Sensitivity**: It indicates the price sensitivity of a product, highlighting how raising prices can eventually lead to no demand.
• **Consumer Preferences**: Zero demand might occur if what is demanded greatly exceeds the perceived value of the product to consumers.
• **Strategic Pricing**: To avoid zero demand, businesses must carefully choose their pricing strategies, ensuring prices remain within consumers' willingness to pay.

For example, as shown in the demand equation \(D(p) = 50 - 2p\), when the market price reaches \(25\) USD per unit, demand hits zero, indicating that no customer is prepared to buy the units at this price.

Deriving the Demand Equation

A demand equation is a mathematical formula that captures the relationship between the price of a good and the quantity demanded. It is a key tool used by economists to model and study consumer behavior.
A basic demand equation is often linear, as seen with \(D(p) = 50 - 2p\), where the term '50' is the intercept and '-2p' represents the slope.

• **Components of the Equation**:
  • The intercept (50) depicts the initial level of demand when price \(p\) is zero. This is the maximum quantity demanded.
  • The slope (-2) shows how demand decreases as price increases. Here, for each dollar increase in price, demand reduces by 2 units.
• **Graphical Representation**: On a graph, the demand equation appears as a straight line descending from left to right, highlighting the inverse relationship between price and demand.
• **Practical Application**: By altering the price variable \(p\), the demand equation helps determine the optimal price and forecast sales.

This equation allows businesses to predict how changes in price might impact the quantity of goods sold, using knowledge of where zero demand points occur.