Problem 19 - 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. \(D(p)=3.6 p^{-0.8}\) units

Step-by-Step Solution

Answer

There is no specific finite market price above which demand is zero.

Step 1: Define the Condition for Zero Demand

To determine the price at which demand becomes zero, observe that demand function \( D(p) = 3.6 p^{-0.8} \) involves \( p^{-0.8} \). The demand function approaches zero as the price \( p \) approaches infinity because the exponent of \( p \) is \(-0.8\), which indicates an inverse relationship with \( p \).

Step 2: Evaluate the Function for Maximum Price

Since \( D(p) = 3.6 p^{-0.8} \) never actually reaches zero for any finite value of \( p \), the demand approaches zero as \( p \) approaches infinity. Therefore, there is no single finite market price above which demand is exactly zero.

Step 3: Analyze Price Bound for Unbounded Demand

To find the price above which demand is unbounded, set \( p \to 0^+ \) in the function \( D(p) = 3.6 p^{-0.8} \). As \( p \to 0^+ \), the term \( p^{-0.8} \) tends towards infinity, suggesting that the demand is unbounded as price approaches zero. Therefore, there is no specific upper bound where demand becomes unbounded in the positive price range.

Key Concepts

Inverse Relationship

In the study of demand functions, an inverse relationship is a critical concept to understand. Simply put, an inverse relationship occurs when two variables move in opposite directions. In the context of the demand function given in the exercise, \( D(p) = 3.6p^{-0.8} \), the price \( p \) and the demand \( D(p) \) exhibit this inverse relationship.

• As the price \( p \) increases, the demand \( D(p) \) will decrease.
• Conversely, as the price \( p \) decreases, the demand \( D(p) \) will increase.

This inverse relationship is illustrated through the negative exponent \(-0.8\), which indicates that demand is inversely proportional to price. Understanding this relationship helps in predicting how changes in market price can influence the quantity demanded by consumers. Such dynamics are foundational in economics, affecting pricing strategies and consumer purchasing decisions. By examining the exponent's value, we can gain insights into the sensitivity of demand to price changes, known as the price elasticity of demand.

Market Price

The market price is essentially the cost at which goods are sold to consumers in the market. Market dynamics, such as supply and demand, largely influence this price. In the exercise's context, understanding the maximum possible market price where demand approaches zero is key. For the function \( D(p) = 3.6p^{-0.8} \), the demand never actually becomes zero for any finite market price. This is because the demand continually decreases as price increases but never truly vanishes. Mathematically, this means that as \( p \to \infty \), \( D(p) \) approaches zero but does not equal zero. Thus, there isn't a specific market price where demand entirely ceases. Instead, prices can increase infinitely, causing demand to dwindle toward zero without ever reaching it completely. Recognizing that no finite market price leads to zero demand is important for businesses and economists trying to set prices optimally.

Price-Demand Analysis

Price-demand analysis involves examining how changes in market price affect the demand for a product. This analysis is crucial for businesses to make informed pricing decisions. In our example with \( D(p) = 3.6p^{-0.8} \), understanding how demand behaves at extreme price points is part of detailed price-demand analysis.

• As the price \( p \to 0^+ \), the demand \( D(p) \) becomes unbounded, meaning it can theoretically become infinitely large.
• This indicates very high demand at very low prices, a common observation used in strategies like penetration pricing.

Such analysis helps businesses predict customer behavior and optimize pricing strategies to balance profit and demand. With the demand function provided, businesses can better understand their demand curve and adjust their strategies accordingly. Recognizing the continuous nature of demand over varying prices also assists in enhancing market competitiveness.