Problem 18 - 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. \(D(p)=1.5 p^{-0.8}\) units

Step-by-Step Solution

Answer

No finite maximum market price; demand approaches zero as price increases indefinitely.

Step 1: Finding Demand Function Behavior

We are given the function \( D(p) = 1.5 p^{-0.8} \), where \( p \) represents price and \( D(p) \) the demand. The negative exponent suggests that the demand decreases as price increases.

Step 2: Analyzing Demand at High Prices

To determine if there is a price above which demand is zero, consider \( \lim_{{p \to \infty}} D(p) \). As \( p \) increases, \( p^{-0.8} \) approaches 0, making the demand approach 0.

Step 3: Absence of Finite Market Cap

There is no specific finite market price at which the demand is zero. The demand approaches zero asymptotically as price increases indefinitely.

Step 4: Concluding Maximum Market Price

Since demand never actually reaches zero, there is no maximum possible market price. The demand decreases continuously but never hits zero.

Key Concepts

Understanding Market Price

The term 'market price' refers to the current price at which a good or service is bought and sold in the marketplace. In the context of demand function analysis, it is crucial to understand that market price is determined by the interplay of supply and demand. If the market price is set too high, buyers may be less inclined to purchase, leading to decreased demand. Conversely, if the price is too low, demand may exceed supply, causing a shortage.
In the given exercise with the demand function, we explore whether there exists a specific market price beyond which the demand for units drops to zero. It's important to understand that the demand function provided, which shows how demand changes with price, does not necessarily specify a set market price. Instead, it allows us to analyze how changes in price affect the quantity demanded.

Exploring Demand Elasticity

Demand elasticity measures how sensitive the quantity demanded of a good is to a change in its price. In simpler terms, it tells us how much the demand for a product will fluctuate with price changes. The elasticity of demand can be categorized as elastic, inelastic, or unitary:

• If demand is elastic, a small change in price leads to a significant change in quantity demanded.
• Inelastic demand implies that changes in price have little effect on the quantity demanded.
• Unitary elasticity signifies that a change in price leads to a proportional change in quantity demanded, retaining the same total revenue.

In our function, where demand decreases as price rises due to the negative exponent, we can infer that there is some degree of elasticity. As the price increases significantly, demand steadily approaches zero but never becomes zero, indicating a fairly inelastic response for very high prices.

Understanding the Limit of Demand

The concept of the 'limit of demand' refers to the behavior of the demand function as the price variable approaches extreme values, such as zero or infinity. To determine the limit of demand in this exercise, we calculate \[\lim_{{p \to \infty}} D(p) = \lim_{{p \to \infty}} 1.5 p^{-0.8}\]The formula describes that as the price \( p \) approaches infinity, the term \( p^{-0.8} \) approaches zero. Therefore, demand \( D(p) \) approaches zero, but does not reach it at any finite price. This mathematical analysis helps us understand the potential ceiling on sales without having an actual maximum market price. It emphasizes that the demand function continues indefinitely, showing that demand reduces continuously with price increments.

Asymptotic Behavior in Demand Functions

Asymptotic behavior refers to how a function behaves as the input approaches a certain point, often infinity. In our exercise, we examined the demand function \[D(p) = 1.5 p^{-0.8}\]and its asymptotic behavior as price \( p \) becomes very large. The demand function indicates that the demand decreases steadily but never truly reaches zero, even as price tends towards infinity. This reflects a type of asymptotic behavior where the curve approaches, but never crosses, the horizontal axis.
Understanding such behavior is important in economics as it shows that despite any incremental price increase, demand remains non-zero, implying that complete elimination of demand through pricing alone might be impossible with this function. This insight aids businesses and analysts in strategizing about pricing and assessing market responses.