91Ó°ÊÓ

In economics, the demand function is a mathematical representation that shows the relationship between the quantity demanded of a good and its price. It helps us understand how consumers will respond to changes in price.
The demand function for potatoes given is:\[ D(p) = 461 + 47 \ln(17 - p) \]Here, \(D(p)\) represents the quantity of potatoes demanded in million hundred-weight (cwt), and \(p\) is the wholesale price in dollars per cwt. The function reveals how demand varies as the price changes. The term \(47 \ln(17 - p)\) indicates that as the price approaches 17, the demand increases logarithmically. This form shows non-linear responses of demand to price changes.

To determine how changes in price affect the quantity demanded, we need to compute the derivative of the demand function. The derivative represents the rate of change of the function concerning the price.
Given the demand function: \[ D(p) = 461 + 47 \ln(17 - p) \] The derivative, \( \frac{dD(p)}{dp} \), is calculated as:\[ \frac{dD(p)}{dp} = \frac{-47}{17-p} \] This derivative tells us that for small changes in price, the change in demand will be inversely related to the remaining difference between the price and 17. Specifically, as the price increases, the rate of change of the demand decreases, reflecting less sensitivity in demand with increasing prices.

Demand is considered inelastic when the quantity demanded is relatively insensitive to price changes. This can be quantitatively represented by a demand elasticity value between 0 and -1. An elasticity value closer to 0 indicates more inelastic demand.
In this exercise, when the wholesale price is $14 per cwt, we calculate the elasticity as:\[ E_d = \frac{-47/3}{40.1} \approx -0.39 \]The calculated elasticity \(-0.39\) falls within the inelastic range. This implies that a change in price has a limited effect on the quantity of potatoes demanded. Consumers are not particularly responsive to price changes at this range, often due to the necessity of the product or lack of substitutes.

Mathematical modeling in economics allows us to represent and analyze the behavior of markets through equations and inequalities. It helps in predicting consumer behavior and making data-driven decisions.
This exercise uses the demand function \(D(p) = 461 + 47 \ln(17 - p)\) as a model to derive insights into how the quantity of potatoes demanded changes with different prices. By calculating derivatives and elasticity, we can understand how responsive demand is to price fluctuations.

• The model simplifies complex consumer behavior into measurable quantities.
• It aids in policy-making and strategic pricing by predicting outcomes of price changes.

Mathematical models are essential in economics as they enable researchers and economists to translate real-world scenarios into comprehensible and quantifiable analyses.