Problem 30
Question
As wholesale prices for potatoes increase from 11 to 16.5 dollars per cwt (hundred pounds), the quantity of potatoes demanded can be modeled as $$ D(p)=461+47 \ln (17-p) \text { million cwt } $$ where \(p\) is the wholesale price in dollars per cwt. a. What is the price elasticity of demand for a wholesale price of \(\$ 14\) per cwt? b. Is demand elastic or inelastic at \(\$ 14\) per cwt? c. Explain in context what elasticity (or inelasticity) at \(\$ 14\) per cwt means.
Step-by-Step Solution
Verified Answer
Demand is inelastic with elasticity -0.39 at $14 per cwt, meaning quantity demanded doesn't change much with price changes.
1Step 1: Understanding the Price Elasticity of Demand
Price elasticity of demand measures how sensitive the quantity demanded is to a change in price. It is given by the formula:\[ E_d = \frac{dD(p)/dp}{D(p)/p} \]where \(E_d\) is the elasticity of demand, \(dD(p)/dp\) is the derivative of the demand function, \(D(p)\) is the demand at price \(p\), and \(p\) is the price.
2Step 2: Find the Derivative of the Demand Function
The demand function is \(D(p) = 461 + 47 \ln (17-p)\). First, find the derivative with respect to \(p\):\[ \frac{dD(p)}{dp} = 47 \cdot \frac{-1}{17-p} = \frac{-47}{17-p} \].
3Step 3: Calculate the Demand at $14 per cwt
Substitute \(p = 14\) into the demand function to find \(D(14)\):\[ D(14) = 461 + 47 \ln (17-14) = 461 + 47 \ln 3 \approx 561.39 \text{ million cwt} \].
4Step 4: Substitute Values into Elasticity Formula
Substitute \(p = 14\), \(D(14)\), and \(\frac{dD}{dp}\) into the elasticity formula:\[ E_d = \frac{-47/(17-14)}{561.39/14} = \frac{-47/3}{40.1} \approx -0.39 \].
5Step 5: Interpret the Elasticity Value
The elasticity value is \(-0.39\), which is between 0 and -1. This indicates that the demand is inelastic at a price of \$14 per cwt. Inelastic demand means the quantity demanded is relatively insensitive to price changes.
Key Concepts
Demand FunctionDerivative of DemandInelastic DemandMathematical Modeling
Demand Function
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.
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.
Derivative of Demand
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.
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.
Inelastic Demand
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.
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
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.
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.
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