Problem 81
Question
If \(q(p)\) is the demand for a product at price \(p,\) that is, the number of units of the product that are sold at price \(p\), then $$ E(p)=-q^{\prime}(p) \cdot \frac{p}{q(p)} $$ is defined to be the elasticity of demand. a. Let \(\Delta p\) denote a small change in \(p,\) and let \(\Delta q=\) \(q(p+\Delta p)-q(p)\) be the corresponding change in \(q\) Show that, for \(\Delta p\) sufficiently small, we have the approximation $$ E(p) \approx-\frac{\Delta q / q(p)}{\Delta p / p} $$ b. Elasticity of demand for potatoes at standard supermarket prices has been estimated to be 0.31 . Approximately what percentage change in potato consumption will result from a \(2 \%\) increase in the price of potatoes? c. Over a fixed period, a television repair service recorded the number of repairs that it made at three different prices: $$ \begin{array}{|l|c|} \hline \text { Price } & \text { Contracts Obtained } \\ \hline \$ 90 & 96 \\ \hline \$ 80 & 200 \\ \hline \$ 70 & 300 \\ \hline \end{array} $$ Estimate the elasticity of demand at \(p=\$ 70\) and at \(p=\$ 90 .\) Use a central difference quotient to estimate elasticity of demand at \(p=\$ 80\). (Elasticity of demand for television repair was estimated to be 3.84 in the 1970s.)
Step-by-Step Solution
Verified Answer
a) Elasticity is approximated as \(-\frac{\Delta q / q(p)}{\Delta p / p}\). b) 0.62% decrease in demand. c) Elasticities: 2.33 at $70; 0.975 at $90; 3.264 at $80.
1Step 1: Approximate Elasticity Formula Derivation
To derive the approximation for elasticity of demand, start from the definition of elasticity: \[ E(p) = -q'(p) \cdot \frac{p}{q(p)} \]Using the definition of a derivative, the change in demand \( \Delta q \approx q'(p) \cdot \Delta p \) for small \( \Delta p \). Substitute this into the formula:\[ E(p) = -\frac{q'(p) \cdot \Delta p}{q(p)} \cdot \frac{p}{\Delta p} = -\frac{\Delta q}{q(p)} \cdot \frac{p}{\Delta p} \]Thus,\[E(p) \approx -\frac{\Delta q / q(p)}{\Delta p / p}\].
2Step 2: Elasticity Impact Calculation for Potatoes
Given the elasticity \( E(p) = 0.31 \), we find the percentage change in demand due to a price change using the formula \(-E(p) \cdot \text{Percentage Change in Price}\). A \(2\%\) price increase gives:\[ \text{Percentage Change in Demand} = -E(p) \cdot 2\% = -0.31 \times 2\% = -0.62\% \]Therefore, a \(2\%\) increase in price results in a \(-0.62\%\) change in demand.
3Step 3: Elasticity at $70 and $90 using Central Difference
Using the data, calculate elasticity at \(70\) and \(90\). For \(p=70\), use the neighboring point \(p=80\):\[ \Delta q = 200 - 300 = -100, \quad \Delta p = 80 - 70 = 10 \]\[ q(70) = 300, \; E(70) \approx -\frac{-100/300}{10/70} = 2.33 \]For \(p=90\), with neighbor \(p=80\):\[ \Delta q = 200 - 96 = 104, \quad \Delta p = 80 - 90 = -10 \]\[ q(90) = 96, \; E(90) \approx -\frac{104/96}{-10/90} = 0.975 \]
4Step 4: Central Difference Estimation at $80
Using central difference for elasticity at \(p=80\), consider prices \(70\) and \(90\):\[ q(90) - q(70) = 96 - 300 = -204 \text{, with } \Delta p = 20 \]\[ E(80) = -\frac{-204/200}{20/80} = 3.264 \]
Key Concepts
DerivativePrice ChangesConsumer BehaviorMathematical Modeling
Derivative
A derivative represents the rate of change of one quantity in relation to another. In the context of elasticity of demand, the derivative, denoted as \( q'(p) \), measures how the quantity demanded \( q \) changes with a small change in price \( p \).
When we talk about demand elasticity, the derivative comes into play as it shows the sensitivity of quantity demanded to price changes. This sensitivity helps businesses and economists make informed decisions about pricing strategies. A higher derivative value indicates that demand is highly responsive to price changes, while a lower value suggests less responsiveness.
Understanding derivatives allows economists to model consumer behavior mathematically, making it possible to predict how small alterations in price can impact demand.
In practical terms, the derivative offers a snapshot of how fluctuations in price affect consumer purchase patterns at a given point. This serves as a foundation for calculating elasticity and assists in making data-driven decisions.
When we talk about demand elasticity, the derivative comes into play as it shows the sensitivity of quantity demanded to price changes. This sensitivity helps businesses and economists make informed decisions about pricing strategies. A higher derivative value indicates that demand is highly responsive to price changes, while a lower value suggests less responsiveness.
Understanding derivatives allows economists to model consumer behavior mathematically, making it possible to predict how small alterations in price can impact demand.
In practical terms, the derivative offers a snapshot of how fluctuations in price affect consumer purchase patterns at a given point. This serves as a foundation for calculating elasticity and assists in making data-driven decisions.
Price Changes
Price changes can significantly affect the quantity of products that consumers buy. In the study of elasticity of demand, price changes are crucial because they offer insights into how sensitive consumers are to price fluctuations.
When the price of a product increases or decreases, it directly impacts consumer behavior. A small increase in price might lead to a significant drop in quantity demanded if consumers are price-sensitive. Conversely, a price decrease might not always result in proportionally large increases in demand, depending on the product's nature and consumer necessity.
This sensitivity to price changes is quantitatively determined by the elasticity of demand. In simple terms, if a small increase in price (e.g., 2%) results in a considerable drop in demand, the product is said to have high elasticity. By understanding how price changes influence demand, businesses can strategically set prices to maximize revenue and meet consumer needs effectively.
When the price of a product increases or decreases, it directly impacts consumer behavior. A small increase in price might lead to a significant drop in quantity demanded if consumers are price-sensitive. Conversely, a price decrease might not always result in proportionally large increases in demand, depending on the product's nature and consumer necessity.
This sensitivity to price changes is quantitatively determined by the elasticity of demand. In simple terms, if a small increase in price (e.g., 2%) results in a considerable drop in demand, the product is said to have high elasticity. By understanding how price changes influence demand, businesses can strategically set prices to maximize revenue and meet consumer needs effectively.
Consumer Behavior
Consumer behavior refers to the decision-making processes and actions of consumers when they buy goods and services. It is influenced by several factors, including price, preferences, necessity, and income.
Elasticity of demand is closely tied to consumer behavior because it measures how changes in price affect the quantity consumers are willing to purchase. Some products, like luxury goods, might have high elasticity, meaning consumer behavior is highly responsive to changes in price. Essential goods, however, may be inelastic as demand remains relatively steady regardless of price fluctuations.
Understanding consumer behavior through elasticity allows businesses to predict how consumers will react to changes in their pricing strategies. It can also help them tailor marketing efforts and inventory management according to consumer demand patterns.
By analyzing and anticipating consumer behavior, companies can ensure they align product offerings with market demand, improving customer satisfaction and business performance.
Elasticity of demand is closely tied to consumer behavior because it measures how changes in price affect the quantity consumers are willing to purchase. Some products, like luxury goods, might have high elasticity, meaning consumer behavior is highly responsive to changes in price. Essential goods, however, may be inelastic as demand remains relatively steady regardless of price fluctuations.
Understanding consumer behavior through elasticity allows businesses to predict how consumers will react to changes in their pricing strategies. It can also help them tailor marketing efforts and inventory management according to consumer demand patterns.
By analyzing and anticipating consumer behavior, companies can ensure they align product offerings with market demand, improving customer satisfaction and business performance.
Mathematical Modeling
Mathematical modeling plays a crucial role in understanding and predicting economic patterns and behaviors. In the context of elasticity of demand, mathematical models use equations and variables to represent the relationships between price changes and consumer demand.
These models bring a quantitative focus to economics, where variables such as price, quantity demanded, and elasticity are represented numerically. Formulas, like the elasticity of demand formula \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \), allow economists and analysts to estimate how changes in one variable (price) will impact another (demand).
Through mathematical modeling, we can simulate different scenarios, test hypotheses, and make predictions about consumer reactions to various price levels. This aids in crafting more effective business strategies and economic policies. It offers a precise, data-driven approach that complements qualitative insights in understanding economic dynamics.
Overall, mathematical models are essential not just for historical analysis but for forecasting future outcomes and crafting strategies that are well-adjusted to market realities.
These models bring a quantitative focus to economics, where variables such as price, quantity demanded, and elasticity are represented numerically. Formulas, like the elasticity of demand formula \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \), allow economists and analysts to estimate how changes in one variable (price) will impact another (demand).
Through mathematical modeling, we can simulate different scenarios, test hypotheses, and make predictions about consumer reactions to various price levels. This aids in crafting more effective business strategies and economic policies. It offers a precise, data-driven approach that complements qualitative insights in understanding economic dynamics.
Overall, mathematical models are essential not just for historical analysis but for forecasting future outcomes and crafting strategies that are well-adjusted to market realities.
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Problem 81
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