Problem 10 - Applied Calculus

Question

The demand for a product is given by \(p=90-10 q .\) Find the elasticity of demand when \(p=50 .\) If this price rises by \(2 \%,\) calculate the corresponding percentage change in demand.

Step-by-Step Solution

Answer

The elasticity of demand is -1.25, indicating a 2% price increase leads to a 2.5% decrease in demand.

Step 1: Understanding the Price-Demand Function

The demand function is given by \( p = 90 - 10q \). Here, \( p \) is the price and \( q \) is the quantity demanded. The function shows the relationship between price and quantity demanded.

Step 2: Find the Initial Quantity Demanded

We need to determine the quantity \( q \) when \( p = 50 \). Substituting \( p = 50 \) into the demand function: \( 50 = 90 - 10q \). Solving for \( q \), we get: \( 10q = 40 \), so \( q = 4 \).

Step 3: Calculate the Derivative of the Demand Function

The derivative of the demand function with respect to \( q \) is \( \frac{dp}{dq} = -10 \). This shows how the price changes with a small change in quantity demanded.

Step 4: Calculate Elasticity of Demand

Elasticity of demand, \( E_d \), is calculated using the formula: \( E_d = \frac{dq}{dp} \times \frac{p}{q} \). Since \( \frac{dq}{dp} = \frac{1}{\frac{dp}{dq}} = -\frac{1}{10} \), we have: \( E_d = -\frac{1}{10} \times \frac{50}{4} = -\frac{50}{40} = -\frac{5}{4} = -1.25 \).

Step 5: Interpret Elasticity Result

An elasticity of \(-1.25\) indicates that the demand is elastic. A 1% increase in price will result in a 1.25% decrease in quantity demanded.

Step 6: Calculate Percentage Change in Demand

If the price rises by 2%, the percentage change in demand is given by elasticity times the percentage change in price. Thus, percentage change in demand = \(-1.25 \times 2\% = -2.5\% \).

Step 7: Conclusion

When the price rises by 2%, the quantity demanded decreases by 2.5%, as indicated by our elasticity calculation.

Key Concepts

Price-Demand Function

The price-demand function is a fundamental concept in economics that describes the relationship between the price of a product and the quantity demanded by consumers. In mathematical terms, this relationship is often expressed as a function where price (\( p \)) is a function of quantity demanded (\( q \)). In our example, the function is given by \( p = 90 - 10q \), indicating that the price decreases as quantity demanded increases. This inverse relationship is typical; as more units of a product are available, consumers are usually only willing to purchase them at lower prices.

The equation \( p = 90 - 10q \) helps us understand, at a glance, how many units consumers will buy at different prices. By substituting different values of \( p \), we can solve for \( q \) to find out the quantity demanded at that price. This function is crucial for businesses to decide pricing strategies and understand market behavior.

Derivative of Demand

The derivative of the demand function is a mathematical tool that helps us understand how the price changes in response to changes in the quantity demanded. In our example, the demand function is \( p = 90 - 10q \), and its derivative with respect to \( q \) is \( \frac{dp}{dq} = -10 \).

This derivative value of -10 shows that the price decreases by 10 units for each additional unit increase in quantity demanded. This constant rate highlights how sensitive the price is to changes in demand. Understanding this sensitivity is important for analyzing how changes in market conditions can impact pricing strategies.

By using derivatives, businesses can respond effectively to demand fluctuations, adjusting pricing to optimize revenue and market share.

Quantity Demanded

Quantity demanded is the amount of a product that consumers are willing and able to purchase at a given price. It is a key metric derived from the price-demand function. In our example, when the price \( p \) is set at 50, we find the quantity demanded \( q \) by solving the demand function \( 50 = 90 - 10q \). This calculation results in \( q = 4 \), meaning that at a price of 50, buyers demand 4 units of the product.

Understanding quantity demanded allows businesses to make informed decisions about production levels, inventory management, and marketing strategies. It provides insights into consumer preferences and economic conditions.

By monitoring changes in quantity demanded, businesses can anticipate shifts in consumer behavior and adjust their operations accordingly to meet market demands.

Percentage Change in Demand

The percentage change in demand is a measure that indicates how much the quantity demanded of a product changes in response to a change in price. This concept is intricately linked to the elasticity of demand, which we calculated as \( -1.25 \) in the original problem.

Elasticity shows us that for a 1% increase in price, the quantity demanded decreases by 1.25%. Therefore, if the price of a product rises by 2%, the corresponding percentage change in demand can be calculated as \( -1.25 \times 2\% = -2.5\% \).

This result tells us that an increase in price will lead to a 2.5% decrease in the quantity demanded. Understanding the percentage change in demand helps businesses evaluate how price adjustments might influence sales volume and revenue. It is an essential consideration in strategic pricing, allowing firms to predict and adapt to changes in market conditions effectively.