Problem 38 - Single Variable Calculus

Question

Effect of Price Increase on Quantity Demanded The quantity \(x\) demanded per week of the Alpha Sports Watch (in thousands) is related to its unit price of \(p\) dollars by the equation $$ x=f(p)=10 \sqrt{\frac{50-p}{p}} \quad 0

Step-by-Step Solution

Answer

In summary, given the demand function \(x = f(p) = 10 \sqrt{\frac{50-p}{p}} \quad 0 < p \leq 50 \), when the price of the Alpha Sports Watch increased from $30 to $35, the quantity demanded per week decreased from approximately 8.2 thousand to 6.5 thousand units. This demonstrates the inverse relationship between price and quantity demanded, a common observation in economics.

Step 1: Understand the relationship between price and quantity demanded

The given equation describes the quantity demanded per week (x, in thousands) as a function of the unit price of the watch (p, in dollars): \[ x = f(p) = 10 \sqrt{\frac{50-p}{p}} \quad 0 < p \leq 50 \] This function represents the behavior of the market in response to various prices, capturing the demand for these watches.

Step 2: Calculate the quantity demanded at a certain price

To calculate the demand for the watch at a specific price point, we can plug the unit price into the equation. Let's say we want to determine the quantity demanded per week when the unit price is $30. \[ x = f(30) = 10 \sqrt{\frac{50-30}{30}} \]

Step 3: Simplify the equation

Simplify the equation to find the quantity demanded at \(p = 30\): \[ x = 10 \sqrt{\frac{20}{30}} = 10 \sqrt{\frac{2}{3}} \] Approximately, we get: \[ x \approx 10 \times 0.82 = 8.2 \] Thus, at a price of $30 per unit, the quantity demanded per week is approximately 8.2 thousand.

Step 4: Observe the effect of a price increase on quantity demanded

Let's examine how an increase in price by \(5 will affect the quantity demanded. To do this, we will plug in the new price, \)35, into the function: \[ x = f(35) = 10 \sqrt{\frac{50-35}{35}} \] Simplify the equation to find the quantity demanded at \(p = 35\): \[ x = 10 \sqrt{\frac{15}{35}} = 10 \sqrt{\frac{3}{7}} \] Approximately, we get: \[ x \approx 10 \times 0.65 = 6.5 \] Thus, at a price of $35 per unit, the quantity demanded drops to approximately 6.5 thousand.

Step 5: Analyze the results

Comparing the results from Step 3 and Step 4, we can notice the following: - When the price was $30, the quantity demanded was around 8.2 thousand watches per week - When the price increased to $35, the quantity demanded decreased to around 6.5 thousand watches per week Therefore, an increase in the unit price led to a decrease in the quantity demanded, which is a common observation in economics - as the price of a product goes up, the demand for it tends to decrease.

Key Concepts

Demand Function

In economics, the demand function is a mathematical model that describes how the quantity demanded of a product varies with its price. It essentially captures the relationship between price and demand, showing how consumers' buying habits change as the price rises or falls.

For instance, the demand function for the Alpha Sports Watch can be expressed as:
\[ x = f(p) = 10 \sqrt{\frac{50-p}{p}} \quad 0 < p \leq 50 \]
In this equation, \( x \) represents the quantity demanded (in thousands), and \( p \) is the price per unit. With a given price, we can calculate how many units consumers are willing to buy. As the price increases, the function typically shows a decrease in quantity demanded, reflecting the law of demand which states that there is an inverse relationship between the price of a good and the quantity demanded.

Economics of Price Changes

The relationship between the price of a product and the demand for it is fundamental in understanding market dynamics. In the context of the Alpha Sports Watch, when the price increases from \(30 to \)35, demand decreases from 8.2 thousand to 6.5 thousand units per week. This illustrates a core concept in economics: the price elasticity of demand, which measures how sensitive the quantity demanded is to a change in price.

Price changes can impact demand in various ways, depending on the elasticity of the product. A product is considered inelastic when a price change has little effect on the quantity demanded, often because these goods are necessities. Conversely, a product is elastic if a small change in price leads to a significant change in quantity demanded, which is typical for luxury items or goods with many substitutes.

Quantity Demanded Calculation

Calculating the quantity demanded at different price points helps businesses and economists predict consumer behavior and plan accordingly. Using the demand function for the Alpha Sports Watch, we can determine the quantity demanded at any given price. For example, to find the quantity demanded at a price of \(30:
\[ x = f(30) = 10 \sqrt{\frac{50-30}{30}} \]
Simplifying, we get approximately 8.2 thousand units. Similarly, if the price rises to \)35, following the same calculation process leads to approximately 6.5 thousand units demanded.

Understanding how to perform these calculations is a valuable skill in economics, aiding in price-setting strategies, market analysis, and financial forecasting. It's also important to recognize the limitations of the demand function, as real-world demand might be affected by factors beyond price, like consumer preferences, income levels, and market competition.