Problem 7
Question
An economist is interested in how the price of a certain item affects its sales. At a price of \(\$ p\), a quantity, \(q\), of the item is sold. If \(q=f(p)\), explain the meaning of each of the following statements: (a) \(f(150)=2000\) (b) \(f^{\prime}(150)=-25\)
Step-by-Step Solution
Verified Answer
(a) 2000 items are sold at \$150. (b) Sales decrease by 25 units for each \$1 increase at \$150.
1Step 1: Understanding the Function
The function \( q = f(p) \) represents the relationship between the price \( p \) of an item and its quantity sold \( q \). The function \( f(p) \) shows how the sales quantity \( q \) depends on the price \( p \) of the item.
2Step 2: Analyzing Statement (a)
The statement \( f(150)=2000 \) tells us that when the price \( p \) of the item is \\(150, the quantity \( q \) sold is 2000 units. This means at a price of \\)150, 2000 items are sold.
3Step 3: Analyzing Statement (b)
The statement \( f^{\prime}(150)=-25 \) provides information about the rate of change of quantity sold with respect to price when the price is \\(150. The derivative \( f^{\prime}(150) = -25 \) signifies that for every 1 unit increase in price (i.e., increasing the price by \\)1) from \$150, the quantity sold decreases by 25 units.
Key Concepts
Price-Quantity RelationshipRate of ChangeSales Function
Price-Quantity Relationship
The Price-Quantity Relationship is a fundamental concept in economics that visualizes how the price of a product affects its sales volume. Consider the function \( q = f(p) \), which represents this relationship. Here, \( q \) is the quantity of items sold, and \( p \) is the price of each item. This relationship can help businesses understand consumer behavior and adjust pricing strategies accordingly.
When you see a statement like \( f(150)=2000 \), it means that at a price of \( \$150 \), exactly 2000 units of the product are sold. This information is crucial for businesses to determine the optimal price point that maximizes sales without underselling the product. A clear understanding of how quantity varies with price enables businesses to forecast demand and plan supply efficiently.
When you see a statement like \( f(150)=2000 \), it means that at a price of \( \$150 \), exactly 2000 units of the product are sold. This information is crucial for businesses to determine the optimal price point that maximizes sales without underselling the product. A clear understanding of how quantity varies with price enables businesses to forecast demand and plan supply efficiently.
- The function captures the market's response to different price levels.
- By analyzing past data, future sales outcomes can be predicted.
Rate of Change
In economics, the Rate of Change is examined through the derivative of a function. It tells us how one quantity changes in response to changes in another. For our function \( q = f(p) \), the derivative \( f'(p) \) indicates the Rate of Change of quantity with respect to price.
For example, if \( f'(150) = -25 \), it signifies that increasing the price from \( \\(150 \) by \( \\)1 \) causes the quantity sold to decrease by 25 units. This type of information is crucial for businesses, as it provides insight into consumer price sensitivity, also known as price elasticity.
For example, if \( f'(150) = -25 \), it signifies that increasing the price from \( \\(150 \) by \( \\)1 \) causes the quantity sold to decrease by 25 units. This type of information is crucial for businesses, as it provides insight into consumer price sensitivity, also known as price elasticity.
- Price elasticity measures how responsive the quantity demanded is to a change in price.
- A negative derivative means sales decrease with an increase in price.
- This knowledge aids in pricing strategy, aiming for a balance between price increments and minimizing loss in sales volume.
Sales Function
The Sales Function, represented as \( q = f(p) \), is key to understanding sales dynamics in economics. It gives a mathematical representation of how sales quantities react to changes in pricing.
By utilizing the function, businesses can simulate different pricing scenarios and predict possible outcomes on sales volume. Hence, when analyzing sales strategies, this function becomes instrumental.
By utilizing the function, businesses can simulate different pricing scenarios and predict possible outcomes on sales volume. Hence, when analyzing sales strategies, this function becomes instrumental.
- The sales function helps to identify high-demand price points.
- It aids in revenue forecasting by predicting sales based on potential price adjustments.
- It also assists in strategic decision-making about inventory and production levels.
Other exercises in this chapter
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