Problem 7 - Applied Calculus

Question

Suppose that \(q=f(p)\) is the demand curve for a product, where \(p\) is the selling price in dollars and \(q\) is the quantity sold at that price. (a) What does the statement \(f(12)=60\) tell you about demand for this product? (b) Do you expect this function to be increasing or decreasing? Why?

Step-by-Step Solution

Answer

(a) At price $12, 60 units are demanded. (b) Decreasing, as higher prices typically reduce demand.

Step 1: Understanding Demand at a Given Price

The statement \(f(12) = 60\) tells us that when the product is sold at a price of \(12, the quantity sold is 60 units. This means that at the price level of \)12, consumers demand 60 units of the product.

Step 2: Analyzing the Trend of the Demand Function

Typically, demand functions are decreasing as price increases. As the price \(p\) of a product increases, the quantity demanded \(q\) usually decreases because of the inverse relationship between price and quantity demanded in economics. Thus, we expect \(f(p)\) to be a decreasing function.

Key Concepts

Selling Price

The selling price is a crucial component in determining demand. It refers to the amount consumers are required to pay to purchase a product. In the context of a demand curve, the selling price is often denoted by the variable \(p\). For instance, in the given exercise, when the selling price is \(12\) dollars, it directly influences how much of the product is bought. Understanding the selling price helps businesses and economists know how a change in price can affect demand. If set too high, it might lower demand; if too low, it may not cover production costs.

• The selling price affects customer purchasing decisions.
• A balance in selling price is key to optimizing sales and profits.

Knowing the optimal selling price helps companies to align their pricing strategy with market demand effectively.

Quantity Sold

Quantity sold refers to the number of units consumers purchase at a given selling price. On a demand curve, this is represented by the variable \(q\). Using the example from our exercise, where \(q = 60\) when \(p = 12\), it shows that at a selling price of 12 dollars, consumers demand 60 units. This insight is important because:

• It shows the level of consumer interest at various prices.
• Gives businesses clear data to forecast future sales and demand.
• Informs inventory and production decisions to avoid overproduction or stock shortages.

Understanding the quantity sold at different price points is essential for managing supply and demand effectively.

Inverse Relationship

In economics, the term inverse relationship describes how two variables move in opposite directions. Specifically, in demand curves, there's typically an inverse relationship between selling price and quantity sold. As the price \(p\) of a product increases, the quantity demanded \(q\) generally decreases, and vice versa.

• This relationship is fundamental in understanding consumer behavior.
• Explains why businesses might reduce prices during sales to boost purchase volumes.
• Helps in price elasticity analysis.

By recognizing this pattern, businesses can make informed decisions about pricing strategies to balance profit and sales volumes.

Economics

Economics is the study of how people manage resources and make choices. The demand curve, a concept from economics, is a graphical representation showing the relationship between the selling price of a product and the quantity sold. It highlights core economic principles such as supply, demand, and market equilibrium.

• Economics helps us understand market dynamics.
• It provides tools for predicting consumer response to changes in pricing.
• Guides businesses in setting strategic prices based on market analysis.

The principles of demand curves are fundamental for any business aiming to optimize resources and maximize profit while meeting consumer needs.