Problem 7 - Calculus Concepts

Question

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. When coffee beans sell for \(p\) dollars per pound, consumers demand \(q\) million pounds. (5,16)\(;(15,3)\)

Step-by-Step Solution

Answer

The input variable is measured in dollars per pound, and the output in million pounds. At $5 per pound, demand is 16 million pounds; at $15, demand is 3 million pounds.

Step 1: Identify Units for Input Variable

The input variable is the price of coffee beans, denoted as \( p \). Since it represents the cost per pound, the units for \( p \) are 'dollars per pound'.

Step 2: Identify Units for Output Variable

The output variable is the quantity demanded, denoted as \( q \). It corresponds to the quantity in millions of pounds. Therefore, the units for \( q \) are 'million pounds'.

Step 3: Interpret the Point (5,16) from Demand Function

The point (5,16) indicates that when the price of coffee beans \( p \) is \(5\) dollars per pound, the demand \( q \) is \(16\) million pounds. In other words, at a selling price of $5, consumers purchase 16 million pounds of coffee beans.

Step 4: Interpret the Point (15,3) from Demand Function

The point (15,3) reveals that when the price \( p \) is \(15\) dollars per pound, the demand \( q \) reduces to \(3\) million pounds. This implies that at a price of $15, consumers are willing to buy only 3 million pounds of coffee beans.

Key Concepts

Input Variables

In the world of economics, an input variable is something we control or measure to see how it affects other things. In the case of the demand function for coffee beans, the input variable is the price of coffee beans. This price is symbolized by \( p \) and is expressed in 'dollars per pound'. It's crucial to remember that the input variable is often what businesses and economists adjust or analyze to understand how it affects sales or consumer behavior. By altering the price, businesses can predict changes in consumer demand. So, when you see \( p \) in this context, think of it as the lever we pull to observe different outcomes in buying patterns.

Output Variables

Output variables represent the results or outcomes in response to changes in input variables. For our demand function, the output variable is the quantity of coffee beans demanded by consumers, denoted as \( q \). The unit for this variable is 'million pounds'. When economists talk about quantity demanded, they are looking at the total amount consumers are willing to purchase at various prices. This function helps to calculate how changes in price (the input) influence the amount of product sold (the output). Knowing the quantity demanded is essential for businesses to determine production levels, inventory needs, and financial forecasts.

Price Elasticity

Price elasticity is a concept that measures how much the quantity demanded responds to a change in price. It's like the sensitivity detector between price and demand. In our demand function, you can observe this through the given data points, like (5,16) and (15,3). When the price per pound of coffee rises from 5 to 15 dollars, the quantity demanded drops from 16 million pounds to just 3 million pounds. This significant change showcases high price elasticity because a small increase in price results in a large decrease in demand. Understanding price elasticity helps businesses to set pricing strategies; knowing how sensitive consumers are to price changes is critical for maximizing revenue.

Units of Measurement

Units of measurement provide a standard way to express and quantify input and output variables, making them crucial for accuracy and understanding. In our demand function for coffee beans, we use 'dollars per pound' for the input variable \( p \), portraying the cost of a pound of coffee. For the output variable \( q \), we use 'million pounds', representing the quantity of coffee beans demanded. Using consistent units ensures everyone interprets data the same way, crucial for comparing, calculating, and analyzing economic scenarios. When you see a demand function, knowing the units involved is key to properly applying the insights gathered from that data.