Problem 5 - 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. The function \(D\) gives the quantity of rice (in million pounds) purchased by consumers at a market price of \(p\) cents per pound. (16,5)\(;(160,0.05)\)

Step-by-Step Solution

Answer

Input: cents per pound; Output: million pounds. At 16 cents, 5 million lbs; at 160 cents, 0.05 million lbs.

Step 1: Identify Units of Measure

The input variable is the market price of rice, given in cents per pound. The output variable is the quantity of rice purchased by consumers, given in million pounds.

Step 2: Interpret the First Point (16, 5)

The point (16, 5) indicates that when the price of rice is 16 cents per pound, the quantity of rice purchased by consumers is 5 million pounds.

Step 3: Interpret the Second Point (160, 0.05)

The point (160, 0.05) shows that when the price of rice is 160 cents per pound, the quantity of rice purchased by consumers is 0.05 million pounds.

Key Concepts

Input and Output Variables

In any mathematical function, like the demand function discussed here, it is essential to understand the roles of input and output variables. The input variable in our case is the **market price of rice**, which is measured in **cents per pound**. This variable is designed to capture changes in price levels.

The output variable is the **quantity of rice purchased by consumers**, expressed in **million pounds**. This variable represents consumers' demand or the amount they are willing to buy at a given price. Recognizing which variable is input and which is output helps in analyzing how changes in one affect the other.

Units of Measure

In mathematics and economics, units of measure are critical for interpreting results correctly. For our demand function:

• **Input Unit**: The price of the rice is measured in **cents per pound**. This unit tells us the cost consumers pay for each pound of rice they purchase.
• **Output Unit**: The quantity of rice is measured in **million pounds**. This indicates the scaling factor of rice bought, making it easier to understand large-scale consumer behavior.

Consistently using these units ensures accurate communication of economic models and real-world dealings. It enables economists and students alike to interpret and compare data accurately.

Interpretation of Data Points

Data points depicted in graphs or equations of a demand function offer insights into consumer behavior. For instance, when examining the point **(16, 5)**:
The first number, **16**, shows the price of rice at **16 cents per pound**. The second number, **5**, indicates that consumers are buying **5 million pounds** of rice. This suggests a higher demand as the price is relatively low.
Another data point, **(160, 0.05)**, reveals different information:
At a price of **160 cents per pound**, only **0.05 million pounds** of rice is purchased. This implies a steep decline in demand, likely because the price is much higher, showing that price increase can deter consumers. Each data point in a demand function is a piece of a larger puzzle, illustrating the relationship between price and consumer purchasing habits.