Problem 6 - Applied Calculus

Question

Refer to Table \(9.2\), which shows \({ }^{1}\) the weekly beef consumption, \(C\), (in lbs) of an average household as a function of \(p\), the price of beef (in $$\$ / \mathrm{lb}$$ ) and \(I\), annual household income (in $$\$ 1000 \mathrm{~s}$$ ). $$\begin{array}{|r|c|c|c|c|} \hline & 3.00 & 3.50 & 4.00 & 4.50 \\ \hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\ \hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\ \hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\ \hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\ \hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\ \hline \end{array}$$ Make a table of the proportion, \(P\), of household income spent on beef per week as a function of price and income. (Note that \(P\) is the fraction of income spent on beef.)

Step-by-Step Solution

Answer

The proportion \( P \) is the weekly beef expenditure divided by the weekly income for each price and income pair in the table.

Step 1: Understand the Given Data

The table provides the weekly beef consumption, \( C \), in pounds, based on the price of beef, \( p \), and annual income, \( I \). Each row corresponds to an income level \( I \) in \( \\(1000s \) and each column corresponds to a beef price \( p \) in \( \\)/lb \).

Step 2: Define the Proportion Formula

The proportion, \( P \), of household income spent on beef weekly is given by the formula:\[ P = \frac{\text{Total weekly expenditure on beef}}{\text{Weekly income}} \]where total weekly expenditure is \( C \times p \), and weekly income is \( \frac{I}{52} \) because \( I \) is the annual income.

Step 3: Calculate Weekly Beef Expenditure

For each combination of \( p \) and \( I \), calculate the weekly expenditure on beef: \( C \times p \). For instance, at \( p = 3.00 \) and \( I = 20 \), the expenditure is \( 2.65 \times 3.00 = 7.95 \).

Step 4: Calculate Weekly Income

Convert annual income to weekly income by dividing by 52. For example, if \( I = 20 \), the weekly income is \( \frac{20,000}{52} \approx 384.62 \).

Step 5: Compute the Proportion \( P \)

For each combination of \( p \) and \( I \), use the formula from Step 2 to calculate \( P \). Continuing the example from Step 3 and Step 4, \[ P = \frac{7.95}{384.62} \approx 0.0207. \]

Step 6: Fill the Proportion Table

Repeat Steps 3-5 for each cell in the given data to fill out the table of \( P \). Each cell in the new table corresponds to swapping values based on Steps 1-5.

Key Concepts

Price Elasticity

Price elasticity measures how the quantity demanded of a good, like beef, changes with its price. It's an important concept for understanding consumer behavior. In this exercise, we analyze beef consumption by viewing how changes in beef prices affect the amount consumed.

Price elasticity is calculated using the formula:

• Price Elasticity = \( \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}} \).

If beef is elastic, consumption will significantly decrease as prices rise. If it's inelastic, price changes have little effect on how much beef people buy. In our table, as the price of beef rises, the consumption in pounds tends to fall slightly, suggesting beef may be relatively inelastic. This means consumers might still purchase similar amounts even if prices increase.

Understanding these concepts helps in predicting consumer responses to price fluctuations, valuable to households, businesses, and policymakers.

Income Effect

The income effect shows how changes in a consumer's income influence the quantity of a good consumed. In this case, we're examining how much beef consumption changes with shifts in household income. Our table offers a clear look into this effect.

Given the table data:

• For lower incomes (\( I = 20 \), \(1000s\)), beef consumption is relatively low.
• As incomes rise (\( I = 60 \), \( I = 100 \)), beef consumption increases significantly.

This pattern indicates that beef is a normal good, where consumption grows with higher income. The income effect is essential for understanding how economic fluctuations impact consumer habits, especially concerning staple or luxury food items like beef.

With higher incomes, households allocate more of their budget to purchasing goods like beef, thus increasing consumption. This insight helps businesses forecast demand based on economic conditions.

Data Tables

Data tables provide an organized way to present the relationship between different variables, like price, income, and beef consumption in our exercise. They make it easy to analyze changes and spot trends within the data.

Key parts of the table include:

• Rows representing different income levels and columns for varying beef prices.
• The intersection of rows and columns shows beef consumption in pounds.

Tables simplify complex datasets, helping users visualize and compute further statistics, such as the proportion of income spent on beef.

When working with such tables, it's vital to pay attention to the variables' scale and units. Also, calculate necessary parameters accurately, like translating annual income to a weekly basis, to ensure correct interpretations. By leveraging data tables, users can draw meaningful conclusions and predict future consumption patterns effectively.