Question

Use the following data to work Problems 1 and 2 . Sara's income is \(\$ 12\) a week. The price of popcorn is \(\$ 3\) a bag, and the price of a smoothie is \(\$ 3\). Calculate the equation for Sara's budget line (with bags of popcorn on the left side). Draw a graph of Sara's budget line with the quantity of smoothies on the \(x\) -axis. What is the slope of Sara's budget line? What determines its value?

Short answer

Sara's budget line equation is \( P + S = 4 \). The slope is -1, indicating a one-to-one trade-off between popcorn and smoothies.

Step-by-step solution

Title - Understand the Budget Equation

Sara's total income per week is \(12. She can buy bags of popcorn and smoothies, each costing \)3. The budget line equation represents all combinations of these two goods that she can afford within her budget.

Title - Formulate the Budget Equation

Let the quantity of popcorn be denoted by P and the quantity of smoothies by S. The budget equation can be formulated as: \( 3P + 3S = 12 \).

Title - Simplify the Equation

Divide the entire equation by 3 to simplify it: \( P + S = 4 \). This is the equation of Sara's budget line with populations of popcorn on the left side.

Title - Determine the Budget Line Graph Intercepts

To draw the graph, find the intercepts: - If P = 0: \( S = 4 \). - If S = 0: \( P = 4 \). So, the intercepts are (4, 0) for popcorn and (0, 4) for smoothies.

Title - Graph the Budget Line

Plot the points (4, 0) and (0, 4) on a graph with the quantity of smoothies on the x-axis and the quantity of popcorn on the y-axis. Draw a straight line connecting these points to represent the budget line.

Title - Calculate the Slope of the Budget Line

The slope of the budget line is calculated as the change in y (popcorn) over the change in x (smoothies). Thus, the slope \( = \frac{-4}{4} = -1 \).

Title - Interpret the Slope

The slope of the budget line is -1. This means that for every smoothie Sara buys, she must give up one bag of popcorn. It reflects the trade-off between the two goods due to their equal monetary value.

Key concepts

Budget Constraint

Sara's budget constraint is a fundamental concept that helps us understand the limits of her spending. This constraint represents the combinations of goods she can purchase with her limited income, given the prices.
In Sara's case, her budget is \(12 per week, with popcorn and smoothies costing \)3 each. The budget equation, representing her budget constraint, is derived as follows:
Let's denote the quantity of popcorn as P and the quantity of smoothies as S. The equation will then be: \[ 3P + 3S = 12 \]
To simplify, we divide by 3: \[ P + S = 4 \]
This equation tells us that the total quantity of these goods she can buy should sum up to 4. When represented graphically, any point on this line shows a feasible combination of the two goods within her budget.

Trade-offs

Trade-offs represent the idea that to gain something, you have to give up something else. In Sara's scenario, since both goods cost the same (\$3 each), the trade-off is straightforward.
For every extra bag of popcorn Sara wants to buy, she must give up one smoothie. This is seen clearly in the simplified budget line equation \( P + S = 4 \). The coefficient ratios indicate that one good exchanges directly for the other at a 1:1 rate.
Understanding trade-offs helps Sara make informed choices about how to allocate her limited income effectively. For instance:

• If she buys 2 bags of popcorn, she can afford only 2 smoothies (2 popcorn bags + 2 smoothies = 4).
• If she chooses 3 smoothies, she can only buy 1 bag of popcorn (1 popcorn bag + 3 smoothies = 4).

Visualizing trade-offs on a graph as a slope can make grasping these concepts easier.

Consumer Choice

Consumer choice deals with how consumers decide to allocate their money among different goods. Sara needs to decide how many bags of popcorn and smoothies to buy, given her budget constraint and the prices.
When making her decision, Sara must consider her preferences and the trade-offs she faces. She might prefer smoothies over popcorn or vice versa, which influences her purchase choices.
A graphical representation aids in visualizing these choices. The budget line on the graph shows the possible combinations of popcorn and smoothies Sara can purchase. Any point on this line is a potential choice she can make, staying within her budget. Thus:

• If Sara is located anywhere on the budget line, she is maximizing her consumption without exceeding her budget.
• Points inside the budget line mean Sara is under-spending and can afford more.
• Points outside the budget line are unattainable as they exceed her budget.

Slope of Budget Line

The slope of the budget line is a crucial aspect of understanding consumer decisions and trade-offs.
In Sara's case, taking the general form of a budget line \( P + S = 4 \), we can derive the slope by considering the rate of substitution between the two goods.
The slope is calculated as the change in the quantity of popcorn (\Delta P ) over the change in the quantity of smoothies (\Delta S ). Thus:
\[ \text{slope} = \frac{\text{change in popcorn (P)}}{\text{change in smoothies (S)}} = \frac{-4}{4} = -1 \]
This -1 slope indicates that for every additional smoothie Sara buys, she must give up one bag of popcorn. Its value depends on the price ratio of the two goods. Here, each good costing \$3 ensures a 1-to-1 trade-off.
A negative slope is natural, reflecting that increasing one good decreases the other due to limited resources. Understanding this helps consumers, like Sara, anticipate how changes in prices or income affect their ability to purchase goods.