Question

Use the following data to work Problems 3 and 4 Sara's income falls from \(\$ 12\) to \(\$ 9\) a week, while the price of popcorn is unchanged at \(\$ 3\) a bag and the price of a smoothie is unchanged at \(\$ 3\). What is the effect of the fall in Sara's income on her real income in terms of (a) smoothies and (b) popcorn?

Short answer

Sara's real income decreases from 4 to 3 in terms of both smoothies and popcorn.

Step-by-step solution

Identify Sara's original income

Sara's original income is given as \( \$ 12 \) per week.

Identify Sara's new income

Sara's new income is given as \( \$ 9 \) per week.

Determine the unchanged prices

The price of both popcorn and smoothies remains at \( \$ 3 \) per bag.

Calculate real income in terms of smoothies (original income)

With her original income of \( \$ 12 \), the maximum number of smoothies Sara could afford was obtained by dividing her income by the price of a smoothie: \[ \text{Maximum smoothies} = \frac{12}{3} = 4 \]

Calculate real income in terms of smoothies (new income)

With her new income of \( \$ 9 \), the maximum number of smoothies Sara can now afford is: \[ \text{Maximum smoothies} = \frac{9}{3} = 3 \]

Find the effect on real income for smoothies

The fall in Sara's income reduces the maximum number of smoothies she can afford from 4 to 3.

Calculate real income in terms of popcorn (original income)

With her original income of \( \$ 12 \), the maximum number of popcorn bags Sara could afford was: \[ \text{Maximum popcorn} = \frac{12}{3} = 4 \]

Calculate real income in terms of popcorn (new income)

With her new income of \( \$ 9 \), the maximum number of popcorn bags Sara can now afford is: \[ \text{Maximum popcorn} = \frac{9}{3} = 3 \]

Find the effect on real income for popcorn

The fall in Sara's income reduces the maximum number of popcorn bags she can afford from 4 to 3.

Key concepts

Income Elasticity

Understanding income elasticity helps in knowing how the demand for goods changes with income. When Sara's income dropped from \( \text{\$12} \) to \( \text{\$9} \), we can see its impact on her purchasing power. Income elasticity of demand is a measure of how much the quantity demanded of a good responds to a change in consumers' income. For normal goods, when income increases, the quantity demanded increases. In Sara's case, her demand for both smoothies and popcorn decreased as her income decreased. This perfectly illustrates how sensitive her consumption patterns are to changes in her income. With her budget shrinking, she can buy fewer goods.

Budget Constraint

A budget constraint represents all the combinations of goods and services that a consumer can afford given their income and the prices of the goods. For Sara, her budget constraint has shifted due to the decrease in her income. Initially, with \( \text{\$12} \) and prices of \( \text{\$3} \) each for popcorn and smoothies, she could buy up to 4 of each. However, when her income fell to \( \text{\$9} \), her budget constraint tightened, limiting her ability to only 3 of each if she spends her entire income on either good. This exercise shows how changes in income shift the budget line inward, indicating that she can afford less of both goods under the new income constraint.

Consumer Choice

Consumer choice is about how individuals decide to allocate their limited resources (income) among various goods and services. Sara must decide how best to spend her reduced income to maximize her satisfaction. She needs to consider the utility, or satisfaction, she gets from each unit of popcorn or smoothie. If Sara prefers smoothies over popcorn, she might choose to buy more smoothies even with her limited budget. This choice is influenced by the principle of diminishing marginal utility, which means each additional unit of a good consumed provides less utility than the one before. In practice, Sara's decision will hinge on her preferences and the trade-offs she is willing to make, balancing the two goods within her budget constraint.