Question

Each week, Bill, Mary, and Jane select the quantity of two goods, \(x_{1}\) and \(x_{2}\), that they will consume in order to maximize their respective utilities. They each spend their entire weekly income on these two goods. a. Suppose you are given the following information about the choices that Bill makes over a three-week period: $$\begin{array}{|cccccc|} \hline & x_{1} & x_{2} & p_{1} & p_{2} & 1 \\ \hline \text { Week 1 } & 10 & 20 & 2 & 1 & 40 \\ \hline \text { Week 2 } & 7 & 19 & 3 & 1 & 40 \\ \hline \text { Week 3 } & 8 & 31 & 3 & 1 & 55 \\ \hline \end{array}$$ Did Bill's utility increase or decrease between week 1 and week \(2 ?\) Between week 1 and week 3 ? Explain using a graph to support your answer. b. Now consider the following information about the choices that Mary makes: $$\begin{array}{|cccccc|} \hline & X_{1} & X_{2} & P_{1} & P_{2} & 1 \\ \hline \text { Week 1 } & 10 & 20 & 2 & 1 & 40 \\ \hline \text { Week 2 } & 6 & 14 & 2 & 2 & 40 \\ \hline \text { Week 3 } & 20 & 10 & 2 & 2 & 60 \\ \hline \end{array}$$ Did Mary's utility increase or decrease between week 1 and week \(3 ?\) Does Mary consider both goods to be normal goods? Explain. "c. Finally, examine the following information about Jane's choices: $$\begin{array}{|lccccc|} \hline & X_{1} & X_{2} & P_{1} & P_{2} & 1 \\ \hline \text { Week 1 } & 12 & 24 & 2 & 1 & 48 \\ \hline \text { Week 2 } & 16 & 32 & 1 & 1 & 48 \\ \hline \text { Week 3 } & 12 & 24 & 1 & 1 & 36 \\ \hline \end{array}$$ Draw a budget line-indifference curve graph that illustrates Jane's three chosen bundles. What can you say about Jane's preferences in this case? Identify the income and substitution effects that result from a change in the price of good \(x_{1}\).

Short answer

Based on the analysis, Bill's utility possibly decreased between week 1 and 2 and increased between week 1 and 3. Mary's utility seems to have increased from week 1 to week 3. Mary considers \(x_{1}\) to be a normal good and \(x_{2}\) could be an inferior good. Jane's analysis shows that she considers both goods as normal and her preferences are consistent.

Step-by-step solution

Analyze Bill's Choices

In the first week, Bill spends $40 (his total income) to buy 10 units of \(x_{1}\) and 20 units of \(x_{2}\) at respective prices of $2 and $1 each. In the second week, the price of \(x_{1}\) rises to $3 but Bill's income remains the same. This forces him to purchase less of both goods, bringing into concern that he might be worse off than before. For third week, Bill's income increases to $55 and so does his consumption for \(x_{2}\) thus suggesting he might be better off than Week 1.

Analyze Mary's Choices

In week one, Mary spends her entire $40 on 10 units of \(x_1\) and 20 units of \(x_2\) at their respective prices of $2 and $1.For week two, the price of \(x_2\) doubles to $2, and Mary reduces her consumption of both goods, which implies both goods could be normal goods. In week three, her income substantially increases to $60. She doubles the quantity of \(x_1\) and halves the quantity of \(x_2\). The income increase leads to a rise in the consumption of \(x_1\) but a decrease in \(x_2\), which suggest \(x_1\) is a normal good but \(x_2\) may be an inferior good.

Analyze Jane's Choices

For week one and two, the price of \(x_1\) halves while \(x_2\) remains constant. Despite having the same income, Jane increases her consumption of both goods, indicating that they are both normal goods. In week three, despite a decrease in her income to $36, and constant prices, Jane's consumption remains identical to week one. We plot three points representing the three weeks' bundles on the budget line-indifference curve graph. The points will provide a visual representation of Jane’s preferences and how they relate to her budget constraints and the prices of the two goods. The movement from week one to two shows the substitution effect, and the movement from week two to three illustrates the income effect.

Key concepts

Budget Constraint

The budget constraint is a fundamental concept within consumer choice theory. It represents the combinations of goods and services that a consumer can purchase given their income and the prices of the goods. The straight line on a graph, where the X-axis represents the quantity of one good and the Y-axis represents the quantity of another, illustrates all the possible purchases a consumer can make. For instance, in the given exercise about Bill, Mary, and Jane's weekly purchases, the budget constraint can be depicted based on their incomes and the prices they face for goods x₁ and x₂.

When a consumer's income changes or the price of a good changes, the budget line correspondingly shifts. If the income increases, the line shifts outward, allowing for more combinations of goods to be purchased; conversely, if the income decreases, it shifts inward, indicating fewer options. Similarly, if the price of a good decreases, the budget constraint pivots outward, making that good more affordable relative to the other. This dynamic can be observed in the exercise, where changes in income and prices lead to shifts in the budget constraints faced by each individual across the weeks.

Indifference Curve

Indifference curves illustrate a consumer's level of satisfaction with various bundles of goods. These curves show combinations of products that provide the consumer with the same level of utility, meaning they are indifferent to any combination lying on the same curve. The farther from the origin an indifference curve lies, the higher the utility it represents. Jane's choices, showcased in the exercise, can be analyzed by plotting her weekly bundles on a graph with indifference curves. These selected bundles should lie on successively higher curves as they move from lower utility to higher utility with income or price changes.

Indifference curves do not cross and are typically convex to the origin, reflecting the principle that consumers are willing to substitute between goods, but this substitution comes with diminishing returns. That is, as a consumer has more of one good, they are willing to give up less of another good to get even more of the first good.

Normal Goods

Normal goods are goods for which demand increases as a consumer's income rises, holding prices constant. The typical response when a person earns more money is to buy more of these goods because they are preferable as their purchasing power grows. As seen in the analysis from Mary’s choices, the quantity of X₁ she purchases increases significantly when her income rises, identifying X₁ as a normal good in her case.

This pattern is important for understanding how the consumption of goods changes with economic conditions. Companies that produce normal goods can expect their sales to grow when the general income level of the population increases.

Inferior Goods

Inferior goods are the opposite of normal goods; they are goods for which demand decreases as income increases. This is typically because these goods are considered to be of lower quality or a less desirable choice compared to alternatives that become affordable at higher income levels. In the exercise, Mary’s behavior suggests that X₂ may be an inferior good since she buys less of it as her income increases from \(40 to \)60.

Understanding the difference between normal and inferior goods helps businesses and policymakers predict consumer behavior in response to changes in the economy such as wage increases or economic downturns.

Substitution Effect

The substitution effect occurs when the price of a good changes relative to other goods, and the consumer adjusts their consumption to maintain the same level of utility. It is one of the components of the total effect of a price change, alongside the income effect. If the price of x₁ decreases, the substitution effect would typically cause a consumer to buy more of x₁ because it's relatively cheaper compared to x₂. This phenomenon is demonstrated within Jane’s choice from the exercise, where she increases her consumption of x₁ when its price halves, substituting it for x₂ despite her income remaining constant.

Income Effect

The income effect describes how a consumer's purchasing choices change in response to changes in their purchasing power or real income, holding all prices constant. When the price of a good falls, the consumer can buy the same quantity as before while spending less, effectively increasing their real income and allowing them to buy more in total or save the difference. In Jane’s case from the exercise, when the price of x₁ drops, and her purchasing power increases, she demonstrates the income effect by buying more items in week 2 compared to week 1. However, when her actual income falls in week 3, she reduces her overall consumption, indicating a negative income effect.