Practice Worksheet

Chapter 21 — The Theory of Consumer Choice

Part 1

Short Answer — The Dining Hall vs. Cup O' Soup Budget

Given A college student has two options for meals: eating at the dining hall for $6 per meal, or eating a Cup O' Soup for $1.50 per meal. Her weekly food budget is $60. Put dining-hall meals on the horizontal axis and Cups O' Soup on the vertical axis.

Income I = $60 P_meal = $6 P_soup = $1.50

Question a

What are the two intercepts of the budget constraint — that is, the most meals she can afford if she spends her entire budget on meals, and the most Cups O' Soup if she spends her entire budget on soup?

Max meals

Max Cups O' Soup

Hint To find each intercept, divide the total budget by the price of that good. Max meals = I / P_meal; max soups = I / P_soup.

Explanation Max meals = $60 ÷ $6 = 10 meals.
Max Cups O' Soup = $60 ÷ $1.50 = 40 soups.

The budget line runs from (0 meals, 40 soups) on the vertical axis down to (10 meals, 0 soups) on the horizontal axis. Every bundle on the line costs exactly $60.

Question b

What is the slope of the budget constraint, in "Cups O' Soup given up per extra meal"? Enter the magnitude (a positive number).

|slope| (soups per meal)

Hint The slope of the budget line with meals on the horizontal axis is −P_meal/P_soup. It tells you how many soups you must give up to afford one more meal.

Explanation Slope = −P_meal / P_soup = −$6 / $1.50 = −4.

So the magnitude is 4. Giving up 1 meal frees up $6, which buys 4 Cups O' Soup. Equivalently, to buy 1 more meal she must give up 4 soups. This slope is the relative price of meals in terms of soups.

Part 2

Short Answer — Optimum and a Price Change

Given Suppose the student's indifference curves are such that her initial optimum has her spending equal dollar amounts on dining-hall meals and Cup O' Soup (so $30 on each).

Question c

At the initial prices ($6 per meal, $1.50 per soup), how many meals and how many Cups O' Soup does she buy at her optimum? Label this bundle point A.

Meals at A

Soups at A

Hint Equal dollar amounts means $30 on meals and $30 on soup. Divide each by its price.

Explanation Meals: $30 ÷ $6 = 5 meals.
Soups: $30 ÷ $1.50 = 20 soups.

Verify feasibility: 5 × $6 + 20 × $1.50 = $30 + $30 = $60 ✓. The bundle (5, 20) lies exactly on the budget line — it's affordable and spends every dollar.

Question d

Now suppose the price of a Cup O' Soup rises from $1.50 to $2. Her budget and the price of a dining-hall meal are unchanged. What are the new intercepts of the budget constraint (max meals, max soups)?

New max meals

New max soups

Hint Only the soup price changes. The meals intercept depends on meal price, so it stays at $60 / $6. The soup intercept drops because soup is now more expensive: $60 / $2.

Explanation Max meals = $60 ÷ $6 = 10 meals (unchanged, since the price of meals is unchanged).
Max soups = $60 ÷ $2 = 30 soups (fell from 40).

The budget line pivots inward around the meals intercept (10). The new slope is −$6 / $2 = −3, so the line is now flatter in "soups per meal."

Question e

After the soup price rises, suppose the student's new optimum has her spending 30% of her budget on dining-hall meals (and the remaining 70% on Cup O' Soup). How many meals and how many soups does she buy at the new optimum? Label this bundle point B.

Meals at B

Soups at B

Hint 30% of $60 = $18 on meals. The remaining 70% = $42 on soup. Divide each dollar amount by the new price of that good.

Explanation Spending on meals: 0.30 × $60 = $18. Meals bought: $18 ÷ $6 = 3 meals.
Spending on soup: 0.70 × $60 = $42. Soups bought: $42 ÷ $2 = 21 soups.

Verify: 3 × $6 + 21 × $2 = $18 + $42 = $60 ✓.

Pay attention to this result. Her consumption of Cup O' Soup went from 20 → 21 even though the price of soup rose. If this 30/70 split came from her actual preferences (rather than being stipulated by the prompt), this would be the signature of a Giffen good: a good whose demand curve slopes upward because the income effect (poorer → buy more cheap calories) dominates the substitution effect. See Question 9 below for the formal definition.

Part 3

Multiple Choice

Figure 1 Use the budget constraint below to answer Questions 1–4. Good Y is on the vertical axis; Good X is on the horizontal axis.

Figure 1. Budget constraint with Y-intercept = 30 and X-intercept = 100.

Question 1 — Refer to Figure 1

If the consumer has $600 in income, what is the price of good X?

$3 $6 $10 $20

Hint The X-intercept tells you how many units of X the consumer can buy if she spends all her income on X. So P_X = Income / X-intercept.

Explanation Answer: B ($6). The X-intercept is 100 units, meaning the consumer can buy 100 units of X if she spends all $600 on X. Therefore P_X = $600 / 100 = $6.

Question 2 — Refer to Figure 1

If the consumer has $600 in income, what is the price of good Y?

$20 $6 $30 $10

Hint Same logic as Question 1, but use the Y-intercept. P_Y = Income / Y-intercept.

Explanation Answer: A ($20). The Y-intercept is 30 units. P_Y = $600 / 30 = $20.

Question 3 — Refer to Figure 1

If the price of good Y is $5, what is the price of good X?

$5 $3 $0.30 $1.50

Hint First find income from the Y-intercept: I = P_Y x Y-intercept. Then use the X-intercept to find P_X = I / X-intercept.

Explanation Answer: D ($1.50). If P_Y = $5, then I = $5 x 30 = $150. Then P_X = $150 / 100 = $1.50.

Question 4 — Refer to Figure 1

If the price of good X is $15, what is the price of good Y?

$15 $50 $30 $100

Hint Find income from the X-intercept: I = P_X x X-intercept. Then P_Y = I / Y-intercept.

Explanation Answer: B ($50). If P_X = $15, then I = $15 x 100 = $1,500. Then P_Y = $1,500 / 30 = $50.

Figure 2 Use the budget constraint and indifference curves below to answer Questions 5–7. The budget line is tangent to I₂ at point C. Points A and B are also on the budget line. Point D is below the budget line on I₁. Point E is above the budget line on I₃.

Figure 2. Budget line tangent to I₂ at point C (the optimum). I₁ is lower; I₃ is higher but unaffordable.

Question 5 — Refer to Figure 2

Given the budget constraint, the consumer's optimal choice will be point:

A B C E

Hint The optimum is the point on the budget line that reaches the highest indifference curve. Look for the tangency.

Explanation Answer: C. Point C is where the budget line is tangent to I₂, the highest indifference curve the consumer can reach. At the tangency, MRS = P_X/P_Y. Points A and B are on the budget line but sit on lower indifference curves. Point E is on I₃ but lies above the budget line — unaffordable.

Question 6 — Refer to Figure 2

It would be possible for the consumer to reach I₃ if:

the price of X increases the consumer's income decreases the price of X increases and the price of Y increases the price of Y decreases

Hint I₃ is currently unaffordable. The consumer needs the budget set to expand. Which change makes the affordable set larger?

Explanation Answer: D — the price of Y decreases. A price decrease for either good expands the budget set outward, potentially making I₃ reachable. Options A and C involve price increases, which shrink the budget set. Option B decreases income, which also shrinks the budget set.

Question 7 — Refer to Figure 2

Bundle B represents a point where:

the consumer is maximizing utility the consumer is not spending all of their income MRS_XY < P_X/P_Y MRS_XY > P_X/P_Y

Hint Point B is on the budget line but above and to the left of the optimum C. At B the indifference curve is steeper than the budget line. Is the IC slope (MRS) greater or less than the budget-line slope (P_X/P_Y)?

Explanation Answer: D — MRS_XY > P_X/P_Y. At point B the consumer has relatively little X and lots of Y, so she values one more unit of X more highly than the market does (the IC is steeper than the budget line). Since MRS > price ratio, she could increase utility by buying more X and less Y — moving down the budget line toward the tangency at C.

(A) is wrong because B is not the optimum. (B) is wrong because B is on the budget line, so all income is spent. (C) has the inequality backward.

Question 8

Assume that a college student purchases only Ramen noodles and textbooks. If Ramen noodles are an inferior good and textbooks are a normal good, then the income effect associated with an increase in the price of a textbook will result in:

an increase in textbooks purchased and an increase in Ramen noodles purchased a decrease in textbooks purchased and an increase in Ramen noodles purchased a decrease in textbooks purchased and a decrease in Ramen noodles purchased an increase in textbooks purchased and a decrease in Ramen noodles purchased

Hint The question asks about the income effect only. A price increase makes the consumer effectively poorer. When you get poorer, what happens to purchases of a normal good vs. an inferior good?

Explanation Answer: B — fewer textbooks, more Ramen noodles.

A textbook price increase makes the student effectively poorer (lower real income). The income effect of becoming poorer means:
Textbooks (normal good): less income → buy fewer.
Ramen noodles (inferior good): less income → buy more (that's what "inferior" means — demand rises when income falls).

Note: the question isolates the income effect. The total effect on textbooks also includes a substitution effect (buy fewer since they're relatively more expensive), reinforcing the same direction.

Question 9

When Joshua's income increases, he purchases more prime-rib dinners. This means that, for Joshua, prime-rib dinners are:

a normal good an inferior good a Giffen good a complement to some other good

Hint The definitions: a normal good is one you buy more of when income rises; an inferior good is one you buy less of when income rises.

Explanation Answer: A — a normal good. By definition, a normal good is one whose quantity demanded increases when income increases, holding prices constant. Joshua buys more prime-rib dinners when his income rises, which is exactly the definition of a normal good.

Part 4

True or False

Question 10

For a typical consumer, most indifference curves are bowed inward (convex to the origin).

Hint "Bowed inward" means convex to the origin. Think about what this shape implies about how willing the consumer is to trade one good for the other as quantities change.

Explanation True. The bowed-inward shape reflects diminishing marginal rate of substitution: as a consumer accumulates more of good X, each additional unit of X is worth less to her (in terms of Y she would give up). People generally prefer balanced bundles to extremes, which is why most indifference curves are convex to the origin.

Question 11

If goods A and B are perfect substitutes, then the marginal rate of substitution (MRS) of good A for good B is constant.

Hint What do the indifference curves look like for perfect substitutes — are they curved or straight lines?

Explanation True. Perfect substitutes have straight-line indifference curves. The slope of a straight line is constant everywhere, so the MRS (the slope of the IC) is the same at every point. For example, if nickels and dimes are perfect substitutes, the consumer is always willing to trade 2 nickels for 1 dime, regardless of how many of each she has.

Question 12

At a consumer's optimal choice, the consumer chooses the combination of goods that equates the marginal rate of substitution (MRS) and the price ratio.

Hint The optimum is the tangency of the budget line and the highest reachable indifference curve. What does "tangent" mean about the two slopes?

Explanation True. At the optimum, the indifference curve is tangent to the budget line. Tangent means "same slope." The slope of the IC is the MRS; the slope of the budget line is P_X/P_Y. So MRS = P_X/P_Y at the optimum. This is the fundamental tangency condition for consumer optimization.

Part 5

Refer to Figure 21-1

Figure 21-1 A consumer chooses between two goods, X and Y. The figure shows her budget constraint (solid black) along with three of her indifference curves (I₁, I₂, I₃) and four labeled bundles (W, X, Y, Z). P_x = $5 and P_y = $10. Use the figure to answer the next two questions.

Figure 21-1. Budget line at I = $500, P_x = $5, P_y = $10. Three indifference curves and four labeled bundles.

Question 13 — Refer to Figure 21-1

Which point represents the consumer's utility-maximizing bundle (the optimum)?

Point W (inside the budget set, on I₁) Point X (on the budget line, on I₁) Point Y (tangent to I₂ on the budget line) Point Z (on I₃, above the budget line)

Hint The optimum must be (1) affordable (on or below the budget line) and (2) on the highest indifference curve that meets that test. The tangency point is the only point where the budget line touches — but never crosses — the highest reachable IC.

Explanation Answer: C (Point Y). Point Y is where the budget line is tangent to I₂, the highest reachable indifference curve. At the tangency, MRS = P_x/P_y.

Why not the others?
W is affordable but leaves money unspent — the consumer could move to a higher IC by spending more.
X is on the budget line but on a lower IC (I₁). The consumer could slide along the budget line toward the middle and reach I₂, a higher curve.
Z is on I₃ (most preferred), but sits above the budget line — unaffordable.

Question 14 — Refer to Figure 21-1

At the optimum in Figure 21-1, what is the consumer's marginal rate of substitution (MRS), in units of Y per unit of X?

MRS = 0.2 (Y per X) MRS = 0.5 (Y per X) MRS = 2 (Y per X) MRS = 5 (Y per X)

Hint At the optimum the IC is tangent to the budget line, so MRS equals the relative price: MRS = P_x / P_y. Plug in $5 and $10.

Explanation Answer: B (MRS = 0.5). At the optimum, MRS = P_x/P_y = $5 / $10 = 0.5 Y per X.

Sanity check on the graph: the budget line runs from (0, 50) to (100, 0), so its slope magnitude is 50/100 = 0.5 units of Y per unit of X. At point Y the IC has the same slope as the budget line (that's what "tangent" means), so MRS = 0.5. Intuitively: Y costs twice as much as X, so one unit of Y must be worth two units of X at the optimum — meaning one X is only worth half a Y.

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