Affordable Bundles and the Budget Set

Introduction

A consumer can prefer one bundle to another and still be unable to afford either one. To separate preference from feasibility, first ask a more basic question: does the total cost of this bundle exceed the consumer’s available income? The collection of all bundles that pass this test is the budget set.

This distinction matters when reading a graph of possible consumption bundles. A point represents quantities of goods; its location on the graph does not by itself say whether the consumer can purchase it. Affordability depends on the prices of the goods and the income available for spending. This tutorial defines the budget set and develops a reliable method for classifying bundles, including plotted points.

Definition of the Budget Set

Suppose a consumer chooses quantities of $n$ goods. Let $x_i$ be the quantity of good $i$, $p_i$ its price per unit, and $m$ the consumer’s income available for these purchases. Assume prices are fixed during the decision, quantities cannot be negative, and the consumer cannot borrow or receive extra funds for the purchases. Any taxes or required charges are included in the prices or in the stated spending resources.

Definition: The budget set is the collection of all nonnegative consumption bundles whose total expenditure does not exceed available income. In symbols, it is the set of bundles satisfying $x_i \geq 0$ for every good $i$, and $\sum_{i=1}^{n} p_i x_i \leq m$. A bundle is affordable if it belongs to this set.

The expression $p_i x_i$ is expenditure on good $i$: price per unit multiplied by the number of units purchased. Adding those expenditures across goods gives the bundle’s total cost. The inequality includes bundles that cost exactly $m$, as well as bundles that cost less. A bundle costing more than $m$ is not affordable under the stated assumptions.

For two goods, a plotted point $(x_1,x_2)$ records the quantity of the first good and the quantity of the second. To classify it, calculate $p_1x_1+p_2x_2$ and compare that amount with $m$. A point that spends exactly all available income is on the affordability boundary; a point that costs less is inside the budget set; a point that costs more is outside it. This classification does not require knowing which bundle the consumer likes best.

Why the Expenditure Test Works

The budget-set condition follows directly from how a purchase is paid for. If the consumer buys $x_i$ units of good $i$, each at a price $p_i$, the payment for that good is $p_ix_i$. The consumer must pay for every good in the bundle, so the total payment is the sum of these amounts. With no borrowing or additional funds, a bundle can be purchased exactly when that sum is no greater than available income.

\text{Total expenditure}=\sum_{i=1}^{n} p_i x_i \qquad\Longrightarrow\qquad \text{affordable exactly when } \sum_{i=1}^{n} p_i x_i \leq m.

The “exactly when” is important. If total expenditure is below income, the consumer can pay for the bundle and retain the difference. If it equals income, the consumer can pay for it but has no income left from this budget. If it exceeds income, the consumer would need more resources than are available. This argument establishes the test; it does not depend on a particular preference ranking.

Classifying Bundles from Prices and Quantities

Worked Example: Comparing Three Grocery Bundles

A shopper has $42 available for two goods. Good A costs $4 per unit and good B costs $7 per unit. Classify bundles $(6,2)$, $(5,4)$, and $(7,2)$ as affordable, unaffordable, or exactly exhausting the budget. Use the total-expenditure method and identify a common mistake it prevents.

Step 1: Calculate the cost of each good in each bundle. For $(6,2)$, good A costs $6 \times \$4=\$24$, and good B costs $2 \times \$7=\$14$. Total expenditure is $\$24+\$14=\$38$. Since $\$38 \leq \$42$, the bundle is affordable, with $\$42-\$38=\$4$ remaining.

For $(5,4)$, good A costs $5 \times \$4=\$20$, and good B costs $4 \times \$7=\$28$. Total expenditure is $\$20+\$28=\$48$. Since $\$48>\$42$, this bundle is unaffordable; it exceeds available income by $\$48-\$42=\$6$.

For $(7,2)$, good A costs $7 \times \$4=\$28$, and good B costs $2 \times \$7=\$14$. Total expenditure is $\$28+\$14=\$42$. The bundle exactly exhausts the budget and is affordable because equality is included in the budget-set condition.

Check. The three total expenditures are $38, $48, and $42. Comparing each with $42 gives, respectively, affordable, unaffordable, and affordable at the limit.

Common mistake avoided: Comparing the number of units in each bundle with income would mix quantities with dollars. The correct method converts each quantity into dollars using its own price before adding the costs.

Reading Affordability on a Plot

On a two-good graph, first read each point’s coordinates using the axis labels and scale. Then treat those coordinates as quantities and calculate expenditure. A point can be visually close to another point yet have a different affordability status because the two goods may have different prices. Do not classify points by their distance from the origin or by whether they appear “high” or “low” on the page.

If a graph marks the set of combinations that spend exactly all available income, that set is the boundary of the budget set. The points satisfying the expenditure test with a strict inequality lie within the affordable region; points failing the test lie outside it. The next step in studying budget constraints is to express the two-good condition as an equation, but classification itself only requires the expenditure comparison.

Worked Example: Classifying Plotted Points

A graph has good A on the horizontal axis and good B on the vertical axis. Each coordinate is measured in units, not dollars. The price of A is $6 per unit, the price of B is $9 per unit, and available income is $63. The plotted points are $P=(6,3)$, $Q=(4,4)$, and $R=(8,2)$. Classify each point and state which one lies on the budget boundary. Use the coordinate-and-cost method, avoiding the mistake of treating a coordinate as a dollar amount.

Step 1: Read the coordinates as quantities. Point $P=(6,3)$ means 6 units of A and 3 units of B. Its cost is $6 \times \$6=\$36$ for A plus $3 \times \$9=\$27$ for B, for a total of $\$36+\$27=\$63$. Since this equals income, $P$ is affordable and lies on the boundary.

Step 2: Apply the same calculation to $Q$. Point $Q=(4,4)$ costs $4 \times \$6=\$24$ for A and $4 \times \$9=\$36$ for B. Total expenditure is $\$24+\$36=\$60$. Because $\$60<\$63$, $Q$ is affordable and lies inside the budget set, with $\$63-\$60=\$3$ unspent.

Step 3: Apply the test to $R$. Point $R=(8,2)$ costs $8 \times \$6=\$48$ for A and $2 \times \$9=\$18$ for B. Total expenditure is $\$48+\$18=\$66$. Because $\$66>\$63$, $R$ is unaffordable and lies outside the budget set by $\$66-\$63=\$3$.

Check. The totals are $63, $60, and $66; compared with income of $63, they classify as boundary, inside, and outside. Only $P$ spends exactly all available income.

Common mistake avoided: A coordinate such as 6 is a quantity of good A, not a cost of $6. Multiply each coordinate by the price of the good on that axis before adding.

More Than Two Goods and Remaining Income

The same test applies when a bundle contains three or more goods: calculate each good’s expenditure separately, then add all amounts. Keeping a line for each good is useful because it makes it easier to catch an omitted purchase or a multiplication error. If the total is affordable, the difference between income and total expenditure is the amount left over from the stated budget.

Worked Example: A Three-Good Equipment Bundle

A technician has $75 available to buy three types of equipment. A sensor costs $5 per unit, a cable costs $8 per unit, and a housing costs $12 per unit. Consider bundle $A=(3,2,2)$, bundle $B=(5,3,2)$, and bundle $C=(4,3,2.5)$, where the coordinates give quantities of sensors, cables, and housings in that order. Assume fractional quantities are permitted for this hypothetical purchase. Classify each bundle and calculate any amount left over or the shortfall. Use the itemized-cost method and avoid rounding a quantity before calculating its cost.

Step 1: Calculate bundle $A$. Sensors cost $3 \times \$5=\$15$, cables cost $2 \times \$8=\$16$, and housings cost $2 \times \$12=\$24$. Total expenditure is $\$15+\$16+\$24=\$55$. Since $\$55\leq\$75$, the bundle is affordable and leaves $\$75-\$55=\$20$.

Step 2: Calculate bundle $B$. Sensors cost $5 \times \$5=\$25$, cables cost $3 \times \$8=\$24$, and housings cost $2 \times \$12=\$24$. Total expenditure is $\$25+\$24+\$24=\$73$. Since $\$73\leq\$75$, the bundle is affordable and leaves $\$75-\$73=\$2$.

Step 3: Calculate bundle $C$. Sensors cost $4 \times \$5=\$20$, cables cost $3 \times \$8=\$24$, and housings cost $2.5 \times \$12=\$30$. Total expenditure is $\$20+\$24+\$30=\$74$. Since $\$74\leq\$75$, the bundle is affordable and leaves $\$75-\$74=\$1$.

Check. The totals $55, $73, and $74 are all no greater than the $75 budget. The corresponding remainders of $20, $2, and $1 add back to the totals to give $75 in each case.

Common mistake avoided: Rounding 2.5 housings down to 2 before multiplying would incorrectly reduce bundle $C$’s cost by $0.5 \times \$12=\$6$. Use the quantities stated in the problem; do not change them unless the purchasing rules require whole units.

Affordability Is Not the Same as Preference

The budget set answers a feasibility question: which bundles can be purchased with the specified resources? Preferences answer a different question: how does the consumer rank bundles? A bundle can be affordable but less preferred than another affordable bundle. It can also be preferred to an affordable option while remaining unaffordable. Indifference curves and preference rankings describe comparisons between bundles, but they do not change prices or increase income.

This separation is especially useful when a graph shows both preferences and a budget set. First identify which plotted bundles are feasible using expenditure and income. Only then compare the preferences among feasible bundles if the problem asks for a choice. Do not infer affordability from an indifference curve, a ranking, or a claim that a bundle would be desirable.

Common Errors to Avoid

Excluding equality. A bundle with total cost exactly equal to income is affordable. The condition is “does not exceed,” not “is strictly less than.”
Adding quantities instead of costs. Quantities of different goods cannot be compared with a dollar budget until each quantity is multiplied by its own price.
Using one good’s price for every good. Calculate $p_i x_i$ separately for each good using the price that belongs to it.
Misreading graph axes. Coordinates usually represent quantities. Read the axis labels and scale before calculating expenditure.
Confusing feasibility with desirability. The budget set classifies what the consumer can afford; it does not say which affordable bundle the consumer prefers.

Key takeaway: To test a bundle, read its quantities, multiply each quantity by that good’s price, add the expenditures, and compare the total with available income. A total no greater than income is affordable; a total greater than income is not.

Check Your Understanding

For each problem, calculate total expenditure before deciding whether the bundle is affordable.

A student has $50 for two goods priced at $4 and $7 per unit. Classify bundles $(5,4)$, $(6,4)$, and $(3,5)$. Which bundle, if any, uses exactly all the income? (Answer: $(5,4)$ costs $48 and is affordable; $(6,4)$ costs $52 and is unaffordable; $(3,5)$ costs $47 and is affordable; none uses exactly all $50.)
On a graph, the horizontal quantity is good A and the vertical quantity is good B. Prices are $2.50 and $6 per unit, and income is $45. Classify points $(6,5)$, $(8,4)$, and $(9,4)$, and identify any boundary point. (Answer: costs are $45, $44, and $46.50, respectively; the first is on the boundary, the second is affordable inside it, and the third is unaffordable.)
A bundle contains three goods with prices $3, $5, and $9 per unit, and income is $68. Classify $(4,5,3)$, $(5,4,3)$, and $(4,4,4)$; state any remaining income. (Answer: expenditures are $64, $62, and $68; all are affordable, with $4 and $6 remaining for the first two, and the third exactly on the boundary.)
A consumer has $60 available and faces prices of $7 per unit for good A and $11 per unit for good B. A plotted point is $(4,3)$. Calculate its total cost, determine its affordability, and state the amount left or the shortfall. (Answer: $4 \times \$7+3 \times \$11=\$28+\$33=\$61$; it is unaffordable by $1.)
A hypothetical buyer has $80 for two goods priced at $8 and $5 per unit. Compare bundles $(5,8)$, $(6,7)$, and $(4,9)$. Classify each and identify which bundle leaves the least income unspent among the affordable bundles. (Answer: expenditures are $80, $83, and $77; the first is affordable at the boundary, the second is unaffordable, and the third is affordable with $3 left; the first leaves the least unspent.)

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