Introduction
A budget line separates the bundles a consumer can afford from those that cost more than the consumer’s available income. To classify a bundle, the key calculation is its total expenditure: multiply the quantity of each good by its price, then add. Comparing that amount with income tells us whether the bundle leaves money unspent, uses the full budget, or is unaffordable.
The distinction matters because a bundle can be affordable without using all available income. The budget line represents bundles that use the full budget; the budget set also includes less expensive bundles. Here we focus on classifying bundles using a given pair of prices and a given income.
The Expenditure Test
Consider a consumer choosing quantities \(x_1\) and \(x_2\) of two goods. Let \(p_1\) and \(p_2\) be their respective prices per unit, and let \(m\) be the consumer’s available income. Assume prices are positive, income is nonnegative, and quantities cannot be negative. The consumer can afford the bundle if its total expenditure is no greater than income.
“Inside” refers to the affordable region that costs less than income; it does not mean the bundle is unaffordable. The consumer can buy an inside bundle and have money left over. A bundle on the line is also affordable, but it spends the entire budget. A bundle outside the budget set costs more than the available income and is not affordable under the stated prices and income.
| Comparison | Classification | Budget interpretation |
|---|---|---|
| \(p_1x_1+p_2x_2<m\) | Inside | Affordable; some income remains unspent |
| \(p_1x_1+p_2x_2=m\) | On the budget line | Affordable; all income is spent |
| \(p_1x_1+p_2x_2>m\) | Outside the budget set | Unaffordable at the stated prices and income |
The amount left over, or the shortfall, can be calculated with the same comparison. Define the budget slack as \(m-E\). A positive slack is money remaining, zero slack means the bundle lies on the line, and negative slack indicates that expenditure exceeds income. The size of the negative amount is the shortfall.
Why the Test Works
The expenditure formula follows from the meaning of a unit price. Buying \(x_1\) units of good 1 at \(p_1\) dollars per unit costs \(p_1x_1\) dollars. Buying \(x_2\) units of good 2 at \(p_2\) dollars per unit costs \(p_2x_2\) dollars. If these are the only purchases, total expenditure is the sum of those two costs.
Income is the maximum amount available for those purchases. Subtracting total expenditure from income gives the amount remaining. If that difference is positive, the bundle costs less than income; if it is zero, it costs exactly income; and if it is negative, the bundle costs more than income. These three possibilities produce the inside, on, and outside classifications.
This test assumes that the two stated goods are the relevant purchases and that the stated prices and income are the amounts to use. It does not account for additional fees, taxes, discounts, or other goods unless those are included in the problem data. It also requires consistent units: if a price is quoted per kilogram, for example, the quantity must be measured in kilograms.
Worked Example: Classifying Three Shopping Bundles
A shopper has $60 to spend on two goods. Good 1 costs $6 per unit and good 2 costs $4 per unit. Classify bundles \(A=(5,6)\), \(B=(4,9)\), and \(C=(7,6)\). For each bundle, calculate the amount left over or the shortfall.
Method. Calculate each bundle’s total expenditure and compare it with $60. This avoids the common mistake of checking just one good’s cost instead of adding the costs of both goods.
Bundle A. The cost of good 1 is \(5\times \$6=\$30\). The cost of good 2 is \(6\times \$4=\$24\). Total expenditure is \(\$30+\$24=\$54\). Since \(\$54<\$60\), bundle A is inside the budget line. The amount left over is \(\$60-\$54=\$6\).
Bundle B. The cost of good 1 is \(4\times \$6=\$24\). The cost of good 2 is \(9\times \$4=\$36\). Total expenditure is \(\$24+\$36=\$60\). Since expenditure equals income, bundle B is on the budget line, with no income left over.
Bundle C. The cost of good 1 is \(7\times \$6=\$42\). The cost of good 2 is \(6\times \$4=\$24\). Total expenditure is \(\$42+\$24=\$66\). Since \(\$66>\$60\), bundle C is outside the budget set. Its shortfall is \(\$66-\$60=\$6\).
Check. The three budget slacks are \(60-54=6\), \(60-60=0\), and \(60-66=-6\), respectively. Their signs agree with the classifications.
Finding the Quantity That Exhausts the Budget
A classification problem may ask for a missing quantity rather than give a complete bundle. To find the quantity of one good that puts a bundle on the line, set total expenditure equal to income and solve for the unknown quantity. The other good’s expenditure must be accounted for first; only the remaining income is available for the unknown quantity.
Worked Example: Finding a Budget-Line Quantity
A consumer has $100. Good 1 costs $12 per unit and good 2 costs $5 per unit. The consumer buys 5 units of good 1. Find the quantity of good 2 that uses the full budget, then classify bundles with 7.5, 8, and 9 units of good 2.
Method. First calculate how much the 5 units of good 1 cost. Then divide the remaining income by the price of good 2. This avoids treating all $100 as available for good 2 after some income has already been spent.
The cost of good 1 is \(5\times \$12=\$60\). Income remaining for good 2 is \(\$100-\$60=\$40\). At $5 per unit, the quantity of good 2 that costs $40 is \(\$40/\$5=8\) units. Therefore, the bundle \((5,8)\) is on the budget line.
For 7.5 units of good 2, its cost is \(7.5\times \$5=\$37.50\). Total expenditure is \(\$60+\$37.50=\$97.50\), so the bundle is inside the line, with \(\$100-\$97.50=\$2.50\) left over.
For 8 units, total expenditure is \(\$60+(8\times \$5)=\$60+\$40=\$100\). This bundle is on the line.
For 9 units, total expenditure is \(\$60+(9\times \$5)=\$60+\$45=\$105\). This bundle is outside the budget set, with a shortfall of \(\$105-\$100=\$5\).
Check. The required quantity of 8 units costs exactly the $40 remaining after purchasing good 1, so total expenditure equals the $100 budget.
Expenditure Equality and Careful Classification
A bundle lies on the budget line only when its total expenditure equals income, not merely when one good’s cost equals some portion of income. Equality should be checked using the complete expenditure calculation. When quantities include fractions, retain those fractions during the calculation and compare the resulting total with income; do not round early and accidentally turn a bundle just inside or outside the line into an apparent equality.
The classification describes affordability, not whether a consumer prefers a bundle. Preferences and budget constraints answer different questions: the expenditure test determines which bundles are feasible at the stated prices and income, while a preference ranking concerns which bundles the consumer likes better. A bundle on the line is not automatically the consumer’s most-preferred bundle.
Worked Example: Checking Equality With Fractional Quantities
A consumer has $84. Good 1 costs $9 per unit and good 2 costs $6 per unit. Classify the bundles \((6,5)\), \((6,4.5)\), and \((7,5)\), and state the slack or shortfall for each.
Method. Multiply each quantity by its own price, add the two costs, and compare the total with $84. This avoids a common mistake: rounding a fractional quantity before checking whether expenditure exactly equals income.
For \((6,5)\), good 1 costs \(6\times \$9=\$54\), and good 2 costs \(5\times \$6=\$30\). Total expenditure is \(\$54+\$30=\$84\). The bundle is on the line, and its slack is \(\$84-\$84=\$0\).
For \((6,4.5)\), good 1 still costs $54. Good 2 costs \(4.5\times \$6=\$27\). Total expenditure is \(\$54+\$27=\$81\). The bundle is inside the line, with \(\$84-\$81=\$3\) left over.
For \((7,5)\), good 1 costs \(7\times \$9=\$63\), and good 2 costs \(5\times \$6=\$30\). Total expenditure is \(\$63+\$30=\$93\). The bundle is outside the budget set, with a shortfall of \(\$93-\$84=\$9\).
Check. The expenditure totals are $84, $81, and $93, so only the first bundle exactly exhausts the $84 budget.
Common Mistakes
- Confusing affordable with on the line. A bundle costing less than income is affordable, but it is inside the line, not on it. Equality is required for a bundle to be on the line.
- Comparing only one good’s cost with income. The test uses the sum \(p_1x_1+p_2x_2\), not either cost by itself.
- Mixing up a shortfall and an amount left over. If expenditure exceeds income, report the positive difference \(E-m\) as the shortfall. The signed slack \(m-E\) is negative.
- Using mismatched units. A price per package must be multiplied by the number of packages, not by a quantity measured in individual items unless the price is converted appropriately.
- Assuming full expenditure is required. The budget constraint permits spending less than income; it only rules out spending more than income.
Check Your Understanding
For each problem, use the stated prices, quantities, and income to classify the bundle and calculate any remaining amount or shortfall.
- A consumer has $50. Good 1 costs $5 per unit and good 2 costs $10 per unit. Classify bundle \((4,2)\) and calculate its budget slack. (Answer: Inside; expenditure is $40, so $10 remains.)
- A household has $58 to spend. Good 1 costs $3.50 per unit and good 2 costs $8 per unit. Classify bundle \((6,4)\) and find the amount left over or the shortfall. (Answer: Inside; expenditure is $53, so $5 remains.)
- Income is $96, and the prices are $12 for good 1 and $8 for good 2. Classify \((5,4)\), \((6,3)\), and \((7,3)\). (Answer: \((5,4)\) is inside with $4 left; \((6,3)\) is on the line; \((7,3)\) is outside with a $12 shortfall.)
- A consumer has $63. Good 1 costs $7 per unit and good 2 costs $4 per unit. If the consumer buys 5 units of good 1, find the quantity of good 2 that exhausts the budget. Then classify the bundles with 6.5 and 8 units of good 2. (Answer: 7 units exhaust the budget; 6.5 units is inside with $2 left, and 8 units is outside with a $4 shortfall.)
- A consumer has $120. Good 1 costs $15 per unit and good 2 costs $10 per unit. Classify bundle \((4.5,5.25)\) and verify the result by calculating its budget slack. (Answer: On the budget line; expenditure is $67.50 + $52.50 = $120, so slack is $0.)