Proportional Changes in Income and All Prices

Why Proportional Changes Matter

A household’s income and the prices it faces are measured in money. If all those dollar amounts rise together in the same proportion, the numbers change, but the household may be able to buy exactly the same bundles as before. This distinction between money amounts and purchasing power is central to understanding budget constraints.

The key condition is that one common positive factor multiplies both income and every good’s price. A change in income and just one price, or different percentage changes for different prices, does not generally meet that condition. The results below are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Suppose a consumer chooses quantities $x_1,\ldots,x_n$, where each $x_i\geq 0$. Good $i$ has positive price $p_i$, and the consumer has income $m\geq 0$. The budget set is the set of all affordable bundles: $$ B(\mathbf{p},m)=\left\{\mathbf{x}\in\mathbb{R}_+^n:\sum_{i=1}^{n}p_i x_i\leq m\right\}. $$ A proportional change uses one factor $\lambda>0$ to set every new price to $p_i'=\lambda p_i$ and new income to $m'=\lambda m$.

The budget set includes both bundles that use all income and bundles that leave some income unspent. A bundle is on the budget boundary when total spending equals income; it is strictly inside the budget set when spending is less than income. Both kinds of bundles matter when comparing the original and changed constraints.

The Invariance Result

When income and all prices are scaled by the same positive factor, the budget set is unchanged:

Theorem: For $\lambda>0$, if $p_i'=\lambda p_i$ for every good $i$ and $m'=\lambda m$, then $B(\mathbf{p}',m')=B(\mathbf{p},m)$. Every bundle affordable before the change remains affordable afterward, and every bundle affordable afterward was affordable before.

The conclusion is about the set of affordable bundles, not about the numerical dollar amounts in the budget equation. To justify it, take any nonnegative bundle $\mathbf{x}$. Under the changed prices, its total cost is $\sum_i p_i'x_i=\sum_i \lambda p_i x_i$. Since $\lambda>0$, multiplying or dividing an inequality by $\lambda$ does not reverse its direction.

\begin{aligned} \mathbf{x}\in B(\mathbf{p}',m') &\iff \sum_{i=1}^{n}p_i'x_i\leq m'\\ &\iff \sum_{i=1}^{n}\lambda p_i x_i\leq \lambda m\\ &\iff \lambda\sum_{i=1}^{n}p_i x_i\leq \lambda m\\ &\iff \sum_{i=1}^{n}p_i x_i\leq m\\ &\iff \mathbf{x}\in B(\mathbf{p},m). \end{aligned}

The same reasoning applies to bundles on the boundary: equality in the original budget equation becomes equality after multiplying both sides by $\lambda$. For bundles below the boundary, the inequality remains strict in the same direction. The positivity of $\lambda$ is essential to this argument.

Worked Example: Checking a Two-Good Bundle After a Price-Level Change

A student has a weekly budget of $100 for two goods. Their prices are $5 per unit of good 1 and $10 per unit of good 2. Both prices and the budget are multiplied by 1.4. Is the bundle $(8,6)$ affordable before and after the change? Find the two budget-line intercepts in each case.

Method. Calculate the bundle’s total cost in each budget equation, then find each intercept by dividing income by the corresponding price. This checks both bundle affordability and the consequence of scaling for the boundary.

Before the change. Prices are $p_1=5$ and $p_2=10$, and income is $m=100$. The budget equation is $5x_1+10x_2=100$. At $(8,6)$, spending is

$$ 5(8)+10(6)=40+60=100. $$

The bundle is exactly on the budget boundary. The good 1 intercept is $100/5=20$ units, and the good 2 intercept is $100/10=10$ units.

After the change. The new prices are $1.4(5)=7$ and $1.4(10)=14$, and the new budget is $1.4(100)=140$. The budget equation is $7x_1+14x_2=140$. The bundle’s new cost is

$$ 7(8)+14(6)=56+84=140. $$

It is still exactly on the boundary. The new intercepts are $140/7=20$ and $140/14=10$, the same as before. Check: every new price and the new budget is 1.4 times its original value, so both the bundle’s cost and available income have increased by that factor. The common mistake this method avoids is comparing the new $140 budget with the old $100 budget without also recalculating prices.

Budget-Line Interpretation

For two goods, the boundary of the budget set is $p_1x_1+p_2x_2=m$. Solving for $x_2$, when $p_2>0$, gives $x_2=m/p_2-(p_1/p_2)x_1$. If every price and income are multiplied by the same $\lambda$, the intercept $m/p_2$ and the price ratio $p_1/p_2$ are unchanged:

\frac{\lambda m}{\lambda p_2}=\frac{m}{p_2}, \qquad \frac{\lambda p_1}{\lambda p_2}=\frac{p_1}{p_2}.

Thus, in the two-good diagram, the budget boundary occupies the same position and has the same slope. More generally, for any number of goods, the theorem says the entire affordable set is unchanged, not merely that a particular boundary point remains affordable. This follows from the equivalence of the budget inequalities proved above.

Worked Example: Verifying the Whole Set for Three Goods

A household has $96 to spend on three goods priced at $2, $4, and $8 per unit. Prices and income are all multiplied by 0.75. Check whether $(10,8,3)$ is affordable before and after the change, calculate the unspent budget in each case, and compare the maximum affordable quantity of each good when spending on that good alone.

Method. Apply the common factor to every price and income, then compare total cost and available income. For the single-good quantities, divide the budget by the relevant price. This method avoids scaling the budget while accidentally leaving one good’s price unchanged.

Before the change. The bundle costs

$$ 2(10)+4(8)+8(3)=20+32+24=76. $$

Since $76\leq 96$, the bundle is affordable, with $96-76=20$ dollars unspent.

After the change. The new prices are $0.75(2)=1.5$, $0.75(4)=3$, and $0.75(8)=6$. The new budget is $0.75(96)=72$. The same bundle costs

$$ 1.5(10)+3(8)+6(3)=15+24+18=57. $$

Since $57\leq 72$, it remains affordable, with $72-57=15$ dollars unspent. The difference in unspent dollar amounts is consistent with the factor: $0.75(20)=15$.

Single-good checks. Before the change, the maximum quantities are $96/2=48$, $96/4=24$, and $96/8=12$. Afterward, they are $72/1.5=48$, $72/3=24$, and $72/6=12$. Check: each maximum is unchanged. The common mistake this example guards against is concluding that affordability changes because both the cost and the remaining budget are smaller dollar amounts.

When the Change Is Not Proportional

A common factor must apply to every price as well as to income. If the ratios $p_i'/p_i$ are not all equal to $m'/m$, the invariance result cannot be used. The budget set may change, even if some prices and income happen to rise by the same percentage.

Worked Example: Detecting a Nonproportional Change

A consumer has income of $120 and faces prices of $3 for good 1 and $6 for good 2. Income rises to $144, while the prices change to $3.60 and $6.60. Determine whether the budget set is unchanged. Use the bundle $(0,21)$ to verify your conclusion, and compare the good 2 intercepts.

Method. First compare each price’s scaling factor with the income’s scaling factor. Then test a bundle whose affordability could differ. This avoids assuming that a budget set is preserved just because income and at least one price rise by the same percentage.

The income factor is $144/120=1.2$. Good 1’s price factor is $3.60/3=1.2$, but good 2’s price factor is $6.60/6=1.1$. There is no single common factor for all prices and income, so the invariance theorem’s condition fails.

At the original prices, the bundle $(0,21)$ costs $3(0)+6(21)=126$ dollars. Since $126>120$, it is not affordable. At the new prices, it costs $3.60(0)+6.60(21)=138.60$ dollars. Since $138.60\leq144$, it is affordable.

The good 2 intercept was $120/6=20$ units; it is now $144/6.60\approx21.82$ units, rounded to two decimal places. The affordable set has changed. Check: the good 1 intercept stays at $120/3=40$ and $144/3.60=40$, but that single unchanged intercept does not establish that the whole budget set is unchanged. The common mistake is to notice the 20% increases in income and the first price, then incorrectly extend that factor to the second price.

How to Apply the Result

When comparing two budget constraints, use the same test regardless of the number of goods. Check whether the ratio of each new price to its original price equals the ratio of new income to original income. If the ratios match and are positive, the affordable sets are identical. If they do not match, do not claim that the set is unchanged; compare the actual constraints or test relevant bundles.

Record the original constraint.
Write $\sum_i p_i x_i\leq m$, with each price and income clearly identified.

Calculate each change factor.
Compare $p_i'/p_i$ for every good with $m'/m$.

Compare the factors.
If all equal one positive $\lambda$, the budget set is unchanged by the theorem. Otherwise, the theorem does not apply.

Verify with a bundle or intercept.
Substitute quantities into the relevant budget inequalities; for a two-good intercept, divide income by the corresponding price.

Common Mistakes and Economic Meaning

Scaling only income and one price. The result requires the same factor for every good’s price.
Using percentage changes without checking the base amounts. Compare ratios such as $p_i'/p_i$ and $m'/m$, not dollar increases alone.
Checking only a single bundle. One bundle being affordable in both situations does not prove the entire budget sets are identical. The theorem proves equality for every bundle.
Forgetting that the factor must be positive. The standard result assumes $\lambda>0$, so prices and income are scaled in the same direction without reversing inequalities.
Confusing money values with purchasing power. If all prices and income double, the consumer has more dollars, but each bundle also costs twice as many dollars. The set of affordable bundles does not expand.

This result is useful whenever a change can be represented as a common rescaling of all monetary amounts. It separates a change in the dollar scale from a change in the consumer’s available choices. If relative prices change, the budget set generally changes; if income and every price move together proportionally, affordability does not.

Check Your Understanding

For each problem, determine whether the stated bundle is affordable and whether the budget set is preserved when the information asks about a proportional change.

A budget of $72 faces prices of $4 and $6. Both prices and income are multiplied by 2. Check the bundle $(6,8)$ before and after the change. (Answer: It costs $4(6)+6(8)=72$ before and $8(6)+12(8)=144$ after; it is on the boundary in both cases, and the set is unchanged.)
Prices are $3, $5, and $10, with income of $150. All are multiplied by 0.6. Check $(10,12,4)$ and its unspent budget in both cases. (Answer: Original cost is $30+60+40=130$, leaving $20; new cost is $18+36+24=78$ against $90, leaving $12; it is affordable both times, and the set is unchanged.)
A two-good budget is $160, with prices $8 and $4. All three amounts rise by 25%. Find the two intercepts before and after, and check $(12,16)$. (Answer: Intercepts are $(20,40)$ both before and after; the bundle costs $96+64=160$ before and $120+80=200$ after, so it is on both boundaries and the set is unchanged.)
Income is $140 and prices are $2 and $7. They change to income $175 and prices $2.50 and $8.75. Is $(30,10)$ affordable before and after, and is the set preserved? (Answer: The common factor is 1.25; costs are $60+70=130$ and $75+87.50=162.50$, leaving $10 and $12.50; it is affordable in both cases and the set is unchanged.)
Income rises from $180 to $225, while prices change from $4 and $9 to $5 and $9. Use $(0,21)$ to test whether the set is unchanged. (Answer: The income and first-price factors are 1.25, but the second-price factor is 1; the bundle costs $189, so it is unaffordable before and affordable afterward. The set is not unchanged.)

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