Budget Lines After Simultaneous Price and Income Changes

Introduction

A budget line can change because income changes, because prices change, or because several of these changes happen together. Looking at only one change at a time can give the wrong picture: a higher income may expand the affordable set, while a price increase may reduce it or alter the trade-off between goods. The reliable method is to write the complete new budget equation, then compare its intercepts and slope with those of the original line.

This tutorial treats a consumer choosing quantities of two goods. Its aim is to calculate the new line and interpret how it has changed—not to choose the consumer's preferred bundle. A point on the budget line spends the entire budget; points below it are affordable, and points above it are not.

Definition: Let $x$ and $y$ be quantities of two goods, $p_x>0$ and $p_y>0$ their respective prices per unit, and $m>0$ the consumer's income available for these purchases. The budget set is the collection of nonnegative bundles satisfying $p_xx+p_yy\leq m$. The budget line is its boundary, $p_xx+p_yy=m$. After income or prices change, denote the new values by $m'$, $p_x'$, and $p_y'$; the new line is $p_x'x+p_y'y=m'$.

The model assumes that the consumer can buy any nonnegative quantity of either good at the stated, constant per-unit prices; that income is available for the purchases; and that the two goods are the only uses of this budget. Taxes, quantity discounts, rationing, and other constraints are not included. A formula based on this model need not describe a situation in which those features matter.

Find the New Line Before Interpreting It

The original and new equations should be kept distinct. The intercept on the horizontal $x$-axis is the quantity of $x$ affordable if the entire budget is spent on $x$, so set $y=0$. The vertical intercept is found by setting $x=0$. Rearranging the equation to make $y$ the subject gives the slope.

\begin{aligned} p_xx+p_yy&=m &\Longrightarrow\quad y&=\frac{m}{p_y}-\frac{p_x}{p_y}x,\\ x\text{-intercept}&=\frac{m}{p_x}, & y\text{-intercept}&=\frac{m}{p_y}, & \text{slope}&=-\frac{p_x}{p_y}.\\[4pt] p_x'x+p_y'y&=m' &\Longrightarrow\quad y&=\frac{m'}{p_y'}-\frac{p_x'}{p_y'}x,\\ x\text{-intercept}&=\frac{m'}{p_x'}, & y\text{-intercept}&=\frac{m'}{p_y'}, & \text{slope}&=-\frac{p_x'}{p_y'}. \end{aligned}

These quantities answer different questions. Income appears in both intercepts, but not in the slope. The slope depends on the two prices, not on income. Thus an income change alone moves both intercepts in the same direction without changing the slope. A price change can alter an intercept, the slope, or both. When income and prices change together, calculate the new values rather than assuming the effect of any one change determines the whole result.

Derivation: On the budget line, any additional expenditure on $x$ must be offset by a reduction in expenditure on $y$. Solving $p_xx+p_yy=m$ for $y$ gives $y=m/p_y-(p_x/p_y)x$. The coefficient of $x$ is therefore the slope, $-p_x/p_y$, while setting either quantity to zero gives the corresponding intercept. Replacing all three parameters with their new values proves the same results for the new line.

A change in the line is parallel if and only if its slope is unchanged, which requires $p_x'/p_y'=p_x/p_y$. The income values do not enter this condition. If the slopes differ, the line has changed its price trade-off and is not parallel to the original. Its intercepts show how the affordable quantities at the axis extremes have changed. A line can have one unchanged intercept even when its slope changes.

Worked Calculations

Worked Example: Income Rises While One Price Falls

A consumer has a monthly budget of $120 for two goods. Initially, $p_x=\$6$ and $p_y=\$4$. The budget rises to $150, $p_x$ falls to $5, and $p_y$ stays at $4. Write both budget-line equations, calculate their intercepts and slopes, and describe the graph.

Method. Use the original values together in the original equation, then use all the new values together in the new equation. This prevents an income or price from one situation being accidentally carried into the other.

Original line. The equation is $6x+4y=120$. Setting $y=0$ gives $x=120/6=20$; setting $x=0$ gives $y=120/4=30$. Solving for $y$, $y=30-1.5x$, so the slope is $-6/4=-1.5$.

New line. The equation is $5x+4y=150$. Its $x$-intercept is $150/5=30$, and its $y$-intercept is $150/4=37.5$. Solving for $y$, $y=37.5-1.25x$, so its slope is $-5/4=-1.25$.

The new line runs between $(30,0)$ and $(0,37.5)$, compared with $(20,0)$ and $(0,30)$ for the original. Both intercepts increase, but the slope changes from $-1.5$ to $-1.25$. The line is therefore not a parallel shift: the relative price trade-off has changed as well as the affordable quantities.

Check. At the new $x$-intercept, the bundle $(30,0)$ costs $5(30)+4(0)=150$. At the new $y$-intercept, $(0,37.5)$ costs $5(0)+4(37.5)=150$. Both satisfy the new budget exactly.

Common mistake avoided: Keeping the old income would give the incorrect new $x$-intercept $120/5=24$. The new line must use the new budget of $150.

Worked Example: One Intercept Stays Fixed

A consumer initially has $m=\$120$, $p_x=\$6$, and $p_y=\$3$. Later, income becomes $150, the price of $x$ becomes $7.50, and the price of $y$ becomes $5. Find both lines and explain why one intercept is unchanged even though the line is not parallel.

Original line. The budget equation is $6x+3y=120$. Its intercepts are $120/6=20$ units of $x$ and $120/3=40$ units of $y$. In slope-intercept form, $y=40-2x$, with slope $-6/3=-2$.

New line. The equation is $7.50x+5y=150$. Its $x$-intercept is $150/7.50=20$, and its $y$-intercept is $150/5=30$. Solving for $y$ gives $y=30-1.5x$, with slope $-7.50/5=-1.5$.

The $x$-intercept stays at 20 because the budget and $x$-price both rise by the same factor: $150/120=1.25$ and $7.50/6=1.25$. Consequently, $m'/p_x'=m/p_x=20$. The $y$-intercept falls from 40 to 30, and the slope changes from $-2$ to $-1.5$. The two lines share the horizontal intercept but are not parallel.

Check. The new endpoint bundles cost $7.50(20)+5(0)=150$ and $7.50(0)+5(30)=150$, as required.

Common mistake avoided: An unchanged intercept does not prove that the whole line is unchanged or parallel. Check the other intercept and the price ratio that determines the slope.

Worked Example: A Price Change Can Offset an Income Change

A consumer has $m=\$240$, $p_x=\$8$, and $p_y=\$6$. The budget later falls to $180, while $p_x$ falls to $6 and $p_y$ rises to $9. Calculate both lines, determine whether they are parallel, and test whether the bundle $(15,10)$ is affordable under each budget.

Original line. The equation is $8x+6y=240$. The intercepts are $240/8=30$ and $240/6=40$. Rearranging gives $y=40-(8/6)x=40-(4/3)x$, so the slope is $-4/3$.

New line. The equation is $6x+9y=180$. Its intercepts are $180/6=30$ and $180/9=20$. Rearranging gives $y=20-(6/9)x=20-(2/3)x$, so the slope is $-2/3$. The lines are not parallel because $-4/3\ne-2/3$. The $x$-intercept is unchanged, while the $y$-intercept falls by 20 units.

Affordability under the original budget. The bundle costs $8(15)+6(10)=120+60=180$. Since $180\leq240$, it is affordable, with $60 left unspent.

Affordability under the new budget. The same bundle costs $6(15)+9(10)=90+90=180$. Since the new budget is $180, the bundle lies exactly on the new line and is affordable.

Check. The bundle has not changed, but its expenditure falls from $180 to $180? Rechecking the arithmetic: the original cost is $8(15)+6(10)=120+60=180$, and the new cost is $6(15)+9(10)=90+90=180$. Thus it costs $180 in both situations; initially this is below the $240 budget, and later it exhausts the $180 budget.

Common mistake avoided: Do not conclude that lower income must make every previously affordable bundle unaffordable. Prices have changed too; evaluate the bundle's cost using the applicable prices and compare it with that situation's income.

A Reliable Comparison Procedure

For a graph or calculation problem, keep the original and new situations in separate rows of working. This makes the source of each intercept and slope visible and makes it easier to catch a mixed-up parameter.

Record the two parameter sets.
Write $(m,p_x,p_y)$ and $(m',p_x',p_y')$ separately. Do not assume that an unmentioned variable changed.

Write each equation.
Use $p_xx+p_yy=m$ for the original line and $p_x'x+p_y'y=m'$ for the new line.

Calculate both intercepts and slopes.
Use $m/p_x$, $m/p_y$, and $-p_x/p_y$, then repeat with the new parameters. Keep enough precision to compare the results reliably.

Describe the change and check it.
Compare slopes to decide whether the lines are parallel. Compare intercepts to describe the endpoints, then substitute each endpoint into its own budget equation.

To draw the result, mark each line's two intercepts on the appropriate axes and connect them. For instance, a new $x$-intercept larger than the original means the consumer can buy more $x$ if choosing no $y$; it does not, by itself, tell you what happens to the $y$-intercept or the slope. Likewise, two lines with different slopes may cross outside the nonnegative quadrant, so the crossing alone does not establish which budget set contains the other throughout the relevant region.

Common Errors and Why the Distinction Matters

The simultaneous-change case matters whenever a comparison involves both a change in purchasing power and a change in relative prices. Intercepts summarize the extreme quantities available if the consumer buys only one good; the slope summarizes how many units of $y$ must be given up to obtain one more unit of $x$ while staying on the line. These are separate economic meanings, and a complete comparison should not substitute one for the other.

Mixing parameter sets. The old equation uses the old income and old prices; the new equation uses the new income and new prices. Combining the new price with old income does not describe either situation unless the problem explicitly asks for that hypothetical comparison.
Using income to infer slope. Income changes both intercepts when prices stay fixed, but it does not alter $-p_x/p_y$. A slope change requires a change in the relative prices.
Calling every outward movement parallel. An increase in one or both intercepts does not establish that slopes match. Compare the price ratios directly.
Ignoring units. If one price is dollars per unit and another is dollars per package, state what each quantity measures. Intercepts are measured in their respective goods, not in dollars.
Testing affordability against the wrong budget. Calculate $p_xx+p_yy$ using the prices for the situation being tested, then compare the result with that situation's income.

Key takeaway: When income and prices change together, rebuild the budget equation from the new values. The new intercepts are $m'/p_x'$ and $m'/p_y'$, and the new slope is $-p_x'/p_y'$. Compare all three with their original counterparts before describing the graph.

Check Your Understanding

For each problem, show the equations or comparisons used, then check that your final description agrees with the calculated intercepts and slope.

A budget is initially $96, with prices $p_x=\$4$ and $p_y=\$8$. It changes to $120, with $p_x'=\$6$ and $p_y'=\$8$. Find both intercepts and slopes, and state whether the lines are parallel. (Answer: Original intercepts 24 and 12, slope $-0.5$; new intercepts 20 and 15, slope $-0.75$; not parallel.)
Initially, $m=\$180$, $p_x=\$9$, and $p_y=\$3$. New values are $m'=\$144$, $p_x'=\$6$, and $p_y'=\$6$. Find each line's intercepts and slope. (Answer: Original intercepts 20 and 60, slope $-3$; new intercepts 24 and 24, slope $-1$; not parallel.)
A consumer's initial budget is $200, with prices $5 for $x$ and $10 for $y$. The new budget is $240; the new prices are $6 for $x$ and $10 for $y$. Which intercept stays unchanged, and are the lines parallel? (Answer: The $x$-intercept stays at 40; the $y$-intercept rises from 20 to 24; slopes change from $-0.5$ to $-0.6$, so the lines are not parallel.)
A budget of $300 with prices $10 for $x$ and $15 for $y$ changes to $420, with prices $12 and $18. Find the intercepts and slopes, then describe the geometric change. (Answer: Original intercepts 30 and 20, slope $-2/3$; new intercepts 35 and $70/3$, slope $-2/3$; the lines are parallel, with the new line shifted outward.)
Initially, a consumer has a $100 budget and prices $2 for $x$ and $5 for $y$. Later, the budget is $120 and prices are $4 and $3. Is the bundle $(20,10)$ affordable in each situation? Find both slopes and say whether the lines are parallel. (Answer: Original cost is $90, so it is affordable; new cost is $110, so it is also affordable. Slopes are $-0.4$ and $-4/3$; the lines are not parallel.)

What You'll Learn