How an Income Change Shifts the Budget Line

Why Income Changes Move a Budget Line

A budget line describes the combinations of two goods that use exactly all of a consumer’s income at given prices. If income changes while both prices stay the same, the consumer can afford a different range of bundles. The key graphical result is simple: both intercepts change, but the slope does not. The new line is parallel to the old one.

This distinction matters when reading a budget diagram. A changed intercept shows that the amount of one good affordable when none of the other is purchased has changed. An unchanged slope shows that the rate at which one good must be given up to afford more of the other, as represented by the budget line, is still determined by the same prices.

Definition: For two goods $x$ and $y$, let $p_x>0$ and $p_y>0$ be their prices per unit, and let $I\geq 0$ be income available for these purchases. The budget line is the set of nonnegative bundles that spend exactly all income: $$ p_xx+p_yy=I. $$ The budget set is the set of nonnegative bundles that cost no more than income: $$ p_xx+p_yy\leq I. $$

Throughout this tutorial, assume there are exactly two goods, prices are positive and fixed, and all income $I$ is available to spend on these goods. Quantities cannot be negative. If income changes from $I_0$ to $I_1$, the prices remain $p_x$ and $p_y$; only the right-hand side of the budget equation changes. The results below concern this fixed-price situation.

Deriving the Intercepts and Slope

The intercept on the $x$-axis is found by setting $y=0$: all income is spent on $x$, so $p_xx=I$ and $x=I/p_x$. The $y$-intercept is found by setting $x=0$, giving $y=I/p_y$. Solving the budget equation for $y$ gives the line’s slope-intercept form.

y=\frac{I}{p_y}-\frac{p_x}{p_y}x

The vertical intercept is $I/p_y$, while the slope is $-p_x/p_y$. Income appears in the intercept, but not in the slope. This algebra already suggests why changing income moves the line without changing its direction.

Result: At fixed positive prices, the $x$-intercept is $I/p_x$, the $y$-intercept is $I/p_y$, and the slope when $y$ is on the vertical axis is $-p_x/p_y$. Changing income changes both intercepts in proportion to income and leaves the slope unchanged.

Why the Shift Is Parallel

Let the original income be $I_0$ and the new income be $I_1$. The original and new lines, written with $y$ on the vertical axis, are

y=\frac{I_0}{p_y}-\frac{p_x}{p_y}x \qquad\text{and}\qquad y=\frac{I_1}{p_y}-\frac{p_x}{p_y}x.

Their slopes are both $-p_x/p_y$. Therefore the lines are parallel. The change in the $x$-intercept is $(I_1-I_0)/p_x$, and the change in the $y$-intercept is $(I_1-I_0)/p_y$. These changes need not be equal: each is measured in units of a different good and depends on that good’s price.

Justification: Subtract the original line equation from the new one at the same value of $x$. The terms involving $x$ cancel because prices are unchanged: $$ \left(\frac{I_1}{p_y}-\frac{p_x}{p_y}x\right) - \left(\frac{I_0}{p_y}-\frac{p_x}{p_y}x\right) = \frac{I_1-I_0}{p_y}. $$ Thus the vertical difference between the lines is constant for every $x$. A constant vertical difference means the lines are parallel. If income rises, this difference is positive; if income falls, it is negative.

Graphically, an increase in income moves both intercepts outward from the origin. A decrease moves both inward. Since the slope stays fixed, draw the new line parallel to the old line through the new intercepts. With positive prices, the new budget set contains the old budget set after an income increase: every bundle that cost no more than $I_0$ also costs no more than $I_1$ when $I_1>I_0$. If income decreases, the reverse containment holds.

Write the original budget equation.
Record the two prices and the original income: $p_xx+p_yy=I_0$.

Substitute the new income only.
At fixed prices, the new equation is $p_xx+p_yy=I_1$.

Calculate both intercepts for each line.
Divide the relevant income by $p_x$ for the $x$-intercept and by $p_y$ for the $y$-intercept.

Compare slopes and sketch.
Both slopes are $-p_x/p_y$. Plot the intercepts and draw parallel lines, outward for higher income and inward for lower income.

Worked Examples

Worked Example: A Rise in Monthly Spending Income

A consumer allocates a monthly budget between transit rides $x$ and prepared meals $y$. A ride costs \$6 and a meal costs \$3. Available income rises from \$60 to \$90. Find the two budget equations, both intercepts before and after the change, and the slope. State how the diagram changes.

Method. Use the budget equation $p_xx+p_yy=I$. Prices are fixed at $p_x=6$ and $p_y=3$.

Original line. With income $I_0=60$, the equation is $6x+3y=60$. Set $y=0$ to get $6x=60$, so the $x$-intercept is $x=10$. Set $x=0$ to get $3y=60$, so the $y$-intercept is $y=20$. Solving for $y$ gives $y=20-2x$, with slope $-6/3=-2$.

New line. With income $I_1=90$, the equation is $6x+3y=90$. Setting $y=0$ gives $x=90/6=15$; setting $x=0$ gives $y=90/3=30$. Solving for $y$ gives $y=30-2x$, again with slope $-6/3=-2$.

The intercepts move from $(10,0)$ and $(0,20)$ to $(15,0)$ and $(0,30)$. Both move outward, and the unchanged slope means the new line is parallel to the original. The vertical difference at each fixed $x$ is $30-20=10$, which also equals $(90-60)/3=10$.

Check. The original intercept bundles each spend \$60: $6(10)=60$ and $3(20)=60$. The new intercept bundles each spend \$90: $6(15)=90$ and $3(30)=90$. The slope in both equations is $-2$. This avoids the mistake of changing the slope just because income increased.

Worked Example: A Fall in Income and a Bundle’s Affordability

A household divides its spending between two services, $x$ and $y$, priced at \$7 and \$3 per unit. Its available income falls from \$84 to \$63. Calculate both lines and intercepts, then determine whether the bundle $(3,14)$ is affordable before and after the fall.

Method. Calculate the lines and intercepts using the two income levels, then compare the bundle’s total cost with each income.

Original line. The equation is $7x+3y=84$. If $y=0$, then $x=84/7=12$; if $x=0$, then $y=84/3=28$. Solving for $y$, $3y=84-7x$, so $y=28-(7/3)x$. The slope is $-7/3$.

New line. The equation is $7x+3y=63$. The intercepts are $x=63/7=9$ and $y=63/3=21$. Solving for $y$ gives $y=21-(7/3)x$, so the slope remains $-7/3$. Both intercepts move inward.

Bundle check. The bundle $(3,14)$ costs $7(3)+3(14)=21+42=63$. It is affordable before the income fall because $63\leq84$, leaving $84-63=21$ dollars unspent. It is exactly on the new budget line because its cost equals the new income, $63$.

Check. Substitution into the new line gives $y=21-(7/3)(3)=21-7=14$, confirming that $(3,14)$ lies on it. The slope is unchanged, and the bundle is still affordable, but it now uses all available income.

Worked Example: Tracking the Intercept Changes

A worker’s weekly budget for two goods changes from \$100 to \$130. Good $x$ costs \$5 per unit and good $y$ costs \$10 per unit. Calculate the original and new intercepts and equations. Then, if the worker keeps $x=8$, find the maximum affordable quantity of $y$ under each income.

Method. Use fixed prices to find the intercepts and solve the budget equation for $y$, first at the original income and then at the new income.

Original line. The equation is $5x+10y=100$. Its intercepts are $x=100/5=20$ and $y=100/10=10$. Solving for $y$: $10y=100-5x$, so $y=10-0.5x$. The slope is $-5/10=-0.5$.

New line. The equation is $5x+10y=130$. Its intercepts are $x=130/5=26$ and $y=130/10=13$. Solving gives $y=13-0.5x$, again with slope $-0.5$. The $x$-intercept increases by $26-20=6$, while the $y$-intercept increases by $13-10=3$. The unequal changes make sense because the prices differ.

At $x=8$. Under the original income, $5(8)=40$ dollars are spent on $x$, leaving $100-40=60$ dollars. The maximum $y$ is $60/10=6$. Under the new income, the amount left is $130-40=90$, so the maximum $y$ is $90/10=9$.

Check. At the original bundle $(8,6)$, spending is $5(8)+10(6)=40+60=100$. At the new bundle $(8,9)$, spending is $40+90=130$. The same $x$-quantity is on each line with the calculated $y$-quantity, and both equations have slope $-0.5$.

Common Mistakes and Why the Distinction Matters

A frequent error is to treat an income change as if it changed the relative prices. For example, in the first example a student might reason that income rose from \$60 to \$90, so the slope should change from $-2$ to $-3$. That calculation incorrectly uses the income ratio $90/60=1.5$ as a slope adjustment. The correct slopes are $-6/3=-2$ for both lines, since the prices did not change. The income change affects the intercepts, not the price ratio.

Another error is to add the same number to both intercepts. The intercept changes are $(I_1-I_0)/p_x$ and $(I_1-I_0)/p_y$, so they will generally differ when the prices differ. In the third example, income rose by \$30. That adds $30/5=6$ units to the $x$-intercept but only $30/10=3$ units to the $y$-intercept.

For a graph, calculate both intercepts rather than estimating where a shifted line should go. Then check that the two equations have the same slope. This is especially useful when a diagram has no scale or when a question provides income and prices but asks for a verbal description of the change. A higher income at unchanged prices expands the budget set; a lower income contracts it. Neither change alters the price-based slope.

Key takeaway: When only income changes, replace $I_0$ by $I_1$ in the budget equation and leave both prices alone. Each intercept changes according to the relevant price, while the slope remains $-p_x/p_y$. On a graph, the result is a parallel shift.

Check Your Understanding

For each problem, assume two nonnegative goods, positive fixed prices, and that all stated income is available for the purchases.

Good $x$ costs \$4 and good $y$ costs \$8. Income rises from \$64 to \$96. Give the old and new intercepts and state the slope of both lines. (Answer: Old intercepts: $x=16,\ y=8$; new intercepts: $x=24,\ y=12$; both slopes are $-1/2$.)
At prices $p_x=\$6$ and $p_y=\$2$, income falls from \$72 to \$48. Write both budget equations solved for $y$, and state the vertical difference between the lines at any fixed $x$. (Answer: $y=36-3x$ and $y=24-3x$; the new line is 12 units below the old line.)
Good $x$ costs \$9 and good $y$ costs \$3. Income rises from \$54 to \$81. Is bundle $(5,12)$ affordable at each income, and where does it lie after the increase? (Answer: Its cost is $9(5)+3(12)=81$, so it is not affordable at \$54; at \$81 it lies on the new budget line.)
A budget line has prices $p_x=\$5$, $p_y=\$15$, and income \$150. Income then falls by \$30 with prices unchanged. Find the change in each intercept and the slope before and after. (Answer: The $x$-intercept falls by $30/5=6$, from 30 to 24; the $y$-intercept falls by $30/15=2$, from 10 to 8; both slopes are $-1/3$.)
With prices $p_x=\$8$ and $p_y=\$4$, a consumer’s income rises from \$80 to \$120. For $x=6$, calculate the maximum affordable $y$ before and after the increase, and verify that the two lines are parallel. (Answer: Before, $y=(80-8(6))/4=8$; after, $y=(120-8(6))/4=18$. The equations are $y=20-2x$ and $y=30-2x$, both with slope $-2$.)

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