Why a Price Change Pivots the Budget Line
A consumer’s budget line records the combinations of two goods that use exactly all available income. If the price of one good changes while income and the other good’s price stay fixed, the affordable quantity of that good changes. The maximum quantity of the other good does not. The budget line therefore rotates around the unchanged intercept rather than shifting parallel.
This geometric effect is useful to identify before analyzing any consumer’s choices. A price change alters the set of affordable bundles in a particular direction; it does not, by itself, tell us which bundle the consumer will choose. The focus here is the budget constraint and how its graph changes.
The analysis assumes two divisible goods, a fixed amount of income, and fixed prices for all goods except the one whose price is changing. Quantities cannot be negative. The intercept formulas require positive prices; if a good is free, the usual finite intercept for that good does not apply. The budget line describes exact expenditure of income, while the budget set includes bundles that leave some income unspent.
Reading the Pivot From the Equation
Solving the budget equation for \(y\) gives its vertical-intercept form:
The vertical intercept is \(m/p_y\), and the slope is \(-p_x/p_y\). If only \(p_x\) changes, the vertical intercept stays fixed because neither \(m\) nor \(p_y\) changed. The horizontal intercept \(m/p_x\) and the slope change. Thus the line rotates around the unchanged vertical intercept. If instead only \(p_y\) changes, the horizontal intercept stays fixed, while the vertical intercept and slope change; the line rotates around the horizontal intercept.
The word “pivot” describes the geometry, not a change in income. A parallel shift occurs when income changes with both prices fixed: both intercepts change in the same proportion, while the slope stays the same. With a single-good price change, one intercept stays put and the slope generally changes.
If \(p_x\) falls, \(m/p_x\) rises: the consumer can afford more \(x\) when spending all income on \(x\). With \(p_y\) unchanged, the \(y\)-intercept remains the same. The budget line becomes less steep in absolute value because \(p_x/p_y\) falls. The budget set expands: every previously affordable bundle remains affordable, and some bundles that were not affordable may become affordable. If \(p_x\) rises, the reverse occurs: the horizontal intercept moves inward, the line becomes steeper in absolute value, and the budget set contracts.
These directions follow directly from the constraint. For any fixed bundle with \(x\geq0\), a lower \(p_x\) weakly lowers expenditure \(p_xx+p_yy\); a higher \(p_x\) weakly raises it. The same reasoning applies if it is \(p_y\) that changes, with the roles of the goods reversed.
Why the Pivot Holds
The fixed intercept gives a direct justification for the pivot. If \(p_x\) changes but \(m\) and \(p_y\) stay fixed, consider the bundle that contains no \(x\) and all affordable \(y\):
Its expenditure is \(p_x(0)+p_y(m/p_y)=m\). That calculation does not depend on \(p_x\), so this bundle remains on the budget line at either price of \(x\). Meanwhile, the all-\(x\) bundle has \(y=0\) and \(x=m/p_x\), which changes as \(p_x\) changes. A straight line through the fixed \(y\)-intercept and the new all-\(x\) intercept is the new budget line, so it pivots around the fixed point.
The same reasoning works when \(p_y\) changes. The bundle \((m/p_x,0)\) still costs exactly \(m\), because its expenditure is \(p_x(m/p_x)+p_y(0)=m\). It remains the fixed horizontal intercept while the vertical intercept moves.
State that income and the price of the other good do not change. This determines which intercept is unchanged.
Use \(m/p_x\) for the horizontal intercept and \(m/p_y\) for the vertical intercept, before and after the price change.
Use \(-p_x/p_y\) when writing \(y\) as a function of \(x\). Check whether its absolute value rises or falls.
Name the fixed pivot, say which intercept moves, and distinguish a rotation of the boundary from expansion or contraction of the affordable set.
Worked Price-Change Calculations
Worked Example: A Fall in the Price of \(x\)
A consumer has $120 of income. Good \(x\) costs $6 per unit and good \(y\) costs $3 per unit. The price of \(x\) falls to $4, with income and the price of \(y\) unchanged. Find both budget lines, their intercepts and slopes, and determine what happens to the bundle \((15,10)\).
Method. Write the budget equation at each price, then calculate each intercept and slope. The common mistake to avoid is changing the \(y\)-intercept even though neither income nor \(p_y\) changed.
Initially, the budget line is \(6x+3y=120\). Setting \(y=0\) gives \(x=120/6=20\), and setting \(x=0\) gives \(y=120/3=40\). Solving for \(y\) gives \(y=40-2x\), so the slope is \(-6/3=-2\).
After the price fall, the budget line is \(4x+3y=120\). The new horizontal intercept is \(120/4=30\). The vertical intercept remains \(120/3=40\). Solving for \(y\) gives \(y=40-(4/3)x\), so the slope is \(-4/3\). Its absolute value falls from \(2\) to \(4/3\), so the line is less steep.
At the old prices, bundle \((15,10)\) costs \(6(15)+3(10)=90+30=120\), so it lies on the original budget line. At the new prices, it costs \(4(15)+3(10)=60+30=90\), which is below $120. The bundle is now affordable but is inside, not on, the new budget line.
Check. Both lines have \(y=40\) when \(x=0\), confirming the fixed pivot. The \(x\)-intercept rises from 20 to 30, as expected when \(p_x\) falls.
Worked Example: A Rise in the Price of \(x\)
A shopper has $210 to spend. Initially, \(p_x=$7\) and \(p_y=$5\). The price of \(x\) rises to $10, while income and \(p_y\) remain fixed. Find the intercepts and slopes before and after the change, and test whether bundle \((20,14)\) is affordable at each set of prices.
Method. Calculate the line and expenditure at each price pair. The trap is assuming that a bundle affordable before a price rise must remain affordable afterward.
Initially, the budget equation is \(7x+5y=210\). The horizontal intercept is \(210/7=30\), the vertical intercept is \(210/5=42\), and the slope is \(-7/5=-1.4\). In slope-intercept form, the line is \(y=42-(7/5)x\).
After the increase, the equation is \(10x+5y=210\). The horizontal intercept is \(210/10=21\), while the vertical intercept remains 42. The slope is \(-10/5=-2\), and the line is \(y=42-2x\). The line is steeper, and its horizontal intercept has moved inward.
At the original prices, \((20,14)\) costs \(7(20)+5(14)=140+70=210\), so it lies on the original line. At the new prices, it costs \(10(20)+5(14)=200+70=270\). Since \(270>210\), the bundle is no longer affordable.
Check. The bundle uses exactly $210 initially and exceeds the budget by $60 after the price rise. The unchanged vertical intercept is 42 in both cases.
Worked Example: A Fall in the Price of \(y\)
A consumer has $96 of income. Good \(x\) costs $8 per unit, and good \(y\) initially costs $4 per unit. The price of \(y\) falls to $3. Find the two budget lines, identify the pivot, and determine whether \((6,18)\) lies on, inside, or outside the new budget set.
Method. Because \(p_y\) changes, calculate the vertical intercept afresh while holding the horizontal intercept fixed. The common error is to keep the vertical intercept fixed simply because it was fixed in the previous kind of price change.
Initially, \(8x+4y=96\). The horizontal intercept is \(96/8=12\), and the vertical intercept is \(96/4=24\). The slope is \(-8/4=-2\), giving \(y=24-2x\).
After the price fall, \(8x+3y=96\). The horizontal intercept remains \(96/8=12\). The vertical intercept becomes \(96/3=32\), and the slope is \(-8/3\), giving \(y=32-(8/3)x\). The line pivots around the fixed horizontal intercept \((12,0)\).
At the new prices, bundle \((6,18)\) costs \(8(6)+3(18)=48+54=102\). Since \(102>96\), it lies outside the new budget set and is unaffordable. As a check, the budget-line quantity of \(y\) when \(x=6\) is \(32-(8/3)(6)=32-16=16\), less than 18.
Check. The bundle’s expenditure exceeds income by $6, and the line’s \(x\)-intercept remains 12. Both checks confirm the classification and the pivot.
Common Mistakes and What the Picture Does Not Say
A frequent error is to treat a price change as if it changed income. For example, when \(p_x\) changes, it is incorrect to change both intercepts: \(m/p_x\) changes, but \(m/p_y\) does not if income and \(p_y\) are fixed. Another error is to report the slope as positive. In a graph with \(x\) on the horizontal axis and \(y\) on the vertical axis, the budget-line slope is negative, \(-p_x/p_y\). Its absolute value, \(p_x/p_y\), describes the steepness.
Also distinguish the boundary from the whole budget set. A bundle on the budget line spends all income; one strictly below the line is affordable with income left over; one above the line is unaffordable. After a price decrease, a previously affordable bundle remains affordable, but it may move from the boundary to the interior. After a price increase, a previously affordable bundle may become unaffordable.
A budget-line pivot does not establish how much of either good the consumer will buy. It identifies which bundles can be chosen at the new prices. To predict a choice requires information about preferences as well as the budget constraint. Keeping those questions separate prevents a graph of affordability from being mistaken for a conclusion about behavior.
Check Your Understanding
For each problem, state the relevant intercepts and describe the price-change effect. Assume nonnegative quantities and that the stated income and unchanged price remain fixed.
- A consumer has $150, with \(p_x=$5\) and \(p_y=$6\). The price of \(x\) rises to $10. Find the old and new intercepts and slopes, and identify the pivot. (Answer: Old intercepts \(x=30,\ y=25\), slope \(-5/6\); new intercepts \(x=15,\ y=25\), slope \(-10/6=-5/3\); the line pivots around \((0,25)\).)
- Income is $84, \(p_x=$7\), and \(p_y=$3\). The price of \(x\) falls to $4. Find the new intercepts and classify bundle \((9,7)\) as on, inside, or outside the new budget set. (Answer: New intercepts \(x=21,\ y=28\); expenditure is \(4(9)+3(7)=36+21=57<84\), so the bundle is inside the budget set.)
- Income is $180, \(p_x=$9\), and \(p_y\) falls from $6 to $4. Give both equations in slope-intercept form and name the fixed intercept. (Answer: Initially \(y=30-(3/2)x\); afterward \(y=45-(9/4)x\); the fixed horizontal intercept is \((20,0)\).)
- Income and both goods’ quantities are measured consistently. If \(p_x\) rises by 20% while income and \(p_y\) stay fixed, state the proportional change in each intercept and in the slope’s absolute value. (Answer: The \(x\)-intercept becomes \(1/1.2=5/6\) of its original value, the \(y\)-intercept is unchanged, and the slope’s absolute value rises by 20%.)
- With income $96 and \(p_y=$4\), the original \(x\)-intercept is 16. The price of \(x\) then rises from $6 to $8. Find the new line’s intercepts and test whether bundle \((8,12)\) is affordable afterward. (Answer: The new \(x\)-intercept is \(96/8=12\), the \(y\)-intercept remains \(96/4=24\), and the bundle costs \(8(8)+4(12)=112>96\), so it is unaffordable.)