Why a Price Change Rotates the Budget Line
A consumer’s chosen bundle depends on both preferences and the opportunities available at current prices and income. When the price of one good changes while income and the other good’s price stay fixed, the budget line pivots: one intercept changes, the other stays put. The consumer then chooses a new best affordable bundle. Comparing those choices at different prices traces the consumer’s individual demand relationship.
The key distinction is between a movement along an individual demand curve and a shift of that curve. A change in the good’s own price, with other demand determinants held fixed, changes the quantity demanded along the existing curve. A change in income, preferences, or another determinant can instead change the quantity demanded at each own price and shift the curve.
If \(p_x\) falls while \(m\) and \(p_y\) remain fixed, the \(y\)-intercept \(m/p_y\) does not change, but the \(x\)-intercept \(m/p_x\) increases. The line pivots outward around its \(y\)-intercept and becomes flatter because the absolute value of its slope, \(p_x/p_y\), falls. The consumer can buy more \(x\) with all income, but can still buy the same maximum amount of \(y\). A rise in \(p_x\) reverses these changes: the line pivots inward around the unchanged \(y\)-intercept and becomes steeper.
This geometric description identifies the new feasible set; it does not by itself identify the consumer’s new choice. That choice also depends on preferences. An interior optimum often occurs where the marginal rate of substitution equals the price ratio, but a consumer may instead choose a boundary bundle. A price change can therefore produce a smooth change in quantities or, with some preferences, a sharp switch between corners.
From a Budget Rotation to Individual Demand
An individual demand function gives the consumer’s chosen quantity of a good as a function of its own price, holding income, other prices, and preferences fixed. If \(x^*(p_x;m,p_y)\) denotes the chosen quantity of \(x\), then evaluating it at two different values of \(p_x\) gives two points on the same individual demand curve, provided \(m\), \(p_y\), and preferences have not changed.
For example, the point \((p_x,x^*)\) records the price of \(x\) and the quantity chosen at that price. Changing only \(p_x\) changes both the budget line and the chosen quantity. It is a movement from one point to another on the same demand curve, even though the consumer’s entire bundle may change. By contrast, if income changes, the budget line shifts rather than rotating around the intercept for the unchanged good. The consumer’s quantity at a given own price may then differ, which is a change in demand rather than a movement along the original curve.
A Cobb-Douglas Derivation
A useful way to see the relationship between budget rotation and choice is to derive demand for a standard preference form. Assume positive prices and income, \(0<\alpha<1\), and utility \(U(x,y)=x^\alpha y^{1-\alpha}\). This model has an interior optimum. Since maximizing \(U\) is equivalent to maximizing \(\ln U=\alpha\ln x+(1-\alpha)\ln y\), the first-order condition equates the marginal rate of substitution to the price ratio:
Rearranging gives \(p_xx=\frac{\alpha}{1-\alpha}p_yy\). At the optimum the budget binds, so \(p_xx+p_yy=m\). Substituting the relationship into the budget equation shows that spending on \(x\) is the fraction \(\alpha\) of income. The remaining fraction is spent on \(y\):
The own-price demand for \(x\) in this model is \(x^*=\alpha m/p_x\). Holding \(m\), \(p_y\), and \(\alpha\) fixed, a change in \(p_x\) changes the chosen quantity along this demand function. This formula is specific to the stated Cobb-Douglas preferences and assumptions; it should not be applied automatically to other preferences or to a problem whose optimum is at a corner.
Worked Example: A Price Increase with an Interior Optimum
A consumer has income \(m=\$120\) and Cobb-Douglas utility \(U(x,y)=x^{1/2}y^{1/2}\). Good \(y\) costs \(p_y=\$6\). Good \(x\) initially costs \(p_x=\$4\), then rises to \(\$8\). Find the budget-line intercepts and optimal bundle at each price, and identify what happens on the individual demand curve for \(x\).
Method and assumptions. The consumer chooses the most preferred affordable bundle, and the positive Cobb-Douglas preferences give an interior optimum. Here \(\alpha=1/2\), so \(x^*=m/(2p_x)\) and \(y^*=m/(2p_y)\). The income, preference parameter, and price of \(y\) remain fixed.
At \(p_x=\$4\). The \(x\)-intercept is \(120/4=30\), and the \(y\)-intercept is \(120/6=20\). The optimal quantities are \(x^*=120/(2\cdot4)=15\) and \(y^*=120/(2\cdot6)=10\). Spending is \(4(15)+6(10)=60+60=\$120\), so the bundle lies on the budget line.
At \(p_x=\$8\). The \(x\)-intercept is \(120/8=15\); the \(y\)-intercept remains \(120/6=20\). The new optimum is \(x^*=120/(2\cdot8)=7.5\) and \(y^*=120/(2\cdot6)=10\). Spending is \(8(7.5)+6(10)=60+60=\$120\), again exhausting the budget.
The \(x\)-intercept falls from 30 to 15 while the \(y\)-intercept stays at 20, so the budget line pivots inward around the \(y\)-intercept. On the individual demand curve, the consumer moves from \((p_x,x^*)=(4,15)\) to \((8,7.5)\). The common mistake is to say that the demand curve shifts just because the chosen quantity changes. Since only \(p_x\) changed, these are two points on the same demand curve.
Price Changes Can Alter Corner Choices
A rotating budget line does not guarantee an interior optimum. With perfect substitutes, a consumer may spend all income on whichever good gives more utility per dollar. Consider \(U(x,y)=a x+b y\), where \(a\) and \(b\) are positive constants. One dollar spent on \(x\) buys \(a/p_x\) units of utility; one dollar spent on \(y\) buys \(b/p_y\). If one ratio is higher, spending everything on that good is optimal. If the ratios are equal, every bundle on the budget line gives the same utility.
Thus a price change can rotate the budget line and cause the consumer to switch from one corner to another. The points on the individual demand curve still correspond to different own prices with the other determinants fixed, but the chosen quantity may jump rather than change gradually.
Worked Example: A Price Change Switches the Chosen Corner
A consumer has \(U(x,y)=3x+2y\), income \(m=\$60\), and \(p_y=\$4\). Good \(x\) initially costs \(\$8\), then its price falls to \(\$3\). Find the chosen bundles and budget intercepts at both prices. At the initial price, account for any non-unique optimum.
Method and assumptions. With perfect substitutes, compare marginal utility per dollar. For \(x\), this is \(3/p_x\); for \(y\), it is \(2/p_y=2/4=0.5\). A strictly higher ratio identifies the good receiving all spending.
At \(p_x=\$8\). The utility per dollar is \(3/8=0.375\) for \(x\), less than \(0.5\) for \(y\). The consumer buys only \(y\): \(x^*=0\) and \(y^*=60/4=15\). The intercepts are \(60/8=7.5\) on the \(x\)-axis and \(60/4=15\) on the \(y\)-axis.
At \(p_x=\$3\). The utility per dollar for \(x\) is \(3/3=1\), greater than \(0.5\) for \(y\). The consumer buys only \(x\): \(x^*=60/3=20\) and \(y^*=0\). The intercepts are \(60/3=20\) on the \(x\)-axis and \(60/4=15\) on the \(y\)-axis.
The budget line pivots outward around the unchanged \(y\)-intercept, and the chosen bundle switches from \((0,15)\) to \((20,0)\). At \(p_x=\$6\), the two utility-per-dollar ratios would both equal \(0.5\), so every affordable bundle would yield the same utility. The common mistake is to apply an interior tangency condition even when the optimum is a corner; comparing utility per dollar correctly handles both the corner choices and the tie.
Distinguishing Movement from a Shift
To classify a change, compare the conditions under which the consumer makes each choice. A change in \(p_x\) alone changes the budget line’s slope and its \(x\)-intercept, and produces a movement along the individual demand curve for \(x\). A change in income changes both intercepts proportionally while leaving the slope unchanged, if prices are fixed. That change can alter the chosen quantity at every given \(p_x\), shifting the demand curve.
The distinction is about what changed, not simply whether the quantity changed. A consumer can choose a different quantity after either a price change or an income change. To diagnose the result, ask whether the own price changed while preferences, income, and other prices were held fixed. If so, it is movement along demand. If another determinant changed, demand may shift. If multiple conditions change together, identify each change rather than attributing the whole outcome to the own price alone.
Worked Example: Separating a Price Movement from an Income Shift
A consumer has utility \(U(x,y)=x^{0.4}y^{0.6}\). Initially, income is \(m=\$100\), \(p_x=\$5\), and \(p_y=\$2\). First, only \(p_x\) falls to \(\$4\). Later, income rises to \(\$125\) while prices remain at \(\$4\) and \(\$2\). Calculate the optimal \(x\)-quantity in each situation and classify each change.
Method and assumptions. For these Cobb-Douglas preferences, \(x^*=0.4m/p_x\). The preferences and positive prices imply an interior optimum. Applying the formula separately makes clear which determinant changed.
Initial bundle’s \(x\)-quantity. At \(m=100\) and \(p_x=5\), \(x^*=0.4(100)/5=40/5=8\). The budget-line intercepts are \(100/5=20\) for \(x\) and \(100/2=50\) for \(y\).
After the price fall. With income still 100 and \(p_x=4\), \(x^*=0.4(100)/4=40/4=10\). The intercepts are now \(100/4=25\) and \(100/2=50\). Only the own price changed, so the change from 8 to 10 is movement along the same demand curve.
After the income rise. At income 125 with prices fixed at 4 and 2, \(x^*=0.4(125)/4=50/4=12.5\). The intercepts are \(125/4=31.25\) and \(125/2=62.5\). Both intercepts have increased proportionally from their levels at income 100, so the budget line shifts outward in parallel. At the same \(p_x=4\), the quantity rises from 10 to 12.5 because income changed; this is a shift in demand. The common mistake is to call both quantity changes movements along demand. Only the first change came from a change in own price with the other determinants fixed.
Common Mistakes and a Reliable Check
- Rotating around the wrong intercept. When \(p_x\) changes and \(m,p_y\) stay fixed, \(m/p_y\), the \(y\)-intercept, is unchanged. The \(x\)-intercept changes.
- Confusing the budget line with demand. The budget line describes feasible bundles at a particular set of prices and income. The individual demand curve relates the own price to the chosen quantity while other determinants are held fixed.
- Assuming every price change gives an interior choice. Preferences such as perfect substitutes can yield corner bundles or multiple optimal bundles. Check the relevant optimum conditions before using an interior formula.
- Calling every quantity change a demand shift. A change in own price gives movement along demand when other determinants stay fixed. Changes in income, preferences, or other prices can shift demand.
A useful final check is to write down what is held constant before comparing choices. Then recalculate both intercepts, the budget-line slope, and the optimum under each set of conditions. The intercepts show how opportunity sets changed; the optimization identifies the chosen bundle; the list of changed determinants tells whether the result is movement along demand or a shift.
Check Your Understanding
For each problem, identify the budget-line change and calculate the requested choice or demand comparison.
- A consumer has \(U(x,y)=x^{1/3}y^{2/3}\), income \(\$90\), and \(p_y=\$3\). The price of \(x\) rises from \(\$6\) to \(\$9\). Find the optimal \(x\)-quantity and both budget intercepts at each price. (Answer: \(x^*=5\) then \(10/3\); the intercepts change from \((15,30)\) to \((10,30)\), where the first entry is the \(x\)-intercept.)
- A consumer has \(U(x,y)=x^{1/4}y^{3/4}\), income \(\$144\), and \(p_y=\$6\). Find the optimum at \(p_x=\$8\), then at \(p_x=\$12\), and state which intercept remains fixed. (Answer: bundles \((4,18)\) then \((3,18)\); the \(y\)-intercept remains 24.)
- A consumer has fixed-proportion preferences requiring \(x=2y\), income \(\$84\), and \(p_y=\$4\). Find the chosen bundle when \(p_x=\$5\) and when it rises to \(\$6\). (Answer: \((12,6)\) then \((10.5,5.25)\); the budget line pivots around its unchanged \(y\)-intercept of 21.)
- A consumer has \(U(x,y)=x^{0.6}y^{0.4}\). Income is initially \(\$100\), with \(p_x=\$10\) and \(p_y=\$5\). The price of \(x\) falls to \(\$8\), then income rises to \(\$120\) with prices unchanged. Find the three optimal quantities of \(x\) and classify both changes. (Answer: \(6\), \(7.5\), and \(9\); the first change is movement along demand, and the second is a demand shift.)
- A consumer has \(U(x,y)=4x+3y\), income \(\$72\), and \(p_y=\$6\). Compare choices at \(p_x=\$12\), \(\$8\), and \(\$5\), accounting for ties. (Answer: at \(\$12\), choose \((0,12)\); at \(\$8\), any budget-line bundle is optimal because both goods yield \(0.5\) utility per dollar; at \(\$5\), choose \((14.4,0)\).)