Why a Price Change Has Two Effects
When the price of a good changes, a consumer faces two changes at once. The good’s price changes relative to other goods, so the consumer has a reason to adjust the mix of goods purchased. At the same time, the consumer’s income buys a different amount of goods than before. The first response is the substitution effect; the second is the income effect, also called the purchasing-power effect.
To separate them, compare three choices: the consumer’s original optimum, a hypothetical intermediate optimum that isolates the change in relative prices, and the actual new optimum. The intermediate choice is not an observed purchase. It is a device for organizing the total change into two parts.
The split requires a benchmark because relative prices and purchasing power change together in an actual price change. One useful budget-line construction keeps the original bundle just affordable at the new prices. It adjusts income by enough to pay for that old bundle at the new prices; it does not keep the consumer’s utility fixed. This construction identifies the substitution component first and the income component second. Other compensation benchmarks can define a different intermediate point and thus a different numerical split, even though the total change is unchanged.
Reading the Components on a Budget-Line Diagram
Let \(x\) be the good whose price changes, \(y\) another good, \(p_x\) and \(p_y\) their prices, and \(m\) income. A budget line is \(p_xx+p_yy=m\), with slope \(-p_x/p_y\) when \(x\) is on the horizontal axis. If \(p_x\) rises while \(m\) and \(p_y\) stay fixed, the line becomes steeper and pivots inward around its \(y\)-intercept. The consumer moves from the original optimum to the new optimum, but that single movement mixes the two effects.
For the intermediate construction, draw a line parallel to the new budget line through the original bundle. It has the new relative prices, but enough adjusted income for the original bundle to remain affordable. The intermediate optimum on this line shows the response to the relative-price change under this benchmark. The movement from that intermediate optimum to the actual optimum on the new budget line represents the income effect.
In the diagram, the original optimum is where the original budget line reaches the highest attainable indifference curve. The intermediate line passes through that original bundle and has the slope of the new budget line. The new budget line is parallel to the intermediate line but lies inward, because it reflects the consumer’s actual, unchanged income. The consumer’s optimum on the intermediate line need not lie on the original indifference curve: the construction preserves affordability of the original bundle, not utility.
Here \(x^0\) is the original chosen quantity, \(x^c\) the intermediate chosen quantity, and \(x^1\) the actual new chosen quantity. The first difference is the substitution component under the stated benchmark; the second is the income component. The identity follows by adding and subtracting the intermediate quantity: \(x^1-x^0=(x^c-x^0)+(x^1-x^c)\). The same accounting applies to \(y\).
Worked Examples
Worked Example: A Cobb–Douglas Consumer Facing a Higher Price
A consumer has utility \(U(x,y)=\sqrt{xy}\), income \(m=120\), and price \(p_y=2\). The price of \(x\) rises from \(p_x=2\) to \(p_x=4\). Find the original and actual choices, then separate the changes using the intermediate budget that keeps the original bundle affordable. Assume both goods are desirable and the consumer chooses an interior optimum.
Method. For Cobb–Douglas utility with equal exponents, the consumer spends half of income on each good. Thus \(x=m/(2p_x)\) and \(y=m/(2p_y)\).
Original choice. At \(p_x=2\), \(x^0=120/(2\cdot2)=30\). Also, \(y^0=120/(2\cdot2)=30\). The budget check is \(2(30)+2(30)=120\).
Actual choice after the price increase. At \(p_x=4\), \(x^1=120/(2\cdot4)=15\), while \(y^1=120/(2\cdot2)=30\). The new budget check is \(4(15)+2(30)=120\).
Intermediate choice. Income must be \(m^c=4(30)+2(30)=180\) for the original bundle to remain affordable at the new prices. Applying the same demand formulas gives \(x^c=180/(2\cdot4)=22.5\) and \(y^c=180/(2\cdot2)=45\). Check: \(4(22.5)+2(45)=180\).
Separate the effects. For \(x\), the substitution effect is \(22.5-30=-7.5\), and the income effect is \(15-22.5=-7.5\). Their sum, \(-15\), equals the total change \(15-30=-15\). For \(y\), the substitution effect is \(45-30=15\), and the income effect is \(30-45=-15\). Their sum is zero, matching the total change \(30-30=0\).
Check and trap. The intermediate line has the new slope and passes through the original bundle; it is not the consumer’s actual post-change budget line. Using the actual income of 120 to calculate the intermediate choice would fail to preserve affordability of the original bundle. All quantities are shown exactly, with no rounding needed.
Worked Example: Perfect Complements and No Substitution
A consumer’s preferences are \(U(x,y)=\min(x,y)\), so the goods are consumed in a fixed one-to-one proportion. Income is \(m=12\), \(p_y=1\), and \(p_x\) rises from 1 to 2. Find the original, intermediate, and actual bundles and explain the two effects. Assume the consumer maximizes utility and can buy fractional quantities.
Method. With one-to-one perfect complements, an optimal bundle has \(x=y=q\). The consumer buys \(q=m/(p_x+p_y)\), because each unit of utility requires one unit of each good.
Original choice. Initially \(q=12/(1+1)=6\), so \((x^0,y^0)=(6,6)\). Its cost is \(1(6)+1(6)=12\).
Intermediate choice. At the new prices, preserving affordability of the original bundle requires income \(m^c=2(6)+1(6)=18\). The fixed-proportion quantity is \(q^c=18/(2+1)=6\), so \((x^c,y^c)=(6,6)\). The intermediate budget check is \(2(6)+1(6)=18\).
Actual choice. With actual income still 12, the new quantity is \(q^1=12/(2+1)=4\), giving \((x^1,y^1)=(4,4)\). Its cost is \(2(4)+1(4)=12\).
Separate the effects. For each good, the substitution effect is \(6-6=0\). The income effect is \(4-6=-2\), which is also the total change. With these preferences, changing relative prices does not change the required one-to-one mix; the lower purchasing power reduces the amount of both goods. The component sums match the observed changes, and no rounding is needed.
Worked Example: Quasilinear Preferences and a Zero Income Effect
A consumer has utility \(U(x,y)=10\ln x+y\), income \(m=100\), and \(p_y=1\). The price of \(x\) rises from 2 to 5. Find the three bundles and the effects, assuming \(x>0\), \(y\geq0\), and an interior optimum with positive \(y\).
Method. At an interior optimum, the marginal rate of substitution equals the price ratio. The marginal utilities are \(MU_x=10/x\) and \(MU_y=1\), so \(10/x=p_x/p_y\), or \(x=10p_y/p_x\). The budget then determines \(y=m-p_xx\).
Original choice. At \(p_x=2\), \(x^0=10/2=5\), and \(y^0=100-2(5)=90\). The budget check is \(2(5)+90=100\).
Intermediate choice. Income at the new prices that preserves affordability of the old bundle is \(m^c=5(5)+1(90)=115\). At \(p_x=5\), \(x^c=10/5=2\), and \(y^c=115-5(2)=105\). Check: \(5(2)+105=115\).
Actual choice. At the new price and actual income, \(x^1=10/5=2\), and \(y^1=100-5(2)=90\). Check: \(5(2)+90=100\).
Separate the effects. For \(x\), substitution is \(2-5=-3\), while the income effect is \(2-2=0\). For \(y\), substitution is \(105-90=15\), while the income effect is \(90-105=-15\). The components sum to the total changes of \(-3\) for \(x\) and zero for \(y\). The positive-\(y\) assumption holds at all three bundles. A common error is to infer that a price change must create a nonzero income effect for every good: here the income change leaves the chosen \(x\) unchanged, though it changes \(y\).
Interpreting the Income Effect and Avoiding Misreads
The substitution effect reflects changed relative prices: a consumer responds to the changed trade-off between goods. Under the intermediate-budget construction, when \(p_x\) rises, the \(x\)-intercept of the budget line falls relative to its \(y\)-intercept, and the consumer generally substitutes away from \(x\). The direction of the response in \(y\) depends on preferences; a consumer may substitute toward or away from it.
The income effect reflects the change from the intermediate budget to the actual budget. A price increase reduces purchasing power when money income is fixed, but the direction of the quantity response depends on the good. For a normal good, lower purchasing power reduces its chosen quantity. For an inferior good, lower purchasing power can increase its chosen quantity. Therefore, do not assume that the income effect always reinforces the substitution effect for the good whose price rose.
A reliable diagram-reading procedure is to identify the three choices before assigning labels. First mark the original optimum. Then draw the new-price parallel line through the original bundle and find its optimum. Finally find the optimum on the actual new budget line. Label the first-to-intermediate movement “substitution” and the intermediate-to-final movement “income.” Reversing these movements, or treating the intermediate line as the actual budget, changes the interpretation.
Check Your Understanding
For each problem, use the intermediate budget that makes the original bundle just affordable at the new prices. Treat any stated optimum formulas or preference conditions as part of the problem data.
- A consumer has \(U(x,y)=\sqrt{xy}\), \(m=108\), and \(p_y=3\). The price of \(x\) rises from 3 to 6. Find the original and actual bundles, the intermediate bundle, and the substitution and income effects for both goods. (Answer: original \((18,18)\), actual \((9,18)\), intermediate \((13.5,27)\); effects for \(x\): \(-4.5,-4.5\), and for \(y\): \(+9,-9\), in substitution-then-income order.)
- Preferences are \(U(x,y)=\min(x,y)\), income is 15, and \(p_y=1\). The price of \(x\) rises from 1 to 2. Find all three bundles and both effects for each good. (Answer: original \((7.5,7.5)\), intermediate \((7.5,7.5)\), actual \((5,5)\); substitution effects are zero and income effects are \(-2.5\) for each good.)
- A consumer has \(U(x,y)=8\ln x+y\), \(m=50\), and \(p_y=1\). The price of \(x\) increases from 2 to 4. At an interior optimum, \(x=8/p_x\). Find all three bundles and separate effects. (Answer: original \((4,42)\), intermediate \((2,50)\), actual \((2,42)\); effects for \(x\): \(-2,0\), and for \(y\): \(+8,-8\).)
- A consumer’s preferences imply the following choices: the original bundle is \((10,20)\), the intermediate optimum is \((7,24)\), and the actual new optimum is \((8,21)\). Calculate the substitution effect, income effect, and total change for each good, then check the component sums. (Answer: for \(x\), substitution \(-3\), income \(+1\), total \(-2\); for \(y\), substitution \(+4\), income \(-3\), total \(+1\); each pair of effects sums to its total.)
- At the original prices, a consumer chooses \((6,12)\). After a price increase, the intermediate optimum is \((4,15)\) and the actual new optimum is \((5,13)\). For each good, find the two components and state whether the income effect reinforces or offsets the substitution effect. (Answer: for \(x\), substitution \(-2\), income \(+1\), total \(-1\), so income offsets substitution; for \(y\), substitution \(+3\), income \(-2\), total \(+1\), so income also offsets substitution.)