Income Effects for Normal and Inferior Goods

Why Income Effects Matter

A price change alters the trade-off between goods, but it also changes what a consumer can afford. These are distinct channels. The substitution effect captures the response to a changed relative price after compensation; the income effect captures the response to the change in purchasing power. Their signs explain whether they reinforce or offset one another.

The key classification is whether demand for a good rises or falls as income changes while prices are held fixed. This tutorial uses that classification to determine the sign of the income effect, then combines it with the compensated substitution effect. The distinction is useful whenever an observed change in demand must be separated into a relative-price response and a purchasing-power response.

Definition: A good is normal over a stated range if its quantity demanded increases when income increases, holding prices and other relevant conditions fixed. It is inferior over a stated range if its quantity demanded decreases when income increases, with prices and other relevant conditions fixed. If demand does not respond to income over the range, the income effect there is zero.

These are descriptions of demand over a range, not permanent labels that must apply at every income and price. A good may be inferior for one consumer or income range and normal for another. To infer an income effect, use the classification that applies between the compensated and actual income levels being compared.

Price Changes and the Income Effect

Consider a consumer choosing quantities $x_1$ and $x_2$, with income $m$ and prices $p_1$ and $p_2$. Let $x_1(p_1,p_2,m)$ denote the consumer’s ordinary, or uncompensated, demand for good 1. Suppose only $p_1$ changes, from $p_1$ to $p_1'$, while $p_2$ and $m$ remain fixed.

For a Slutsky decomposition, define compensated income $m^c$ as the income that would let the consumer afford the original bundle at the new prices. With only $p_1$ changing, this is $m^c=m+(p_1'-p_1)x_1(p_1,p_2,m)$. If the price rises, $m^c$ exceeds actual income; if the price falls, $m^c$ is below actual income. Demand at the new prices and this compensated income is compared with the original demand to isolate the substitution effect.

Slutsky decomposition: For this own-price change, the total change in demand can be written as the sum of a Slutsky substitution effect and an income effect: $$ \begin{aligned} \Delta x_1 &=x_1(p_1',p_2,m)-x_1(p_1,p_2,m)\\ &=\underbrace{x_1(p_1',p_2,m^c)-x_1(p_1,p_2,m)}_{\text{substitution effect}}\\ &\quad+\underbrace{x_1(p_1',p_2,m)-x_1(p_1',p_2,m^c)}_{\text{income effect}}. \end{aligned} $$

The equation is an exact decomposition: adding and subtracting $x_1(p_1',p_2,m^c)$ on the right leaves the original total change. The compensation also has an economic interpretation. At the new prices, the original bundle costs $m^c$, so that bundle is affordable with compensated income. The substitution effect compares the original choice with the choice at new prices and compensated income; the income effect then compares compensated income with actual income at the new prices.

For an own-price rise, the compensated substitution effect is nonpositive; for an own-price fall, it is nonnegative. These effects can be zero as well as strictly signed. The sign follows from the compensated choice: the original bundle is affordable at the new prices with $m^c$, and the consumer chooses a bundle at least as preferred. The standard revealed-preference argument therefore does not require the consumer to change the quantity of good 1 strictly.

The income-effect sign follows from comparing $m$ with $m^c$ at the new prices. For a normal good, demand rises with income; for an inferior good, it falls with income. This gives the following sign guide, provided the classification holds over the relevant income range.

Own-price change	Income comparison	Normal good	Inferior good
Price rises	Actual income is below compensated income	Income effect is nonpositive	Income effect is nonnegative
Price falls	Actual income is above compensated income	Income effect is nonnegative	Income effect is nonpositive

A zero price change or zero income response can make the income effect zero. For finite changes, the table describes the sign when the good keeps the same normal or inferior classification across the comparison. If its income response changes over that range, the endpoints alone do not determine the sign without examining demand over the intervening incomes.

Worked Applications

In each example, the substitution and income effects are calculated using the same compensation convention. The numerical method is to find the original quantity, calculate compensated income, evaluate compensated demand at the new price, and then compare it with actual demand at that price.

Worked Example: A Normal Good After a Price Rise

A consumer has Cobb–Douglas preferences with demand $x_1=0.4m/p_1$ for good 1 and $x_2=0.6m/p_2$ for good 2. Income is $m=\$100$, $p_2=\$1$, and $p_1$ rises from $\$2$ to $\$4$. Calculate the total change in $x_1$, its Slutsky substitution effect, and its income effect.

Method and classification. Cobb–Douglas demand for good 1 is increasing in income because $\partial x_1/\partial m=0.4/p_1>0$. It is normal. First calculate the original quantity: $$ x_1(2,1,100)=\frac{0.4(100)}{2}=20. $$

Compensated income. The price increase is $\$4-\$2=\$2$ per unit. Restoring affordability of the original 20 units requires $$ m^c=100+(4-2)(20)=100+40=140. $$

Substitution effect. At the new price and compensated income, demand is $$ x_1(4,1,140)=\frac{0.4(140)}{4}=14. $$ Thus the substitution effect is $14-20=-6$ units.

Income effect and total check. At the new price but actual income, demand is $x_1(4,1,100)=0.4(100)/4=10$. The income effect is $10-14=-4$ units, and the total change is $10-20=-10$ units. The check is $-6+(-4)=-10$. The two effects reinforce one another, as expected for a normal good after a price rise. This method avoids treating the entire observed fall as a substitution effect.

Worked Example: An Inferior Good After a Price Rise

At fixed $p_2=\$1$, suppose a hypothetical consumer’s locally valid demand schedule for good 1 is $x_1(p_1,m)=12+8/p_1-0.04m$. Income is $\$100$, and $p_1$ rises from $\$2$ to $\$3$. Quantities are measured in units and income in dollars. Calculate the two effects and the total change.

Method and classification. At fixed prices, the income coefficient is negative: an extra dollar of income reduces demand by $0.04$ units over the stated range. The good is inferior over this range. Initial demand is $$ x_1(2,100)=12+\frac{8}{2}-0.04(100)=12+4-4=12. $$

Compensated income. The price rises by $\$1$ per unit, so the income needed to afford the original 12 units at the new price is $$ m^c=100+(3-2)(12)=112. $$

Substitution effect. At the new price and compensated income, $$ x_1(3,112)=12+\frac{8}{3}-0.04(112) =12+2.6667-4.48 =10.1867. $$ The substitution effect is $10.1867-12=-1.8133$ units.

Income effect and total check. At the new price and actual income, $$ x_1(3,100)=12+\frac{8}{3}-4=10.6667. $$ The income effect is $10.6667-10.1867=0.4800$ units. The total change is $10.6667-12=-1.3333$ units, and the check is $-1.8133+0.4800=-1.3333$. The inferior-good income effect is positive after a price rise, so it offsets some—but not all—of the negative substitution effect in this example. Values are rounded to four decimal places at the final step.

Worked Example: Perfect Complements and a Zero Substitution Effect

A consumer’s utility is $u(x_1,x_2)=\min(x_1,x_2)$, so the goods are consumed in fixed one-to-one proportions. Income is $\$120$, $p_2=\$2$, and $p_1$ rises from $\$2$ to $\$4$. Find the change in demand for good 1 and separate it into Slutsky effects.

Method. To obtain one unit of utility, the consumer needs one unit of each good, costing $p_1+p_2$. The optimal bundle has $x_1=x_2=m/(p_1+p_2)$. Initially, each quantity is $120/(2+2)=30$ units.

Compensation and substitution effect. The cost increase for the original 30 units of good 1 is $(4-2)(30)=\$60$, so $m^c=120+60=180$. At the new prices and compensated income, each quantity is $180/(4+2)=30$. The substitution effect for good 1 is $30-30=0$. The change in relative prices does not alter the one-to-one bundle required for utility.

Income effect and total check. At the new prices and actual income, each quantity is $120/(4+2)=20$. The income effect is $20-30=-10$ units, and the total change is also $20-30=-10$ units. The check is $0+(-10)=-10$. This example shows why the compensated substitution effect is nonpositive for a price rise, not necessarily strictly negative.

Combining the Effects and Avoiding Traps

For a price rise, a normal good has a nonpositive substitution effect and a nonpositive income effect, so the effects reinforce or one may be zero. An inferior good has a nonpositive substitution effect and a nonnegative income effect, so the effects oppose one another or one may be zero. For a price fall, the signs reverse in the corresponding way: normal-good effects are nonnegative, while the inferior-good income effect is nonpositive and opposes the nonnegative substitution effect.

Opposing signs mean the total response must be calculated, not guessed from the good’s classification alone. The income effect may be smaller or larger in magnitude than the substitution effect. In either case, the Slutsky equation gives the exact total as their sum. Determining the total requires both effect magnitudes, not just their directions.

Do not swap the classifications. Inferior means demand falls as income rises; it does not mean demand must fall when its own price rises.
Do not assign the income-effect sign from the price change alone. First compare actual income with compensated income, then apply the normal-or-inferior classification.
Do not assume a strict substitution response. The compensated substitution effect can be zero, as with the fixed-proportion preferences in the third example.
Do not add effects with inconsistent baselines. Calculate both using the same compensation and price-change convention, then verify that their sum equals final demand minus initial demand.

These distinctions matter in demand analysis because an observed price response combines two mechanisms. An actuary examining a hypothetical insurance product, for example, might need to distinguish a change in relative attractiveness from a change in customers’ purchasing power. The economics does not depend on the product being insurance: the definitions, compensation, and sign logic are those of consumer choice.

Key takeaway: A price change’s income-effect sign depends on whether actual income lies above or below compensated income and on whether the good is normal or inferior over that income range. Add that effect to the compensated substitution effect to obtain the total demand change; either effect can be zero.

Check Your Understanding

For each problem, identify the relevant income-effect direction and use the stated demand information to determine the requested response.

A normal good has demand $x_1=0.3m/p_1$. Income is $\$80$, $p_1$ rises from $\$4$ to $\$5$, and the other price is fixed. Find compensated income, the substitution effect, the income effect, and the total change. (Answer: $m^c=\$86$; substitution effect $=-0.84$ units; income effect $=-0.36$ units; total change $=-1.20$ units.)
At fixed $p_2=\$1$, an inferior good has local demand $x_1=9+6/p_1-0.02m$. Income is $\$100$, and $p_1$ rises from $\$2$ to $\$4$. Calculate compensated income and the three quantity changes. (Answer: $m^c=\$120$; substitution effect $=-1.90$ units; income effect $=+0.40$ units; total change $=-1.50$ units.)
For $u(x_1,x_2)=\min(x_1,x_2)$, income is $\$80$, $p_2=\$1$, and $p_1$ rises from $\$1$ to $\$3$. Find the compensated substitution and income effects for good 1. (Answer: substitution effect $=0$ units; income effect $=-20$ units; total change $=-20$ units.)
A normal good has demand $x_1=0.25m/p_1$. Income is $\$120$, and $p_1$ falls from $\$6$ to $\$3$. Find compensated income and the effects on demand. (Answer: $m^c=\$105$; substitution effect $=+3.75$ units; income effect $=+1.25$ units; total change $=+5$ units.)
An inferior good faces an own-price fall. Its compensated substitution effect is $+2$ units and its income effect is $-0.7$ units. Find the total response and state whether the effects reinforce or oppose one another. (Answer: total change $=+1.3$ units; the effects oppose one another.)

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