5 Comparative Statics and Demand

Consumer choice and the budget constraint

A consumer’s optimal choice depends on preferences, prices, and income. Comparative statics examines how that choice changes when one of these conditions changes while the others are held fixed. The resulting choices show how individual demand responds to income and prices.

With two goods, the budget constraint is

where $p_1$ and $p_2$ are the goods’ prices, $x_1$ and $x_2$ are their quantities, and $m$ is income. The consumer chooses the most-preferred bundle they can afford. At an interior optimum, the marginal rate of substitution equals the price ratio:

How income changes affect choices

When income rises while prices stay fixed, the budget line shifts outward in parallel; when income falls, it shifts inward. The consumer then reselects the most-preferred affordable bundle. The direction in which a particular good’s consumption changes depends on preferences.

A normal good is one whose demand rises when income rises, holding prices fixed. An inferior good is one whose demand falls as income rises. An income change need not affect all goods in the same direction. The relationship between income and the chosen quantity of a good, with prices held fixed, is its Engel curve.

Price changes and individual demand

A change in one price changes the slope of the budget line, rotating it around the intercept for the other good. The consumer re-optimizes at the new prices. For example, if $p_1$ rises while income and $p_2$ remain fixed, good 1 becomes relatively more expensive and the budget set shrinks in the direction of good 1.

Marshallian (uncompensated) demand for good $i$ is the quantity chosen at given prices and income, written $x_i(p,m)$. Varying the price of good 1 while holding income, the other price, and preferences fixed traces the consumer’s individual demand curve for good 1. A change in its own price produces a movement along this curve; a change in income or preferences generally changes the demand relationship itself.

Substitution and income effects

A price change affects demand through two channels. The substitution effect is the change in choice caused by changed relative prices. For an own-price increase, its effect on demand for that good is nonpositive: the good becomes relatively less attractive.

The income effect is the change in choice caused by the effect of the price change on purchasing power. For a normal good, an own-price increase reduces real purchasing power and therefore reduces demand. For an inferior good, the income effect on demand runs in the opposite direction. If this positive income effect is large enough to outweigh the substitution effect, the good is Giffen: its quantity demanded rises when its own price rises.

The Slutsky decomposition

The Slutsky decomposition separates the effects by introducing a compensated intermediate budget. After a price change, income is adjusted just enough that the consumer can still afford the original bundle. The move from the original choice to the choice on this compensated budget is the substitution effect. The move from that compensated choice to the actual choice at the new prices and original income is the income effect. This compensation preserves the ability to buy the original bundle, not necessarily the original utility level.

In differential form, the Slutsky equation is

Here, $h_i(p,u)$ is Hicksian demand at fixed utility $u$. The first term is the compensated substitution response; the second is the income-effect adjustment. For an own-price change, the compensated substitution term is nonpositive. The income term depends on whether the good is normal or inferior.

A Slutsky decomposition example

Suppose a consumer has utility $u(x,y)=xy$, income $m=100$, and prices $p_x=2$ and $p_y=5$. The optimal Cobb–Douglas bundle is $x=m/(2p_x)=25$ and $y=m/(2p_y)=10$. If $p_x$ rises to $4$, the new optimum at the original income is $(12.5,10)$.

To isolate the substitution effect using Slutsky compensation, raise income enough to afford the original bundle at the new prices:

At prices $(4,5)$ and compensated income $150$, the optimum is $(18.75,15)$. For $x$, the substitution effect is $18.75-25=-6.25$, and the income effect is $12.5-18.75=-6.25$. Together they produce the total change $12.5-25=-12.5$. The example shows that a price change’s total effect is not only the response to a change in relative price; purchasing power changes as well.

Putting the effects together

Income changes shift the budget line in parallel, while price changes rotate it. Re-optimizing after these changes generates individual demand. The Slutsky decomposition divides a price-driven demand change into a substitution effect and an income effect, with the latter determined by how demand responds to purchasing power.

For a normal good, both effects reinforce the usual fall in demand after an own-price increase. For an inferior good, the effects oppose each other; in the Giffen case, the income effect dominates.