Substitution & Income Effect

Why decompose a price change at all

A standard demand curve answers a simple question: at this price, how much do you buy? But that single number conflates two very different stories. When the price of gasoline jumps from $4 to $5, the buyer behaves differently for two distinct reasons. First, gasoline has become more expensive relative to everything else — substitute goods like the bus or carpooling now look better. Second, the buyer is poorer in real terms — for any fixed quantity of gasoline, less is left for groceries. These two channels — relative-price substitution and real-income reduction — can be told apart, and they have to be told apart to answer almost every interesting policy question.

Cost-of-living adjustments use the decomposition explicitly. So does the analysis of fuel-price spikes (does a carbon tax burden the poor disproportionately?), welfare programs (does a food-stamp benefit really increase food consumption, or just free up cash?), and the textbook puzzles of Giffen goods and backward-bending labor supply. The substitution-income decomposition is the workhorse machinery behind all of them.

The diagram: indifference curves and a pivoted budget line

Start with a consumer choosing between two goods X and Y. Their preferences are captured by a family of indifference curves; their constraint is a budget line Px·X + Py·Y = M. The optimal bundle A is the tangency between the highest reachable indifference curve U₁ and the budget line.

Now drop the price of X from Px to Px'. The budget line pivots outward around the Y-intercept: at the same income M the consumer can now afford more X for any level of Y. A new tangency C appears on a higher indifference curve U₂, with more X (almost always) and possibly more or less Y.

Total effect on X: change from A to C. The decomposition splits this into two legs:

• Substitution effect (A → B). Imagine simultaneously taking enough income away that the consumer is just barely on the original indifference curve U₁ but at the new price ratio. That gives a hypothetical budget line tangent to U₁ at a new point B. The move from A to B is a slide along U₁ — pure substitution, no real-income change. It is always toward X (the cheaper good).
• Income effect (B → C). Now give the taken-away income back. The budget line shifts parallel outward to the actual new line through C. The move from B to C reflects the change in real income at fixed relative prices. Its direction depends on whether X is normal (income up → more X) or inferior (income up → less X).

For a normal good, both legs of the journey go in the same direction — both push X up when Px falls. For an inferior good, the substitution leg pushes X up but the income leg pushes it down. For a Giffen good, the income leg dominates and total demand for X falls when Px falls — equivalently, demand rises when Px rises.

The Slutsky equation

The algebraic version of the diagram is the Slutsky equation, named after Eugen Slutsky (1915) and made famous by John Hicks and R.G.D. Allen in 1934. It writes the derivative of Marshallian (ordinary, uncompensated) demand as a sum of two pieces:

∂q_i / ∂p_j  =  ∂q_i^h / ∂p_j   −   q_j · ∂q_i / ∂I

       │              │                       │
       │              │                       └── income effect
       │              └─ Hicksian (compensated) substitution effect
       └─── Marshallian (ordinary) own- or cross-price effect

Specialised to the own-price case (i = j):

∂q / ∂p   =   ∂q^h / ∂p    −    q · ∂q / ∂I
              ≤ 0 (always)        sign depends on
                                  normal vs inferior

Three implications worth memorising:

• The first term — the Hicksian own-price slope — is always non-positive. This is the substitution effect's iron law: at constant utility, a higher relative price never raises compensated demand.
• The second term — the income-effect correction — has a sign determined by ∂q/∂I. If q is a normal good (∂q/∂I > 0) the correction is subtracted as a negative contribution, reinforcing the substitution piece. If q is inferior (∂q/∂I < 0) the correction adds positively, fighting the substitution piece.
• The size of the income-effect correction scales with q. For goods consumed in tiny quantities the income effect is small no matter the income elasticity; for staples the income effect can be enormous.

Hicks vs Slutsky: two ways to "compensate"

The decomposition above is the Hicks version: take away just enough income to leave the consumer on the original indifference curve U₁. Slutsky's original 1915 paper used a different compensation rule: take away just enough income to leave the original bundle A barely affordable at the new prices. The two rules coincide only at the original bundle itself; for any displacement they differ, with Slutsky compensation leaving the consumer slightly better off than Hicks compensation because at the new prices the consumer would substitute toward the cheaper good and reach a higher indifference curve.

Rule	What's held fixed	Definition	Practical use
Hicks (compensating variation)	Utility u(x,y) = U₁	Minimum income to reach U₁ at new prices	Welfare theory; CES demand systems
Slutsky (compensating bundle)	Original bundle (x₁, y₁) affordable	p_new · (x₁, y₁) — income needed	Observable; CPI bias estimation

For small price changes the two agree to first order — that's why the Slutsky equation, derived from either definition, is the same. For finite price changes the difference shows up empirically as the upper-bound bias in a Laspeyres price index (Slutsky-compensated) versus the true cost-of-living index (Hicks-compensated). The bias is what Boskin-commission-style estimates of CPI overestimation are trying to quantify.

The sign table: normal, inferior, Giffen

For an own-price increase (price up), the substitution effect always reduces consumption. The income effect's sign and magnitude determine the total response:

Type of good	∂q/∂I	Substitution effect (price ↑)	Income effect (price ↑)	Total effect	Demand slope
Normal good	> 0	q falls	q falls (real income ↓ → buy less of a normal good)	q falls	Downward
Inferior good (non-Giffen)	< 0, small	q falls	q rises (real income ↓ → buy more of an inferior good)	q falls (sub wins)	Downward
Giffen good	< 0, large	q falls	q rises	q rises (income wins)	Upward

The Giffen case sits at the extreme corner. It requires (1) the good is inferior (rare in rich economies), (2) the inferiority is strong (the income elasticity is very negative), and (3) the good takes a large enough share of the budget that the income effect is sizeable. The triple condition is hard to satisfy outside very poor economies and staple foods.

Worked example: a price drop, walked through

Aiden spends $100 per week on apples (X) and bread (Y). Initial prices Px = $4, Py = $5; initial bundle A = (10, 12). Cobb-Douglas preferences u = X^0.4 · Y^0.6 mean Aiden spends 40% on apples (10 × $4 = $40) and 60% on bread (12 × $5 = $60). Suppose Px drops to $2.

New optimum C. With Cobb-Douglas demand X = 0.4·M/Px = 0.4·100/2 = 20 apples; Y = 0.6·100/5 = 12 loaves. So C = (20, 12). Total effect on X: +10 apples.

Hicks-compensated bundle B. To leave Aiden on the original indifference curve U₁ = 10^0.4 · 12^0.6 ≈ 11.27 at the new prices, find the bundle that minimises expenditure subject to X^0.4·Y^0.6 = 11.27 at Px = 2, Py = 5. The Hicksian demand is X^h = (α/(1−α))·(Py/Px)·(1−α)·U/(...) — or by direct optimisation: X^h = 11.27·(0.4/0.6 · Py/Px)^0.6 = 11.27·(0.4/0.6 · 5/2)^0.6 = 11.27·1.6667^0.6 ≈ 11.27·1.367 ≈ 15.4. Similarly Y^h ≈ 11.27·(0.6/0.4 · 2/5)^0.4 ≈ 11.27·0.6^0.4 ≈ 11.27·0.815 ≈ 9.2. So B ≈ (15.4, 9.2). Substitution effect on X: +5.4 apples (A → B). Income effect on X: +4.6 apples (B → C). Both legs are positive — apples are a normal good.

Slutsky-compensated bundle B'. To leave Aiden's original bundle (10, 12) barely affordable at the new prices, give him income M' = 2·10 + 5·12 = $80. Cobb-Douglas demand at (Px=2, Py=5, M=$80): X' = 0.4·80/2 = 16, Y' = 0.6·80/5 = 9.6. So Slutsky B' = (16, 9.6). Slutsky substitution effect on X: +6 apples (vs Hicks +5.4). The two compensations agree on the sign and roughly the magnitude; they differ slightly because Slutsky overcompensates relative to Hicks.

The labor-supply Slutsky equation

One of the most-cited applications is labor supply. Replace the consumer's two goods with consumption C and leisure L, treat the wage w as the price of leisure (every hour of leisure costs you w in foregone wages), and run the Slutsky decomposition:

∂L / ∂w   =   ∂L^h / ∂w     +     (T − L) · ∂L / ∂Y
              < 0                 > 0 if L normal

               (substitution                (income — more
                effect: leisure              wage income →
                more expensive →             more leisure)
                less leisure)

Total: ∂L/∂w is the sum of a negative substitution effect (work more) and a positive income effect (work less). The sign is ambiguous, and that ambiguity is the source of the famous backward-bending labor-supply curve: at low wages workers work more as wages rise (substitution wins); at high wages they work less (income wins). The threshold depends on the worker's income elasticity of leisure.

Hours = T − L, so ∂Hours/∂w = −∂L/∂w. The labor-supply elasticity that empirical work tries to measure — the percentage change in hours from a one-percent wage change — has been estimated repeatedly, with prime-age men landing around 0.0–0.2, prime-age women around 0.3–0.6, and secondary earners more elastic still. The dominant cited estimate from the modern Chetty-Saez line of work is a compensated (Hicksian) elasticity around 0.25 for the U.S. broadly.

Saving and the interest rate

The intertemporal version runs the same machinery across two periods of consumption C₀ and C₁. The "price" of present consumption in terms of future is 1/(1+r); when r rises, present consumption becomes relatively more expensive. The Slutsky decomposition gives:

• Substitution effect. Higher r → present consumption more expensive → substitute toward future consumption → save more.
• Income effect. For a net saver, higher r → more lifetime wealth → both periods' consumption rise → save less. For a net borrower, higher r → less lifetime wealth → both periods' consumption fall → borrow less / save more.

So a rate hike unambiguously raises borrower saving but ambiguously affects saver saving. Aggregate household saving is the net of millions of such positions; empirical estimates of the elasticity of intertemporal substitution land between 0.1 and 0.5 for the U.S., implying that the substitution effect is small enough that income effects often dominate at the household level. That's why "raise rates to stimulate saving" is much weaker than first-year theory suggests.

Variants and extensions

• Cross-price Slutsky. The equation generalises to ∂q_i/∂p_j = ∂q_i^h/∂p_j − q_j·∂q_i/∂I. Cross-Hicksian terms are symmetric (∂q_i^h/∂p_j = ∂q_j^h/∂p_i — Young's theorem on the expenditure function). Marshallian cross-effects are typically not symmetric because income elasticities differ.
• Endowment income (Slutsky-Hurwicz). If the consumer is endowed with the goods rather than just income, ∂q_i/∂p_j gets an additional term reflecting that the endowment's value changes with prices. The classic case is a worker whose "endowment" is total time T at the wage w: a wage rise both pivots the budget and raises the wealth from the time endowment.
• Random utility / discrete choice. McFadden's discrete-choice models recover an analogous decomposition: a price rise on alternative A reduces its choice probability via a substitution-like share-reallocation channel and a residual income-effect channel that depends on income elasticities of the alternatives.
• Frisch demand and lifecycle. Marginal-utility-of-wealth-constant (Frisch) demand is yet another compensated concept used in intertemporal models. The Frisch elasticity is generally larger in magnitude than the Hicksian, because it nets out the wealth-effect channel completely.
• Almost Ideal Demand System (AIDS, Deaton-Muellbauer 1980). A functional form that allows the Slutsky equation to be estimated from observed budget shares and prices. Industry standard for empirical demand work in food economics, energy economics, and consumer-finance studies.

Where the decomposition matters in practice

• CPI bias. The Boskin Commission (1996) and subsequent BLS work concluded that the U.S. CPI overstates inflation by 0.3-1.1 percentage points per year, partly because a Laspeyres index (fixed-bundle, Slutsky-compensated) misses the substitution toward goods whose relative prices have fallen. Chained indices that approximate Hicks compensation correct most of this.
• Carbon-tax incidence. A tax on gasoline raises Px (gasoline) and the burden is the sum of a substitution effect (consumers switch to public transit, EVs, denser housing) and an income effect (real income falls, disproportionately for low-income households who spend a higher share on energy). Optimal-tax design uses the decomposition to rebate the income effect via lump-sum transfers while keeping the substitution-incentive intact.
• Giffen-good empirics. Robert Jensen and Nolan Miller's 2008 QJE study randomized rice subsidies in two Chinese provinces and found that for the poorest households a price decrease for rice (their staple) led to less rice consumption — the income effect freed up money for meat and vegetables, displacing rice. The mirror-image result is Giffen behavior for a price increase: rare, but empirically real.
• SNAP and cash-out experiments. Whether food stamps raise food consumption more than equivalent cash depends on whether the program-induced income effect on food is the same as the substitution effect of an in-kind transfer. Field experiments (Hoynes-Schanzenbach 2009; Bruich 2014) find a modest extra-marginal effect — food stamps increase food spending by about 17 cents per dollar more than cash would, suggesting most of the effect is income-channel, not substitution-channel.
• Tax-elasticity of taxable income. The labor-supply Slutsky equation underpins the estimation of how labor supply responds to marginal-tax-rate changes — central to the optimal-income-tax literature (Mirrlees-Saez). Compensated elasticities of taxable income near 0.25 imply Laffer-curve peaks for top marginal rates around 60–70%, much higher than current U.S. rates.

Common pitfalls

• Confusing Marshallian with Hicksian effects in welfare calculations. Consumer-surplus changes ∫q(p)dp are exact only for Hicksian (compensated) demand. Using Marshallian demand introduces income-effect contamination — usually small for narrow goods but large for staples. Hausman's 1981 paper shows how to back out the Hicksian demand from Marshallian estimates.
• Mistaking inferior for Giffen. Inferior goods are common (cheap noodles, used cars); Giffen goods are exotic. An inferior good has income effect opposing substitution effect; for it to be Giffen, the income effect must win the magnitude contest. Most inferior goods don't get there.
• Forgetting that substitution sign depends on which good's price moves. Own-price compensated demand is always non-positive in own price. Cross-price compensated demand can be positive (gross substitutes) or negative (gross complements). Many beginning treatments blur this and end up with sign errors.
• Applying the Slutsky equation when income isn't fixed. If the consumer is endowed with goods rather than receiving lump-sum income, the relevant decomposition includes an endowment-revaluation term. Labor supply is the most-cited case; ignoring it overstates the substitution effect of a wage rise.
• Reading ∂q/∂I from time-series alone. Income elasticities for the Slutsky equation are conceptually cross-sectional: how does q change across consumers with different incomes at fixed prices. Time-series increases in income usually come with price changes, so naive time-series estimates conflate the two and bias the decomposition.
• Forgetting that the equation is local. The Slutsky equation is an exact statement about derivatives at one point. For finite price changes, the income effect varies along the path of integration, and naive multiplication of the local equation by Δp overstates the precision of the decomposition.