Slutsky Compensation and the Slutsky Equation

Why Adjust Income After a Price Change?

When a price changes, a consumer’s purchasing power changes even if nominal income does not. To identify the substitution effect in a controlled way, economists adjust income just enough that the consumer can still afford the original bundle at the new prices. This is Slutsky compensation. It does not hold utility fixed; its defining condition is that the original bundle remains affordable.

That affordability condition gives a precise intermediate budget. Comparing the original choice with the choice on this compensated budget isolates the substitution effect. Comparing the compensated choice with the actual choice at the new prices isolates the income effect. The two changes add to the total demand change.

Definition: Let the original prices be $p^0$, original income be $m^0$, and original optimal bundle be $x^0=x(p^0,m^0)$. After prices change to $p^1$, Slutsky-compensated income is $$ m^S=p^1\cdot x^0. $$ At the new prices, the original bundle costs exactly $m^S$, so it is just affordable. The Slutsky-compensated bundle is $x^S=x(p^1,m^S)$. If actual income after the price change is $m^1$, the actual new bundle is $x^1=x(p^1,m^1)$.

For a consumer spending all income initially, $m^0=p^0\cdot x^0$. Therefore, the compensated income can also be written $$ m^S=m^0+(p^1-p^0)\cdot x^0. $$ The adjustment is the change in the cost of the original quantities, not the change in the cost of a new or compensated choice. If some goods become more expensive and others cheaper, the net adjustment depends on all the original quantities and all the price changes.

The Finite Slutsky Decomposition

The actual demand change can be split by adding and subtracting the compensated bundle:

$$ x^1-x^0=(x^S-x^0)+(x^1-x^S). $$

The term $x^S-x^0$ is the Slutsky substitution effect: prices change from $p^0$ to $p^1$, while income is adjusted so the original bundle remains affordable. The term $x^1-x^S$ is the income effect: prices stay at $p^1$, while income changes from $m^S$ to actual income $m^1$. This is an exact accounting identity for finite changes, provided the three bundles are defined as above.

For good $i$, the same relationship applies to its quantity:

$$ x_i^1-x_i^0=(x_i^S-x_i^0)+(x_i^1-x_i^S). $$

The decomposition describes how the total change is allocated between two comparisons; it does not, by itself, say that either component must have a particular sign. The substitution comparison uses a compensated budget, while the income comparison uses the consumer’s demand response to a change in income at the new prices.

Worked Example: Cobb–Douglas Demand After a Price Increase

A consumer has utility $u(x_1,x_2)=\sqrt{x_1x_2}$, faces initial prices $p^0=(2,1)$, and has income $m^0=120$. The price of good 1 rises to 3, while the price of good 2 stays at 1. Actual income remains 120. Find the compensated income and decompose the changes in both quantities.

Method. For this equal-exponent Cobb–Douglas utility, the consumer spends half of income on each good. Thus $x_1=m/(2p_1)$ and $x_2=m/(2p_2)$.

1. Find the original bundle. At the initial prices and income, $x_1^0=120/(2\cdot2)=30$ and $x_2^0=120/(2\cdot1)=60$. The original bundle costs $2\cdot30+1\cdot60=120$, as required by the budget.

2. Reprice the original bundle. At new prices $p^1=(3,1)$, that bundle costs $3\cdot30+1\cdot60=90+60=150$. Therefore $m^S=150$. The increase in compensated income is $150-120=30$, exactly the increase in the cost of the original bundle.

3. Find the compensated bundle. At prices $(3,1)$ and income 150, $x_1^S=150/(2\cdot3)=25$ and $x_2^S=150/(2\cdot1)=75$. Its expenditure is $3\cdot25+1\cdot75=75+75=150$.

4. Find actual demand and decompose. At the same new prices but actual income 120, $x_1^1=120/6=20$ and $x_2^1=120/2=60$. For good 1, the total change is $20-30=-10$: substitution is $25-30=-5$, and income is $20-25=-5$. For good 2, the total change is $60-60=0$: substitution is $75-60=15$, and income is $60-75=-15$.

Check and trap avoided. In each case, substitution plus income equals the total change. The common error is to calculate compensation using the post-change demand. The required income is instead the new-price cost of the original bundle: 150, not the cost of either later bundle.

Deriving the Differential Slutsky Equation

The finite decomposition compares bundles. For small price changes, it leads to a differential equation that separates a price response into a compensated price response and an income response. Assume demand is differentiable near the original prices and income, the consumer chooses an optimum, and the compensated income is defined by keeping the original bundle affordable.

Write compensated demand.
For a small change in prices from $p$, compensated income is $m^S=p'\cdot x(p,m)$. The compensated demand for good $i$ is $x_i(p',m^S)$.

Differentiate with respect to $p_j$.
At the starting point $p'=p$, a small increase $dp_j$ changes compensated income by $dm^S=x_j\,dp_j$, because the original quantity of good $j$ is $x_j$.

Apply the chain rule.
The change in compensated demand for good $i$, per unit change in $p_j$, is the direct price response plus the response to the compensating income change.

Let $S_{ij}$ denote the change in compensated demand for good $i$ per unit change in price $p_j$, evaluated at the original prices and income. The chain rule in Step 3 gives

S_{ij}=\frac{\partial x_i}{\partial p_j} +x_j\frac{\partial x_i}{\partial m}.

Rearranging yields the differential Slutsky equation:

\frac{\partial x_i}{\partial p_j} =S_{ij}-x_j\frac{\partial x_i}{\partial m}.

The factor $x_j$ appears because a small increase in $p_j$ raises the cost of the original bundle by approximately $x_j\,dp_j$. The income response to that cost change is therefore $x_j(\partial x_i/\partial m)\,dp_j$. Subtracting it from the total price response leaves the compensated price response. This differential form is a local result under the stated differentiability assumptions; the finite-bundle identity does not require taking derivatives.

Worked Example: Separating Local Price and Income Responses

At a consumer’s current choice, good 1 has quantity $x_1=4$. A one-unit increase in the price of good 1 produces a compensated demand response $S_{11}=-2$ units for good 1. The income derivative is $\partial x_1/\partial m=0.5$ units per dollar. Use the Slutsky equation to find the uncompensated own-price derivative.

Method. Use the differential Slutsky equation with $i=j=1$: $\partial x_1/\partial p_1=S_{11}-x_1(\partial x_1/\partial m)$.

Substitute the values. The income-effect term is $4\cdot0.5=2$. Hence $\partial x_1/\partial p_1=-2-2=-4$. The total local response consists of a compensated price response of $-2$ plus an income-related response of $-2$.

Check and trap avoided. Substituting the result back gives $-4+4(0.5)=-2=S_{11}$. A common sign error is to add the income term when solving for the uncompensated derivative. The equation defines $S_{11}$ as the total derivative plus the income term, so rearrangement requires subtraction.

Compensation Can Depend on Every Good

A price change to one good does not mean compensation should be calculated using only that good if other prices also change. Each original quantity must be multiplied by its own price change. The resulting adjustment can be positive, negative, or zero, depending on the combined change in the original bundle’s cost.

Worked Example: Two Prices Change in Opposite Directions

A consumer initially buys $x^0=(4,5)$ at prices $p^0=(3,2)$, spending all income $m^0=22$. New prices are $p^1=(4,1)$. The consumer’s actual income remains 22. Calculate Slutsky-compensated income and write the exact decomposition for each good. Do not assume a particular utility function.

Method. Reprice the original quantities at the new prices. The given data determine the compensation and the identity’s structure, but without preferences they do not determine the new optimal bundles.

1. Calculate the new cost of the original bundle. It is $4\cdot4+1\cdot5=16+5=21$, so $m^S=21$. Equivalently, the price-change adjustment is $(4-3)4+(1-2)5=4-5=-1$, giving $m^S=22-1=21$.

2. State the decomposition with the information available. Write $x^S=x((4,1),21)$ and $x^1=x((4,1),22)$. For good 1, the total change is $x_1^1-4=(x_1^S-4)+(x_1^1-x_1^S)$. For good 2, it is $x_2^1-5=(x_2^S-5)+(x_2^1-x_2^S)$.

Check and trap avoided. At the new prices, the old bundle costs 21, one dollar less than initial income. It would be wrong to compensate by adding only the four-dollar increase in the first good’s cost: the five-dollar reduction in the second good’s cost must also be included. Numerical demand effects cannot be computed without preferences.

Common Exam Traps

Using the wrong bundle in the compensation calculation. Always calculate $m^S=p^1\cdot x^0$. The compensated budget is built around the original quantities, not the actual new bundle.
Reversing the sign of the price adjustment. The equivalent formula is $m^S=m^0+(p^1-p^0)\cdot x^0$. A higher cost of the old bundle requires more compensated income; a lower cost requires less.
Mixing up the two parts of the decomposition. The substitution comparison is $x^S-x^0$; the income comparison is $x^1-x^S$. Their sum must equal $x^1-x^0$.
Assuming the new compensated choice equals the original bundle. The original bundle is made affordable, not automatically chosen. The consumer reoptimizes at the new prices.
Applying the differential equation to a large change as if it were exact. The derivative equation is local. For finite changes, use the compensated-bundle identity and calculate the relevant bundles.

Key takeaway: Slutsky compensation sets income equal to the new-price cost of the original bundle. The resulting compensated choice creates an intermediate point that separates the total demand change into a compensated price response and an income response.

Check Your Understanding

For each problem, identify the compensated income before using the decomposition.

A consumer with $u(x_1,x_2)=\sqrt{x_1x_2}$ has prices $(1,2)$ and income 80. The first price rises to 2, while actual income stays 80. Find the original bundle, compensated income, compensated bundle, and actual bundle; decompose each quantity change. (Answer: $x^0=(40,20)$, $m^S=120$, $x^S=(30,30)$, $x^1=(20,20)$; good 1 changes by $-20=-10-10$, and good 2 changes by $0=10-10$.)
For $u(x,y)=\ln x+y$, let $p_x=1$, $p_y=1$, and income be 15. The price of $x$ rises to 2, with actual income unchanged. Assume the optimum is interior with nonnegative $y$. Find the compensated income and decompose the changes in $x$ and $y$. (Answer: $x^0=1,y^0=14$, $m^S=16$, $x^S=0.5,y^S=15$, $x^1=0.5,y^1=14$; for $x$, $-0.5=-0.5+0$, and for $y$, $0=1-1$.)
A consumer has fixed-proportion utility $u(x,y)=\min\{x,y/2\}$. Initial prices are $(1,1)$, income is 36, and the price of $y$ rises to 2. Actual income remains 36. Find the compensated and actual bundles and decompose the changes. (Answer: $x^0=12,y^0=24$, $m^S=60$, $x^S=(12,24)$, $x^1=(7.2,14.4)$; substitution effects are zero for both goods, and income effects are $-4.8$ for $x$ and $-9.6$ for $y$.)
An original bundle is $(4,5)$, initial prices are $(3,2)$, and new prices are $(4,1)$. Calculate the compensated income if initial income equals the original bundle’s cost. Explain why the income adjustment is not the first good’s price increase times its quantity alone. (Answer: initial income is 22 and compensated income is $4(4)+5(1)=21$; the net adjustment is $4-5=-1$, because both price changes affect the original bundle’s cost.)
At a current choice, $x_j=3$, $\partial x_i/\partial m=0.4$, and $S_{ij}=-1.7$. Use the differential Slutsky equation to find $\partial x_i/\partial p_j$, and verify by substitution. (Answer: $\partial x_i/\partial p_j=-1.7-3(0.4)=-2.9$; $-2.9+3(0.4)=-1.7$.)

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