Income and substitution effects

Decompose a supplied demand change using a clearly defined compensated comparison.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will decompose a supplied demand change using a clearly defined compensated comparison, showing the calculation and stating the assumptions that make the conclusion valid.

2. Starting point

Use the supplied definitions and units. Separate an accounting identity, a behavioral assumption, and a normative criterion before drawing conclusions.

3. Terms to use precisely

4. A price change alters both relative prices and purchasing power

When the price of x changes while nominal income and the price of y stay fixed, two features of the consumer's opportunity set change. The market trade-off between x and y changes, and the original income can afford a different collection of bundles. A demand response can therefore reflect both substitution between goods and a change in purchasing power. A decomposition defines a comparison that separates those channels within a model.

The decomposition needs three bundles or three corresponding x quantities: the original choice, a compensated choice at the new prices, and the final choice at the new prices with the actual post-change income. The compensated comparison is hypothetical. It is introduced to organize the reasoning, not because the consumer necessarily receives an actual payment matching that calculation.

The total change in x is final quantity minus original quantity. The substitution component is compensated quantity minus original quantity. The income component is final quantity minus compensated quantity. Adding the last two expressions cancels the compensated quantity and recovers the total change. This identity checks the arithmetic, while the economic interpretation depends on how the compensated bundle was defined.

Do not describe an observed total response as purely a substitution effect merely because a price changed. The relative-price and purchasing-power changes occur together. Likewise, the two components are not directly observed just because the original and final bundles are observed. Identifying them requires a preference model, suitable data and assumptions, or another justified method for constructing the compensated comparison.

Term	What it means
Compensated comparison	A hypothetical choice at new prices with an explicitly defined purchasing-power adjustment.
Slutsky compensation	Income at new prices that makes the original bundle exactly affordable.
Hicks compensation	Minimum income at new prices needed to reach the original utility level.
Substitution component	Compensated quantity minus original quantity under the stated convention.
Income component	Final quantity minus compensated quantity.
Giffen response	An own-price demand response opposite to the usual direction because the income effect outweighs substitution under the relevant model.

Another way: Slutsky compensation preserves an original bundle's affordability

One finite-change method adjusts income at the new prices so that the original chosen bundle is exactly affordable. This is a Slutsky-style purchasing-power compensation. If the original bundle is (x0,y0), the compensated income is new px times x0 plus py times y0 when only px changes. The consumer then reoptimizes at the new prices with that compensated income.

The compensated choice need not equal the original bundle. The original bundle remains affordable, but the new relative prices can make another bundle preferable. This is the point of the comparison: it allows a response to changed trade-offs while preserving a specific measure of purchasing power. The resulting utility can be above the original utility because the old bundle is available and the consumer can choose something better.

For example, U=xy, income thirty-two, original px four and py two imply x0 four and y0 eight. If px falls to two, the original bundle costs eight plus sixteen, or twenty-four, at the new prices. Compensated income is therefore twenty-four. Reoptimizing product utility at px two and py two gives compensated x six and y six, rather than returning to the original four and eight.

At the unchanged actual income thirty-two and new equal prices two, the final choice is eight units of each good. The substitution component in x is six minus four, or two. The income component is eight minus six, also two. The total increase is four. All three quantities follow from the same demand model, and the compensation convention is stated rather than hidden.

Another way: Hicks compensation preserves the original preference level instead

Another method adjusts income to reach the original utility level as cheaply as possible at the new prices. This is Hicksian compensation. It preserves the original indifference level rather than the affordability of the exact original bundle. The two methods are closely connected, but their finite-change compensated incomes and bundles generally differ.

Suppose U=xy, original income twenty-four, px four and py one. The original choice is x three and y twelve, with product utility thirty-six. If px falls to one, the Hicksian compensated bundle minimizing expenditure at utility thirty-six is x six and y six. It reaches the same utility and costs twelve. The Slutsky compensated income instead prices the original bundle at the new prices: three plus twelve, or fifteen.

At equal new prices and Slutsky income fifteen, product-utility demand is 7.5 units of each good, giving utility 56.25. The original bundle is affordable, but the compensated consumer can reach a higher preference level. This is not an error in the Slutsky method. It shows that preserving bundle affordability differs from preserving utility. A finite decomposition must identify which comparison it uses.

The infinitesimal relationships developed in more advanced consumer theory connect these approaches through the Slutsky equation under appropriate differentiability conditions. This lesson focuses on finite quantities so the components can be inspected directly. It does not use the name substitution effect as if it selected one unique compensated bundle without specifying a convention.

Another way: Use signs to distinguish normal, inferior and Giffen responses

For an own-price reduction, the compensated substitution response normally moves toward the now relatively cheaper good under the standard preference and feasibility conditions. The relevant compensated law of demand restricts the direction of this response. The income component can reinforce or oppose it depending on how demand for the good responds to purchasing power.

For a normal good, the purchasing-power increase associated with a price reduction tends to raise demand, so the income component reinforces the substitution component. In the product-utility example, both are positive. That result belongs to the specified preferences; it should not be assumed for every good merely because the initial and final quantities are positive.

For an inferior good, the income component can be negative after a price reduction. If original x is four, compensated x six, and final x five, substitution contributes plus two while income contributes minus one, leaving a total increase of one. The good is inferior in the relevant response, but total demand still rises when its price falls. Inferiority alone does not imply Giffen behavior.

A sufficiently strong negative income component can outweigh the substitution component. With original four, compensated six and final three, the components are plus two and minus three, giving a total decline of one after the price reduction. This supplied pattern is Giffen-like own-price behavior under the stated model. The decomposition identifies the needed relationship; it does not establish that any real good has that pattern without evidence.

Another way: Reconcile the arithmetic without overclaiming causality

Calculate each component using the same order of subtraction throughout: comparison quantity minus starting quantity. If the price falls, do not force every result to be positive. A negative income component is meaningful. Report signed effects and check that substitution plus income equals the total. An answer that gives magnitudes only can conceal whether the components reinforce or offset each other.

Keep the good and unit fixed. A change of two units of x cannot be added to an income amount of ten dollars as if both were effects on quantity. Compensation is an income adjustment used to construct a bundle; the decomposition components are changes in the chosen quantity. The exercise can ask for both, but they belong in separately labeled fields.

If nominal income, the other good's price, preferences, or constraints also change, the simple three-bundle interpretation may need revision. A before-and-after comparison with several simultaneous changes is not automatically an identified own-price experiment. Specify which quantities come from a controlled model comparison and which are observed under changing conditions.

The final explanation should name the compensation convention, the three quantities, and the signed decomposition. It should also state what evidence would be needed to use the result as a claim about actual behavior. Consumer theory supplies a disciplined counterfactual structure, while empirical identification determines whether a proposed application credibly isolates the mechanism. Keeping those tasks separate avoids turning an exact accounting identity into an unsupported causal conclusion.

Another way: The local Slutsky equation uses derivatives rather than finite gaps

Under appropriate differentiability, ordinary own-price demand has derivative dx/dpx equal to the compensated own-price derivative minus x times the income derivative dx/dm. The compensated derivative holds the relevant preference level fixed, while ordinary demand holds nominal income fixed. For a normal good, dx/dm is positive, so the income term makes the ordinary response to a price increase more negative. For an inferior good, that term has the opposite sign and can partly or, in a Giffen case, more than offset substitution.

This equation is a local relationship. Its derivatives are rates of change, not the finite quantity differences entered in this lesson's worksheets. A large price change may pass through regions with different response rates or corner conditions, so multiplying one initial derivative by the whole price change need not give the exact final response. Keep the local analytical result separate from the finite decomposition whose intermediate bundle has been explicitly constructed.

5. Decomposing a fictional price reduction

A classroom model represents a consumer by U=xy, with original income thirty-two, px four and py two. The original bundle is (4,8). The price of x falls to two, while nominal income and py remain unchanged. The final bundle under the same utility model is (8,8). The total response of x is therefore plus four units.

To make a Slutsky-style finite decomposition, price the original bundle at the new prices. Four x now cost eight and eight y still cost sixteen, so compensated income is twenty-four. At this hypothetical income and the new prices, the consumer chooses (6,6). The compensated bundle is a modeled comparison, not a reported payment actually made to the consumer.

The substitution component is six minus four, or plus two x units. The income component is eight minus six, also plus two. Adding them gives the observed-model total of four. The twenty-four-unit compensation income is not itself one of these quantity effects; it is the budget used to construct the middle choice.

If a researcher instead wanted a Hicksian decomposition, the middle bundle would have to preserve the original utility level, not merely make (4,8) affordable. If actual income or tastes also changed during a real price reduction, further evidence would be needed before attributing the observed demand change to these two channels alone. The example demonstrates a transparent model decomposition and its limits, rather than claiming that every before-and-after price comparison identifies consumer responses.

6. Check the tempting inference

Name the compensation convention: preserving original-bundle affordability differs from preserving utility for finite price changes. Keep signed effects and quantity units; compensation income is not a quantity effect. Inferiority need not imply Giffen behavior. The decomposition identity is not by itself evidence that an observed demand change is causally identified.

7. A Slutsky income adjustment

8. Components reinforce for the supplied normal-good case

9. An opposing income component can dominate

10. Finish an offsetting decomposition

11. Guided practice

A finite Slutsky comparison after an own-price reduction gives original x4, compensated x6 and final x8. Other prices and preferences are fixed. Produce the signed substitution, income and total changes.

12. Guided practice

Original x3, compensated x5 and final x8 are supplied under a valid fixed compensation convention. Complete the signed decomposition.

13. Guided practice

A valid compensated comparison after a price fall gives original x5, compensated x9 and final x7. Report signed substitution, income and total quantity changes.

	Your result
Signed substitution component
Signed income component
Signed total quantity change