Difficulty: Intermediate-Advanced | Prerequisites: Consumer Theory Foundations (Doc 1) and Demand and Consumption Curves (Doc 2). You must be comfortable finding optimal bundles and deriving demand before tackling decomposition and welfare measures.
When a price changes, the consumer's optimal bundle shifts. This document teaches you to split that shift into two economically distinct pieces (the substitution effect and the income effect) and to put a dollar value on how the price change affects the consumer's wellbeing (compensating and equivalent variation). These are the most calculation-heavy topics on Exam 1 and appear in two of the four long problems.
A price change does two things at once: it changes the relative cost of goods (substitution effect) and it changes the consumer's purchasing power (income effect). Separating these requires a hypothetical "compensated" budget. Once you have the expenditure function, you can calculate CV (how much income to give or take after the change to restore the old utility) and EV (how much income to give or take before the change to reach the new utility). Both are tested as full exam problems.
Substitution effect
The change in quantity demanded due purely to the change in relative prices, holding utility constant. You isolate it by asking: "At the new prices, what bundle would the consumer choose if they had just enough income to reach their original utility?" The substitution effect always goes in the opposite direction to the price change (if p₁ rises, the substitution effect reduces x₁).
Income effect
The change in quantity demanded due to the change in purchasing power caused by the price change. After isolating the substitution effect, the income effect is whatever is left. For a normal good, the income effect reinforces the substitution effect. For an inferior good, it works against it.
Indirect utility function V(p₁, p₂, m)
The maximum utility the consumer can achieve given prices and income. You obtain it by substituting the demand functions back into the utility function. It is "indirect" because utility is expressed as a function of prices and income rather than quantities.
Expenditure function E(p₁, p₂, u̅)
The minimum income required to achieve utility level u̅ at given prices. It is the inverse of the indirect utility function (solve V = u̅ for m). This is the key tool for computing CV and EV.
Compensating variation (CV)
The amount of income that must be taken away from (or given to) the consumer after a price change to return them to their original utility level. CV = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) − m. If a price falls, the consumer is better off, and CV is the income you could take away to leave them just as well off as before.
Equivalent variation (EV)
The amount of income that must be given to (or taken from) the consumer at the original prices to give them the same utility as the price change would. EV = m − E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ). It answers: "How much cash would be equivalent to this price change?"
Compensated budget constraint
The budget line at the new prices but with income adjusted so the consumer can just reach the original utility level. Its equation is p₁ⁿᵉʷx₁ + p₂x₂ = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ). On a graph, it has the slope of the new prices but is tangent to the original indifference curve.
Equivalent budget constraint
The budget line at the original prices but with income adjusted so the consumer can just reach the new utility level. Its equation is p₁ᵒˡᵈx₁ + p₂x₂ = E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ). On a graph, it has the slope of the old prices but is tangent to the new indifference curve.
Suppose p₁ falls from p₁ᵒˡᵈ to p₁ⁿᵉʷ, with p₂ and m unchanged.
Step 1: find the original optimum. Solve for (x₁*, x₂*) at the old prices. Call the utility at this bundle u̅ᵒˡᵈ.
Step 2: find the compensated bundle. Using the new prices (p₁ⁿᵉʷ, p₂), find the cheapest bundle that achieves u̅ᵒˡᵈ. This is the bundle on the original indifference curve where MRS = p₁ⁿᵉʷ/p₂. Call it (x₁ᶜ, x₂ᶜ).
The income needed for this bundle is E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) = p₁ⁿᵉʷx₁ᶜ + p₂x₂ᶜ.
Step 3: find the new optimum. Solve for (x₁**, x₂**) at the new prices with the original income m. Call the utility at this bundle u̅ⁿᵉʷ.
Decomposition:
Substitution effect = x₁ᶜ − x₁* (the move along the original indifference curve from the old tangent point to the compensated tangent point).
Income effect = x₁** − x₁ᶜ (the move from the compensated bundle to the new optimum, driven by the change in real purchasing power).
Total effect = x₁** − x₁* = substitution effect + income effect.
For a normal good when p₁ falls: the substitution effect is positive (buy more x₁) and the income effect is also positive (higher real income, buy more of a normal good). For an inferior good: the substitution effect is still positive but the income effect is negative.
Indirect utility function: substitute the demand functions into the utility function.
For Cobb-Douglas U = x₁^a · x₂^b with a + b = 1:
Demands: x₁* = am/p₁, x₂* = bm/p₂.
V(p₁, p₂, m) = (am/p₁)^a · (bm/p₂)^b = aᵃbᵇ · m/(p₁ᵃp₂ᵇ).
Expenditure function: invert the indirect utility function. Set V = u̅ and solve for m.
E(p₁, p₂, u̅) = u̅ · p₁ᵃp₂ᵇ / (aᵃbᵇ).
The expenditure function tells you the minimum cost of reaching utility u̅ at prices (p₁, p₂). It is increasing in prices and in u̅, and it is homogeneous of degree 1 in prices (double all prices and the expenditure doubles).
CV answers the question: after the price change has happened, how much income would we need to take from (or give to) the consumer to put them back on their original indifference curve?
CV = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) − m
If p₁ falls (consumer is better off), CV is negative: you could take income away and they would still be no worse off than before. The magnitude of CV is the dollar value of the gain, measured at the new prices.
The compensated budget constraint has:
Slope = −p₁ⁿᵉʷ/p₂ (the new price ratio)
Income = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) (just enough to reach the old utility at the new prices)
On the graph, it is tangent to the original indifference curve, with the slope of the new budget line
The compensated bundle is the tangent point of this line with the original indifference curve.
EV answers the question: at the original prices, how much income would we need to give the consumer to make them as well off as the price change would?
EV = m − E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ)
If p₁ falls (consumer is better off), EV is positive: it is the cash equivalent of the price drop. Alternatively, think of it as the maximum the consumer would pay to secure the price reduction.
The equivalent budget constraint has:
Slope = −p₁ᵒˡᵈ/p₂ (the original price ratio)
Income = E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ) (enough to reach the new utility at the old prices)
On the graph, it is tangent to the new indifference curve, with the slope of the old budget line
For a price decrease in a normal good: CV < EV (the consumer needs less income at the cheaper new prices to reach the old utility than they would need at the expensive old prices to reach the new utility).
The exam asks you to draw four budget lines on the same diagram. Here is how they relate when p₁ falls (holding p₂ and m fixed):
Original budget constraint. Slope = −p₁ᵒˡᵈ/p₂. Income = m. Tangent to the original indifference curve at the original optimum.
New budget constraint. Slope = −p₁ⁿᵉʷ/p₂ (flatter, because p₁ fell). Income = m. Same vertical intercept as the original line; the horizontal intercept moves right. Tangent to the new (higher) indifference curve at the new optimum.
Compensated budget constraint. Slope = −p₁ⁿᵉʷ/p₂ (same as new). Income = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) < m. Parallel to the new budget line but shifted inward. Tangent to the original indifference curve. The tangent point is the compensated bundle.
Equivalent budget constraint. Slope = −p₁ᵒˡᵈ/p₂ (same as original). Income = E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ) > m. Parallel to the original budget line but shifted outward. Tangent to the new indifference curve.
Tips for drawing:
Lines 1 and 4 are parallel (same slope, different intercepts).
Lines 2 and 3 are parallel (same slope, different intercepts).
The compensated constraint sits between the original and new constraints (for a price fall).
Label every line, every tangent bundle, and both indifference curves.
Budget constraint: p₁x₁ + p₂x₂ = m
Cobb-Douglas indirect utility (a + b = 1): V = aᵃbᵇ · m / (p₁ᵃp₂ᵇ)
Cobb-Douglas expenditure function: E = u̅ · p₁ᵃp₂ᵇ / (aᵃbᵇ)
CV: E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) − m
EV: m − E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ)
Substitution effect on x₁: x₁ᶜ − x₁* (compensated bundle minus original bundle)
Income effect on x₁: x₁** − x₁ᶜ (new bundle minus compensated bundle)
Slutsky equation (conceptual): total effect = substitution effect + income effect
Compensating variation is the logic behind cost-of-living adjustments. If prices rise, how much extra income does a worker need to maintain their previous standard of living? That is exactly what CV measures.
Equivalent variation is the logic behind willingness-to-pay calculations. When a government considers building a public park, EV measures how much each resident would pay in cash to get the same welfare boost the park provides.
"CV and EV are the same number." They are equal only when there is no income effect (quasilinear utility). For normal goods, CV < EV for a price decrease and CV > EV for a price increase.
"The compensated budget line is tangent to the new indifference curve." It is tangent to the old indifference curve (that is the whole point: it restores the old utility at the new prices). The equivalent budget line is the one tangent to the new indifference curve.
"The substitution effect can go in the same direction as the price change." It cannot. The substitution effect always moves the consumer away from the good that became relatively more expensive. This holds regardless of whether the good is normal, inferior, or Giffen.
"Income and substitution effects are observed separately in the real world." They are not. The decomposition is a conceptual exercise. In the real world, we observe only the total effect. The compensated bundle is hypothetical.
⚠️ One long problem asks you to solve for income and substitution effects and draw the graph. Practise the full three-step decomposition with a Cobb-Douglas utility function until it is automatic.
⚠️ Another long problem asks you to calculate CV and EV. You need the expenditure function. Derive it from the indirect utility function, then plug in the right prices and utility levels.
⚠️ Be prepared to draw all four budget constraints on the same graph: original, new, compensated, and equivalent. Label each one clearly and mark the tangent points.
⚠️ Short-answer questions may ask you to explain the economic interpretation of CV or EV, or to state which is larger for a price decrease of a normal good.
True or false: the substitution effect of a price increase for good 1 always reduces x₁.
Fill in the blank: CV = E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) − ____.
True or false: for a normal good, CV = EV after a price change.
Fill in the blank: the compensated budget constraint has the slope of the ____ prices and is tangent to the ____ indifference curve.
True or false: the income effect for an inferior good works in the opposite direction to the substitution effect.
Answers: 1. True. 2. m (original income). 3. False (CV < EV for a price fall; CV > EV for a price rise). 4. New; old. 5. True.
Q: A consumer has U = x₁^(1/2) · x₂^(1/2), m = 200, p₁ = 4, p₂ = 4. The price of good 1 falls to p₁ = 1. Find the substitution and income effects on x₁.
A: Original optimum: x₁* = (1/2)(200/4) = 25, x₂* = (1/2)(200/4) = 25. Original utility: u̅ᵒˡᵈ = 25^(1/2) · 25^(1/2) = 25.
Compensated bundle at new prices: MRS = x₂/x₁ = p₁ⁿᵉʷ/p₂ = 1/4, so x₂ = x₁/4. Stay on u̅ = 25: x₁^(1/2)(x₁/4)^(1/2) = 25, so x₁/(2) = 25, x₁ᶜ = 50, x₂ᶜ = 12.5.
New optimum: x₁** = (1/2)(200/1) = 100, x₂** = (1/2)(200/4) = 25.
Substitution effect: 50 − 25 = 25. Income effect: 100 − 50 = 50. Total effect: 100 − 25 = 75.
Q: Using the same setup, calculate CV and EV.
A: Expenditure function: E = u̅ · p₁^(1/2) · p₂^(1/2) / (0.5^(1/2) · 0.5^(1/2)) = u̅ · 2p₁^(1/2) · p₂^(1/2) (since a = b = 1/2, aᵃbᵇ = 0.5).
CV = E(1, 4, 25) − 200 = 25 · 2 · 1 · 2 − 200 = 100 − 200 = −100. The consumer gains the equivalent of 100 in income.
New utility: u̅ⁿᵉʷ = 100^(1/2) · 25^(1/2) = 10 · 5 = 50.
EV = 200 − E(4, 4, 50) = 200 − 50 · 2 · 2 · 2 = 200 − 400 = −200. Wait, let me recalculate. E(4, 4, 50) = 50 · 2 · 4^(1/2) · 4^(1/2) = 50 · 2 · 2 · 2 = 400. EV = 200 − 400 = −200. This is negative, which would mean the consumer is worse off, but they should be better off.
Let me correct: EV = m − E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ) = 200 − 400 = −200 is wrong because I have the formula direction inverted for this case.
Actually: for a price fall, the consumer is better off (u̅ⁿᵉʷ > u̅ᵒˡᵈ). EV = E(p₁ᵒˡᵈ, p₂, u̅ⁿᵉʷ) − m = 400 − 200 = 200. The consumer would need 200 extra at the old prices to be as well off as the price drop makes them. CV = m − E(p₁ⁿᵉʷ, p₂, u̅ᵒˡᵈ) = 200 − 100 = 100.
Note that CV (100) < EV (200) for this price decrease with a normal good, as expected.
Q: Draw and label the compensated budget constraint for the scenario above.
A: The compensated budget line has slope −p₁ⁿᵉʷ/p₂ = −1/4. Its income is E(1, 4, 25) = 100. Intercepts: vertical = 100/4 = 25; horizontal = 100/1 = 100. It is tangent to the original indifference curve (u = 25) at the compensated bundle (50, 12.5). It lies between the original and new budget lines.
The substitution and income effects here build directly on the demand curves from Doc 2. Moving along the demand curve in response to a price change is the total effect; the decomposition separates the underlying forces. Later in ECON 302, the same logic extends to labour supply (where the good is leisure and the price is the wage) and to intertemporal choice (where the goods are consumption today and consumption tomorrow). Compensating and equivalent variation also connect to consumer surplus, which you will study when the course turns to welfare economics and market efficiency.
substitution effect, income effect, Slutsky decomposition, Hicksian decomposition, compensating variation, CV, equivalent variation, EV, indirect utility function, expenditure function, cost function, compensated demand, Hicksian demand, Marshallian demand, compensated budget constraint, equivalent budget constraint, welfare measurement, consumer surplus, cost-of-living index, normal good, inferior good, Giffen good, quasilinear utility