Difficulty: Intermediate | Prerequisites: Budget constraints, indifference curves, tangency conditions.
When a price rises, a consumer is hurt. But by how much, exactly? Compensating variation answers that: it is the minimum lump-sum payment that would make the consumer willing to accept the price increase. This topic bridges budget-constraint mechanics and the deeper decomposition of a price change into a substitution effect (the consumer switches away from the now-dearer good) and an income effect (the consumer feels poorer overall). You need to be comfortable drawing budget lines and indifference curves before tackling this, because the entire argument is built on the geometry of those two tools.
Compensating variation is the cash payment that keeps a consumer on their original indifference curve after a price increase. Even with full compensation, the consumer still reduces their consumption of the more expensive good (the substitution effect is always present). The revenue the seller collects from the price increase will exceed the compensation paid, because the consumer drives less after the price hike.
Compensating variation (CV)
The minimum lump-sum transfer that makes a consumer willing to accept a price increase, restoring them to their original utility level. Think of it as the bribe that leaves them exactly as happy as before.
Equivalent variation (EV)
The maximum lump-sum amount a consumer would pay to avoid a price increase. It is measured at the old prices rather than the new ones. Think of it as the opposite direction: how much would you hand over to prevent the change?
Substitution effect
The change in consumption of a good due solely to the change in relative prices, holding utility constant (i.e. sliding along the same indifference curve). In simple terms, goods that get relatively more expensive always see their quantity fall, regardless of income.
Income effect
The change in consumption caused by the change in purchasing power after a price change, holding relative prices at the new level. Think of it as the "you feel richer or poorer" part of a price change.
Conventional (well-behaved) preferences
Preferences that are complete, transitive, continuous, and convex (diminishing marginal rate of substitution). This guarantees smooth, bowed-in indifference curves.
Hicksian (compensated) demand
The demand curve that holds utility constant rather than income constant. It isolates the substitution effect. In simple terms, it answers: if we kept you equally happy, how much less would you buy when the price goes up?
Harry currently pays 20 cents/mile to use the family car. His parents want to raise the fee to 40 cents/mile.
Rather than just imposing the increase, they ask: "What is the minimum weekly allowance increase you would accept in exchange for the higher fee?"
Harry (truthfully) answers $10/week. This $10 is his compensating variation for the price increase from 20 to 40 cents/mile.
Yes, he will drive less. Here is why.
The $10 compensation is defined as the amount that puts Harry back on his original indifference curve at the new, higher price.
At the original price (20 cents/mile), Harry chose an optimal bundle where his indifference curve was tangent to his old budget line.
After the price rise plus $10 compensation, Harry is on the same indifference curve but facing a steeper budget line (driving is now relatively more expensive).
The tangency point on this compensated budget line must involve fewer miles driven, because the indifference curve is convex (conventional preferences) and the slope of the budget line is now steeper.
This reduction in miles is the pure substitution effect. It is always negative for a normal good whose price has risen. There is no income effect here because the compensation has exactly neutralised it.
The additional revenue from the mileage charge equals the price increase (20 cents/mile) times the number of miles Harry drives after the change.
Because Harry drives fewer miles after compensation (let us call the new mileage M1, which is less than his original mileage M0), the revenue is 0.20 × M1.
The question is whether 0.20 × M1 is greater than, less than, or equal to $10.
Key insight: $10 is the amount that would make Harry indifferent at the new price. If we gave him exactly $10 more and charged 40 cents/mile, he would land on his original indifference curve but at a different point (fewer miles, more of other goods).
The compensated budget line (original utility, new prices) is tangent to the indifference curve at a point with fewer miles. The old budget line was also tangent to that curve, but at the old bundle. Because the indifference curve is strictly convex, the compensated budget line lies everywhere below the old budget line except at the single point of tangency.
The old bundle (which costs exactly the old income at old prices) lies above the compensated budget line. This means the old bundle is more expensive than the compensated income at new prices. Put differently, the income needed to reach the old utility at new prices (old income + CV) is less than what it would cost to buy the old bundle at new prices.
Old bundle at new prices would cost: original income + (price increase × old miles) = original income + 0.20 × M0. The CV is $10. So 10 < 0.20 × M0 in general, but what matters is the revenue after the change.
More directly: at the compensated point, Harry's spending on miles is 0.40 × M1. His old spending on miles was 0.20 × M0. The additional revenue is 0.40 × M1 – 0.20 × M0. But the compensation was chosen so that total expenditure rises by exactly $10. Since he substitutes away from driving, the share going to miles rises by less than $10, and the share going to other goods absorbs some of the compensation. The additional mileage revenue is therefore less than $10.
Wait, let us be more careful. The additional mileage charge revenue is (0.40 – 0.20) × M1 = 0.20 × M1 (the extra 20 cents on each mile he still drives). Is this more or less than $10?
At the old optimum, Harry drove M0 miles. The cost of driving those M0 miles at the new price is 0.40 × M0 = 0.20 × M0 + 0.20 × M0. The additional charge on the old mileage would be 0.20 × M0.
But Harry drives M1 < M0 miles after the compensated price change, so the additional revenue is 0.20 × M1 < 0.20 × M0.
The $10 compensating variation is less than 0.20 × M0 (the revenue if Harry had not changed his driving). This is because $10 only needs to restore utility, not purchasing power. The consumer adjusts and substitutes, so the cost of restoring utility is less than the cost of buying the old bundle at new prices.
Therefore, the additional revenue (0.20 × M1) is less than 0.20 × M0, and $10 < 0.20 × M0. But we need to compare 0.20 × M1 to $10.
The correct conclusion: the additional revenue is less than $10/wk. The parents come out behind on net. Here is the intuition: they gave Harry $10, but because he drives fewer miles at the higher price, the extra 20 cents/mile yields less than $10 in revenue. The substitution effect means they cannot fully recoup the compensation through the surcharge.
Actually, let me reconsider using the graphical logic more carefully:
The compensating variation is defined so that (income + CV) at new prices puts Harry on his original indifference curve U0.
At the compensated bundle, Harry consumes fewer miles (M1 < M0) and more of other goods.
The parents' net position: they pay $10 more in allowance and receive an additional 0.20 × M1 in mileage fees.
Because the compensated budget line is tangent to U0 at a point to the left of the original tangency (fewer miles), and because $10 is the minimum compensation (i.e. it exactly funds the tangency), the additional mileage revenue 0.20 × M1 must be less than $10.
Think of it this way: if Harry drove the same number of miles, the extra revenue would be 0.20 × M0, and Harry would need more than $10 to be compensated (because he would not be substituting at all, so the full cost increase hits him). The fact that he substitutes away from driving means he needs less compensation ($10 < 0.20 × M0), but it also means the parents collect less revenue (0.20 × M1 < 0.20 × M0 < what the full cost would be). The net result: revenue < $10.
Corrected conclusion for Part (b): The additional revenue from the mileage charge is less than $10/wk. The parents pay out more in extra allowance than they recoup from the higher mileage fee, because Harry substitutes away from driving.
Compensating variation definition: CV = minimum ΔM such that U(new prices, M + ΔM) = U(old prices, M)
Decomposition of a price change: Total effect = Substitution effect + Income effect
Graphical sketch for the Tom and Karen problem:
Draw axes: miles driven (horizontal) vs. all other goods in dollars (vertical).
Draw the original budget line with slope = –0.20 (price per mile). Mark Harry's optimum at tangency with indifference curve U0.
Draw the new budget line with slope = –0.40 (steeper). Without compensation, Harry would be on a lower indifference curve.
Shift the new budget line outward by $10 (parallel shift upward by $10 on the vertical axis). This compensated budget line is tangent to the original indifference curve U0, but at a point to the left (fewer miles).
The horizontal distance between the old optimum and the compensated optimum is the substitution effect: always negative (fewer miles).
The difference between the compensated optimum and the final optimum (if you were to remove the $10) is the income effect.
Compensating variation is the standard tool for cost-benefit analysis in public policy. When a government raises petrol taxes, analysts estimate how much a lump-sum rebate would be needed to leave households no worse off. Insurance pricing, environmental regulation, and congestion charging all use this framework. The Tom-and-Karen problem is a domestic version of the same logic: a Pigouvian-style fee paired with a transfer.
Students often think that if you fully compensate someone for a price increase, they will not change their behaviour. They will. The substitution effect always operates, even under full compensation.
A frequent error is confusing compensating variation with equivalent variation. CV asks: how much cash after the change? EV asks: how much cash instead of the change? They are measured at different price levels.
Students sometimes assume the additional revenue from the price hike equals the compensation. It does not, precisely because the consumer substitutes away from the more expensive good.
Some students draw the compensated budget line parallel to the old budget line. It should be parallel to the new budget line (same new price ratio) but shifted so it is tangent to the original indifference curve.
⚠️ The two-part question structure ("will he drive less?" then "how does revenue compare to compensation?") is a classic exam pattern. Both parts require the substitution-effect argument.
⚠️ You must show the graphical decomposition: old budget line, new budget line, compensated budget line, indifference curve, and the shift in tangency point.
⚠️ Be prepared to explain in words why the revenue from the price increase is less than the compensation paid. The examiner is testing whether you understand the substitution effect, not just whether you can draw it.
True or false: The substitution effect of a price increase for a normal good always reduces consumption of that good. (True.)
Fill in the blank: Compensating variation restores the consumer to their original ______ . (Indifference curve / utility level.)
True or false: If Harry is fully compensated for the price increase, he will drive exactly the same number of miles. (False; the substitution effect still reduces miles.)
Fill in the blank: The compensated budget line has the same slope as the ______ budget line. (New.)
True or false: The additional mileage revenue will exceed the $10 compensation. (False; it will be less than $10.)
Q: Tom and Karen raise Harry's mileage fee from 20 to 40 cents/mile and increase his allowance by $10/wk (his stated compensating variation). Will Harry drive more, less, or the same number of miles?
A: Less. The $10 compensation places Harry back on his original indifference curve, but at the new (steeper) price ratio. The tangency point on the compensated budget line lies to the left of the original optimum. This leftward shift is the substitution effect, which is always negative for a good whose relative price has risen.
Q: Will the additional revenue from the higher mileage charge be more than, less than, or equal to $10/wk?
A: Less than $10/wk. Harry drives fewer miles at the higher price, so the extra 20 cents/mile is collected on a smaller base of miles. The parents pay out $10 in extra allowance but recoup less than $10 from the surcharge, resulting in a net cost to them.
Q: How would the answer to part (b) change if Harry had perfect-complement (L-shaped) preferences between driving and all other goods?
A: With perfect complements, there is no substitution effect; Harry would drive exactly the same number of miles regardless of the price change (the kink in the indifference curve means the tangency does not slide). The additional revenue would then be exactly 0.20 × M0, and the CV would also equal 0.20 × M0, so revenue would equal the compensation. This is the one edge case where the inequality collapses to equality.
This material builds directly on budget constraints (Doc 1) and is the foundation for deriving Hicksian (compensated) demand curves. It also connects to Slutsky decomposition (an alternative way of separating substitution and income effects using purchasing power rather than utility). In welfare economics, CV and EV are the standard measures of consumer surplus changes from policy interventions.
Compensating variation, equivalent variation, substitution effect, income effect, Hicksian demand, compensated demand, Slutsky decomposition, indifference curve tangency, price increase compensation, welfare analysis, consumer surplus, Pigouvian tax, mileage fee, Tom and Karen problem, NYU microeconomics