How to Calculate Compensating Variation
Compensating variation is the income needed at the new prices to reach the old utility level, minus actual income, so it measures a price change in dollars.
The Compensating Variation formula
Calculator
Enter income, the budget share of the good and the two prices to get compensating variation under Cobb-Douglas preferences.
What the consumer has to spend before any compensation.
The exponent on the good in a Cobb-Douglas utility function. It stays put whatever prices do.
Price of the good the consumer faced originally.
Price of the same good once it has moved. Set it below the old price for a price cut.
Handing the consumer $250 after the price change leaves them at exactly the utility they started with.
- Income needed at the new prices
- $1,250
- Price change
- 56.25%
- Compensation as a share of income
- 25%
- Cost of the old bundle at the new price
- $281.25
- What the sign means
- The consumer needs paying
Reaching the old utility once prices have moved costs $1,250, and compensating variation is that figure minus what the consumer actually has.
The good moved by 56.25%, and only the price of this one good enters the shortcut.
The payment is 25% of income, which is how much of this household's buying power the price change took.
Buying the original quantity at the new price would cost $281.25 more. That is an upper bound, and the gap against the answer above is what switching to other goods saves.
A price rise gives a positive compensating variation, money owed to the consumer. A price fall gives a negative one, money that could be taken away.
How to calculate Compensating Variation, step by step
- 1Fix the utility to be restored. Compensating variation holds the consumer at the utility they had before the price moved, so start from the bundle bought at the old prices.
- 2Cost that utility at the new prices. Find the smallest income that still reaches the old utility now that prices have changed. This is the expenditure function, and it lets the consumer rearrange the bundle rather than forcing the old one.
- 3Subtract actual income. The difference is the compensating variation. A price rise gives a positive answer, money owed to the consumer, and a price fall gives a negative one.
- 4Use the Cobb-Douglas shortcut where it applies. With utility x^α y^(1 − α) the share of income spent on each good never moves, and the whole calculation collapses to income × [(new price ÷ old price)^α − 1].
- 5Check it against the no-substitution bound. The old quantity times the price rise is an upper bound. An answer above it is wrong, because a consumer free to switch goods always needs less than that.
Worked example: Compensating Variation
A household has $1,000 of income and spends half of it on rent, so α = 0.5. Rent per unit rises from $16 to $25. CV = 1,000 × [(25 ÷ 16)^0.5 − 1] = 1,000 × [1.25 − 1] = $250, so it takes $1,250 at the new rent to be as well off as $1,000 was at the old rent.
Compensating Variation questions
What is the difference between compensating variation and equivalent variation?
They use different reference prices. Compensating variation asks how much money at the NEW prices restores the old utility, while equivalent variation asks how much money at the OLD prices would have done the same damage as the price change. The two agree only when the good carries no income effect.
Why is compensating variation smaller than the old quantity times the price rise?
Because the consumer does not have to keep buying the old bundle. In the worked example the household bought 0.5 × 1,000 ÷ 16 = 31.25 units, and repurchasing them at the higher price would cost 31.25 × $9 = $281.25. The exact answer is $250, and the $31.25 gap is what switching toward other goods saves.
Does the shortcut need the price of the other good?
No. Under Cobb-Douglas preferences the unchanged price cancels out of the expenditure ratio, so only the price that moved and the budget share of that good survive in the formula. That is why the calculator asks for one price pair rather than two.
Is compensating variation the same as the change in consumer surplus?
Not exactly. The change in consumer surplus is an approximation that sits between compensating variation and equivalent variation for a normal good, because it uses ordinary demand rather than holding utility fixed. The three coincide when preferences are quasilinear and the good has no income effect.
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