Compensating Variation vs Certainty Equivalent
Compensating Variation and Certainty Equivalent are two Microeconomic Theory concepts in AP Economics that students often mix up. Compensating variation is the amount of money that, after a price change, would return a consumer to exactly the utility they had before it. The certainty equivalent is the guaranteed amount of money that gives a person the same utility as a risky gamble. Here is how they compare side by side.
Compensating variation measures the welfare effect of a price change in dollars, using the new prices as the reference point. After a price rise, it is the payment a consumer would need to be as well off as before; after a price fall, it is the amount you could take away and still leave the consumer as well off as before. It answers a practical question: how much does this change actually cost this household in money terms? Equivalent variation is the mirror image, asking instead how much money at the OLD prices would do the same damage or good as the price change; because the two use different reference prices, they usually differ in size. Both are more precise than the change in consumer surplus, which only approximates them.
The certainty equivalent converts a gamble into a single sure dollar figure the decision maker would accept in its place. You find it by computing the gamble's expected utility, then asking what guaranteed wealth produces that same utility level, which means inverting the utility function. For a risk-averse person the certainty equivalent is below the gamble's expected value, and the difference is the risk premium. A risk-neutral person's certainty equivalent equals the expected value exactly, and a risk-loving person's sits above it. The measure is useful because it puts risky and safe options on one scale in dollars, which is how firms compare an uncertain project against a guaranteed contract.
Compensating Variation vs Certainty Equivalent: Two Money Yardsticks for Utility
| Dimension | Compensating Variation | Certainty Equivalent |
|---|---|---|
| Question it answers | How much income restores the old utility after a price change | How much guaranteed money matches the utility of a gamble |
| What is uncertain | Nothing, the setting is fully known | The outcome, which arrives with a probability attached |
| Preference model used | Ordinary preferences over bundles of goods | Expected utility over uncertain payoffs |
| How it is built | From the expenditure function at two price vectors | By inverting the utility function at expected utility |
| Its companion measure | Equivalent variation, computed at the old prices | The risk premium, expected value minus the certainty equivalent |
| What it is compared against | The change in consumer surplus under a demand curve | The expected value of the gamble in dollars |
| Where it is applied | Welfare effects of taxes, tariffs and price rises | Insurance pricing and portfolio choice |
One offsets a price change, the other offsets risk
Compensating variation prices a movement in prices. The certainty equivalent prices a spread of outcomes. Both come out as dollar amounts that leave a person exactly indifferent, which is why they get blurred together, but what is being neutralised differs. Take the certainty equivalent first, since the arithmetic is short. Someone whose utility equals the square root of wealth faces a coin flip paying $100 on heads and nothing on tails. Expected wealth is $50. Expected utility is half of the square root of 100 plus half of the square root of nothing, which is 5. The certainty equivalent is the sure amount whose square root is 5, so $25. This person would hand over the gamble for $25 in cash, and the $25 gap between the $50 expected value and the $25 certainty equivalent is the risk premium, the price of bearing the risk. No price of any good appeared anywhere in that calculation. Compensating variation poses a question with no probabilities in it at all. The price of a good has risen, the consumer is worse off, and CV is the extra income that would place them back on the original indifference curve while facing the new prices. See /glossary/certainty-equivalent for the risk side alone.
The formulas keep them apart
Written out, the two constructions have almost nothing in common. Compensating variation comes from the expenditure function: the cost of reaching the original utility level at the new prices, minus the cost of reaching that same level at the old prices. Put in a price rise and CV is positive, the compensation owed to the consumer. Put in a price fall and CV is negative, the sum you could take away and still leave them as well off as before, which is one way of reading willingness to pay for the fall. Its sibling, equivalent variation, runs the calculation at the original prices using the new utility level, and the two agree only when the good has no income effect. The change in consumer surplus read off an ordinary demand curve sits between them, which is why surplus is treated as an approximation rather than an exact welfare measure. The certainty equivalent instead inverts a utility function over a probability distribution. It falls below expected value for anyone risk averse, equals it for someone risk neutral, and exceeds it for a risk lover. Insurers live in that gap: a household pays a premium above the expected loss because a certain small loss beats the certainty equivalent of staying uninsured. One measure values a shifted budget line, the other values the removal of a spread.
Frequently asked questions
Is the certainty equivalent a kind of compensating variation?
No. Both are dollar amounts that leave a person indifferent, but they answer different questions. Compensating variation is the income adjustment that offsets a change in prices, while the certainty equivalent is the sure sum matching the expected utility of a risky outcome. One contains no probabilities and the other is defined by them.
How do you calculate a certainty equivalent?
Work out expected utility across the possible outcomes, then invert the utility function to find the sure wealth that delivers that same utility. With utility equal to the square root of wealth and an even chance of $100 or nothing, expected utility is 5, so the certainty equivalent is $25 and the risk premium is $25.
What is the difference between compensating variation and equivalent variation?
Compensating variation is the money needed at the new prices to restore the original utility level. Equivalent variation is the money change at the original prices that would produce the new utility level instead. The two give identical answers only when the good in question has no income effect.
Related comparisons
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