Consumer theory · Lesson 15 of 17
Compensating and equivalent variation
A government cuts the price of a good. By how much is she better off, in rupees? Lesson 9 offered the area under the demand curve, but with an income effect that number is only an approximation. Two exact answers measure the gain with the expenditure function: the compensating variation and the equivalent variation.
A question to hold on to
In general, for normal goods, the relationship between the compensating variation in income (CV), the equivalent variation in income (EV) and a change in consumer surplus (ΔCS) for a price increase is: CV > EV > ΔCS; EV > ΔCS > CV; EV > CV > ΔCS; or CV > ΔCS > EV.
Hyderabad 2012Solved at the end ↓
The problem
What is a price change worth?
Utility is only an ordering, so “she gains 7 utils” means nothing. But money is a common yardstick, so we ask a question with an answer in rupees. There are two natural versions, and they differ in which prices and which utility level they use as the reference.
Take a fall in the price of x from p⁰ to p¹ at fixed income m. Write u⁰ for her utility before and u¹ after. The expenditure function of Lesson 14 turns utilities into rupees: e(p, u) is what it costs, at prices p, to reach utility u.
The definitions
CV and EV
- Compensating variation (CV): after the change, how much income would we have to take away (for a fall) or add (for a rise) to put her back on her old utility? It is measured at the new prices. It is the most she would pay to have the price fall.
- Equivalent variation (EV): instead of the change, how much income would we have to give her (for a fall) or take away (for a rise) at the old prices so that she reaches her new utility? It is the least she would take to give up the price fall.
CV and EV, as changes in income
The convention of Varian and of the JNU papers: the change in income that does the job. After a price fall CV is negative and EV positive; after a rise it is the other way round. Some books flip the sign of CV to call it the gain.
Both use a Hicksian (utility-constant) comparison, so each is exact. Their difference is the reference point. CV asks what the change is worth at her old standard of living and EV at her new one. A third number is the change in consumer surplus (the area to the left of the ordinary demand curve between the two prices), which is not exact, because the ordinary curve mixes the compensation with the income effect.
Pause and predict
The price of x falls. After the fall she could give up Rs 30 of income and be exactly as well off as before. In the convention of this lesson, the CV is
Interactive
Try it: five measures of the same change
Income is Rs 100 and y has price Rs 1, so every rupee is a unit of y and each measure is a vertical distance on the left diagram. The dotted line is the new price, tangent to the old curve (that gap is the CV). The red line is the old price, tangent to the new curve (that gap is the EV). The bars on the right add the change in consumer surplus and the two Slutsky versions of the next chapter.
Try the three tastes. With u = xy, the three gains are different and in a fixed order. With quasi-linear tastes they are the same number.
A normal good: the income effect is real, so the three measures differ. Income Rs 100, price of y Rs 1 (so a rupee is a unit of y). The price of x goes from Rs 2 to Rs 1.
Dashed: old budget and old curveBlack: new budgetDotted: new prices, old curve (CV)Red: old prices, new curve (EV)
Gains are shown as positive; for a price rise they are losses and the bars show negative amounts.
The two Hicks measures
After the fall she could lose Rs 29.29 of income and still be as well off as before (CV, shown as a negative change in income). Instead of the fall, a cash gift of Rs 41.42 at the old prices would do as much (EV).
The order
For a price fall on a normal good the gains run EV (41.42) above consumer surplus (34.66) above CV (29.29).
The cheaper test
The Slutsky versions: the JNU Anand questions
The Hicks measures need the expenditure function. A simpler pair keeps a bundle affordable instead of a utility level, as in Lesson 12:
- Slutsky CV: the cost of her old bundle at the new prices, less her income: p¹·x⁰ − m.
- Slutsky EV: the cost of her new bundle at the old prices, less her income: p⁰·x¹ − m.
JNU 2015 built three questions on one setting. Anand has U = X²Y, pX = 1, pY = 2 and income 100. The price of Y falls to 1.
| Step | Working | Result |
|---|---|---|
| Before | Exponents 2 and 1 give shares ⅔ and ⅓ | X = 200/3, Y = 50/3 (cost 100) |
| After | Same shares, Y now costs 1 | X = 200/3, Y = 100/3 |
| Equally well off (Q28) | Same utility at the new prices: X = 2Y and 4Y³ = U⁰ | X = 2(500,000)1/3/3, Y = (500,000)1/3/3 |
| Slutsky CV (Q29) | Old bundle at new prices: 200/3 + 50/3 = 250/3; less 100 | −50/3 |
| Slutsky EV (Q30) | New bundle at old prices: 200/3 + 2 × 100/3 = 400/3; less 100 | +100/3 |
Question 28 asks for the bundle at the new prices that leaves her exactly as well off as before: that is the Hicksian bundle of Lesson 14, on the old curve. Its cost, about Rs 79.37, gives the exact Hicks CV: 79.37 − 100 = −20.63. The exact EV is +25.99, and the change in consumer surplus is 23.10.
Why the Slutsky versions differ from the Hicks ones
Keeping the old bundle affordable can cost more than reaching the old utility the cheapest way, since the old bundle is only one of the bundles on the old curve. So as changes in income each Slutsky measure is at least as large as its Hicks twin: after this price fall the Slutsky CV is −16.67 against the exact −20.63 (smaller in size), and the Slutsky EV is 33.33 against the exact 25.99. They are easy to compute but never exact.
The comparison
How CV, EV and consumer surplus compare
The two Hicks measures are areas to the left of two different compensated demand curves, one through the old point (utility u⁰, giving CV) and one through the new point (utility u¹, giving EV). The ordinary demand curve passes through both points. For a normal good the compensated curves are steeper than the ordinary one (Lesson 14), so the ordinary curve lies between them and the change in consumer surplus lies between CV and EV.
- Price fall, normal good: EV > ΔCS > |CV|. The gift she would accept instead of the fall is the largest, and what she would pay for it is the smallest.
- Price rise, normal good: |CV| > |ΔCS| > |EV|, measuring each as a loss. So CV > ΔCS > EV in sizes, the answer to the opening question (UOH 2012).
- Inferior good: the order reverses, because the compensated curves are flatter than the ordinary one.
- Quasi-linear tastes: no income effect, so the compensated and ordinary curves are the same and CV, EV and ΔCS are equal.
With u = xy, income 100, py = 1 and px falling from 2 to 1: she buys 25 and then 50 units of x, and u goes from 35.4 to 50 (in the form √(xy)). CV = 70.71 − 100 = −29.29, EV = 141.42 − 100 = 41.42, and the change in consumer surplus is ∫ 50/p dp from 1 to 2, which is 50 ln 2 = 34.66. 41.42 > 34.66 > 29.29.
Watch for
Traps
Watch out for
The mistakes examiners are fishing for
- Mixing the reference prices. CV uses the new prices, EV the old. A way to remember: Compensating is Compared at the Changed prices.
- Sign conventions. Read which sign the question uses. In this lesson, CV and EV are changes in income: CV is negative after a price fall, EV positive.
- Using consumer surplus as exact. It is exact only without income effects.
- Calling Slutsky’s measure the Hicks one. The Slutsky versions keep a bundle, the Hicks versions a utility level, and the first is a little larger in size for EV and smaller for CV after a price fall.
- Reversing the order for a price rise. It is CV > ΔCS > EV in sizes of loss, but the signs of CV and EV flip.
Worked questions
The opening question, and three more
Hyderabad 2012 · advanced
For normal goods and a price increase, how do CV, EV and ΔCS compare?
- Three areas.CV is the area under the Hicksian curve through the old point, EV the area under the one through the new point, ΔCS the area under the ordinary curve.
- Slopes.For a normal good the Hicksian curves are steeper than the ordinary curve. After a price rise she is poorer, so the old-utility curve (CV) lies to the right of the ordinary curve and the new-utility curve (EV) lies to its left.
- Order.So the loss measured by CV is the biggest and by EV the smallest.
Answer
CV > ΔCS > EV.
JNU 2015 Q29 · advanced
U = X²Y, prices (1, 2), income 100. The price of Y falls to 1. Find the Slutsky compensating variation.
- Old bundle.Shares ⅔ and ⅓ of 100: X = 200/3, Y = 50/3.
- At the new prices.It costs 200/3 + 1 × 50/3 = 250/3.
- Compare with income.250/3 − 100 = −50/3.
Answer
−50/3.
JNU 2015 Q30 · advanced
Same setting. Find the Slutsky equivalent variation.
- New bundle.At prices (1, 1): X = 200/3, Y = 100/3.
- At the old prices.It costs 200/3 + 2 × 100/3 = 400/3.
- Compare with income.400/3 − 100 = 100/3.
Answer
+100/3.
Quasi-linear · standard
u = √x + y, py = 1, income large. The price of x falls from 1 to 0.25. Find ΔCS, and say what CV and EV are.
- Demand.x = (1/2p)² = 1/(4p²).
- Area.∫ from 0.25 to 1 of 1/(4p²) dp = (1/4)(1/0.25 − 1/1) = (1/4)(4 − 1) = 0.75.
- Others.No income effect, so −CV = EV = ΔCS = 0.75.
Answer
0.75 for all three.
Revise
The lesson on one screen
The lesson on one screen
- CV
- e(p¹, u⁰) − m. Change in income, at the new prices, that restores the old utility.
- EV
- e(p⁰, u¹) − m. Change in income, at the old prices, that gives the new utility.
- Signs
- Price fall: CV < 0 < EV. Price rise: CV > 0 > EV. Sizes: what she would pay, what she would accept.
- Slutsky versions
- CV: p¹·x⁰ − m. EV: p⁰·x¹ − m. Keep a bundle, not a utility.
- Order, normal good
- Fall: EV > ΔCS > |CV|. Rise: |CV| > |ΔCS| > |EV|. Inferior good: reversed.
- Quasi-linear
- No income effect: CV, EV and ΔCS agree.
- Areas
- CV and EV are areas under Hicksian curves; ΔCS under the ordinary curve.
Practice
Check yourself
Quick questions
1After a price fall, the compensating variation (as a change in income at the new prices that restores the old utility) is
2The equivalent variation is the change in income
3With quasi-linear tastes (both goods bought), CV, EV and the change in consumer surplus are
4After a price fall of a normal good, the largest in size is
5The Slutsky equivalent variation is the cost of
Numerical problems
Compensating variation
u = xy, py = Rs 1, income Rs 100. The price of x falls from Rs 2 to Rs 1. Find the compensating variation (the change in income). Use e = 2√(pxpyu) for utility in the form xy.
Equivalent variation
Same consumer and price change. Find the equivalent variation (the change in income at the old prices that gives the new utility).
Quasi-linear
u = √x + y, py = Rs 1 and income is large. The price of x falls from Rs 1 to Rs 0.25. What is the change in consumer surplus? (It also equals EV and the size of CV.)
The Slutsky versions
Same first consumer (u = xy, py = 1, income 100, px from 2 to 1). Find the Slutsky CV and the Slutsky EV.
Real questions
Past papers, by level
The three JNU questions on Anand (28 to 30) share the setting of Chapter 4, so read that set-up first: questions 29 and 30 refer back to question 28.
Standard
Statements to judge, matching lists, and demands to compute with unusual tastes, up to JAM, GATE and JNU level.
4 questionsPractise →
Delhi School level
This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 17: Income effects, Slutsky splits and duality.
11 questionsLesson 17 →
All 4 past-paper questions for this lesson, easiest first: practise the full set →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
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