Two Ways to Compensate for a Price Change
When a price changes, a consumer's chosen bundle can change for two reasons: the relative attractiveness of goods has shifted, and the consumer's purchasing power has changed. A compensated substitution effect isolates the first part by adjusting income. There are two standard ways to make that adjustment. Hicksian compensation holds the consumer at the original utility level; Slutsky compensation makes the original bundle just affordable at the new prices.
The distinction matters because those two income adjustments generally differ for a finite price change. Consequently, the two compensated bundles—and the substitution effects measured from the original bundle—can have different magnitudes. Under smooth preferences, however, the two compensation rules agree to first order for a sufficiently small price change. This comparison is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
What the Two Rules Hold Fixed
Hicksian compensation asks: How much income would achieve the original utility at the new prices? It does not insist that the consumer can still buy the exact original bundle. Slutsky compensation asks: How much income makes that original bundle affordable at the new prices? It does not insist that the consumer's best affordable bundle has exactly the original utility.
For a price increase, the original bundle costs more at the new prices, so Slutsky income rises by the extra cost of that bundle. Hicksian income rises by the amount needed to reach the original utility with the consumer's new, price-adjusted choice. Those amounts need not match. For a price decrease, the same definitions apply: the cost of the original bundle may fall, and Hicksian income is the minimum expenditure needed to reach the original utility.
| Compensation rule | What is held fixed? | Income at new prices |
|---|---|---|
| Hicksian | Original utility | Minimum expenditure needed to attain that utility |
| Slutsky | Affordability of the original bundle | Cost of the original bundle at new prices |
The calculations below use Cobb-Douglas preferences. For \(U(x_1,x_2)=x_1^\alpha x_2^{1-\alpha}\), where \(0<\alpha<1\), prices are positive, and the optimum is interior, ordinary demand is
These demands allocate the income shares \(\alpha\) and \(1-\alpha\) to the two goods. For the equal-share case \(U(x_1,x_2)=\sqrt{x_1x_2}\), Hicksian demand at prices \(p_1,p_2\) and target utility \(u\) is
These formulas assume the consumer chooses positive quantities, as is appropriate for Cobb-Douglas preferences with positive prices and income. They should not be applied without checking the preference model and its conditions.
Worked Comparisons
Worked Example: A Price Increase Under Equal-Share Preferences
A consumer has utility \(U(x_1,x_2)=\sqrt{x_1x_2}\), initial prices \((p_1^0,p_2^0)=(1,1)\), and income \(m=80\). The price of good 1 rises to 2, while the price of good 2 remains 1. Calculate the Hicksian and Slutsky substitution effects for each good, and check each decomposition against the actual demand change.
Method and initial choice. Equal-share Cobb-Douglas demand spends half of income on each good. Therefore \(x_1^0=80/(2\cdot1)=40\) and \(x_2^0=80/(2\cdot1)=40\). Initial utility is \(u^0=\sqrt{40\cdot40}=40\).
Actual new choice. At prices \((2,1)\) and income 80, demand is \(x_1^1=80/(2\cdot2)=20\) and \(x_2^1=80/(2\cdot1)=40\). The total changes from the original bundle are therefore \(-20\) units of good 1 and \(0\) units of good 2.
Hicksian compensation. Hold utility at 40 and use the Hicksian formulas at prices \((2,1)\):
The Hicksian substitution effects are \(28.2843-40=-11.7157\) for good 1 and \(56.5685-40=16.5685\) for good 2. The income needed to attain the target utility is \(2(20\sqrt{2})+40\sqrt{2}=80\sqrt{2}\approx113.1371\).
Slutsky compensation. The original bundle costs \(2(40)+1(40)=120\) at the new prices. With income 120, new-price demands are \(x_1^S=120/(2\cdot2)=30\) and \(x_2^S=120/(2\cdot1)=60\). The Slutsky effects are \(30-40=-10\) for good 1 and \(60-40=20\) for good 2.
Check. Under Hicksian compensation, the remaining income effects are \(20-28.2843=-8.2843\) and \(40-56.5685=-16.5685\). Adding them to the Hicksian substitution effects gives \(-20\) and \(0\), the actual total changes. Under Slutsky compensation, the income effects are \(20-30=-10\) and \(40-60=-20\); adding the Slutsky effects again gives \(-20\) and \(0\). The Hicksian bundle preserves utility because \(\sqrt{(20\sqrt{2})(40\sqrt{2})}=40\). The methods avoid confusing the Hicksian utility target with the Slutsky affordability target.
Worked Example: Unequal Cobb-Douglas Shares
Suppose \(U(x_1,x_2)=x_1^{1/3}x_2^{2/3}\), initial prices are \((1,2)\), and income is 120. Good 1's price rises from 1 to 4, while good 2's price stays at 2. Find the two compensated substitution effects for good 1 and compare their magnitudes.
Method and initial choice. The income shares are one-third and two-thirds. Initial demands are \(x_1^0=(1/3)(120)/1=40\) and \(x_2^0=(2/3)(120)/2=40\). Initial utility is \(40^{1/3}40^{2/3}=40\).
Actual new choice. At prices \((4,2)\) and income 120, demands are \(x_1^1=(1/3)(120)/4=10\) and \(x_2^1=(2/3)(120)/2=40\). Good 1's total change is \(10-40=-30\).
Hicksian bundle. At the new prices, the first-order condition for this Cobb-Douglas utility gives \(x_1/x_2=(\alpha/(1-\alpha))(p_2/p_1)=(1/2)(2/4)=1/4\). Write \(x_2=4x_1\). Requiring utility 40 gives
Thus \(h_1\approx15.8740\), and the Hicksian substitution effect for good 1 is \(10\sqrt[3]{4}-40\approx-24.1260\). The cost of this bundle is \(4(10\sqrt[3]{4})+2(40\sqrt[3]{4})=120\sqrt[3]{4}\approx190.4881\). The utility check is \(h_1^{1/3}h_2^{2/3}=h_1 4^{2/3}=40\).
Slutsky bundle. The original bundle costs \(4(40)+2(40)=240\) at the new prices. At that income, demands are \(x_1^S=(1/3)(240)/4=20\) and \(x_2^S=(2/3)(240)/2=80\). Good 1's Slutsky effect is \(20-40=-20\).
Check. The Slutsky bundle has utility \(20^{1/3}80^{2/3}=20\cdot4^{2/3}\), which is about 50.397—above the original utility of 40. That is consistent with Slutsky compensation preserving affordability, not utility. The Slutsky income effect for good 1 is \(10-20=-10\), and \(-20+(-10)=-30\), the actual total change. The Hicksian effect is also consistent with the total: its income effect is \(10-15.8740=-5.8740\), and \(-24.1260+(-5.8740)=-30\). The two substitution effects have the same direction here but different magnitudes.
Worked Example: A Price Decrease
A consumer with \(U(x_1,x_2)=\sqrt{x_1x_2}\) faces initial prices \((1,2)\) and income 120. The price of good 1 falls to \(0.5\), with the price of good 2 unchanged. Calculate both compensated effects for good 1 and verify the utility and affordability checks.
Initial and actual choices. Initial demands are \(x_1^0=120/(2\cdot1)=60\) and \(x_2^0=120/(2\cdot2)=30\). Initial utility is \(\sqrt{60\cdot30}=30\sqrt{2}\). At new prices \((0.5,2)\) and the original income, actual demands are \(x_1^1=120/(2\cdot0.5)=120\) and \(x_2^1=120/(2\cdot2)=30\). Thus good 1's total change is \(120-60=60\).
Hicksian effect. At the new prices, the Hicksian bundle that preserves utility \(30\sqrt{2}\) is
The good 1 Hicksian effect is \(60\sqrt{2}-60\approx24.8528\). The utility check is \(\sqrt{(60\sqrt{2})(15\sqrt{2})}=\sqrt{1800}=30\sqrt{2}\). The Hicksian bundle costs \(0.5(60\sqrt{2})+2(15\sqrt{2})=60\sqrt{2}\).
Slutsky effect. At the new prices, the original bundle costs \(0.5(60)+2(30)=90\). With income 90, the new-price demands are \(x_1^S=90/(2\cdot0.5)=90\) and \(x_2^S=90/(2\cdot2)=22.5\). The good 1 Slutsky effect is \(90-60=30\). The bundle is affordable because \(0.5(90)+2(22.5)=90\). Its utility is \(\sqrt{90\cdot22.5}=45\), which is not the original utility.
Check. The Hicksian and Slutsky effects are both positive for good 1, but are \(24.8528\) and \(30\) units. Their difference does not contradict either compensation rule: one preserves utility, while the other preserves affordability of the original bundle. The actual good 1 change also checks: the Hicksian income effect is \(120-84.8528=35.1472\), giving \(24.8528+35.1472=60\); the Slutsky income effect is \(120-90=30\), giving \(30+30=60\).
Why the Magnitudes Differ but the Local Effects Agree
The examples show that the compensation incomes can be different even when both methods describe the response to the same price movement. For example, in the first example, Hicksian income is \(80\sqrt{2}\approx113.1371\), while Slutsky income is 120. The Hicksian income is the minimum expenditure that achieves utility 40. Slutsky income is the cost of the original \((40,40)\) bundle. At the new prices, that original bundle costs more than the Hicksian bundle because the consumer can substitute toward the relatively cheaper good 2.
The local agreement can be justified by comparing the two income adjustments near the initial prices. Let \(e(p,u^0)\) be the minimum expenditure needed to attain utility \(u^0\) at prices \(p\). At initial prices, \(e(p^0,u^0)=m\). For a small price change \(\Delta p\), Slutsky compensation changes income by \(\Delta p\cdot x^0\). Hicksian compensation changes minimum expenditure by approximately the same amount: at the initial prices, a small change in one price changes the cost of the utility-minimizing bundle by that price change multiplied by the quantity chosen. In differential form, at a smooth interior optimum,
The equality on the right is the change in the cost of the original bundle, which determines the Slutsky income adjustment. The approximation says the two compensation rules match to first order around the initial choice. For a finite price change, the consumer's Hicksian bundle can shift substantially as relative prices change, so the expenditure needed to maintain utility need not equal the cost of the original bundle. The resulting substitution effects can then differ in magnitude. In the examples, the own-good effects have the same local direction for a given price movement, while their numerical sizes differ.
Common Exam Trap
Do not confuse the Hicksian target with the Slutsky affordability condition. A common error is to calculate the Hicksian first coordinate correctly and then put the reciprocal price ratio into the second coordinate. In the first example, that mistake would give \((20\sqrt{2},40/\sqrt{2})\). Its utility is
not 40. The correct Hicksian bundle is \((20\sqrt{2},40\sqrt{2})\), with utility \(\sqrt{(20\sqrt{2})(40\sqrt{2})}=\sqrt{3200}=40\sqrt{2}\)? This expression is not 40; checking the product reveals that the proposed coordinates do not preserve utility. The correct equal-share Hicksian quantities at prices \((2,1)\) are instead \(x_1=20\sqrt{2}\) and \(x_2=40\sqrt{2}\) only if their product is 800? To attain utility 40, their product must be 1600. Therefore the correct bundle is \(x_1=40/\sqrt{2}=20\sqrt{2}\) and \(x_2=40\sqrt{2}=56.5685\), whose product is 1600 and utility is 40.
A second frequent error is to use the Hicksian expenditure as the Slutsky income, or vice versa. Keep the tests separate: verify Hicksian utility with \(U(h)=u^0\); verify Slutsky affordability with \(p^1\cdot x^0=m^S\). In either method, subtract the original quantity from the compensated quantity to obtain the substitution effect.
A Reliable Calculation Sequence
Record the initial prices, income, bundle, and utility. These identify the reference point for both compensation rules.
This gives the total change against which the substitution and income effects can be checked.
For Hicksian compensation, impose the initial utility target. For Slutsky compensation, calculate the new-price cost of the original bundle.
Subtract the original quantity to get each compensated effect, then add the corresponding income effect and check that the result equals the actual total change.
Check Your Understanding
For each problem, identify the relevant compensation rule and check the resulting bundle against its defining condition.
- For \(U=\sqrt{x_1x_2}\), initial prices are \((1,1)\) and income is 100. Price 1 rises to 4. Find the initial bundle and utility, then calculate the Hicksian bundle at the new prices. (Answer: Initial bundle \((50,50)\), utility 50; Hicksian bundle \((25,100)\).)
- For \(U=\sqrt{x_1x_2}\), initial prices are \((2,1)\), income is 120, and good 1's price rises to 3. Find Slutsky income and the Slutsky compensated bundle. (Answer: Slutsky income 180; compensated bundle \((30,90)\).)
- For \(U=x_1^{1/3}x_2^{2/3}\), initial prices are \((2,1)\) and income is 90. Find the original bundle. If the price of good 1 falls to 1, calculate Slutsky income and the Slutsky compensated bundle. (Answer: Original bundle \((15,60)\); Slutsky income 75; compensated bundle \((25,100)\).)
- For \(U=\sqrt{x_1x_2}\), initial prices are \((1,2)\), income is 120, and the price of good 1 rises to 4. Calculate the Hicksian bundle that preserves initial utility and the good 1 Hicksian substitution effect. (Answer: Hicksian bundle \((30,60)\); good 1 effect \(-30\) units.)
- In the setting of question 4, calculate the Slutsky compensated bundle and good 1 Slutsky effect. State which bundle preserves utility and which preserves affordability of the original bundle. (Answer: Slutsky income 180; compensated bundle \((22.5,60)\); good 1 effect \(-7.5\) units; Hicksian preserves utility and Slutsky preserves original-bundle affordability.)