Why Compensation Matters
When a good’s price rises, its actual chosen quantity reflects both substitution and income effects. The compensated substitution effect isolates the response to changed relative prices by adjusting income according to a specified compensation rule. Under standard assumptions, that compensated quantity of the now more expensive good cannot be higher. The conclusion is a weak decrease: the quantity may fall or remain unchanged, and the result concerns compensated—not necessarily actual—demand.
Two common compensation rules are Hicksian compensation, which holds utility at its original level, and Slutsky compensation, which makes the original bundle just affordable at the new prices. Both yield the nonpositive direction for the compensated response to a rise in one price, though their compensated bundles need not be the same.
The Hicksian Result: Cost-Minimization Proof
For Hicksian compensation, fix the original utility level \(u_0\). At any positive price vector \(p\), a Hicksian bundle minimizes expenditure while achieving at least that utility. Write \(h\) for a minimizing bundle at the original prices \(p\), and \(h'\) for a minimizing bundle at the new prices \(p'\). The argument requires that these cost-minimizing bundles exist. It does not require differentiable utility, an interior optimum, or strictly convex preferences.
The proof compares each minimizing bundle with the other bundle. Because \(h'\) reaches the target utility, it is feasible for the old expenditure-minimization problem. Because \(h\) reaches that same target utility, it is feasible for the new problem. Optimality therefore gives two inequalities:
Adding them cancels the expenditure terms that use the old prices against the corresponding terms using the new prices:
Only the first price changed, so \(p'-p=(p_1'-p_1,0,\ldots,0)\). The inequality becomes \((p_1'-p_1)(h_1'-h_1)\le 0\). Since \(p_1'-p_1>0\), division by that positive change gives \(h_1'-h_1\le 0\), or \(h_1'\le h_1\). This is the desired result. Notice that the proof establishes a weak inequality: cost-minimizing choices can leave the quantity unchanged.
Worked Example: Hicksian Demand for a Cobb–Douglas Preference
A consumer has utility \(u(x_1,x_2)=\sqrt{x_1x_2}\), and the target utility is \(u_0=6\). Prices change from \((p_1,p_2)=(2,8)\) to \((p_1',p_2')=(8,8)\). Find the Hicksian quantities of good 1 before and after the price increase, and check the direction of the effect. Use cost minimization; the common mistake to avoid is treating the target utility as if it were the product \(x_1x_2\).
Method and constraint. At utility 6, the requirement is \(\sqrt{x_1x_2}\ge 6\), or \(x_1x_2\ge 36\). Since both prices are positive, a cost-minimizing bundle meets the utility requirement exactly. For this utility function, the cost-minimizing bundle satisfies \(p_1x_1=p_2x_2\). Combining this condition with \(x_1x_2=36\) gives the Hicksian quantities \(x_1=6\sqrt{p_2/p_1}\) and \(x_2=6\sqrt{p_1/p_2}\).
At the original prices. \(x_1=6\sqrt{8/2}=6\cdot2=12\), and \(x_2=6\sqrt{2/8}=6\cdot\frac12=3\). The bundle reaches the target because \(\sqrt{12\cdot3}=\sqrt{36}=6\). Its expenditure is \(2(12)+8(3)=24+24=48\).
At the new prices. \(x_1'=6\sqrt{8/8}=6\), and \(x_2'=6\sqrt{8/8}=6\). The bundle still reaches utility 6 because \(\sqrt{6\cdot6}=6\). Its new-price expenditure is \(8(6)+8(6)=48+48=96\).
Check. Good 1’s compensated quantity falls from 12 to 6 when its price rises from 2 to 8. The bundles and expenditures satisfy the utility target and the stated prices. The quantity falls, as the theorem permits; it need not fall for every preference.
Slutsky Compensation: Revealed Affordability
The Slutsky rule sets compensated income equal to the cost of the original bundle at the new prices. Thus the original bundle is affordable after the price change. Under locally nonsatiated preferences, the Slutsky-compensated choice of good 1 also cannot exceed its original quantity when only \(p_1\) rises. Here locally nonsatiated means that from any bundle, there are bundles arbitrarily close to it that the consumer strictly prefers. This assumption ensures that a bundle strictly inside a budget set cannot be utility-maximizing.
Let the original choice be \(x\), with original income \(m=p\cdot x\). Slutsky-compensated income at the new prices is \(m'=p'\cdot x\), so \(x\) is affordable after the price change.
Let \(x'\) maximize utility at prices \(p'\) and income \(m'\). Since \(x\) is affordable there, optimality implies that \(x'\) is at least as good as \(x\).
If \(x_1'>x_1\), then the new bundle costs less than \(m\) at the old prices, as shown by the inequality below. Local nonsatiation would then allow a strictly preferred affordable bundle at the original prices, contradicting that \(x\) was originally optimal.
To verify the last step, put \(\Delta=p_1'-p_1>0\). The new choice is affordable at the new prices, so \(p'\cdot x'\le m'=p'\cdot x\). If \(x_1'>x_1\), then:
Thus \(x'\) has strict slack in the original budget. Local nonsatiation provides a strictly preferred bundle sufficiently close to \(x'\) that it still fits within that budget. This contradicts the original optimality of \(x\). Therefore \(x_1'\le x_1\). The proof depends on changing just one price; when multiple prices change, the cost comparison does not isolate the change in one quantity in this way.
Worked Example: Slutsky Compensation with Cobb–Douglas Demand
A consumer has utility \(u(x_1,x_2)=\sqrt{x_1x_2}\), income \(m=24\), and initial prices \((3,1)\). The price of good 1 rises to 5 while the price of good 2 remains 1. Calculate the Slutsky-compensated income and the new compensated quantity of good 1. The method is to first make the original bundle affordable at the new prices; the common mistake is to keep income at 24.
Original choice. For this utility, expenditure shares are equal, so each good receives half of income. Equivalently, maximizing \(x_1x_2\) subject to the budget gives \(x_1=m/(2p_1)\) and \(x_2=m/(2p_2)\). Hence \(x_1=24/(2\cdot3)=4\) and \(x_2=24/(2\cdot1)=12\). The original bundle is \((4,12)\), and its cost is \(3(4)+1(12)=12+12=24\), as required.
Compensated income. At the new prices \((5,1)\), the original bundle costs \(5(4)+1(12)=20+12=32\). Therefore \(m'=32\), not 24.
Compensated choice. Using the same demand rule at the new prices and compensated income, \(x_1'=32/(2\cdot5)=32/10=3.2\), and \(x_2'=32/(2\cdot1)=16\). Its new-price cost is \(5(3.2)+1(16)=16+16=32\). The original bundle also costs 32 at those prices, confirming the Slutsky compensation.
Check. Good 1’s quantity falls from 4 to 3.2. The original bundle is affordable at the compensated budget, and the compensated choice exhausts that budget. Using the uncompensated income of 24 would answer a different question.
Why the Result May Be an Equality
The theorem rules out an increase, not a tie. For example, with perfect complements \(u(x_1,x_2)=\min(x_1,x_2/2)\), attaining utility \(u_0\) requires at least \(x_1=u_0\) and \(x_2=2u_0\). With positive prices, the least-cost bundle is exactly \((u_0,2u_0)\), regardless of the relative prices.
Worked Example: Perfect Complements Leave the Hicksian Quantity Unchanged
Take \(u(x_1,x_2)=\min(x_1,x_2/2)\) and target utility 5. Prices change from \((3,2)\) to \((7,2)\). Find both Hicksian bundles and compare their quantities of good 1. The method is to impose the fixed proportions; the common mistake is to assume that every compensated price increase must cause a strict quantity decrease.
Required proportions. To reach utility 5, the bundle must satisfy \(x_1\ge5\) and \(x_2\ge10\). With positive prices, any extra amount of either good raises expenditure without increasing the target utility, so the least-cost bundle is \((5,10)\) at either price vector.
Expenditures and check. At the original prices, expenditure is \(3(5)+2(10)=15+20=35\). At the new prices, expenditure is \(7(5)+2(10)=35+20=55\). Both bundles provide utility \(\min(5,10/2)=5\). Good 1’s Hicksian quantity is 5 before and after the price increase: the substitution effect is zero, which satisfies the weak inequality.
Scope, Interpretation, and Exam Traps
The result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Its importance is that it gives a reliable sign for the compensated substitution effect without requiring a differentiable demand function or an interior solution. It applies to Hicksian compensation by the expenditure inequalities and to Slutsky compensation under the stated utility-maximization and local-nonsatiation conditions.
- Do not claim a strict fall. Perfect complements can leave the Hicksian quantity unchanged. At corners or with non-smooth preferences, a strict response is not guaranteed.
- Do not confuse compensated and actual demand. The result concerns a choice with income adjusted by a compensation rule. Actual demand also reflects the income effect and is not covered by this sign result alone.
- Do not change several prices and claim the same coordinate result. The Hicksian inequalities then imply \((p'-p)\cdot(h'-h)\le0\), a restriction on a weighted sum of quantity changes. They do not, in general, determine the sign of each individual change.
- Do not treat affordability as strict preference. Under Slutsky compensation, the original bundle is affordable, so the new choice is at least as good. To rule out a larger quantity, the proof also uses the strict budget slack that would arise at the old prices and local nonsatiation.
Check Your Understanding
For each problem, identify the compensation rule and compare the compensated quantity with its original value.
- A consumer has \(u(x_1,x_2)=\sqrt{x_1x_2}\), target utility 6, and prices change from \((1,9)\) to \((4,9)\). Find the Hicksian quantity of good 1 at each price vector and the change. (Answer: It falls from 18 to 9, a decrease of 9.)
- For \(u(x_1,x_2)=\min(x_1,x_2/3)\), target utility 4, and prices changing from \((2,1)\) to \((6,1)\), find the Hicksian bundles and the quantity change for good 1. (Answer: The bundle is \((4,12)\) at both prices; good 1’s quantity is unchanged at 4.)
- A consumer with \(u(x_1,x_2)=\sqrt{x_1x_2}\) has income 30 and prices \((3,2)\). Price \(p_1\) rises to 6. Find the original bundle, Slutsky-compensated income, and new compensated quantity of good 1. (Answer: The original bundle is \((5,7.5)\); compensated income is \(6(5)+2(7.5)=45\); the new quantity of good 1 is \(45/(2\cdot6)=3.75\), down from 5.)
- At two price vectors, Hicksian minimizers \(h\) and \(h'\) achieve the same utility. Only \(p_2\) rises, from 4 to 7. If \(h_2=10\), what is the greatest possible value of \(h_2'\) consistent with the cost-minimization inequality? (Answer: \(h_2'\le10\); the greatest possible value is 10.)
- For a Hicksian response, suppose \((p'-p)=(2,0)\) and \(h'-h=(1,-3)\). Does this satisfy the cost-minimization inequality, and what does it say about the first good? (Answer: Yes; the dot product is \(2(1)+0(-3)=2\), which is positive, so it does not satisfy the required inequality. A Hicksian response with these price and quantity changes is impossible under the theorem.)