Why a Price Change Can Affect Demand for Another Good
A consumer chooses a bundle of goods subject to prices, income, and preferences. If the price of good B changes, the consumer may adjust the quantity of good A as well as the quantity of B. The change in A's demand is a cross-price effect. Its direction is informative: a price increase for B that raises demand for A is consistent with the goods acting as substitutes in observed consumer choice; a price increase for B that lowers demand for A is consistent with their acting as complements.
To identify the effect, change one price and hold income and the other price fixed. This is an uncompensated, or Marshallian, response: the consumer faces the new price with the same money income. It therefore includes both the incentive to substitute between goods and the change in purchasing power caused by the price change.
The cross-price elasticity measures the percentage response of demand for A to a percentage change in the price of B. At a point where prices and quantity are positive, it is the cross-price effect multiplied by \(p_B/x_A\). A positive elasticity indicates a positive cross-price effect; a negative elasticity indicates a negative one. Unlike an absolute-value elasticity, its sign is essential to interpretation.
When demand is given by a discrete schedule or a finite change rather than a differentiable function, calculate the change in quantity while holding the specified variables fixed. If a percentage response is requested for a finite change, state the percentage-change convention being used; do not silently treat a finite change as a derivative.
Reading the Sign of the Cross-Price Effect
In the standard terminology for Marshallian demand, goods A and B are gross substitutes at a given choice when an increase in \(p_B\), with income and \(p_A\) fixed, increases demand for A. They are gross complements when that price increase decreases demand for A. The terms describe a demand response, not merely whether two goods can serve similar purposes or are often bought together.
A zero cross-price effect means that the specified demand for A does not respond locally to \(p_B\), under the stated conditions. It does not by itself prove that the goods have no relationship in preferences. In particular, the uncompensated response combines substitution and purchasing-power effects, which may offset one another.
Worked Example: A Positive Cross-Price Response
Suppose a hypothetical local demand schedule for good A is \(x_A=42-3p_A+2p_B+0.1m\), valid over the price and income range considered. At income \(m=200\), with \(p_A=4\) and \(p_B=3\), calculate demand for A. Then find its response and cross-price elasticity when \(p_B\) rises to 5, holding income and \(p_A\) fixed.
Method. Substitute the initial values into the demand schedule, then change only \(p_B\). The coefficient of \(p_B\) is the cross-price effect per unit of price over this linear schedule.
Initially, \(x_A=42-3(4)+2(3)+0.1(200)=42-12+6+20=56\) units.
When \(p_B=5\), demand is \(x_A=42-12+2(5)+20=60\) units. Thus the price increase of \(5-3=2\) raises demand by \(60-56=4\) units. Equivalently, the schedule gives \(\partial x_A/\partial p_B=2\) units per dollar.
At the initial choice, the cross-price elasticity is \(2(3/56)=6/56=0.1071\), to four decimal places. Its positive sign means the two goods are gross substitutes at this choice according to the uncompensated response.
Check. Increasing \(p_B\) by 2 dollars adds \(2(2)=4\) units to the schedule, matching the direct calculation. A common mistake is to use the coefficient \(-3\), which describes the effect of \(p_A\) on A's own demand, not the effect of \(p_B\).
Why Compensated and Uncompensated Effects Can Differ
An uncompensated cross-price effect holds money income fixed. A compensated effect instead adjusts income so the consumer can attain the original utility level after the price change. Write \(h_A(p_A,p_B,u)\) for compensated demand for A at utility level \(u\). The compensated cross-price effect is \(\partial h_A/\partial p_B\), holding \(p_A\) and \(u\) fixed. It isolates the response after removing the purchasing-power change.
For an interior choice with differentiable demand, the cross-price form of the Slutsky equation relates these two effects. Here \(x_B\) is the initial quantity of B, and \(\partial x_A/\partial m\) is the income response of demand for A.
The equation says that the uncompensated response equals the compensated substitution response minus the purchasing-power term. For example, if A is normal, then \(\partial x_A/\partial m>0\). A rise in \(p_B\) reduces what the initial income can buy, which tends to reduce demand for A; this can offset a positive compensated response.
The compensated effect asks how demand changes at constant utility. The uncompensated effect asks what changes at constant money income.
A small change in \(p_B\) changes the cost of the original bundle by approximately \(x_B\,dp_B\).
Compensating income for that cost change gives \(\partial h_A/\partial p_B=\partial x_A/\partial p_B+x_B(\partial x_A/\partial m)\), which rearranges to the displayed equation.
This result is local: it compares marginal responses at a particular choice. It assumes differentiable demand and an interior choice where the derivatives are defined. At a corner, a kink, or a discontinuity, a derivative may not describe the response; compare the actual choices before and after the price change instead. The sign of a compensated cross-price effect need not match the sign of the uncompensated one.
Worked Example: Fixed Proportions and a Negative Cross-Price Effect
A consumer has hypothetical perfect-complements preferences \(U(x_A,x_B)=\min(x_A,x_B)\), so the chosen bundle uses one unit of each good per matched pair. Income is \(m=120\), and initially \(p_A=4\), \(p_B=2\). Find the initial quantity of A and its change when \(p_B\) rises to 3, with income and \(p_A\) fixed. Then calculate the initial point cross-price elasticity.
Method. At the fixed-proportion choice, set \(x_A=x_B=q\). Spending on a bundle with \(q\) units of each good is \(4q+2q=6q\). The budget is exhausted at the affordable matched quantity \(q=120/6=20\), so initially \(x_A=20\).
After the price change, the cost per matched pair is \(4+3=7\). The new quantity is \(q=120/7=17.1429\) units, to four decimal places. The change in demand for A is \(120/7-20=(120-140)/7=-20/7=-2.8571\) units, to four decimal places.
For this choice rule, \(x_A=120/(p_A+p_B)\), so \(\partial x_A/\partial p_B=-120/(p_A+p_B)^2\). At the initial prices this is \(-120/6^2=-120/36=-10/3\) units per dollar. The point elasticity is \((-10/3)(2/20)=-1/3=-0.3333\), to four decimal places.
Check. The negative sign agrees with the direct calculation: a higher price for B makes each matched pair more expensive, reducing demand for A as well. A common mistake is to calculate the effect using only \(p_B\) in the denominator. The budget pays for both goods, so the relevant total price is \(p_A+p_B\).
A Zero Observed Effect Can Hide Substitution
Consider a consumer with utility \(U(x_A,x_B)=\sqrt{x_Ax_B}\), positive prices, and an interior choice. The consumer spends half of income on each good, so \(x_A=m/(2p_A)\). This demand for A does not depend on \(p_B\): its uncompensated cross-price effect is zero. Yet at constant utility, a higher \(p_B\) makes A relatively less expensive, and compensated demand for A rises.
Worked Example: A Zero Marshallian Effect but Positive Compensation Effect
At \(m=40\), \(p_A=4\), and \(p_B=1\), calculate the initial bundle and utility. Then raise \(p_B\) to 1.21. Compare demand for A at fixed money income with compensated demand for A at the original utility level.
Method. Use the stated Marshallian demand for the fixed-income calculation. For the compensated calculation, use \(h_A=u\sqrt{p_B/p_A}\), obtained by minimizing expenditure at a target utility \(u\) for this utility function.
Initially, \(x_A=40/(2\cdot4)=5\), and \(x_B=40/(2\cdot1)=20\). The initial utility is \(u=\sqrt{5\cdot20}=\sqrt{100}=10\).
At the new price and unchanged income, \(x_A=40/(2\cdot4)=5\) still. Thus the finite uncompensated change in A is zero; consistently, \(\partial x_A/\partial p_B=0\).
At the new price but holding utility at 10, compensated demand is \(h_A=10\sqrt{1.21/4}=10\sqrt{0.3025}=10(0.55)=5.5\). The compensated increase is \(5.5-5=0.5\) units. It is positive even though the fixed-income response is zero.
Check with Slutsky. At the initial choice, \(\partial h_A/\partial p_B=u/[2\sqrt{p_Ap_B}]=10/[2\sqrt{4}]=10/4=2.5\). Also, \(x_B=20\) and \(\partial x_A/\partial m=1/(2p_A)=1/8\), so \(x_B(\partial x_A/\partial m)=20/8=2.5\). Therefore \(\partial x_A/\partial p_B=2.5-2.5=0\), matching the direct demand formula.
The finite compensated change of 0.5 is not the same as the local derivative multiplied by the finite price change; the derivative is a marginal measure at the initial prices. The common mistake is to infer from the zero Marshallian effect that the compensated substitution response must also be zero. The calculation shows that the income effect can exactly offset the substitution effect.
Interpreting Cross-Price Results Carefully
A cross-price calculation is only meaningful when the changed price and the held-fixed conditions are clear. In a demand schedule with several prices and income, changing more than one argument prevents you from attributing the result to a single cross-price effect. In a word problem, identify which good's price changes and which good's quantity is measured before selecting a derivative or calculating a difference.
For a differentiable schedule, the derivative \(\partial x_A/\partial p_B\) gives the local response in units of A per unit of B's price. The elasticity converts that response into a percentage scale. For finite changes, calculate both endpoints from the same demand rule and state the resulting difference; do not confuse that finite difference with the point derivative.
Finally, use the terminology precisely. A positive uncompensated cross-price effect indicates gross substitutes at the stated choice, while a negative one indicates gross complements. A compensated effect describes substitution at constant utility and may have a different sign. These measures answer related but distinct questions about consumer choice.
Check Your Understanding
For each problem, identify the response while respecting which variables are held fixed.
- A hypothetical demand schedule is \(x_A=36-2p_A+1.25p_B+0.05m\). At \(p_A=6\), \(p_B=4\), and \(m=200\), calculate initial demand, demand when \(p_B\) rises to 4.8, and the initial cross-price elasticity. (Answer: Initial demand 39 units; new demand 40 units; elasticity \(5/39=0.1282\), positive.)
- A hypothetical schedule is \(x_B=70-p_B-0.8p_A+0.02m\). At \(p_B=10\), \(p_A=5\), and \(m=500\), calculate \(x_B\). Then raise \(p_A\) to 7.5 and calculate the new demand and initial cross-price elasticity of B with respect to \(p_A\). (Answer: Initial demand 66 units; new demand 64 units; elasticity \(-0.8(5/66)=-0.0606\), negative.)
- A consumer chooses \(x_A=x_B=q\) under \(U=\min(x_A,x_B)\). With income 90 and prices \(p_A=3\), \(p_B=2\), calculate \(q\). Then raise \(p_B\) to 3 and calculate the new \(q\) and initial point cross-price elasticity of A with respect to \(p_B\). (Answer: Initial \(q=18\); new \(q=15\); elasticity \(-90/25\cdot(2/18)=-0.4000\).)
- For \(U=\sqrt{x_Ax_B}\), income is 60, \(p_A=5\), and \(p_B=3\). Find the initial \(x_A\), then calculate compensated demand for A at the original utility if \(p_B\) rises to 4.32. (Answer: Initial \(x_A=6\), original utility \(\sqrt{60}\), and compensated \(h_A=\sqrt{60}\sqrt{4.32/5}=\sqrt{51.84}=7.2\), an increase of 1.2 units.)
- At a particular interior choice, the compensated cross-price derivative is \(\partial h_A/\partial p_B=1.8\), \(x_B=12\), and \(\partial x_A/\partial m=0.1\). Use the Slutsky equation to find the uncompensated cross-price derivative and classify the observed response. (Answer: \(1.8-12(0.1)=0.6\) units per dollar; positive, so A and B are gross substitutes at this choice.)