Complement Prices and Demand Shifts

Why the Price of a Related Good Matters

A household deciding whether to buy a printer may also consider the cost of the ink cartridges needed to use it. If cartridge prices rise, some buyers may be less willing to purchase printers, even when the printer’s own price has not changed. The price change affects demand for the printer through the relationship between the two goods.

This is a change in demand, not simply a change in the quantity demanded caused by a change in the printer’s own price. The distinction matters: a change in the printer’s own price is represented by movement along its demand curve, while a change in the price of a related good can shift the printer’s demand curve. This tutorial explains the complement relationship and how to analyze that shift, aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Two goods are complements when, holding other relevant factors fixed, an increase in the price of one good reduces demand for the other. Equivalently, a decrease in the price of one increases demand for the other. The relationship describes how buyers respond; it does not require that the goods always be used together or in fixed proportions.

Reading the Relationship in a Demand Function

Suppose the demand for good A depends on both its own price and the price of good B. Write market demand as $Q_A=f(P_A,P_B)$, where $Q_A$ is the quantity of A demanded, $P_A$ is A’s price, and $P_B$ is B’s price. Other influences on demand are held fixed. To decide whether A and B are complements, ask what happens to $Q_A$ when $P_B$ changes while $P_A$ and those other influences remain unchanged.

A simple linear demand equation makes this relationship visible. In the model below, $\alpha$, $\beta$, and $\gamma$ are positive constants, and prices and quantities are in the stated units. The negative coefficient on A’s own price represents the usual downward-sloping demand relationship. The coefficient on B’s price indicates how B’s price relates to demand for A.

Q_A=\alpha-\beta P_A-\gamma P_B,\qquad \beta>0,\quad \gamma>0

Here, an increase in $P_B$ lowers $Q_A$, so A and B are complements in this model. More generally, in a linear equation written as $Q_A=\alpha-\beta P_A+\theta P_B$, a negative $\theta$ indicates complements, a positive $\theta$ indicates substitutes, and a zero $\theta$ means the equation specifies no direct effect of B’s price on demand for A. The sign describes the modeled relationship with other factors held fixed; it is not established just because two goods could be used together.

Why a Complement’s Price Shifts Demand

The basic result follows directly by comparing demand before and after the related good’s price changes. Keep A’s own price and all other demand influences fixed. If B’s price rises by $\Delta P_B$, the linear model gives the following change in quantity demanded at each fixed price of A:

\Delta Q_A = \left(\alpha-\beta P_A-\gamma(P_B+\Delta P_B)\right) -\left(\alpha-\beta P_A-\gamma P_B\right) =-\gamma\Delta P_B

Because $\gamma>0$, a price increase for B means $\Delta P_B>0$, so $\Delta Q_A<0$. At every fixed $P_A$, buyers demand less A than before. That is a leftward shift of A’s demand curve. If B’s price falls, $\Delta P_B<0$, so $\Delta Q_A>0$ and A’s demand shifts right.

The comparison at a fixed $P_A$ is essential. It isolates the effect of B’s price. If A’s own price changes at the same time, the observed quantity demanded can reflect both the demand shift and a movement along the shifted curve. The change in observed quantity alone may therefore not tell you the direction of the shift.

Worked Example: A Higher Cartridge Price and Printer Demand

In a hypothetical market, monthly printer demand is $Q_P=180-4P_P-2P_I$, where $Q_P$ is the number of printers demanded, $P_P$ is the printer price in dollars, and $P_I$ is the price of an ink cartridge in dollars. Assume the equation applies over the prices considered and that other factors are unchanged. Find printer demand at a printer price of $25 when cartridge prices are $10 and $20. State the shift and check the result.

Method. Hold the printer price fixed, substitute each cartridge price into the demand equation, and compare the quantities. The negative coefficient on $P_I$ is consistent with printers and cartridges being complements in this model.

At $P_P=25$ and $P_I=10$:

$$ Q_P=180-4(25)-2(10)=180-100-20=60 $$

At the same printer price, with $P_I=20$:

$$ Q_P=180-4(25)-2(20)=180-100-40=40 $$

Thus, when the cartridge price rises by $10, monthly printer demand at a fixed printer price falls by 20 printers. More generally, the equation can be written separately for each cartridge price:

\begin{aligned} P_I=10:&\quad Q_P=160-4P_P\\ P_I=20:&\quad Q_P=140-4P_P \end{aligned}

At every fixed printer price, the second equation gives 20 fewer printers demanded. The demand curve shifts left; the printer’s own price did not change, so this is not a movement along the original demand curve. Check: at $P_P=25$, the two equations give $160-100=60$ and $140-100=40$, matching the direct calculations. The common mistake is to treat the $10 increase in cartridge price as though it were a $10 increase in printer price; the equation gives the two prices different coefficients, so they cannot be interchanged.

Using a Schedule to Identify the Shift

A demand schedule can show the same relationship without an equation. Compare quantities demanded at the same target-good prices under two prices for the complementary good. If the complementary good becomes more expensive and the quantity of the target good is lower at each listed target price, demand for the target good has shifted left. If the complementary good becomes cheaper and the target quantities are higher, demand has shifted right.

The schedule gives a set of comparisons, not a rule that every imaginable pair of goods must be complements. It describes the specified market conditions and price range. When only one target-good price is shown, a quantity difference at that price supports the direction of the shift at that point; a full schedule provides comparisons across several prices.

Worked Example: Comparing a Demand Schedule at Two Subscription Prices

Suppose a hypothetical market records monthly demand for smart speakers at three speaker prices. A music subscription is used with the speakers. Complete the comparison and determine how the speaker demand curve shifts when the subscription price rises from $8 to $12.

Speaker price	Quantity when subscription is $8	Quantity when subscription is $12
$30	90	78
$40	70	58
$50	50	38

Method. Compare the two quantities row by row, keeping the speaker price fixed in each comparison.

At a speaker price of $30, the change is $78-90=-12$ speakers per month. At $40, it is $58-70=-12$. At $50, it is $38-50=-12$. The quantity demanded falls by 12 at each listed speaker price when the subscription price rises by $4. The schedule therefore shows a leftward shift in speaker demand over the listed prices.

The constant difference in this schedule is consistent with two parallel demand curves over these observations, but the table alone does not establish what happens at every unlisted price. Check: each quantity in the higher-subscription column is 12 below the corresponding quantity in the lower-subscription column. A common mistake is to compare, for example, the $30 speaker price under one subscription price with the $40 speaker price under the other. That changes both prices being compared and does not isolate the subscription-price effect.

When Both Prices Change

A price change for a complement shifts demand for the target good, while a change in the target good’s own price affects quantity demanded along its demand curve. When both prices change, calculate each effect separately or substitute both new prices into the demand function. Do not infer the direction of the demand shift from the final quantity alone.

Worked Example: A Cheaper Complement but a Higher Target Price

In a hypothetical market, monthly demand for electric toothbrushes is $Q_T=900-10P_T-5P_H$, where $P_T$ is the toothbrush price in dollars and $P_H$ is the price of a replacement-head pack in dollars. Initially, $P_T=50$ and $P_H=12$. Then replacement heads fall to $8 while toothbrushes rise to $55. Calculate initial and final quantity demanded, and identify the demand shift separately from the quantity change.

Method. First compare demand at the old and new replacement-head prices while holding the toothbrush price fixed. Then use the actual new prices to calculate final quantity demanded.

Initially, demand is:

Q_{T,0}=900-10(50)-5(12)=900-500-60=340

To isolate the complement-price effect, keep the toothbrush price at $50 and use the new head-pack price of $8:

$$ Q_T=900-10(50)-5(8)=900-500-40=360 $$

At any fixed toothbrush price, the $4 decrease in the complement’s price increases toothbrush demand by $5(4)=20$. This is a rightward shift. But the actual toothbrush price also rises to $55. At both new prices, final demand is:

Q_{T,1}=900-10(55)-5(8)=900-550-40=310

Final quantity demanded is 310, which is 30 below the initial 340, even though demand shifted right because replacement heads became cheaper. The own-price increase reduces quantity demanded by $10(5)=50$, more than the 20-unit demand increase from the cheaper complement. Check: the two effects combine to $+20-50=-30$, agreeing with $310-340=-30$. The common mistake is to say that demand shifted left because the final quantity is lower. The equation shows that demand shifted right; the higher toothbrush price moved the outcome along that shifted curve.

Common Mistakes and Practical Checks

Complement-price questions are often short, but a reliable answer identifies the target good, identifies which other good’s price changes, states what is held fixed, and gives the shift direction. Use these checks:

Keep the goods in the right roles. If asked about demand for printers when ink prices change, printers are the target good and ink is the related good.
Use the correct sign. A higher complementary-good price reduces demand for the target good; a lower complementary-good price increases it.
Separate demand from quantity demanded. A complement’s price change shifts the target demand curve. A change in the target good’s own price changes quantity demanded along the curve.
Hold other factors fixed for the comparison. A claim about the complement-price effect requires comparing outcomes with the target price and other demand influences unchanged.
Do not assume joint use proves complementarity. The economic definition concerns how demand responds to price. In an application, use the stated demand relationship or evidence rather than relying on a product description alone.

Key takeaway: For complements, a rise in the price of one good shifts demand for the other good left; a fall shifts it right. Make the comparison at a fixed price of the target good to distinguish the shift from movement along its demand curve.

Check Your Understanding

For each problem, identify the relevant price comparison and distinguish a demand shift from a change in quantity demanded where needed.

A hypothetical demand equation is $Q_A=120-2P_A-3P_B$. At $P_A=20$, find demand when $P_B$ rises from 5 to 7, and state the shift direction. (Answer: Quantity falls from 65 to 59; demand for A shifts left by 6 units at that price.)
A demand schedule gives quantities of 80, 60, and 40 at target prices of $10, $20, and $30 when a complementary good costs $6. When that good costs $9, the quantities are 71, 51, and 31. What is the shift at each listed target price? (Answer: Left by 9 units at each listed price.)
In a hypothetical market, $Q_X=240-4P_X-2P_Y$. If $P_Y$ falls from $15 to $10, how much does demand for X change at any fixed $P_X$, and in which direction does the curve shift? (Answer: Demand increases by 10 units at every fixed $P_X$; the curve shifts right.)
Suppose $Q_M=500-5P_M-2P_C$. Initially $P_M=40$ and $P_C=20$; then $P_M$ rises to 44 while $P_C$ falls to 15. Find initial and final quantities, and state the demand shift caused by the complement-price change alone. (Answer: Initial quantity is 260; final quantity is 250; the lower $P_C$ shifts demand right by 10 units, while the higher $P_M$ reduces quantity by 20.)
A hypothetical demand equation is $Q_R=300-3P_R+4P_S$. If $P_S$ rises by $2 while $P_R$ is fixed, what happens to demand for R in this model, and are the goods complements under the stated definition? (Answer: Demand for R rises by 8 units; no, the positive cross-price effect indicates substitutes, not complements.)

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