Cambridge A‑Level Economics 9708 – The Interaction of Demand and Supply: Joint (Complementary) Demand
1. Learning Objectives
- Define joint (complementary) demand and distinguish it from substitute demand.
- Explain how a price change in one market affects the demand curve, equilibrium price and quantity in a related market.
- Calculate and interpret cross‑price elasticity of demand for complements.
- Analyse the welfare effects (consumer surplus, producer surplus, dead‑weight loss) of shifts in joint‑demand.
- Evaluate the impact of the six micro‑intervention tools listed in the syllabus on complementary goods.
- Link micro‑changes in joint‑demand to macro‑economic variables (AD, inflation, balance of payments).
- Apply the concepts to real‑world examples and A‑Level extensions (derived demand, bundling, price‑setting, signalling).
2. Position in the Cambridge Syllabus
Joint demand is examined under the AS‑level blocks 1.1–1.6 (basic ideas), 2.1–2.5 (demand‑supply, elasticities, surplus) and 3.1–3.3 (government micro‑intervention). The A‑Level extensions (7–11) explore derived demand, market structures and price‑setting. The sections below are organised to satisfy each of these requirements while keeping the material concise for revision.
3. Key Definitions
| Joint (complementary) demand | Demand for a good that is used together with another good; the quantity demanded of each depends on the price of the other. |
| Cross‑price elasticity of demand | \[
\varepsilon_{X,Y}= \frac{\%\Delta Q_X}{\%\Delta P_Y}
\]
For complements \(\varepsilon_{X,Y}<0\). |
| Own‑price elasticity of demand | \[
\varepsilon_{X}= \frac{\%\Delta Q_X}{\%\Delta P_X}
\] |
| Market equilibrium | Quantity demanded equals quantity supplied \((Q_d=Q_s)\) at price \(P^*\) and quantity \(Q^*\). |
| Consumer surplus (CS) | Area between the demand curve and the price line up to the quantity bought. |
| Producer surplus (PS) | Area between the supply curve and the price line up to the quantity sold. |
| Dead‑weight loss (DWL) | Loss of total surplus caused by a market distortion (tax, price ceiling, etc.). |
| Private good | Rivalrous and excludable; most complementary goods are private. |
4. The Standard Demand‑Supply Framework (Recap)
- Demand function: \(Q_{dX}=f(P_X,\,P_Y,\,I,\,T)\)
- Supply function: \(Q_{sX}=g(P_X,\,C,\,T_s)\) where \(C\) denotes production costs.
- Movements along a curve are caused by a change in the good’s own price; shifts are caused by changes in any other determinant (including the price of a complement).
- Equilibrium is found where the demand and supply curves intersect.
5. Joint (Complementary) Demand
5.1 Theoretical Basis
When two private goods, \(X\) and \(Y\), are complements:
\[
\frac{\partial Q_{dX}}{\partial P_Y}<0\qquad\text{and}\qquad\varepsilon_{X,Y}<0
\]
This negative relationship means that a fall in the price of \(Y\) raises the quantity demanded of \(X\) at every price of \(X\), and vice‑versa.
5.2 Cross‑Price Elasticity in Practice
Example: A 10 % fall in the price of petrol leads to a 5 % rise in the quantity demanded for cars.
\[
\varepsilon_{\text{car, petrol}}=\frac{5\%}{-10\%}= -0.5\;( <0\;\text{⇒ complements})
\]
5.3 Graphical Illustration (X‑market)
- Initial equilibrium: \(E_1\) at \((P_1,Q_1)\) with demand curve \(D_1\) and supply curve \(S\).
- Price of complement \(Y\) falls → rightward shift of demand to \(D_2\).
- New equilibrium \(E_2\) at \((P_2,Q_2)\) where \(P_2>P_1\) and \(Q_2>Q_1\) (upward‑sloping supply).
- CS expands (area A + B + C); PS expands (area B + D). The rectangle between \(D_1\) and \(D_2\) represents the gain in total surplus.
5.4 Numerical Exercise (Optional)
Suppose the demand for smartphones is \(Q_d = 200 - 20P_s + 10P_d\) where \(P_s\) = price of smartphones and \(P_d\) = price of data plans. If \(P_d\) falls from £30 to £24, calculate the change in quantity demanded of smartphones (holding \(P_s\) constant at £200).
- Initial quantity: \(Q_{d0}=200-20(200)+10(30)=200-4000+300=-3500\) (negative – indicates the functional form is illustrative; in practice we would use realistic parameters).
- New quantity: \(Q_{d1}=200-20(200)+10(24)=200-4000+240=-3560\).
- Δ\(Q_d\) = \(-60\) units → a fall because the coefficient on \(P_d\) is positive (higher data‑plan price raises smartphone demand), confirming the goods are complements.
6. Government Micro‑Intervention Tools (Syllabus 3.1–3.3)
For each tool, indicate the likely effect on the demand for a complement, the resulting equilibrium in the focal market, and the welfare change.
| Tool |
Action on Complement (Y) |
Effect on Demand for X |
Resulting Equilibrium (X‑market) |
Welfare Outcome (CS, PS, DWL) |
| Indirect (excise) tax on Y |
Increases \(P_Y\) |
Leftward shift of \(D_X\) |
Lower \(P_X^*\) and \(Q_X^*\) (if supply upward‑sloping) |
CS ↓, PS ↓, DWL = area between old and new demand curves & supply |
| Subsidy on Y |
Decreases effective \(P_Y\) |
Rightward shift of \(D_X\) |
Higher \(P_X^*\) and \(Q_X^*\) |
CS ↑, PS ↑, government outlay = subsidy × \(Q_Y\) (potential DWL if subsidy > marginal benefit) |
| Price ceiling on Y (binding) |
Sets \(P_Y\) below market level |
Rightward shift of \(D_X\) |
Higher \(P_X^*\), higher \(Q_X^*\) (possible shortage of Y) |
CS ↑ for X‑buyers, but shortage may create rationing loss; DWL from unmet demand for Y. |
| Price floor on Y (binding) |
Sets \(P_Y\) above market level |
Leftward shift of \(D_X\) |
Lower \(P_X^*\), lower \(Q_X^*\) |
CS ↓, PS may rise for Y‑producers but falls for X‑producers; DWL from surplus of Y. |
| Buffer‑stock scheme for Y |
Government buys excess Y when price falls, sells when price rises → stabilises \(P_Y\) |
Demand for X becomes more predictable; small shifts only if price expectations change. |
Equilibrium in X remains relatively stable. |
CS and PS relatively unchanged; fiscal cost = buying‑selling spread. |
| Direct provision of Y (e.g., free public transport) |
Effective price of Y drops to zero for a segment of consumers |
Strong rightward shift of \(D_X\) for that segment |
Higher \(P_X^*\) and \(Q_X^*\) (if supply can respond) |
CS ↑ for users, PS ↑ if firms can expand output; fiscal cost = provision expense; possible DWL if over‑consumption. |
7. Welfare Analysis for Joint‑Demand Shifts
- Rightward demand shift (cheaper complement): CS expands (larger area under demand), PS usually rises (higher equilibrium price) when supply is upward‑sloping.
- Leftward demand shift (more expensive complement): Both CS and PS fall; the loss is represented by the DWL triangle between the old and new demand curves and the supply curve.
- When a tax/subsidy is applied to the complement, the DWL is the area between the pre‑tax and post‑tax demand curves (or supply curves, if the tax is on producers) and the supply curve of the focal good.
8. Connecting Joint Demand to the Macro‑Economy (Syllabus 4.1–4.6)
- Aggregate Demand (AD): A fall in the price of a complement (e.g., fuel) raises consumption of the related good (cars), which contributes to higher household consumption (C) and thus shifts AD rightward.
- Inflation: Increased demand for the complement‑good pair can generate cost‑push pressures if supply is inelastic, feeding into the CPI.
- Balance of Payments: If the complementary good is an import (e.g., foreign‑made smartphones), a price cut in data plans can raise imports, widening the current‑account deficit.
- Fiscal/Monetary Policy Interaction: A fuel tax (government revenue tool) simultaneously reduces car demand (joint‑demand effect) and raises tax receipts, illustrating the dual micro‑macro impact required by the syllabus.
9. International Trade Implications (Syllabus 5–6)
Complementary goods often cross borders:
- Export‑oriented industries (e.g., car manufacturers) depend on imported inputs (petrol, steel). A change in world oil prices alters the demand for cars worldwide.
- Exchange‑rate movements affect the relative price of a complement, thereby shifting demand for the associated good in the domestic market.
- Trade policies (tariffs, quotas) on one good can have spill‑over effects on its complement – e.g., a tariff on imported coffee beans raises coffee prices, reducing demand for sugar in the domestic market.
10. A‑Level Extensions (Syllabus 7–11)
10.1 Derived Demand
The demand for a factor of production (e.g., steel) is derived from the demand for the final good (cars). A fall in car prices raises the derived demand for steel, mirroring the joint‑demand logic.
10.2 Bundling and Price‑Setting in Imperfect Competition
- Monopolistic competition: Firms may set a low price for a complement (e.g., printer) to lock in future sales of a high‑margin product (ink cartridges).
- Oligopoly (price leadership): A dominant firm may coordinate the price of a complement across the industry to maximise joint profits.
10.3 Signalling and Rationing Functions of Price
When the price of a complement rises, the signal to consumers is to reduce consumption of both goods; producers receive information about the optimal scale of production for the related market.
10.4 Coordination in Supply Chains
Vertical integration (e.g., a car maker acquiring an engine supplier) reduces the risk of mismatched supply of complementary components.
11. Real‑World Case Studies of Joint Demand
| Complementary Pair |
Typical Shock to Complement (Y) |
Resulting Shift in Demand for X |
Observed Welfare Change |
| Petrol & Cars (large‑engine) |
Petrol price ↓ 15 % |
Rightward shift of car demand (especially high‑displacement models) |
CS ↑ for car buyers; PS ↑ for manufacturers; possible DWL from congestion & environmental externalities. |
| Smartphones & Data Plans |
Data‑plan price ↓ 20 % |
Rightward shift of smartphone demand |
CS ↑ for consumers; PS ↑ for handset makers; PS ↓ for network operators if revenue per GB falls. |
| Printers & Ink Cartridges |
Ink cartridge price ↑ 30 % |
Leftward shift of printer demand |
CS ↓ for consumers; PS ↑ for cartridge producers, ↓ for printer manufacturers. |
| Coffee & Sugar |
Sugar price ↑ 10 % |
Leftward shift of coffee demand (if sugar is a strong complement) |
CS ↓ for coffee drinkers; PS ↓ for coffee retailers. |
12. Summary Checklist (Revision Aid)
- Identify whether two goods are complements (\(\varepsilon_{X,Y}<0\)) or substitutes (\(\varepsilon_{X,Y}>0\)).
- State the direction of the demand shift for a good when the price of its complement changes (price ↓ Y → demand for X rightward; price ↑ Y → demand leftward).
- Show how the shift moves the equilibrium price and quantity in the focal market (use a diagram).
- Explain the change in consumer surplus, producer surplus and any dead‑weight loss.
- Evaluate a government policy (tax, subsidy, price ceiling/floor, buffer‑stock, direct provision, information) that targets one good and its impact on the complementary market.
- Link the micro‑effect to macro variables (AD, inflation, balance of payments) where relevant.
- Discuss any A‑Level extension (derived demand, bundling, price‑setting, signalling).
13. Practice Questions
13.1 Question 1 (AS Level)
Prompt: The price of coffee falls from $5 to $4 per cup. Assuming sugar is a complement, describe qualitatively how the demand curve for sugar will shift. Then, using a diagram, illustrate the effect on the equilibrium price and quantity in the sugar market and comment on the likely change in consumer surplus.
Guidance Points:
- Lower coffee price → higher coffee consumption → rightward shift of sugar demand.
- New equilibrium: higher price and higher quantity of sugar (upward‑sloping supply).
- Consumer surplus for sugar increases – the area between the new price line and the demand curve expands.
- Optional extension: If a sugar tax is later introduced, the demand curve shifts left, creating a DWL equal to the triangle between the two demand curves and the supply curve.
13.2 Question 2 (A Level – Policy Evaluation)
Prompt: The government imposes an excise tax of £0.10 per litre on petrol. Explain, using the concepts of joint demand, how this tax is likely to affect the market for cars. In your answer, discuss the impact on equilibrium price and quantity, consumer and producer surplus, and any dead‑weight loss. Finally, evaluate whether the tax achieves a likely policy objective.
Key Points to Cover:
- Tax raises the effective price of petrol → leftward shift of car demand.
- Resulting equilibrium: lower price and quantity of cars (if supply upward‑sloping).
- CS falls (area loss), PS falls (price received falls), government revenue = tax × quantity of petrol sold.
- DWL = triangle between old and new demand curves and the supply curve.
- Policy objective (e.g., reducing congestion or emissions) may be met, but the welfare cost (CS + PS loss) must be weighed against environmental benefits.
13.3 Question 3 (Data Response – International Trade)
Prompt: A 20 % fall in the world price of wheat reduces the price of bread in the UK. Discuss the likely effect on the demand for butter, a complement to bread, and explain any possible repercussions for the UK’s current‑account balance.
Structure:
- Lower bread price → higher bread consumption → rightward shift in butter demand.
- Higher butter price and quantity (assuming upward‑sloping supply).
- If butter is largely imported, the increase in imports worsens the current account; if domestic, the effect is neutral.
- Consider exchange‑rate movements and possible policy responses (tariff, subsidy).
13.4 Question 4 (A‑Level Extension – Derived Demand)
Prompt: Explain how a fall in the price of electric cars can affect the derived demand for lithium, a key input for batteries. Include a diagram showing the chain of effects from the car market to the lithium market.
Answer Outline:
- Electric‑car price ↓ → demand for cars (X) rises (rightward shift).
- Higher car sales increase demand for batteries, raising derived demand for lithium (Y).
- In the lithium market, this appears as a rightward shift of demand, raising equilibrium price and quantity.
- Discuss welfare effects for lithium producers (PS ↑) and consumers (CS ↓) and any externalities (environmental).