Difficulty: Introductory to Intermediate | Prerequisites: Price elasticity of demand (see companion notes); basic algebra.
Once you understand own-price elasticity, the natural next questions are: how does the price of one good affect demand for another (cross-price elasticity), how does income affect demand (income elasticity), and how do we add up individual consumers to get market demand? These concepts tie together the consumer theory you have been building. Cross-price elasticity tells you whether goods are substitutes or complements. Income elasticity distinguishes normal goods from inferior goods. And market demand aggregation has subtleties that exam questions love to exploit, particularly around who "counts" as a consumer.
Cross-price elasticity measures how the quantity demanded of one good responds to a price change in another, telling you whether the goods are substitutes (positive) or complements (negative). Market demand is the horizontal sum of individual demand curves, but only from consumers with purchasing power and willingness to pay. Income elasticity and Engel curves describe how demand shifts as income changes, distinguishing normal goods from inferior goods.
Cross-price elasticity of demand (XED)
The percentage change in quantity demanded of good X divided by the percentage change in price of good Y. In simple terms, it tells you whether two goods are friends (complements, negative XED) or rivals (substitutes, positive XED).
Substitute goods
Goods with a positive cross-price elasticity. When the price of one rises, demand for the other increases. Think Coca-Cola and Pepsi.
Complementary goods
Goods with a negative cross-price elasticity. When the price of one rises, demand for the other falls. Think printers and ink cartridges.
Market demand
The horizontal summation of all individual demand curves in the market. At each price, you add up the quantities demanded by every consumer who is both willing and able to buy.
Effective demand
Demand backed by purchasing power. A desire for a good without the income or resources to pay for it does not count towards market demand.
Engel curve
A graph showing the relationship between a consumer's income and the quantity demanded of a good, holding prices constant. Named after the statistician Ernst Engel.
Income elasticity of demand (YED)
The percentage change in quantity demanded divided by the percentage change in income. Positive for normal goods, negative for inferior goods.
Normal good
A good for which demand increases as income rises (positive income elasticity). Most goods fall into this category.
Inferior good
A good for which demand decreases as income rises (negative income elasticity). Think instant noodles or bus travel as income grows.
Cross-price elasticity (XED) = (%ΔQx) / (%ΔPy).
Positive XED: goods are substitutes. A rise in the price of Y pushes consumers towards X.
Negative XED: goods are complements. A rise in the price of Y reduces demand for X as well.
XED near zero: the goods are unrelated.
Worked example from the assignment: In 2001, good X costs $3 and sells 400 units; good Y costs $10 and sells 200 units. In 2002, X still costs $3 but sells 300 units; Y's price rises to $12 and Y sells 150 units.
Since the price of X did not change, the drop in X's sales (400 to 300) must be caused by the change in Y's price (assuming ceteris paribus and a linear demand function for X in terms of Py).
The demand function for X in terms of Py: Qx = a + b(Py). Using the two data points: 400 = a + 10b and 300 = a + 12b. Solving: b = –50, a = 900.
At the 2001 point (Py = 10, Qx = 400): XED = b × (Py / Qx) = –50 × (10 / 400) = –1.25.
The negative sign confirms X and Y are complements in 2001.
Market demand is the horizontal summation of all individual demand curves at each price.
Only consumers with effective demand count. Effective demand requires both willingness to pay and ability to pay.
The classic exam example: a five-year-old child may desperately want ice cream, but if they have no money and no decision-making authority over purchases, their desire does not enter market demand. This is not a flaw in the concept. Market demand measures what people will and can buy, not what they wish they could.
Market demand is not an underestimation of "true" consumer demand. It is an accurate measure of demand as economists define it: desire backed by purchasing power.
An Engel curve plots quantity demanded against income (prices held constant). It slopes upward for normal goods and downward (or bends back) for inferior goods.
Income elasticity of demand (YED) = (%ΔQ) / (%ΔIncome).
YED > 0: normal good. YED > 1: luxury good (income-elastic). 0 < YED < 1: necessity (income-inelastic).
YED < 0: inferior good.
Individual Engel curves can be summed horizontally (at each income level, add up quantities across consumers) to get the market-level Engel curve.
However, the true/false trap: this does not always let you calculate a single market-level income elasticity of demand in a straightforward way. Different consumers may have very different Engel curves. Some may treat a good as normal while others treat it as inferior. The market-level Engel curve could therefore have a complex, non-monotonic shape.
The statement that "we always get to calculate income elasticity from individual Engel curves" is an oversimplification. You can always construct the market Engel curve by summation, but interpreting a single elasticity from it is not guaranteed to be meaningful if the curve is not smooth or monotonic.
\text{XED} = \frac{\%\Delta Q_X}{\%\Delta P_Y} = \frac{dQ_X}{dP_Y} \times \frac{P_Y}{Q_X}\text{YED} = \frac{\%\Delta Q}{\%\Delta I} = \frac{dQ}{dI} \times \frac{I}{Q}For the worked cross-price elasticity problem:
Linear demand for X: Qx = 900 – 50Py
At Py = 10: XED = –50 × (10/400) = –1.25
For market demand (horizontal summation): at each price P, sum Qi(P) across all consumers i who have Qi(P) > 0.
Cross-price elasticity is central to competition policy. If regulators want to define a market (say, for an antitrust case), they look at cross-price elasticities: goods with high positive XED are in the same market. Income elasticity matters for forecasting demand as economies grow: luxury goods (YED > 1) see demand grow faster than income, while necessities (0 < YED < 1) see slower growth. Engel's law, the observation that food's share of spending falls as income rises, is one of the most robust empirical findings in economics.
Students often think market demand understates "true" demand because it excludes people who want things but cannot afford them. It does not understate it; economics defines demand as desire plus ability to pay. That is the definition, not a limitation.
A frequent error is assuming that a negative cross-price elasticity means the goods are substitutes. It is the opposite: negative XED means complements.
Students sometimes confuse the Engel curve (quantity vs. income, prices fixed) with the demand curve (quantity vs. price, income fixed). They measure different relationships.
Another trap: assuming that because individual Engel curves exist, a single market-level income elasticity can always be meaningfully computed. When some consumers treat a good as normal and others as inferior, the aggregated curve can be irregular.
⚠️ Cross-price elasticity calculations with real data points are common exam problems. Practise setting up the linear demand function from two observations and computing XED at a specific point.
⚠️ The "five-year-old wanting ice cream" question tests whether you understand effective demand. The answer centres on the difference between desire and demand.
⚠️ True/false questions on Engel curves and market-level income elasticity test whether you recognise that horizontal summation of Engel curves is possible but that a single meaningful elasticity figure is not always the result.
⚠️ Know the sign conventions cold: positive XED = substitutes, negative XED = complements, positive YED = normal good, negative YED = inferior good.
True or false: A positive cross-price elasticity means the two goods are complements.
Fill in the blank: Market demand is the ____ summation of individual demand curves.
True or false: A child who wants a good but has no money contributes to market demand.
Fill in the blank: An Engel curve plots quantity demanded against ____, holding prices constant.
True or false: If YED is negative, the good is a luxury.
Answers: 1. False (substitutes). 2. Horizontal. 3. False. 4. Income. 5. False (inferior good).
Q: In 2001, good X costs $3 and sells 400 units. Good Y costs $10 and sells 200 units. In 2002, X still costs $3 but sells 300 units. Y's price rises to $12. Assuming demand for X is a linear function of Y's price, what is the cross-price elasticity of demand for X with respect to Y in 2001?
A: Set up the linear function Qx = a + bPy. From the two data points: 400 = a + 10b and 300 = a + 12b. Subtracting gives –100 = 2b, so b = –50 and a = 900. Point elasticity at 2001 values: XED = b × (Py/Qx) = –50 × (10/400) = –1.25. The goods are complements.
Q: Andrew, a five-year-old, has strong demand for ice cream but his demand does not count towards market demand. Why not? Is market demand an underestimation?
A: Andrew has desire but not effective demand. He lacks the purchasing power (money) and likely the legal authority to enter a market transaction independently. Market demand counts only consumers who are both willing and able to buy. This is not an underestimation; it is the correct measure of demand as economics defines it.
Q: True or false and explain: We can always calculate income elasticity of demand for a good as long as individual Engel curves are available, since we can obtain market-level Engel curves through horizontal summation.
A: This is partially true but misleading. You can indeed sum individual Engel curves horizontally to get a market-level Engel curve. However, "always" being able to calculate a meaningful income elasticity from it is not guaranteed. If different consumers have different income elasticities (some treating the good as normal, others as inferior), the market Engel curve may be non-monotonic or kinked, making a single elasticity figure poorly defined or dependent on the income level chosen.
Cross-price elasticity links to the broader theory of consumer choice and the classification of goods. It connects to market structure: in perfect competition, many close substitutes exist, pushing XED high; monopolists sell goods with few substitutes, so XED with alternatives is low. Income elasticity connects to Engel's law, welfare economics, and the study of how economies transition from spending on necessities to spending on luxuries as they develop.
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