Lecture 5 · Week 6 · Besanko & Braeutigam ch. 5 · about 50 min

The Theory of Demand

Where the demand curve actually comes from: tracing out quantity demanded as price varies, splitting a price change into substitution and income effects, Engel curves, and consumer surplus as the dollar value of buying at a single market price.

By the end you can

Derive an individual demand curve by varying price and tracing the price-consumption curve
Decompose the effect of a price change into a substitution effect and an income effect
Classify a good as normal, inferior or Giffen from the sign of its income effect
Compute consumer surplus from a linear demand curve

From consumer choice to a demand curve

Lesson 4 found a single optimal bundle at one set of prices. To get a full demand curve, repeat the tangency calculation at every possible price of X, holding income and the price of Y fixed, and trace out how $X^*$ changes.

Individual demand curveDefinitionFormula sheet →

X^d(P_X) = \arg\max_X\, U(X,Y) \text{ s.t. } P_XX + P_YY = M, \text{ for each value of } P_X

$X^d(P_X)$: the optimal quantity of X at each possible price of X

Use it when you need to explain, conceptually, where a downward-sloping demand curve comes from: the price-consumption curve connects the optimal bundles across all values of P_X, and the demand curve simply re-plots X* against P_X.

Splitting a price change: substitution and income effects

When a price falls, two distinct forces push quantity demanded up. The substitution effect is the pure change in relative attractiveness: the now-cheaper good looks better even holding satisfaction (utility) constant. The income effect is the change in purchasing power: a lower price effectively makes the consumer richer, even though no money literally arrived.

Total price effectSlutsky decompositionFormula sheet →

\Delta X^{\text{total}} = \Delta X^{\text{substitution}} + \Delta X^{\text{income}}

$substitution effect$: holds purchasing power fixed at the new prices, isolating pure relative-price reasoning
$income effect$: the remainder, from purchasing power changing

Use it when a question asks you to decompose the total effect of a price change, or explain why a demand curve could theoretically slope upward (a Giffen good).

Normal, inferior and Giffen goods

Classifying goods by the income effectRulesFormula sheet →

\text{Normal good: income effect and substitution effect same sign} \qquad \text{Inferior good: opposite signs} \qquad \text{Giffen good: income effect so strongly negative it reverses the total effect}

Use it when you need to classify a good from a description of how its consumption responded to a price change, or to explain why the law of demand can (in rare, extreme cases) fail.

Engel curves

An Engel curve plots quantity demanded of a good against income, holding prices fixed — the income-side counterpart of a demand curve.

Engel curveFormula sheet →

X^d(M), \text{ holding } P_X, P_Y \text{ fixed}

Use it when a question asks specifically about how quantity demanded responds to income (not price), or to identify a good as normal or inferior from the slope of its Engel curve.

An upward-sloping Engel curve identifies a normal good; a downward-sloping one identifies an inferior good over that income range. Many goods are normal at low incomes and become inferior at higher incomes (public transport is a common real-world example).

Consumer surplus

Consumer surplusKey formulaFormula sheet →

CS = \int_0^{Q^*}\!\big[P^d(Q) - P^*\big]\,dQ

$P^d(Q)$: inverse demand curve
$P^*$: the market price actually paid
$Q^*$: quantity purchased at P^*

Use it when you need the total dollar benefit consumers get from being able to buy at a single market price, rather than at the maximum price they would each have been willing to pay.

For a linear demand curve, this integral is just the area of a triangle, computed without calculus.

Exam practice

Summary and review

An individual demand curve traces optimal $X^*$ as $P_X$ varies, income and other prices held fixed.
Slutsky decomposition: total price effect = substitution effect + income effect.
The substitution effect always moves opposite to the price change; only the income effect can push the other way.
Normal: both effects agree. Inferior: they conflict. Giffen: income effect so strong it flips the total effect’s sign.
Engel curve: quantity demanded plotted against income, prices fixed; slope identifies normal vs inferior.
Consumer surplus from linear demand: the triangle above price, below the demand curve, $CS = \tfrac12 Q^*(P_{\text{intercept}} - P^*)$ .