M15.L04 — Price Consumption Curves and the Demand Function

Module: Module 15 — Comparative Statics and Demand Theory
Lesson: L04 of 05
Duration: ~30 minutes
Level: Intermediate
Provenance: Intermediate Microeconomics with Excel (Barreto) | University of Utah Intermediate Microeconomics

Learning Objective

Derive the price consumption curve and explain its relationship to the demand function.

The Price Consumption Curve

The price consumption curve (PCC) traces the locus of optimal consumption bundles as the price of one good changes, holding income and the other price constant.

As p₁ falls (budget line rotates outward along x₁ axis), the consumer's optimal bundle moves. Connecting all optimal bundles as p₁ varies traces the PCC in (x₁, x₂) space.

Deriving the demand curve from the PCC:
For each optimal bundle on the PCC, read off x₁ and the corresponding p₁. Plotting (p₁, x₁) gives the Marshallian demand curve for good 1.

PCC Slope and Gross Substitutes/Complements

The slope of the PCC reveals whether good 2 is a gross substitute or gross complement for good 1:

If as p₁ falls, x₂ also falls (PCC slopes downward): goods are gross complements — demand for good 2 moves with demand for good 1. Example: coffee (good 1) and coffee cups (good 2).
If as p₁ falls, x₂ rises (PCC slopes upward): goods are gross substitutes — when good 1 becomes cheaper, consumers buy more good 1 and less good 2. Example: Coles (good 1) and Woolworths (good 2).

For Cobb-Douglas U = x₁^a × x₂^(1-a):
x₂* = (1-a)m/p₂ — independent of p₁! The PCC is horizontal, meaning goods are neither gross substitutes nor complements in the Cobb-Douglas case.

Homogeneity and the Demand Function

An important property: Marshallian demand functions are homogeneous of degree zero in (p₁, p₂, m). Doubling all prices and income leaves demand unchanged — only real purchasing power and relative prices matter, not the price level.

Formally: xᵢ(λp₁, λp₂, λm) = xᵢ(p₁, p₂, m) for all λ > 0.

This has policy implications: if the RBA targets 2-3% inflation and wages rise proportionally, real consumption baskets are unchanged.

Worked Example

Petrol price fall and public transport demand in Australian cities

Let x₁ = kilometres driven (petrol), x₂ = public transport trips, p₁ = fuel cost per km, m = transport budget.

When petrol prices fell 30% in 2020 (COVID demand collapse), if PCC slopes upward (substitutes), public transport demand should fall. PTV and Translink data showed exactly this: car use rebounded faster than public transport after prices fell.

For U = x₁^0.6 × x₂^0.4, p₁ = $0.15/km, p₂ = $3/trip, m = $300/month:
x₁* = 0.6×300/0.15 = 1,200 km; x₂* = 0.4×300/3 = 40 trips

If p₁ falls to $0.10:
x₁* = 0.6×300/0.10 = 1,800 km; x₂* = 0.4×300/3 = 40 trips (unchanged — Cobb-Douglas)

The Cobb-Douglas PCC is horizontal — the goods are income-independent (neither substitutes nor complements in this model). Real demand data suggests they are substitutes, requiring a different utility specification.

Common Misconception

"The demand curve and the PCC are the same thing."

The PCC lives in quantity-quantity space (x₁, x₂), tracing bundles as price varies. The demand curve lives in price-quantity space (p₁, x₁). The demand curve is derived FROM the PCC by reading off (p₁, x₁) coordinates for each bundle on the PCC.

Key Takeaways

The PCC traces optimal bundles as one price varies; the demand curve plots price against optimal quantity
PCC sloping downward → gross complements; PCC sloping upward → gross substitutes; PCC horizontal → neither (Cobb-Douglas)
Marshallian demand is homogeneous of degree zero in prices and income
Australia case: petrol and public transport are empirical gross substitutes — PCC slopes upward

Practice

For U = min(x₁, 2x₂) (perfect complements), derive x₁* and x₂* as functions of p₁, holding p₂=$5 and m=$100. What is the shape of the PCC?
U = x₁ + 2x₂ (perfect substitutes). Derive demand and the PCC. What happens at p₁/p₂ = 1/2?
Australian mobile data prices fell 40% between 2017-2022. Using the PCC framework, predict what happens to demand for fixed broadband (complement or substitute?).

Further Resources

📺 Price Consumption Curve (University of Utah) — intermediate series
📺 Deriving Demand from Indifference Curves (Khan Academy)
📚 Barreto — Intermediate Microeconomics with Excel

Provenance: Intermediate Microeconomics with Excel (Barreto) | University of Utah Intermediate Microeconomics