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M15.L03 ā€” Income Consumption Curves and Engel Curves

Module: Module 15 ā€” Comparative Statics and Demand Theory
Lesson: L03 of 05
Duration: ~30 minutes
Level: Intermediate
Provenance: Intermediate Microeconomics with Excel (Barreto) | Principles of Microeconomics (Curtis and Irvine)

Learning Objective

Derive the income consumption curve and Engel curve from utility maximisation.

The Income Consumption Curve

The income consumption curve (ICC) traces the locus of optimal consumption bundles as income m increases, holding prices constant. For each income level, we solve the utility maximisation problem and plot the optimal bundle. Connecting these points traces the ICC.

For a normal good: as income rises, the optimal quantity of xā‚ rises. The ICC slopes upward and to the right through the origin (for homothetic preferences).

For an inferior good: as income rises beyond some threshold, the optimal quantity actually falls. The ICC bends backward toward the xā‚‚ axis.

For homothetic preferences (like Cobb-Douglas), the ICC is a ray through the origin ā€” income expansion proportionally increases all consumption.

Engel Curves

An Engel curve plots quantity demanded of one good against income, holding prices fixed. It is derived directly from the ICC by projecting the income-quantity relationship.

For Cobb-Douglas U = xā‚^a Ɨ xā‚‚^(1-a):

xā‚* = am/pā‚ ā†’ Engel curve is linear through origin with slope a/pā‚

Normal good: Engel curve slopes upward (dxā‚/dm > 0)
Inferior good: Engel curve slopes backward (dxā‚/dm < 0) above some income level
Luxury good: expenditure share rises with income (income elasticity > 1)
Necessity: expenditure share falls with income (0 < income elasticity < 1)

Engel's Law

Ernst Engel (1857) observed that as household income rises, the share of income spent on food decreases. This Engel's Law holds remarkably consistently across countries and time periods.

Australian ABS Household Expenditure Survey (2015-16): food and non-alcoholic beverages accounted for 17% of low-income household budgets but only 11% of high-income households ā€” consistent with Engel's Law. Housing, recreation, and transport are the spending categories that grow with income.

Worked Example

Deriving Engel curves for an Australian household

Utility: U = xā‚^0.3 Ɨ xā‚‚^0.7 (xā‚ = food, xā‚‚ = other goods)
Prices: pā‚ = $10, pā‚‚ = $1

Marshallian demands:
xā‚* = 0.3m/pā‚ = 0.3m/10 = 0.03m
xā‚‚* = 0.7m/pā‚‚ = 0.7m

Engel curve for food: xā‚ = 0.03m ā€” linear, normal good
Income elasticity of food: Īµ = dxā‚/dm Ɨ m/xā‚ = 0.03 Ɨ m/(0.03m) = 1 (unit elastic)

Expenditure on food: pā‚xā‚* = 10 Ɨ 0.03m = 0.3m = 30% of income (constant share ā€” homothetic)

For a luxury like international travel (income elasticity > 1), the Engel curve would be convex ā€” spending share rises with income.

Common Misconception

"All goods have upward-sloping Engel curves."

Inferior goods have backward-bending Engel curves above some income threshold ā€” demand falls as income rises. Classic examples include budget cuts of meat (replaced by premium cuts), instant noodles (replaced by restaurant meals), and bus travel (replaced by car ownership). In Australia, demand for government housing is an inferior good ā€” higher-income households exit the waiting list.

Key Takeaways

  • The ICC traces optimal bundles as income varies (prices fixed); the Engel curve maps one good's quantity against income
  • Normal goods have upward-sloping Engel curves; inferior goods have backward-bending curves
  • Engel's Law: food budget share falls as income rises ā€” consistently observed in ABS data
  • For homothetic preferences (Cobb-Douglas), all goods have linear Engel curves through the origin

Practice

  1. For U = xā‚^0.4 Ɨ xā‚‚^0.6 with pā‚=\(5, pā‚‚=\)2: derive the Engel curves and calculate income elasticity for each good.
  2. A good has income elasticity 1.8. Is it a normal good, inferior good, luxury, or necessity? What does its Engel curve look like?
  3. Using ABS HES data, food spending by quintile is: Q1=\(220/wk, Q2=\)280, Q3=\(320, Q4=\)380, Q5=\(480. Income by quintile is: Q1=\)600, Q2=\(900, Q3=\)1,200, Q4=\(1,700, Q5=\)3,500. Calculate income elasticity between Q1 and Q5 and verify Engel's Law.

Further Resources

Provenance: Intermediate Microeconomics with Excel (Barreto) | Principles of Microeconomics (Curtis and Irvine)