ECOS2001 · Intermediate Microeconomics
Consumer Choice & Demand
Here the budget and preferences meet: the consumer maximises utility on the budget line. At an interior optimum the indifference curve is tangent to the budget line, |MRS| = p₁/p₂, equivalently the equimarginal rule MU₁/p₁ = MU₂/p₂. Solving the tangency together with the binding budget gives the demand functions. Cobb-Douglas utility yields the clean constant-expenditure-share result; perfect substitutes give corner solutions; perfect complements give a kinked, no-tangency solution; and quasi-linear preferences make demand for one good independent of income. The Lagrangian is the general engine that produces these first-order conditions.
What this chapter covers
- 01Interior optimum: tangency |MRS| = p₁/p₂ and the equimarginal rule MU₁/p₁ = MU₂/p₂
- 02Cobb-Douglas demands and constant expenditure shares
- 03Corner solutions for perfect substitutes (spend on the higher MU-per-dollar good)
- 04Kinked solution for perfect complements (ax₁ = bx₂ plus the budget)
- 05Quasi-linear demand: x₁ independent of income at an interior optimum
- 06The Lagrangian as the general utility-maximisation engine
Cobb-Douglas demand via the share rule
- 2 marksFor Cobb-Douglas U = x₁ᵃx₂ᵇ the expenditure shares are constant: a fraction a/(a+b) of income goes to good 1. Here a = 2/3, b = 1/3, so a/(a+b) = 2/3 and b/(a+b) = 1/3.
- 2 marksDemand for good 1: x₁ = [a/(a+b)]·(m/p₁) = (2/3)·(60/2) = (2/3)·30 = 20.
- 1 markDemand for good 2: x₂ = [b/(a+b)]·(m/p₂) = (1/3)·(60/4) = (1/3)·15 = 5.
- 1 markBudget check: p₁x₁ + p₂x₂ = 2·20 + 4·5 = 40 + 20 = 60 = m, so the budget binds.
Key terms
- Tangency condition
- At an interior optimum the indifference curve just touches the budget line, so |MRS| = p₁/p₂ — the consumer's trade-off matches the market's.
- Equimarginal rule
- MU₁/p₁ = MU₂/p₂: at the optimum the last dollar spent on each good yields the same marginal utility.
- Constant expenditure share
- A Cobb-Douglas property: the fraction of income spent on each good is fixed by the utility exponents, regardless of prices or income.
- Corner solution
- An optimum on an axis, where one good is not bought — typical of perfect substitutes, chosen by comparing the MU-per-dollar (a/p) ratios.
Consumer Choice & Demand FAQ
When does the tangency condition fail?
When the indifference curves are not smooth or not strictly convex. Perfect complements have a kink (no tangent), so you use ax₁ = bx₂ plus the budget; perfect substitutes are linear, giving a corner solution; and low income can force a corner under quasi-linear preferences.
How do I know whether to use the share rule or the Lagrangian?
Use the share rule only for Cobb-Douglas utility. For any other smooth utility, set up the tangency |MRS| = p₁/p₂ (or the Lagrangian first-order conditions) and solve it simultaneously with the binding budget constraint.
Exam move
Drill the Cobb-Douglas share rule until it is automatic, then practise one Lagrangian solve and one perfect-complements solve so you can switch methods by recognising the utility form. Always finish by checking the budget binds.
