M14.L01 — Optimal Consumer Choice: Utility Maximisation

Module: Module 14 — Consumer Choice and Demand Lesson: L01 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

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Learning Objective

Solve the consumer's utility maximisation problem to find optimal consumption bundles.

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Utility Maximisation Problem

In Australian supermarkets, consumers face budget constraints when choosing between goods like Vegemite (good 1) and Tim Tams (good 2). The utility maximisation problem is:

max U(x₁, x₂) subject to p₁x₁ + p₂x₂ ≤ m

At an interior solution, the optimal bundle satisfies the tangency condition:

MRS = p₁/p₂

Where MRS is the marginal rate of substitution (slope of indifference curve). For Cobb-Douglas utility U(x₁,x₂) = x₁ᵃx₂ᵇ, this becomes:

(a/x₁)/(b/x₂) = p₁/p₂

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Worked Example

Optimising Aussie Snack Consumption

Suppose a student has $20 to spend on snacks (Vegemite: $5, Tim Tams: $4) with utility U = x₁⁰·⁵x₂⁰·⁵:

1. Set up Lagrangian: ℒ = x₁⁰·⁵x₂⁰·⁵ + λ(20 - 5x₁ - 4x₂)
2. FOCs: ∂ℒ/∂x₁ = 0.5x₁⁻⁰·⁵x₂⁰·⁵ - 5λ = 0 ∂ℒ/∂x₂ = 0.5x₁⁰·⁵x₂⁻⁰·⁵ - 4λ = 0
3. Solve for λ and set equal: (x₂/x₁) = 5/4
4. Substitute into budget: 5x₁ + 4(1.25x₁) = 20 → x₁* = 2, x₂* = 2.5

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Common Misconception

"Corner solutions are rare in real-world consumer choice"

Many Australian consumers regularly choose corner solutions - like buying only petrol (no public transport) in rural areas or only streaming services (no DVDs) in cities.

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Key Takeaways

• The optimal bundle occurs where the budget line is tangent to the highest indifference curve
• MRS = price ratio at interior solutions
• Lagrangian method systematically solves constrained optimization
• Corner solutions occur when MRS ≠ price ratio at zero consumption
• Geometrically, we seek the highest IC touching the budget line

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Practice

1. Solve for optimal (x₁,x₂) when U = x₁x₂², p₁=2, p₂=3, m=60
2. Graphically show why MRS ≠ p₁/p₂ leads to corner solutions
3. Find conditions when a consumer would buy only good 1

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Further Resources

• 📺 Utility Maximisation (Uni of Adelaide) — Australian lecturer explains with local examples
• 📚 Varian Ch.5: Choice — Textbook reference

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Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics