EC451

This is the micro course for LSE.

Revealed Preference

Choice - Preference - Utility

Open Ball

EC451_Macro

Here are your one-hour final review notes. Focus on the definitions and the core logic of the proofs. Good luck!

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Lecture 2: Choice & Preferences 🧐

WARP (Weak Axiom of Revealed Preference)

• Definition: If bundle x is chosen when y is affordable, then y cannot be chosen when x is affordable.

• Proof Skill: Almost always a proof by contradiction. Assume WARP is violated and show the logical inconsistency. Find two choice sets where the preference between two items is reversed.

Key Preference Definitions

• Rationality: Complete (can always compare any two bundles) AND Transitive (if A ≻ B and B ≻ C, then A ≻ C).

• Continuity: No “jumps.” The set of bundles “at least as good as x” is a closed set.

• Monotonicity: More is better.

• Convexity: Averages are at least as good as extremes. The utility function is quasi-concave.

• Strict Convexity: Averages are strictly better than extremes. Guarantees a unique optimal bundle.

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Lecture 3: Consumer Problem & Duality 🛒

How to Solve Problems

• UMP (find Marshallian Demand x(p,w)):

  1. Tangency: MRS=MU2​MU1​​=p2​p1​​

  2. Budget: p1​x1​+p2​x2​=w

  3. Solve the two equations for x1​ and x2​.

• EMP (find Hicksian Demand h(p,U)):

  1. Tangency: MRS=MU2​MU1​​=p2​p1​​ (This condition is the same!)

  2. Utility: u(x1​,x2​)=U

  3. Solve the two equations for x1​ and x2​.

Key Proofs to Know

• Walras’s Law (Prop 4): Proof by Contradiction.

  • Assume the consumer does not spend their whole budget (p⋅x<w).

  • By local non-satiation, there must be a better bundle nearby that is still affordable.

  • This contradicts that the original choice was optimal.

• Uniqueness of Solution with Strict Convexity (Prop 3): Proof by Contradiction.

  • Assume there are two distinct optimal bundles, x and y.

  • By strict convexity, their average, z = 0.5x + 0.5y, must be strictly preferred.

  • Since the budget set is convex, z is also affordable.

  • This contradicts that x and y were optimal.

• Duality (Prop 15): Proof by Contradiction.

  • UMP solution ⇒ EMP solution: Assume UMP solution x∗ is NOT the cheapest way to get utility U∗=u(x∗). Find a cheaper bundle x′. Show that x′ was affordable in the UMP and that a nearby bundle x′′ would have given higher utility, contradicting that x∗ was the UMP solution.

  • EMP solution ⇒ UMP solution: Assume EMP solution h∗ does NOT maximize utility for budget w∗=p⋅h∗. Find a better bundle x′. Show that a slightly scaled-down bundle x′′ achieves the target utility for a lower cost, contradicting that h∗ was the EMP solution.

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Lecture 4: Expected Utility Theory 🎲

Your only task is to use the axioms to prove simple results.

The Three Axioms

• Rationality: Preferences over lotteries are complete and transitive.

• Continuity: No “heaven or hell” outcomes. Trade-offs are always possible.

• Independence: If you prefer lottery p to q, you also prefer a mix of p and r to the same mix of q and r. The r part is irrelevant and “cancels out.”

Key Proof Skill (for Lemma 5)

The main trick is to cleverly apply the Independence Axiom.

• To prove: If p≻q, then p≻αp+(1−α)q.

• Proof: Start with the premise p≻q. Apply the independence axiom, mixing both sides with the lottery p:

  αp+(1−α)p≻αq+(1−α)p.

  The left side simplifies to p. The result is proven. This pattern is key.