Theory of Income · UChicago · Fall 2026 (Alvarez)

Companion Course: Welfare Theorems, Overlapping Generations & Aggregation

25 lessons covering lecture folders 1 (Welfare Theorems), 2 (OLG Models) and 3 (Aggregation, Social Planner). Each lesson takes about 30–40 minutes and ends with exercises that have hints and full solutions.
How to use this. Do the lessons in order; each one builds on the last. Attempt every exercise before opening the solution, because effortful recall is what makes it stick. The figures are interactive: drag the points and move the sliders. When you return to a topic, start with the quiz rather than re-reading. The cheatsheets are for quick review before exams. Ask your teacher (Claude) about anything unclear.
Part I · Welfare Theorems (Slides 1)
  1. 01The economy — commodities, households, firms, ownership, feasibility
  2. 02Prices and competitive equilibrium — definition, Walras' law, Edgeworth box
  3. 03Pareto optimality and local non-satiation — PO, contract curve, LNS
  4. 04The First Welfare Theorem — the proof, line by line
  5. 05When the 1WT fails — externalities, market power, incomplete markets, beliefs
  6. 06The Second Welfare Theorem — convexity and separating hyperplanes
  7. 07Arrow's remark and decentralization — cheapest points, transfers, non-convex failures
Part II · Overlapping Generations (Class Note II)
  1. 08The OLG environment — generations, endowments, no-trade equilibrium
  2. 09Prices, interest rates and savings — the log example
  3. 10Is the OLG equilibrium efficient? — best symmetric allocation, three cases
  4. 11Pay-as-you-go social security — when it Pareto-improves
  5. 12Why the 1WT fails in OLG — the value of aggregate consumption
  6. 13The growing OLG economy — 1+r vs (1+g)(1+n), PAYG with growth
Part III · OLG Extensions (Class Note III)
  1. 14Privatizing social security — government debt and Ricardian equivalence
  2. 15OLG with capital — capital accumulation and social security
  3. 16Heterogeneity and long lives — aggregation, distribution, N-period agents
Part IV · Perpetual Youth (Blanchard–Yaari slides)
  1. 17Perpetual youth: households — death rates, annuities, consumption rule
  2. 18Perpetual youth: aggregation — human wealth, the C–V system, endowment equilibrium
  3. 19Perpetual youth: capital — phase diagram, steady state, comparative statics
  4. 20Perpetual youth: fiscal policy — debt dynamics, (non-)Ricardian equivalence
Part V · Aggregation and the Social Planner (Class Note 3)
  1. 21The planner problem and Pareto weights — utility possibility set, PO ⇔ weighted planner
  2. 22The representative agent — u(x;λ), exchange economies, planner FOCs
  3. 23Negishi weights — welfare theorems via FOCs, λ = 1/μ, γ = p
  4. 24Gorman aggregation — identical homothetic preferences, prices independent of wealth
  5. 25When aggregation fails, and how to rescue it — counterexamples, Stone–Geary, translated CES
Reference
  1. RGlossary & notation traps
  2. RWelfare theorems cheatsheet
  3. ROLG cheatsheet
  4. RPerpetual youth cheatsheet
  5. RAggregation & planner cheatsheet
📝 Exam prep. The exercises follow the formats used on past Theory of Income exams: "True or false? Justify." blocks and guided multi-part derivations with point weights. Tags show where each one comes from: Past exam adapted from a real exam, Problem set from past problem sets (2022–2025), Exam-inspired written in the same style.