Foundations: preferences, choice, utility. Preferences as the basis for choice. Formalization of preferences as a binary relation endowed with some properties. Induced choice. Utility as a way to represent preferences. Representation theorems for finite and countable sets of choice. Additional requirements for uncountable sets (continuity is sufficient); representation theorems for uncountable sets.

The basics of the consumer problem. The formulation of the consumer problem (CP). The budget set. Marshallian demand and indirect utility. Properties of the solution(s), under appropriate assumptions on preferences. Berge’s theorem; properties of the Marshallian demand and of the indirect utility. Inverting the indirect utility to recover the utility function. Roy’s identity; economic interpretation. From Roy’s identity to Marshallian demand.

The dual consumer problem. The formulation of the dual of the consumer problem (DCP). Hicksian demand and expenditure function. Links between the CP and the DCP, and between the indirect utility and the expenditure function. Shepard’s lemma. The Slutsky matrix.

Classical demand theory and revealed preferences. Hicksian demand as compensated demand. Consistency between observed individual demand functions and the theoretical construct. Consistency between observed individul choices and the theoretical construct; the generalized axiom of revealed preferences (GARP) and Afriat’s theorem.

Uncertainty foundations. What does uncertainty mean. Two approaches to choice under uncertainty. The special structure imparted by uncertainty to the decision problem: independence. The attitude towards risk and the measures of riskiness (specializing to the case of single-dimensional outcomes). Properties of the utility function, risk attitude and riskiness.

The Competitive Firm. The standard formulation of the problem, and the restrictive assumptions that it entails. Various assumptions on the production possibilities set (PPS). Properties of the solution(s) to the problem and of its value, i.e. the profit function. The dual firm problem. The cost function, conditional factor demand. Properties of the cost function. Shepard’s lemma.

Walrasian equilibrium without time and uncertainty: background, definitions and properties • An important notion of “goodness” of economic systems: Pareto efficiency • Where would an unrestricted and costless trading process with a lot of information lead? Barter equilibria • Barter equilibria are Pareto Efficient • Representation of barter equilibria for a two-person, two good economy via the Edgeworth box. The contract curve. • Barter equilibria are Pareto efficient, but not all the Pareto efficient allocations are barter equilibria given the initial endowments. The Core of an exchange economy. • A different trading protocol, with much less information: all trades occur in markets and are mediated by a price. Does it lead to different outcomes compared to barter? • Formal definition of an economy (with production and pure-exchange) and of the Walrasian Equilibrium. • The properties of a Walrasian equilibrium (assuming that there is one).

Walrasian equilibrium without time and uncertainty: examples and existence • Detailed construction of Walrasian equilibrium for two simple economies, with and without production and firms.In an Edgeworth box, a first hint of the relationship between Walrasian equilibria and Pareto efficiency. • Proofs of existence of Walrasian equilibria, following two approaches: with consumers and firms explicitly in the picture, in a generalized game; with consumers and firms implicitly subsumed in an aggregate excess demand correspondence. • The first approach, in three steps: a general theorem of existence for a bounded generalized game; applying the theorem to a bounded economy; showing that, under appropriate conditions, an unbounded economy has the same equilibria of the bounded one. • The second approach. • Appendix: quick reminder of some mathematical notions needed in the proofs: continuity of a correspondence, Berge's theorem, Kakutani fixed-point theorem.

Welfare Analysis: Pareto efficiency of Walrasian equilibrium and implementation of Pareto efficient allocation as Walrasian (quasi) equilibrium • Remember the question: Does a trading process mediated only by prices lead to different (and less efficient) outcomes compared to an unrestricted, costless and information-intensive barter process? We provide here a detailed answer. • The First Theorem of Welfare Economics: under appropriate conditions, any Walrasian equilibrium is Pareto efficient. A proof for a pure-exchange economy and a proof for an economy with production. Discussion of the conditions required. • The Second Theorem of Welfare Economics: under appropriate conditions, and allowing a (costless) redistribution of endowments and firms' ownership, any Pareto efficient allocation can be obtained, using the markets, as Walrasian quasi-equilibrium. Under further conditions, a Walrasian quasi-equilibrium is a Walrasian equilibrium. Discussion of the conditions required. The importance of market completeness. • Walrasian equilibria and the Core: any Walrasian equilibrium is in the Core, but not all Core allocations can be obtained as Walrasian equilibria (without redistribution). • Core allocations shrinking in the (appropriate) limit to Walrasian equilibrium allocation. Replica economies. • Externalities as an example of market incompleteness. Welfare properties of Walrasian equilibrium with externalities: the equilibrium is no longer necessarily Pareto efficient. The Lindahl equilibrium, as a way to restore efficiency. • How to find the set of all Pareto efficient allocations. • Appendix: quick reminder of some mathematical notions needed in the proofs: the Separating Hyperplane theorem.

General Equilibrium with Time and Uncertainty • The specification of a time/uncertainty/information structure. The idea of contingent commodities: interpreting the same physical good or service at different points in time and in different contingencies as different goods. Extending the markets to these contingent commodities. • If there are, all at once, markets for all possible contingent commodities, the properties of the Walrasian equilibrium survive intact: time and uncertainty can be encompassed in the formal theory developed so far, with virtual no changes. • Dynamic economies: not all markets are open at the beginning, but securities can be traded to move resources across time and contingencies, with trading of goods taking place in contingent spot markets. Dynamic economies have a sequence of budget constraints. • Market completeness achieved through securities trading: when this is possible, when it is not. • The simplest case of Arrow-Debreu securities. Equivalence between the economy with all contingent markets at once and a dynamic economy with a number of Arrow-Debreu securities equal to the number of contingencies (minus 1). Security prices indeterminacy if securities payout is in terms of the numeraire (nominal securities); the indeterminacy is solved if securities payout is in terms of a physical good (real securities). • The case of general securities. The notion of market completeness for general securities. Again, the problem of indeterminacy. • The problems that might arise with market incompleteness, in a simple example.

Arrow's Impossibility Theorem • Is it possible to aggregate the preferences of the agents in a society in such a way that they produce a well defined (i.e. complete and transitive) societal preference? • The answer is trivially positive if we impose no restrictions on the societal preference: pick the preference of anyone agent in the society and make her a dictator, so that her preferences are also the societal one! • But suppose we impose some restrictions to the societal preference. For example, we do not want a dictator, and we want that whenever \textit{everybody} in the society prefers x to y, also the societal preference considers x better than y i.e. the aggregation satisfies unanimity. Is the answer still positive? • Under an additional requirement, known as independence of irrelevant alternatives, Arrow proved that the answer is negative. This is a remarkable result in the theory of social choice. • Proof of the theorem. • Some examples of rules to aggregate individual preferences (they necessarily fail at least one of Arrow's requirements). • Appendix: a quick reminder of the notion and properties of orderings.

Recitals and training for the exam