General Equilibrium Notation Pareto Optimality Core Econ 2100, Fall 2020 Lecture 1, 8 October Outline 1 Logistics 2 General Equilibrium notation 3 Pareto Optimality 4 The Core Logistics Instructor: Luca Rigotti Offi ce: WWPH ????, email: luca at pitt.edu Offi ce Hours: Wednesday 1:30pm-3pm, or by appointment (send me an email) Teaching Assistant: Morag Dor Offi ce: WWPH ???, email: [email protected] Offi ce Hours: Thursday 2:30-4pm. Class Time and Location Tuesday & Thursday, 9am to 10:15am on Zoom (please leave your camera on if you can) Recitation Times Friday , 8:30am to 10:15am, on Zoom Class Webpages: http://www.pitt.edu/~luca/ECON3030/ and Canvas Announcements None Today Logistics Handouts: For each class, I will post an handout that you can use to take notes (either print and bring to class, or use computer). Useful textbooks: Kreps, Microeconomic Foundations I: Choice and Competitive Markets, Princeton University Press Mas-Collel, Whinston and Green: Microeconomic Theory. Other useful textbooks by Varian, Jehle and Reny, older Kreps. The best book: Gerard Debreu: Theory of Value. Grading Luca’sHalf: Problem sets 90%, Exam 10% Exam, 2 hours long, November 24th at 8:15 Problem Sets Weekly, some questions appear during class. DO NOT look for answers... These are for you to learn what you can and cannot do; we can help you get better only if we know your limits. Working in teams strongly encouraged, but turn in problem sets individually. Start working on exercises on your own, and then get together to discuss. Typically, a Problem Set will be due at the beginning of each Tuesday class (inlcuding the exam class). Goals for This Half Learn and understand the general equilibrium theory every academic economist needs to know. Stimulate interest in theory as a field (some of you may want to become theorists). Enable you to read papers that use general equilibrium theory, and understand seminars. We will cover the following standard topics: general equilibrium theory: economy of consumers and firms, welfare, Pareto optimality, competitive markets, Walrasian equilibrium, First and Second Welfare Theorems, existence of equilibrium, uniqueness of equilibrium, markets with uncertainty and time, Arrow-Debreu economies. In the Spring, Professor Svetlana Kosterina will cover game theory and information economics. Questions? This will be fun: let’sgo! General Equilibrium General Equilibrium Theory Models interactions between all agents in an economy; GE considers tastes, technology, and resources as the only exogenous items. The name general (as opposed to partial) equilibirum refers to the fact that all outcomes are determined endogenously. 1 How would a social planner allocate resources? 1 The main concept is Pareto effi ciency: outcomes cannot be redistributed to improve welfare. 2 How do competitive markets allocate resources? 1 The main concept is competitive equilibrium: all agents make optimal choices, and prices are such that demand and supply are equal in all markets. 1 The crucial assumption is that agents are price-taking optimizers. 3 Questions one can ask in general equilibrium: 1 when is a competitive equilibirum Pareto effi cient? First Welfare Theorem; 2 when is a Pareto effi cient outcome an equilibirum? Second Welfare Theorem; 3 when does a competitive equilibirum exist? Arrow-Debreu-McKenzie Theorem; 4 when is a competitive equilibrium unique? 5 is a competitive equilibrium robust to small perturbations of the parameters? 6 how do we get to a competitive equilibrium and what happens away from it? 7 what is different with time or uncertainty? The Economy An economy consists of I individuals and J firms who consume and produce L perfectly divisible commodities. L Each firm j = 1, ..., J is described by a production set Yj R . Each consumer i = 1, ..., I is described by four objects: L a consumption set Xi R ; preferences %i ; an individual endowment given by a vector of commodities !i Xi ; and J 2 a non-negative share i [0, 1] of the profits of each firm j. 2 Definition An economy with private ownership is a tuple I J Xi , i ,!i , i , Yj f % gi=1 f gj=1 n o If we do not need to track who owns what, we can define an economy as I J Xi , i , Yj ,! f % gi=1 f gj=1 I L n o where ! = !i R is the aggregate endowment. i=1 2 The aggregateP endowment describes the resources available to the economy. Allocations First, we define a possible outcome for the economy. Definition An allocation specifies a consumption vector xi Xi for each consumer i = 1, ..., I , 2 and a production vector yj Yj for each firm j = 1, ..., J; an allocation is 2 I J L(I +J) (x, y) i=1Xi j=1 Yj R 2 This is a long lists of what everyone consumes and what every firm produces: (x1, ..., xI , y1, ..., yJ ) X1 ... XI Y1 ... YJ 2 Notice that, by definition, consumption must be considered by each consumer (xi Xi ) and production must be possible for each firm (yj Yj ). 2 2 Is any allocation achieveable for the economy? No The definition of allocation does not reflect available resources. Initial endowments constrain what can be achieved. Feasibility Definition An allocation (x, y) is feasible if I I J xi !i + yj ≤ i=1 i=1 j=1 X X X Consumption and production are compatible with the aggregate endowment. Feasibility has nothing to do with choices by consumers or firms; it just states that a particular allocation could be consumed and produced given the economy’sresources. Why the inequality instead of an equality? Definition The set of feasible allocations is denoted by I J I J L(I +J) (x, y) i=1Xi j=1 Yj : xi ! + yj R f 2 ≤ g i=1 j=1 X X Edgeworth Box Economy There is no production. There are two goods, denoted 1 and 2, and two consumers, denoted A and B. An allocation is given by four numbers xA = (x1 A, x2 A) and xB = (x1 B , x2 B ) A feasible allocation is described as 4 2 2 (xA, xB ) R : xA R+, xB R+, and xA + xB !A + !B 2 2 2 ≤ so a feasible allocation must satisfy x1 A + x1 B !1 A + !1 B and x2 A + x2 B !2 A + !2 B ≤ ≤ Since the inequality is important, we can formaly think of the economy as having one firm with the following technology: 2 Y1 = R+ it can destroy some amounts of the goods. We can represent all feasible allocation by means of an Edgeworth Box. Draw it (with all details). Exchange Economy In an exchange economy there is no production; all consumers can do is trade with each other. This is like an Edgeworth Box economy, but there are I consumers and L goods. In an exchange economy there is a single firm that has the ability to dispose freely of any amount of any of the goods. L Formally, J = 1, and Yj = R+. Example In an exchange economy the set of feasible allocations is I LI x X1 ... XI : xi ! R f 2 ≤ g i=1 X This is a simple environment in which many results are easier to prove, and that most of the time has the same intuition as an economy with production. Edgeworth Box An Edgeworth Box is an exchange economy with two consumers and two goods. 2 2 Formally L = 2, I = 2, X1 = X2 = R+, J = 1, and Yj = R+. Representative Agent Economy There are two goods, food F and labor L, one firm, and one consumer (both are Agent). The Agent’sendowment is ! = (1, 0) (one unit of labor and no food). The Agent’stechnology transforms labor into food: 2 Y = (yL, yF ) R : yL 0, and yF f ( yL) 2 ≤ ≤ The function f ( ) is a standard production function. · The technology has the free disposal property: if y Y and y 0 y then y 0 is also an element of Y . 2 ≤ A feasible allocation is described as 4 2 (x, y) R : x R+, y Y , and x ! + y 2 2 2 ≤ Because of free disposal, one can think of the inequality as an equality without loss. Draw the production possibility set and the set of feasible consumption bundles. Feasible Allocations: Conditions Proposition If the following conditions are satisfied, the set of feasible allocations is closed and bounded.a b 1 Xi is closed and bounded below for each i = 1, ..., I. 2 Yj is closed for each j = 1, ..., J. 3 Y = j Yj is convex and satisfies 1 0L Y (inaction), P2 L 2 Y R 0L (no free-lunch), and \ + f g 3 if y Y and y = 0L then y / Y (irreversibility). 2 6 2 L Moreover, if R+ Yj (free-disposal) and there are xi Xi for each i such that 2 xi !, then the set of feasible allocations is also non empty. i ≤ P aClosed and bounded means there exists an r > 0 such that for each l = 1, ..., L, i = 1, ..., I , and j = 1, ..., J, we have xl,i < r and yl,j < r for each (x, y ) A. b 2 Bounded below means there exists an r > 0 such that xl,i > r for each l = 1, ..., L. When these conditions are not satisfied, the latter in particular, we may have nothing to talk about. So we take them for granted. Pareto Optimality (Effi ciency) A minimal effi ciency requirement is that there is no waste of resources. Definition An allocation (x, y) Pareto dominates the allocation (x, y) if x x x x i %i i and i i i for all i = 1, ..., I for some i Definition A feasible allocation (x, y) is Pareto optimal if there is no other feasible allocation (x 0, y 0) such that x x x x i0 %i i and i0 i i for all i = 1, ..., I for some i A feasible allocation is Pareto optimal if no feasible allocation Pareto dominates it.