Notes on Macroeconomic Theory
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Chapter 4 — Complete Markets
Chapter 4 — Complete Markets The Basic Complete-Market Problem There are N primary assets each with price pi and payoff Xsi is state s. There are also a complete set of S pure state securities each of which pays $1 in its own state s and 0 otherwise. These pure state securities are often called Arrow-Debreu securities. Assuming there is no arbi- trage, the price of the pure state s security must be qs, the state price. In a complete market like this, investors will be indifferent between choosing a portfolio from the S state securities and all N + S securities. A portfolio holding n shares of primary assets and η1 shares of the Arrow-Debreu securities will provide consumption c = Xn + η1 in the states at time 1. A portfolio holding η2 ≡ Xn + η1 of just the Arrow-Debreu securities obviously gives the same consumption. The costs of the two portfolios are q′c and p′n + q′η1. From the definition of c and the no-arbitrage result that p′ = q′X, these are obviously the same. Of course in equilibrium the original primary assets must actually be held. However this can be done by financial intermediaries like mutual funds. They buy all the primary shares finan- cing the purchases by issuing the Arrow-Debreu securities. By issuing h = Xn Arrow-Debreu securities where n is the aggregate supply of the primary assets, the intermediary is exactly hedged, and their role can be ignored. So with no loss of generality, we can therefore assume that individual investors only purchases Arrow-Debreu securities ignoring the primary assets. -
Calculating the Value Function When the Bellman Equation Cannot Be
Journal of Economic Dynamics & Control 28 (2004) 1437–1460 www.elsevier.com/locate/econbase Investment under uncertainty: calculating the value function when the Bellman equation cannot besolvedanalytically Thomas Dangla, Franz Wirlb;∗ aVienna University of Technology, Theresianumgasse 27, A-1040 Vienna, Austria bUniversity of Vienna, Brunnerstr.˝ 72, A-1210 Vienna, Austria Abstract The treatment of real investments parallel to ÿnancial options if an investment is irreversible and facing uncertainty is a major insight of modern investment theory well expounded in the book Investment under Uncertainty by Dixit and Pindyck. Thepurposeof this paperis to draw attention to the fact that many problems in managerial decision making imply Bellman equations that cannot be solved analytically and to the di6culties that arise when approximating therequiredvaluefunction numerically.This is so becausethevaluefunctionis thesaddlepoint path from the entire family of solution curves satisfying the di7erential equation. The paper uses a simple machine replacement and maintenance framework to highlight these di6culties (for shooting as well as for ÿnite di7erence methods). On the constructive side, the paper suggests and tests a properly modiÿed projection algorithm extending the proposal in Judd (J. Econ. Theory 58 (1992) 410). This fast algorithm proves amendable to economic and stochastic control problems(which onecannot say of theÿnitedi7erencemethod). ? 2003 Elsevier B.V. All rights reserved. JEL classiÿcation: D81; C61; E22 Keywords: Stochastic optimal control; Investment under uncertainty; Projection method; Maintenance 1. Introduction Without doubt, thebook of Dixit and Pindyck (1994) is oneof themost stimulating books concerning management sciences. The basic message of this excellent book is the following: Considering investments characterized by uncertainty (either with ∗ Corresponding author. -
QR43, Introduction to Investments Class Notes, Fall 2004 I. Prices, Portfolios, and Arbitrage
QR43, Introduction to Investments Class Notes, Fall 2004 I. Prices, Portfolios, and Arbitrage A. Introduction 1. What are investments? You pay now, get repaid later. But can you be sure how much you will be repaid? Time and uncertainty are essential. Real investments, such as machinery, require an input of re- sources today and deliver an output of resources later. Financial investments,suchasstocksandbonds,areclaimstotheoutput produced by real investments. For example: A company issues shares of stock to investors. The proceeds of the share issue are used to build a factory and to hire workers. The factory produces goods, and the sales proceeds give the company revenue or sales. After paying the workers and set- ting aside money to replace the machinery and buildings when they wear out (depreciation), the company is left with earnings.The company keeps some earnings to make new real investments (re- tained earnings), and pays the rest to shareholders in the form of dividends. Later, the company decides to expand production and needs to raisemoremoneytofinance its expansion. Rather than issue new shares, it decides to issue corporate bonds that promise fixed pay- ments to bondholders. The expanded production increases revenue. However, interest payments on the bonds, and the setting aside of 1 money to repay the principal (amortization), are subtracted from revenue. Thus earnings may increase or decrease depending on the success of the company’s expansion. Financial investments are also known as capital assets or finan- cial assets. The markets in which capital assets are traded are known as capital markets or financial markets. Capital assets are commonly divided into three broad categories: Fixed-income securities promise to make fixed payments in the • future. -
A Macroeconomic Model with Financial Panics Gertler, Mark, Nobuhiro Kiyotaki, and Andrea Prestipino
K.7 A Macroeconomic Model with Financial Panics Gertler, Mark, Nobuhiro Kiyotaki, and Andrea Prestipino Please cite paper as: Gertler, Mark, Nobuhior Kiyotaki, and Andrea Prestipino (2017). A Macroeconomic Model with Financial Panics. International Finance Discussion Papers 1219. https://doi.org/10.17016/IFDP.2017.1219 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1219 December 2017 Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1219 December 2017 A Macroeconomic Model with Financial Panics Mark Gertler, Nobuhiro Kiyotaki and Andrea Prestipino NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com. A Macroeconomic Model with Financial Panics Mark Gertler, Nobuhiro Kiyotaki and Andrea Prestipino NYU, Princeton and Federal Reserve Board December, 2017 Abstract This paper incorporates banks and banking panics within a conven- tional macroeconomic framework to analyze the dynamics of a financial crisis of the kind recently experienced. We are particularly interested in characterizing the sudden and discrete nature of the banking panics as well as the circumstances that makes an economy vulnerable to such panics in some instances but not in others. Having a conventional macroeconomic model allows us to study the channels by which the crisis a¤ects real activity and the e¤ects of policies in containing crises. -
The Macroeconomic Model
Briefing paper No. 5 The macroeconomic model October 2013 © Crown copyright 2013 You may re-use this information (excluding logos) free of charge in any format or medium, under the terms of the Open Government Licence. To view this licence, visit http://www.nationalarchives.gov.uk/doc/open- government-licence/ or write to the Information Policy Team, The National Archives, Kew, London TW9 4DU, or e-mail: [email protected]. Any queries regarding this publication should be sent to us at: [email protected] ISBN 978-1-909790-28-5 PU1558 Contents Chapter 1 Introduction.................................................................................. 1 Chapter 2 Expenditure components of GDP ................................................... 9 Chapter 3 The labour market ...................................................................... 63 Chapter 4 Prices, costs and earnings ........................................................... 71 Chapter 5 Balance sheets and income accounts ........................................... 95 Chapter 6 Public sector ............................................................................. 145 Chapter 7 Domestic financial sector........................................................... 183 Chapter 8 North Sea................................................................................. 187 Annex A Glossary of Winsolve notations.................................................. 193 Annex B Winsolve model code............................................................... -
The Soft Budget Constraint
Acta Oeconomica, Vol. 64 (S1) pp. 25–79 (2014) DOI: 10.1556/AOecon.64.2014.S1.2 THE SOFT BUDGET CONSTRAINT An Introductory Study to Volume IV of the Life’s Work Series* János KORNAI** The author’s ideas on the soft budget constraint (SBC) were first expressed in 1976. Much progress has been made in understanding the problem over the ensuing four decades. The study takes issue with those who confine the concept to the process of bailing out loss-making socialist firms. It shows how the syndrome can appear in various organizations and forms in many spheres of the economy and points to the various means available for financial rescue. Single bailouts do not as such gener- ate the SBC syndrome. It develops where the SBC becomes built into expectations. Special heed is paid to features generated by the syndrome in rescuer and rescuee organizations. The study reports on the spread of the syndrome in various periods of the socialist and the capitalist system, in various sectors. The author expresses his views on normative questions and on therapies against the harmful effects. He deals first with actual practice, then places the theory of the SBC in the sphere of ideas and models, showing how it relates to other theoretical trends, including institutional and behav- ioural economics and theories of moral hazard and inconsistency in time. He shows how far the in- tellectual apparatus of the SBC has spread in theoretical literature and where it has reached in the process of “canonization” by the economics profession. Finally, he reviews the main research tasks ahead. -
Lecture 4 " Theory of Choice and Individual Demand
Lecture 4 - Theory of Choice and Individual Demand David Autor 14.03 Fall 2004 Agenda 1. Utility maximization 2. Indirect Utility function 3. Application: Gift giving –Waldfogel paper 4. Expenditure function 5. Relationship between Expenditure function and Indirect utility function 6. Demand functions 7. Application: Food stamps –Whitmore paper 8. Income and substitution e¤ects 9. Normal and inferior goods 10. Compensated and uncompensated demand (Hicksian, Marshallian) 11. Application: Gi¤en goods –Jensen and Miller paper Roadmap: 1 Axioms of consumer preference Primal Dual Max U(x,y) Min pxx+ pyy s.t. pxx+ pyy < I s.t. U(x,y) > U Indirect Utility function Expenditure function E*= E(p , p , U) U*= V(px, py, I) x y Marshallian demand Hicksian demand X = d (p , p , I) = x x y X = hx(px, py, U) = (by Roy’s identity) (by Shepard’s lemma) ¶V / ¶p ¶ E - x - ¶V / ¶I Slutsky equation ¶p x 1 Theory of consumer choice 1.1 Utility maximization subject to budget constraint Ingredients: Utility function (preferences) Budget constraint Price vector Consumer’sproblem Maximize utility subjet to budget constraint Characteristics of solution: Budget exhaustion (non-satiation) For most solutions: psychic tradeo¤ = monetary payo¤ Psychic tradeo¤ is MRS Monetary tradeo¤ is the price ratio 2 From a visual point of view utility maximization corresponds to the following point: (Note that the slope of the budget set is equal to px ) py Graph 35 y IC3 IC2 IC1 B C A D x What’swrong with some of these points? We can see that A P B, A I D, C P A. -
Lecture Notes General Equilibrium Theory: Ss205
LECTURE NOTES GENERAL EQUILIBRIUM THEORY: SS205 FEDERICO ECHENIQUE CALTECH 1 2 Contents 0. Disclaimer 4 1. Preliminary definitions 5 1.1. Binary relations 5 1.2. Preferences in Euclidean space 5 2. Consumer Theory 6 2.1. Digression: upper hemi continuity 7 2.2. Properties of demand 7 3. Economies 8 3.1. Exchange economies 8 3.2. Economies with production 11 4. Welfare Theorems 13 4.1. First Welfare Theorem 13 4.2. Second Welfare Theorem 14 5. Scitovsky Contours and cost-benefit analysis 20 6. Excess demand functions 22 6.1. Notation 22 6.2. Aggregate excess demand in an exchange economy 22 6.3. Aggregate excess demand 25 7. Existence of competitive equilibria 26 7.1. The Negishi approach 28 8. Uniqueness 32 9. Representative Consumer 34 9.1. Samuelsonian Aggregation 37 9.2. Eisenberg's Theorem 39 10. Determinacy 39 GENERAL EQUILIBRIUM THEORY 3 10.1. Digression: Implicit Function Theorem 40 10.2. Regular and Critical Economies 41 10.3. Digression: Measure Zero Sets and Transversality 44 10.4. Genericity of regular economies 45 11. Observable Consequences of Competitive Equilibrium 46 11.1. Digression on Afriat's Theorem 46 11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes 47 11.3. Brown and Matzkin: Testable Restrictions On Competitve Equilibrium 48 12. The Core 49 12.1. Pareto Optimality, The Core and Walrasian Equiilbria 51 12.2. Debreu-Scarf Core Convergence Theorem 51 13. Partial equilibrium 58 13.1. Aggregate demand and welfare 60 13.2. Production 61 13.3. Public goods 62 13.4. Lindahl equilibrium 63 14. -
Tipping Points in Macroeconomic Agent-Based Models
Tipping points in macroeconomic Agent-Based models Stanislao Gualdi,1 Marco Tarzia,2 Francesco Zamponi,3 and Jean-Philippe Bouchaud4 1 Universit´ePierre et Marie Curie - Paris 6, Laboratoire de Physique Th´eoriquede la Mati`ere Condens´ee, 4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France and IRAMIS, CEA-Saclay, 91191 Gif sur Yvette Cedex, France ∗ 2 Universit´ePierre et Marie Curie - Paris 6, Laboratoire de Physique Th´eoriquede la Mati`ere Condens´ee, 4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France 3Laboratoire de Physique Th´eorique, Ecole´ Normale Sup´erieure, UMR 8549 CNRS, 24 Rue Lhomond, 75231 Paris Cedex 05, France 4CFM, 23 rue de l'Universit´e,75007 Paris, France, and Ecole Polytechnique, 91120 Palaiseau, France. The aim of this work is to explore the possible types of phenomena that simple macroeconomic Agent-Based models (ABM) can reproduce. Our motivation is to understand the large macro- economic fluctuations observed in the \Mark I" ABM devised by D. Delli Gatti and collaborators. Our major finding is the existence of a first order (discontinuous) phase transition between a \good economy" where unemployment is low, and a \bad economy" where unemployment is high. We show that this transition is robust against many modifications of the model, and is induced by an asymmetry between the rate of hiring and the rate of firing of the firms. This asymmetry is induced, in Mark I, by the interest rate. As the interest rate increases, the firms become more and more reluctant to take further loans. The unemployment level remains small until a tipping point beyond which the economy suddenly collapses. -
From Single-Agent to Multi-Agent Reinforcement Learning: Foundational Concepts and Methods Learning Theory Course
From Single-Agent to Multi-Agent Reinforcement Learning: Foundational Concepts and Methods Learning Theory Course Gon¸caloNeto Instituto de Sistemas e Rob´otica Instituto Superior T´ecnico May 2005 Abstract Interest in robotic and software agents has increased a lot in the last decades. They allow us to do tasks that we would hardly accomplish otherwise. Par- ticularly, multi-agent systems motivate distributed solutions that can be cheaper and more efficient than centralized single-agent ones. In this context, reinforcement learning provides a way for agents to com- pute optimal ways of performing the required tasks, with just a small in- struction indicating if the task was or was not accomplished. Learning in multi-agent systems, however, poses the problem of non- stationarity due to interactions with other agents. In fact, the RL methods for the single agent domain assume stationarity of the environment and cannot be applied directly. This work is divided in two main parts. In the first one, the reinforcement learning framework for single-agent domains is analyzed and some classical solutions presented, based on Markov decision processes. In the second part, the multi-agent domain is analyzed, borrowing tools from game theory, namely stochastic games, and the most significant work on learning optimal decisions for this type of systems is presented. ii Contents Abstract 1 1 Introduction 2 2 Single-Agent Framework 5 2.1 Markov Decision Processes . 6 2.2 Dynamic Programming . 10 2.2.1 Value Iteration . 11 2.2.2 Policy Iteration . 12 2.2.3 Generalized Policy Iteration . 13 2.3 Learning with Model-free methods . -
2. Budget Constraint.Pdf
Engineering Economic Analysis 2019 SPRING Prof. D. J. LEE, SNU Chap. 2 BUDGET CONSTRAINT Consumption Choice Sets . A consumption choice set, X, is the collection of all consumption choices available to the consumer. n X = R+ . A consumption bundle, x, containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector xx =(1 ,..., xn ) ∈ X n . Commodity price vector pp =(1 ,..., pn ) ∈R+ 1 Budget Constraints . Q: When is a bundle (x1, … , xn) affordable at prices p1, … , pn? • A: When p1x1 + … + pnxn ≤ m where m is the consumer’s (disposable) income. The consumer’s budget set is the set of all affordable bundles; B(p1, … , pn, m) = { (x1, … , xn) | x1 ≥ 0, … , xn ≥ 0 and p1x1 + … + pnxn ≤ m } . The budget constraint is the upper boundary of the budget set. p1x1 + … + pnxn = m 2 Budget Set and Constraint for Two Commodities x2 Budget constraint is m /p2 p1x1 + p2x2 = m. m /p1 x1 3 Budget Set and Constraint for Two Commodities x2 Budget constraint is m /p2 p1x1 + p2x2 = m. the collection of all affordable bundles. Budget p1x1 + p2x2 ≤ m. Set m /p1 x1 4 Budget Constraint for Three Commodities • If n = 3 x2 p1x1 + p2x2 + p3x3 = m m /p2 m /p3 x3 m /p1 x1 5 Budget Set for Three Commodities x 2 { (x1,x2,x3) | x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 and m /p2 p1x1 + p2x2 + p3x3 ≤ m} m /p3 x3 m /p1 x1 6 Opportunity cost in Budget Constraints x2 p1x1 + p2x2 = m Slope is -p1/p2 a2 Opp. -
Some Political Economy of Monetary Rules
SUBSCRIBE NOW AND RECEIVE CRISIS AND LEVIATHAN* FREE! “The Independent Review does not accept “The Independent Review is pronouncements of government officials nor the excellent.” conventional wisdom at face value.” —GARY BECKER, Noble Laureate —JOHN R. MACARTHUR, Publisher, Harper’s in Economic Sciences Subscribe to The Independent Review and receive a free book of your choice* such as the 25th Anniversary Edition of Crisis and Leviathan: Critical Episodes in the Growth of American Government, by Founding Editor Robert Higgs. This quarterly journal, guided by co-editors Christopher J. Coyne, and Michael C. Munger, and Robert M. Whaples offers leading-edge insights on today’s most critical issues in economics, healthcare, education, law, history, political science, philosophy, and sociology. Thought-provoking and educational, The Independent Review is blazing the way toward informed debate! Student? Educator? Journalist? Business or civic leader? Engaged citizen? This journal is for YOU! *Order today for more FREE book options Perfect for students or anyone on the go! The Independent Review is available on mobile devices or tablets: iOS devices, Amazon Kindle Fire, or Android through Magzter. INDEPENDENT INSTITUTE, 100 SWAN WAY, OAKLAND, CA 94621 • 800-927-8733 • [email protected] PROMO CODE IRA1703 Some Political Economy of Monetary Rules F ALEXANDER WILLIAM SALTER n this paper, I evaluate the efficacy of various rules for monetary policy from the perspective of political economy. I present several rules that are popular in I current debates over monetary policy as well as some that are more radical and hence less frequently discussed. I also discuss whether a given rule may have helped to contain the negative effects of the recent financial crisis.