Chapter 3 the Representative Agent Model
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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. -
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 . -
Dynamic Programming for Dummies, Parts I & II
DYNAMIC PROGRAMMING FOR DUMMIES Parts I & II Gonçalo L. Fonseca [email protected] Contents: Part I (1) Some Basic Intuition in Finite Horizons (a) Optimal Control vs. Dynamic Programming (b) The Finite Case: Value Functions and the Euler Equation (c) The Recursive Solution (i) Example No.1 - Consumption-Savings Decisions (ii) Example No.2 - Investment with Adjustment Costs (iii) Example No. 3 - Habit Formation (2) The Infinite Case: Bellman's Equation (a) Some Basic Intuition (b) Why does Bellman's Equation Exist? (c) Euler Once Again (d) Setting up the Bellman's: Some Examples (i) Labor Supply/Household Production model (ii) Investment with Adjustment Costs (3) Solving Bellman's Equation for Policy Functions (a) Guess and Verify Method: The Idea (i) Guessing the Policy Function u = h(x) (ii) Guessing the Value Function V(x). (b) Guess and Verify Method: Two Examples (i) One Example: Log Return and Cobb-Douglas Transition (ii) The Other Example: Quadratic Return and Linear Transition Part II (4) Stochastic Dynamic Programming (a) Some Basics (b) Some Examples: (i) Consumption with Uncertainty (ii) Asset Prices (iii) Search Models (5) Mathematical Appendices (a) Principle of Optimality (b) Existence of Bellman's Equation 2 Texts There are actually not many books on dynamic programming methods in economics. The following are standard references: Stokey, N.L. and Lucas, R.E. (1989) Recursive Methods in Economic Dynamics. (Harvard University Press) Sargent, T.J. (1987) Dynamic Macroeconomic Theory (Harvard University Press) Sargent, T.J. (1997) Recursive Macroeconomic Theory (unpublished, but on Sargent's website at http://riffle.stanford.edu) Stokey and Lucas does more the formal "mathy" part of it, with few worked-through applications. -
The Role of Values of Economists and Economic Agents in Economics: a Necessary Distinction Nestor Nieto
The Role of Values of Economists and Economic Agents in Economics: A Necessary Distinction Nestor Nieto To cite this version: Nestor Nieto. The Role of Values of Economists and Economic Agents in Economics: A Necessary Distinction. 2021. hal-02735869v2 HAL Id: hal-02735869 https://hal.archives-ouvertes.fr/hal-02735869v2 Preprint submitted on 28 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright The Role of Values of Economists and Economic Agents in Economics: A Necessary Distinction Nestor Lovera Nieto1 Abstract The distinction between the value judgments of economists and those of economic agents is not clear in the literature of welfare economics. In this article, I show that this distinction is important because economists have to step back and consider not only their own value judgments but also those of economic agents when they study economic phenomena. I use the subject of Interpersonal Comparisons of Utility to discuss this distinction, more specifically the analysis of Harsanyi’s impartial observer theorem, which provides a framework to justify that value judgments of economists and those of economic agents have to be properly identified. I suggest that it is essential to have a theoretical framework to make this distinction. -
Agent-Based Macroeconomics
Faculty of Business Administration and Economics Working Papers in Economics and Management No. 02-2018 January 2018 Agent-Based Macroeconomics H. Dawid D. Delli Gatti Bielefeld University P.O. Box 10 01 31 33501 Bielefeld − Germany ISSN 2196−2723 www.wiwi.uni−bielefeld.de Agent-Based Macroeconomics Herbert Dawid∗ Domenico Delli Gatti yz January 2018 This paper has been prepared as a chapter in the Handbook of Computational Economics, Volume IV, edited by Cars Hommes and Blake LeBaron. Abstract This chapter surveys work dedicated to macroeconomic analysis using an agent- based modeling approach. After a short review of the origins and general characteristics of this approach a systemic comparison of the structure and modeling assumptions of a set of important (families of) agent-based macroeconomic models is provided. The comparison highlights substantial similarities between the different models, thereby identifying what could be considered an emerging common core of macroeconomic agent-based modeling. In the second part of the chapter agent-based macroeconomic research in different domains of economic policy is reviewed. Keywords: Agent-based Macroeconomics, Aggregation, Heterogeneity, Behavioral Rules, Business Fluctuations, Economic Policy JEL Classification: C63, E17, E32, E70, 1 Introduction Starting from the early years of the availability of digital computers, the analysis of macroe- conomic phenomena through the simulation of appropriate micro-founded models of the economy has been seen as a promising approach for Economic research. In an article in the American Economic Review in 1959 Herbert Simon argued that "The very complexity that has made a theory of the decision-making process essential has made its construction exceedingly difficult. -
The Uncertainty Bellman Equation and Exploration
The Uncertainty Bellman Equation and Exploration Brendan O’Donoghue 1 Ian Osband 1 Remi Munos 1 Volodymyr Mnih 1 Abstract tions that maximize rewards given its current knowledge? We consider the exploration/exploitation prob- Separating estimation and control in RL via ‘greedy’ algo- lem in reinforcement learning. For exploitation, rithms can lead to premature and suboptimal exploitation. it is well known that the Bellman equation con- To offset this, the majority of practical implementations in- nects the value at any time-step to the expected troduce some random noise or dithering into their action value at subsequent time-steps. In this paper we selection (such as -greedy). These algorithms will even- consider a similar uncertainty Bellman equation tually explore every reachable state and action infinitely (UBE), which connects the uncertainty at any often, but can take exponentially long to learn the opti- time-step to the expected uncertainties at subse- mal policy (Kakade, 2003). By contrast, for any set of quent time-steps, thereby extending the potential prior beliefs the optimal exploration policy can be com- exploratory benefit of a policy beyond individual puted directly by dynamic programming in the Bayesian time-steps. We prove that the unique fixed point belief space. However, this approach can be computation- of the UBE yields an upper bound on the vari- ally intractable for even very small problems (Guez et al., ance of the posterior distribution of the Q-values 2012) while direct computational approximations can fail induced by any policy. This bound can be much spectacularly badly (Munos, 2014). tighter than traditional count-based bonuses that For this reason, most provably-efficient approaches to re- compound standard deviation rather than vari- inforcement learning rely upon the optimism in the face of ance. -
Macroeconomics: a Dynamic General Equilibrium Approach
Macroeconomics: A Dynamic General Equilibrium Approach Mausumi Das Lecture Notes, DSE Jan 29-Feb 19, 2019 Das (Lecture Notes, DSE) DGE Approach Jan29-Feb19,2019 1/113 Modern Macroeconomics: the Dynamic General Equilibrium (DGE) Approach Modern macroeconomics is based on a dynamic general equilibrium approach which postulates that Economic agents are continuously optimizing/re-optimizing subject to their constraints and subject to their information set. They optimize not only over their current choice variables but also the choices that would be realized in future. All agents have rational expectations: thus their ex ante optimal future choices would ex post turn out to be less than optimal if and only if their information set is incomplete and/or there are some random elements in the economy which cannot be anticipated perfectly. The agents are atomistic is the sense that they treat the market factors as exogenous in their optimization exercise. The optimal choices of all agents are then mediated through the markets to produce an equilibrium outcome for the macroeconomy (which, by construction, is also consistent with the optimal choice of each agent). Das (Lecture Notes, DSE) DGE Approach Jan29-Feb19,2019 2/113 Modern Macroeconomics: DGE Approach (Contd.) This approach is ‘dynamic’because agents are making choices over variables that relate to both present and future. This approach is ‘equilibrium’because the outcome for the macro-economy is the aggregation of individuals’equilibrium (optimal) behaviour. This approach is ‘general equilibrium’because it simultaneously takes into account the optimal behaviour of diiferent types of agents in different markets and ensures that all markets clear. -
Alternative Microfoundations for Strategic Classification
Alternative Microfoundations for Strategic Classification Meena Jagadeesan 1 Celestine Mendler-Dünner 1 Moritz Hardt 1 Abstract the context of macroeconomic policy making fueled the microfoundations program following the influential critique When reasoning about strategic behavior in a ma- of macroeconomics by Lucas in the 1970s (Lucas Jr, 1976). chine learning context it is tempting to combine Microfoundations refers to a vast theoretical project that standard microfoundations of rational agents with aims to ground theories of aggregate outcomes and popula- the statistical decision theory underlying classifi- tion forecasts in microeconomic assumptions about individ- cation. In this work, we argue that a direct combi- ual behavior. Oversimplifying a broad endeavor, the hope nation of these ingredients leads to brittle solution was that if economic policy were microfounded, it would concepts of limited descriptive and prescriptive better anticipate the response that the policy induces. value. First, we show that rational agents with perfect information produce discontinuities in the Predominant in neoclassical economic theory is the as- aggregate response to a decision rule that we of- sumption of an agent that exhaustively maximizes a util- ten do not observe empirically. Second, when any ity function on the basis of perfectly accurate informa- positive fraction of agents is not perfectly strate- tion. This modeling assumption about agent behavior under- gic, desirable stable points—where the classifier writes many celebrated results on markets, mechanisms, and is optimal for the data it entails—no longer exist. games. Although called into question by behavioral eco- Third, optimal decision rules under standard mi- nomics and related fields (e.g. -
Economics 2010C: Lecture 1 Introduction to Dynamic Programming
Economics 2010c: Lecture 1 Introduction to Dynamic Programming David Laibson 9/02/2014 Outline of my half-semester course: 1. Discrete time methods (Bellman Equation, Contraction Mapping Theorem, and Blackwell’s Sufficient Conditions, Numerical methods) Applications to growth, search, consumption, asset pricing • 2. Continuous time methods (Bellman Equation, Brownian Motion, Ito Process, and Ito’s Lemma) Applications to search, consumption, price-setting, investment, indus- • trial organization, asset-pricing Outline of today’s lecture: 1. Introduction to dynamic programming 2. The Bellman Equation 3. Three ways to solve the Bellman Equation 4. Application: Search and stopping problem 1 Introduction to dynamic programming. Course emphasizes methodological techniques and illustrates them through • applications. We start with discrete-time dynamic optimization. • Is optimization a ridiculous model of human behavior? Why or why not? • Today we’ll start with an -horizon stationary problem: • ∞ The Sequence Problem (cf. Stokey and Lucas) Notation: is the state vector at date (+1) is the flow payoff at date ( is ‘stationary’) is the exponential discount function is referred to as the exponential discount factor The discount rate is the rate of decline of the discount function, so ln = ≡− − h i Note that exp( )= and exp( )= − − Definition of Sequence Problem: Find () such that ∞ (0)= sup (+1) +1 =0 { }∞=0 X subject to +1 Γ() with 0 given. ∈ Remark 1.1 When I omit time subscripts, this implies that an equation holds for all relevant values of . In the statement above, +1 Γ() implies, ∈ +1 Γ() for all =0 1 2 ∈ Example 1.1 Optimal growth with log utility and Cobb-Douglas technology: ∞ sup ln() =0 { }∞=0 X subject to the constraints, 0+ +1 = and 0 given. -
The 2001 Trading Agent Competition
From: AAAI-02 Proceedings. Copyright © 2002, AAAI (www.aaai.org). All rights reserved. The 2001 Trading Agent Competition Michael P. Wellman∗ Amy Greenwald Peter Stone Peter R. Wurman University of Michigan Brown University AT&T Labs — Research North Carolina State University [email protected] [email protected] [email protected] [email protected] Abstract to useful observations, but all conclusions from such investi- gations must be qualified by questions of realism. Since the The 2001 Trading Agent Competition was the second in a effectiveness of one agent’s strategy depends on the strate- series of events aiming to shed light on research issues in gies of others, having one designer choose all strategies in- automating trading strategies. Based on a challenging mar- ket scenario in the domain of travel shopping, the compe- troduces a great source of fragility to the research exercise. tition presents agents with difficult issues in bidding strat- One natural approach is for researchers to cooperate, by egy, market prediction, and resource allocation. Entrants in addressing their design energy to a common problem. The 2001 demonstrated substantial progress over the prior year, Trading Agent Competition (TAC) is an attempt to induce with the overall level of competence exhibited suggesting that this cooperation, by organizing an event providing infras- trading in online markets is a viable domain for highly au- tructure, and promising external attention to the results. tonomous agents. The first TAC (Wellman et al. 2001) was held in July 2000 in Boston, in conjunction with the International Con- ference on Multiagent Systems (ICMAS-00). -
Graduate Macro Theory II: Notes on Investment
Graduate Macro Theory II: Notes on Investment Eric Sims University of Notre Dame Spring 2011 1 Introduction These notes introduce and discuss modern theories of firm investment. While much of this is done as a decision rule problem of the firm, it is easily incorporated into a general equilibrium structure. 2 Tobin's Q Jim Tobin (1969) developed an intuitive and celebrated theory of investment. He reasoned that if the market value of physical capital of a firm exceeded its replacement cost, then capital has more value \in the firm” (the numerator) than outside the firm (the denominator). Formally, Tobin's Q is defined as: Market Value of Firm Capital Q = (1) Replacement Cost of Capital Tobin reasoned that firms should accumulate more capital when Q > 1 and should draw down their capital stock when Q < 1. That is, net investment in physical capital should depend on where Q is in relation to one. How would one measure Q in the data? Typically this is done by using the value of the stock market in the numerator and data on corporate net worth in the denominator { here net worth is defined as the replacement cost of capital goods. For example, suppose that a company is in the business of bulldozing. It owns a bunch of bulldozers. Suppose that it owns 100 bulldozers, which cost 100 each to replace. This means its replacement cost is 10; 000. Suppose it has 1000 outstanding shares of stock, valued at 15 per share. The market value of the firm is thus 15; 000 and the replacement cost is 10; 000, so Tobin's Q is 1.5.