H: Economic Modelling: The Art Of Smooth Pasting (harwood Fundamentals Of Pure And Applied Economics) (volume 1)

This book aims to widen the understanding of stochastic dynamic choice and equilibrium models. It offers a simplified and heuristic exposition of the theory of Brownian motion and its control or regulation, rendering such methods more accessible to economists who do not require a detailed, mathematical treatment of the subject. The main mathematical ideas are presented in a context which with which economists will be familiar. Using a binomial approach to Brownian motion, the mathematics is reduced to simple algebra, progressing to some equally simple limits. The starting point of the calculus of Brownian motion - 'Ito's Lemma' - emerges by analogy with the economics of risk-aversion. Conditions for the optimal regulation of Brownian motion, including the important, but often mysterious, 'smooth pasting' condition, are derived in a similar way. Each theoretical derivation is illustrated by developing a significant economic application, drawn mainly from recent research in macroeconomics and international economics. Cover......Page 1 Title......Page 4 Copyright......Page 5 Original Title......Page 6 Original Copyright......Page 7 Contents......Page 8 Introduction to the Series......Page 10 Preface......Page 12 1. Brownian Motion......Page 14 1.1. Random Walk Representation......Page 15 1.2. Itô's Lemma......Page 17 1.3 Geometric Brownian Motion......Page 19 1.4 Some Generalizations......Page 21 2. Discounted Present Values......Page 22 2.1. Present Values for Exponential and Polynomials......Page 23 2.2. Present Values for Powers of Geometric Brownian Motion......Page 26 2.3. A Basic Differential Equation for Present Value......Page 27 2.4. Derivation by Discrete Approximation......Page 28 2.5. The General Solution......Page 29 2.6. Differential Equation for Geometric Brownian Motion......Page 32 2.7 General Diffusion Processes......Page 34 3. Barriers......Page 35 3.1. The Basic Differential Equation......Page 36 3.3. Stopping......Page 38 3.5. Reflection......Page 39 3.6. Example: Price Ceiling......Page 40 3.7. Example: Exchange Rate Target Zones......Page 41 3.8 Transitional Boundary......Page 43 3.9 Example: Temporary Suspension......Page 44 4. Optimal Control and Regulation......Page 45 4.1. Stopping......Page 47 4.2. Example: Irreversible Investment......Page 50 4.3. Convex Costs: Continuous Control......Page 51 4.4. Lump-Sum Costs: Impulse Control......Page 52 4.5. Example: Menu Costs......Page 54 4.6. Linear Costs: Barrier Control......Page 55 4.7. Some Geometry and Intuition......Page 56 4.8. Example: Competitive Industry......Page 58 5.1 Mean-Reverting Processes......Page 60 5.2 Finite Horizon......Page 62 6. Some Characterization of Optimal Paths......Page 64 6.1 Short Run: Time Until First Action......Page 65 6.2 Long Run: Stationary Distribution and Average Action......Page 71 6.3 Dynamics of Brownian motion: KolmogorovEquations......Page 76 References......Page 82 Index......Page 84 Cover 1 Title 4 Copyright 5 Original Title 6 Original Copyright 7 Contents 8 Introduction to the Series 10 Preface 12 1. Brownian Motion 14 1.1. Random Walk Representation 15 1.2. Itô's Lemma 17 1.3 Geometric Brownian Motion 19 1.4 Some Generalizations 21 2. Discounted Present Values 22 2.1. Present Values for Exponential and Polynomials 23 2.2. Present Values for Powers of Geometric Brownian Motion 26 2.3. A Basic Differential Equation for Present Value 27 2.4. Derivation by Discrete Approximation 28 2.5. The General Solution 29 2.6. Differential Equation for Geometric Brownian Motion 32 2.7 General Diffusion Processes 34 3. Barriers 35 3.1. The Basic Differential Equation 36 3.2. Geometric Brownian Motion 38 3.3. Stopping 38 3.4. Resetting 39 3.5. Reflection 39 3.6. Example: Price Ceiling 40 3.7. Example: Exchange Rate Target Zones 41 3.8 Transitional Boundary 43 3.9 Example: Temporary Suspension 44 4. Optimal Control and Regulation 45 4.1. Stopping 47 4.2. Example: Irreversible Investment 50 4.3. Convex Costs: Continuous Control 51 4.4. Lump-Sum Costs: Impulse Control 52 4.5. Example: Menu Costs 54 4.6. Linear Costs: Barrier Control 55 4.7. Some Geometry and Intuition 56 4.8. Example: Competitive Industry 58 5. Generalizations 60 5.1 Mean-Reverting Processes 60 5.2 Finite Horizon 62 6. Some Characterization of Optimal Paths 64 6.1 Short Run: Time Until First Action 65 6.2 Long Run: Stationary Distribution and Average Action 71 6.3 Dynamics of Brownian motion: Kolmogorov Equations 76 References 82 Index 84 The main mathematical ideas are presented in a context with which economists should be familiar. Using a binomial approximation to Brownian motion, the mathematics is reduced to simple algebra, progressing to some equally simple limits. The starting point of the calculus of Brownian motion - "Ito's Lemma"--Emerges by analogy with the economics of risk-aversion. Conditions for the optimal regulation of Brownian motion, including the important, but often mysterious "smooth pasting" condition, are derived in a similar way. Each theoretical derivation is illustrated by developing a significant economic application, drawn mainly from recent research in macroeconomics and international economics Aims to widen the understanding of stochastic dynamic choice and equilibrium models. The book offers a simplified exposition of the theory of Brownian motion and its regulation, rendering such methods accessible to economists who do not require a detailed mathematical treatment of the subject. The authors present a basic model of the Bayesian implementation problem and then consider its application in areas including classical pure exchange economies, public goods provision, auctions and bargaining. A comprehensive exposition of rational expectations models is provided here, working up from simple univariate models to more sophisticated multivariate and non-linear models The ability to construct, test and explore models is one of the key tasks for the economist and underpins most work in contemporary economics. Avinash Dixit. A Volume In The Stochastic Methods In Economic Analysis Section. Includes Bibliographical References (p. 69-70) And Index. Discusses the theory of labour-managed firms or producers' cooperatives, and of economies companies principally of such firms.