Recursive Macroeconomic Theory, fourth edition

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Publisher : MIT Press
ISBN 13 : 0262038668
Total Pages : 1477 pages
Book Rating : 4.2/5 (62 download)

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Book Synopsis Recursive Macroeconomic Theory, fourth edition by : Lars Ljungqvist

Download or read book Recursive Macroeconomic Theory, fourth edition written by Lars Ljungqvist and published by MIT Press. This book was released on 2018-09-11 with total page 1477 pages. Available in PDF, EPUB and Kindle. Book excerpt: The substantially revised fourth edition of a widely used text, offering both an introduction to recursive methods and advanced material, mixing tools and sample applications. Recursive methods provide powerful ways to pose and solve problems in dynamic macroeconomics. Recursive Macroeconomic Theory offers both an introduction to recursive methods and more advanced material. Only practice in solving diverse problems fully conveys the advantages of the recursive approach, so the book provides many applications. This fourth edition features two new chapters and substantial revisions to other chapters that demonstrate the power of recursive methods. One new chapter applies the recursive approach to Ramsey taxation and sharply characterizes the time inconsistency of optimal policies. These insights are used in other chapters to simplify recursive formulations of Ramsey plans and credible government policies. The second new chapter explores the mechanics of matching models and identifies a common channel through which productivity shocks are magnified across a variety of matching models. Other chapters have been extended and refined. For example, there is new material on heterogeneous beliefs in both complete and incomplete markets models; and there is a deeper account of forces that shape aggregate labor supply elasticities in lifecycle models. The book is suitable for first- and second-year graduate courses in macroeconomics. Most chapters conclude with exercises; many exercises and examples use Matlab or Python computer programming languages.

Recursive Methods in Economic Dynamics

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Publisher : Harvard University Press
ISBN 13 : 0674735188
Total Pages : 607 pages
Book Rating : 4.6/5 (747 download)

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Book Synopsis Recursive Methods in Economic Dynamics by : Nancy L. Stokey

Download or read book Recursive Methods in Economic Dynamics written by Nancy L. Stokey and published by Harvard University Press. This book was released on 1989-10-10 with total page 607 pages. Available in PDF, EPUB and Kindle. Book excerpt: This rigorous but brilliantly lucid book presents a self-contained treatment of modern economic dynamics. Stokey, Lucas, and Prescott develop the basic methods of recursive analysis and illustrate the many areas where they can usefully be applied.

Dynamic Macroeconomic Theory

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Publisher : Harvard University Press
ISBN 13 : 9780674043084
Total Pages : 392 pages
Book Rating : 4.0/5 (43 download)

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Book Synopsis Dynamic Macroeconomic Theory by : Thomas J. Sargent

Download or read book Dynamic Macroeconomic Theory written by Thomas J. Sargent and published by Harvard University Press. This book was released on 2009-06-01 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The tasks of macroeconomics are to interpret observations on economic aggregates in terms of the motivations and constraints of economic agents and to predict the consequences of alternative hypothetical ways of administering government economic policy. General equilibrium models form a convenient context for analyzing such alternative government policies. In the past ten years, the strengths of general equilibrium models and the corresponding deficiencies of Keynesian and monetarist models of the 1960s have induced macroeconomists to begin applying general equilibrium models. This book describes some general equilibrium models that are dynamic, that have been built to help interpret time-series of observations of economic aggregates and to predict the consequences of alternative government interventions. The first part of the book describes dynamic programming, search theory, and real dynamic capital pricing models. Among the applications are stochastic optimal growth models, matching models, arbitrage pricing theories, and theories of interest rates, stock prices, and options. The remaining parts of the book are devoted to issues in monetary theory; currency-in-utility-function models, cash-in-advance models, Townsend turnpike models, and overlapping generations models are all used to study a set of common issues. By putting these models to work on concrete problems in exercises offered throughout the text, Sargent provides insights into the strengths and weaknesses of these models of money. An appendix on functional analysis shows the unity that underlies the mathematics used in disparate areas of rational expectations economics. This book on dynamic equilibrium macroeconomics is suitable for graduate-level courses; a companion book, Exercises in Dynamic Macroeconomic Theory, provides answers to the exercises and is also available from Harvard University Press.

Solutions Manual for Recursive Methods in Economic Dynamics

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Publisher : Harvard University Press
ISBN 13 : 0674038967
Total Pages : 304 pages
Book Rating : 4.6/5 (74 download)

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Book Synopsis Solutions Manual for Recursive Methods in Economic Dynamics by : Claudio IRIGOYEN

Download or read book Solutions Manual for Recursive Methods in Economic Dynamics written by Claudio IRIGOYEN and published by Harvard University Press. This book was released on 2009-06-30 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This solutions manual is a companion volume to the classic textbook Recursive Methods in Economic Dynamics by Nancy L. Stokey and Robert E. Lucas. Efficient and lucid in approach, this manual will greatly enhance the value of Recursive Methods as a text for self-study.

Macroeconomic Theory

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Author :
Publisher : Emerald Group Pub Limited
ISBN 13 : 9780126197518
Total Pages : 510 pages
Book Rating : 4.1/5 (975 download)

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Book Synopsis Macroeconomic Theory by :

Download or read book Macroeconomic Theory written by and published by Emerald Group Pub Limited. This book was released on 1987 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: Macroeconomic Theory, in its first edition, was widely adopted for use as a graduate text; this updated and expanded version should find even greater popularity as a text and as a research reference. It has been substantially revised to include three entirely new chapters: The Consumption Function, Government Debt and Taxes, and Dynamic Optimal Taxation. Significant additions have been made to three of the original chapters dealing with difference equations, stochastic difference equations, and investment under uncertainty. Key Features * This book has been substantially revised to include three entirely new chapters on consumption, government debt and taxes, and dynamic optimal taxation * Significant additions have been made to three of the original chapters dealing with difference equations, stochastic difference equations, and investment under uncertainty

Macroeconomic Theory

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Author :
Publisher : Princeton University Press
ISBN 13 : 1400842476
Total Pages : 617 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Macroeconomic Theory by : Michael Wickens

Download or read book Macroeconomic Theory written by Michael Wickens and published by Princeton University Press. This book was released on 2012-03-12 with total page 617 pages. Available in PDF, EPUB and Kindle. Book excerpt: The definitive graduate textbook on modern macroeconomics Macroeconomic Theory is the most up-to-date graduate-level macroeconomics textbook available today. This revised second edition emphasizes the general equilibrium character of macroeconomics to explain effects across the whole economy while taking into account recent research in the field. It is the perfect resource for students and researchers seeking coverage of the most current developments in macroeconomics. Michael Wickens lays out the core ideas of modern macroeconomics and its links with finance. He presents the simplest general equilibrium macroeconomic model for a closed economy, and then gradually develops a comprehensive model of the open economy. Every important topic is covered, including growth, business cycles, fiscal policy, taxation and debt finance, current account sustainability, and exchange-rate determination. There is also an up-to-date account of monetary policy through inflation targeting. Wickens addresses the interrelationships between macroeconomics and modern finance and shows how they affect stock, bond, and foreign-exchange markets. In this edition, he also examines issues raised by the most recent financial crisis, and two new chapters explore banks, financial intermediation, and unconventional monetary policy, as well as modern theories of unemployment. There is new material in most other chapters, including macrofinance models and inflation targeting when there are supply shocks. While the mathematics in the book is rigorous, the fundamental concepts presented make the text self-contained and easy to use. Accessible, comprehensive, and wide-ranging, Macroeconomic Theory is the standard book on the subject for students and economists. The most up-to-date graduate macroeconomics textbook available today General equilibrium macroeconomics and the latest advances covered fully and completely Two new chapters investigate banking and monetary policy, and unemployment Addresses questions raised by the recent financial crisis Web-based exercises with answers Extensive mathematical appendix for at-a-glance easy reference This book has been adopted as a textbook at the following universities: American University Bentley College Brandeis University Brigham Young University California Lutheran University California State University - Sacramento Cardiff University Carleton University Colorado College Fordham University London Metropolitan University New York University Northeastern University Ohio University - Main Campus San Diego State University St. Cloud State University State University Of New York - Amherst Campus State University Of New York - Buffalo North Campus Temple University - Main Texas Tech University University of Alberta University Of Notre Dame University Of Ottawa University Of Pittsburgh University Of South Florida - Tampa University Of Tennessee University Of Texas At Dallas University Of Washington University of Western Ontario Wesleyan University Western Nevada Community College

Real Analysis with Economic Applications

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Publisher : Princeton University Press
ISBN 13 : 1400840899
Total Pages : 832 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Real Analysis with Economic Applications by : Efe A. Ok

Download or read book Real Analysis with Economic Applications written by Efe A. Ok and published by Princeton University Press. This book was released on 2011-09-05 with total page 832 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are many mathematics textbooks on real analysis, but they focus on topics not readily helpful for studying economic theory or they are inaccessible to most graduate students of economics. Real Analysis with Economic Applications aims to fill this gap by providing an ideal textbook and reference on real analysis tailored specifically to the concerns of such students. The emphasis throughout is on topics directly relevant to economic theory. In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. Efe Ok complements the mathematical development with applications that provide concise introductions to various topics from economic theory, including individual decision theory and games, welfare economics, information theory, general equilibrium and finance, and intertemporal economics. Moreover, apart from direct applications to economic theory, his book includes numerous fixed point theorems and applications to functional equations and optimization theory. The book is rigorous, but accessible to those who are relatively new to the ways of real analysis. The formal exposition is accompanied by discussions that describe the basic ideas in relatively heuristic terms, and by more than 1,000 exercises of varying difficulty. This book will be an indispensable resource in courses on mathematics for economists and as a reference for graduate students working on economic theory.

Recursive Macroeconomic Theory

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Author :
Publisher : Createspace Independent Publishing Platform
ISBN 13 : 9781537653532
Total Pages : 630 pages
Book Rating : 4.6/5 (535 download)

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Book Synopsis Recursive Macroeconomic Theory by : Thomas J. Sargent

Download or read book Recursive Macroeconomic Theory written by Thomas J. Sargent and published by Createspace Independent Publishing Platform. This book was released on 2016-09-15 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recursive methods offer a powerful approach for characterizing and solving complicated problems in dynamic macroeconomics. Recursive Macroeconomic Theory provides both an introduction to recursive methods and advanced material, mixing tools and sample applications. The second edition contains substantial revisions to about half the original material, and extensive additional coverage appears in seven chapters new to this edition. The updated and added material covers exciting new topics that further illustrate the power and pervasiveness of recursive methods.Significant improvements to original chapters include a better treatment of the existence of recursive equilibria, an enhanced account of the supermartingale convergence theorem, and an extended treatment of an optimal taxation problem in an economy in which there are incomplete markets.

Econometrics

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Publisher : Princeton University Press
ISBN 13 : 1400823838
Total Pages : 708 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Econometrics by : Fumio Hayashi

Download or read book Econometrics written by Fumio Hayashi and published by Princeton University Press. This book was released on 2011-12-12 with total page 708 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most authoritative and comprehensive synthesis of modern econometrics available Econometrics provides first-year graduate students with a thoroughly modern introduction to the subject, covering all the standard material necessary for understanding the principal techniques of econometrics, from ordinary least squares through cointegration. The book is distinctive in developing both time-series and cross-section analysis fully, giving readers a unified framework for understanding and integrating results. Econometrics covers all the important topics in a succinct manner. All the estimation techniques that could possibly be taught in a first-year graduate course, except maximum likelihood, are treated as special cases of GMM (generalized methods of moments). Maximum likelihood estimators for a variety of models, such as probit and tobit, are collected in a separate chapter. This arrangement enables students to learn various estimation techniques in an efficient way. Virtually all the chapters include empirical applications drawn from labor economics, industrial organization, domestic and international finance, and macroeconomics. These empirical exercises provide students with hands-on experience applying the techniques covered. The exposition is rigorous yet accessible, requiring a working knowledge of very basic linear algebra and probability theory. All the results are stated as propositions so that students can see the points of the discussion and also the conditions under which those results hold. Most propositions are proved in the text. For students who intend to write a thesis on applied topics, the empirical applications in Econometrics are an excellent way to learn how to conduct empirical research. For theoretically inclined students, the no-compromise treatment of basic techniques is an ideal preparation for more advanced theory courses.

Advanced Macroeconomics

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Publisher : McGraw-Hill/Irwin
ISBN 13 :
Total Pages : 680 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis Advanced Macroeconomics by : David Romer

Download or read book Advanced Macroeconomics written by David Romer and published by McGraw-Hill/Irwin. This book was released on 2001 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text helps lay the groundwork for students to begin doing research in macroeconomics and monetary economics. A series of formal models are used to present and analyse important macroeconomic theories. The theories are supplemented by examples of relevant empirical work, which illustrate the ways that theories can be applied and tested.

An Introduction to Mathematical Analysis for Economic Theory and Econometrics

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Publisher : Princeton University Press
ISBN 13 : 1400833086
Total Pages : 696 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis An Introduction to Mathematical Analysis for Economic Theory and Econometrics by : Dean Corbae

Download or read book An Introduction to Mathematical Analysis for Economic Theory and Econometrics written by Dean Corbae and published by Princeton University Press. This book was released on 2009-02-17 with total page 696 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics. Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra. Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem Focuses on examples from econometrics to explain topics in measure theory

Economic Dynamics, second edition

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Publisher : MIT Press
ISBN 13 : 0262544776
Total Pages : 395 pages
Book Rating : 4.2/5 (625 download)

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Book Synopsis Economic Dynamics, second edition by : John Stachurski

Download or read book Economic Dynamics, second edition written by John Stachurski and published by MIT Press. This book was released on 2022-08-16 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of a rigorous and example-driven introduction to topics in economic dynamics that emphasizes techniques for modeling dynamic systems. This text provides an introduction to the modern theory of economic dynamics, with emphasis on mathematical and computational techniques for modeling dynamic systems. Written to be both rigorous and engaging, the book shows how sound understanding of the underlying theory leads to effective algorithms for solving real-world problems. The material makes extensive use of programming examples to illustrate ideas, bringing to life the abstract concepts in the text. Key topics include algorithms and scientific computing, simulation, Markov models, and dynamic programming. Part I introduces fundamentals and part II covers more advanced material. This second edition has been thoroughly updated, drawing on recent research in the field. New for the second edition: “Programming-language agnostic” presentation using pseudocode. New chapter 1 covering conceptual issues concerning Markov chains such as ergodicity and stability. New focus in chapter 2 on algorithms and techniques for program design and high-performance computing. New focus on household problems rather than optimal growth in material on dynamic programming. Solutions to many exercises, code, and other resources available on a supplementary website.

Modern Macroeconomics

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Publisher : Edward Elgar Publishing
ISBN 13 : 1845424670
Total Pages : 825 pages
Book Rating : 4.8/5 (454 download)

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Book Synopsis Modern Macroeconomics by : Brian Snowdon

Download or read book Modern Macroeconomics written by Brian Snowdon and published by Edward Elgar Publishing. This book was released on 2005-01-01 with total page 825 pages. Available in PDF, EPUB and Kindle. Book excerpt: Snowdon and Vane s book is extremely welcome. Indeed the authors examine, compare, and evaluate the evolution of the major rival stories comprising contemporary macroeconomic thought, but they also trace the development and interaction of key events and ideas as they occurred in the last century. Interviews with leading economists, one or two at the end of each chapter, also greatly help to shed light on this complexity. . . In sum, this is book which is very difficult to put down. Alessio Moneta, Journal of the History of Economic Thought It is not difficult to understand why this volume commands high praise from macroeconomic theorists, practitioners and teachers. It contains many interesting features that make it an excellent companion for both students and teachers of tertiary level macroeconomics. . . The authors present the material in a way that conveys to readers that macroeconomics is a living science , continually developing and still open to debate, controversy and competing policy prescriptions. In this respect it is a book that ought to be required reading for all teachers of the subject. It is also a valuable source of background reading for professional economists involved with economic policy making. Economic Outlook and Business Review . . . a wonderful history of macroeconomic thought from Keynes to the present, with an outstanding bibliography. It should be useful to undergraduates and graduate students as well as professional economists. Highly recommended. Steven Pressman, Choice Brian Snowdon and Howard Vane are well-known for their astute understanding of the main macroeconomic schools of thought and their skilled use of interviews with major figures. Here, they deploy a depth of scholarship in explaining the different schools and their key points of departure from one another. This book will be particularly useful to students looking for a clear, non-technical explanation of the main approaches to macroeconomics. Patrick Minford, Cardiff University, UK There are two steps to learning macroeconomics. First, to see it as it is today. Second, to understand how it got there: to understand the right and the wrong turns, the hypotheses that proved false, the insights that proved true, and the interaction of events and ideas. Only then, does one truly understand macroeconomics. This book is about step two. It does a marvellous job of it. The presentation is transparent, the interviews fascinating. You will enjoy, and you will learn. Olivier Blanchard, Massachusetts Institute of Technology, US In 40 years of teaching macroeconomics, there has been just one textbook that I have assigned year after year after year, namely, A Modern Guide to Macroeconomics by Snowdon, Vane and Wynarczyk. That altogether admirable book made clear to students what were, and are, the main intellectual issues in macroeconomics and did so with just enough formal modeling to avoid distortion by over-simplification. That book is now ten years old and the debate in macro has moved on. So there is good reason to welcome Snowdon and Vane back with this superb updated version. Axel Leijonhufvud, University of Trento, Italy This outstanding book avoids the narrow scope of most textbooks and provides an excellent guide to an unusually broad range of ideas. Thomas Mayer, University of California, Davis, US More than a decade after the publication of the critically acclaimed A Modern Guide to Macroeconomics, Brian Snowdon and Howard Vane have produced a worthy successor in the form of Modern Macroeconomics. Thoroughly extended, revised and updated, it will become the indispensable text for students and teachers of macroeconomics in the new millennium. The authors skilfully trace the origins, development and current state of modern macroeconomics from an historical perspective. They do so by thoroughly appraising the central tenets underlying the main competing schools of macroeconomic thought as well as their diverse policy imp

Structural Macroeconometrics

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Publisher : Princeton University Press
ISBN 13 : 1400840503
Total Pages : 435 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Structural Macroeconometrics by : David N. DeJong

Download or read book Structural Macroeconometrics written by David N. DeJong and published by Princeton University Press. This book was released on 2011-10-03 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: The revised edition of the essential resource on macroeconometrics Structural Macroeconometrics provides a thorough overview and in-depth exploration of methodologies, models, and techniques used to analyze forces shaping national economies. In this thoroughly revised second edition, David DeJong and Chetan Dave emphasize time series econometrics and unite theoretical and empirical research, while taking into account important new advances in the field. The authors detail strategies for solving dynamic structural models and present the full range of methods for characterizing and evaluating empirical implications, including calibration exercises, method-of-moment procedures, and likelihood-based procedures, both classical and Bayesian. The authors look at recent strides that have been made to enhance numerical efficiency, consider the expanded applicability of dynamic factor models, and examine the use of alternative assumptions involving learning and rational inattention on the part of decision makers. The treatment of methodologies for obtaining nonlinear model representations has been expanded, and linear and nonlinear model representations are integrated throughout the text. The book offers a rich array of implementation algorithms, sample empirical applications, and supporting computer code. Structural Macroeconometrics is the ideal textbook for graduate students seeking an introduction to macroeconomics and econometrics, and for advanced students pursuing applied research in macroeconomics. The book's historical perspective, along with its broad presentation of alternative methodologies, makes it an indispensable resource for academics and professionals.

Monetary Theory and Policy

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Publisher : MIT Press
ISBN 13 : 9780262232319
Total Pages : 636 pages
Book Rating : 4.2/5 (323 download)

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Book Synopsis Monetary Theory and Policy by : Carl E. Walsh

Download or read book Monetary Theory and Policy written by Carl E. Walsh and published by MIT Press. This book was released on 2003 with total page 636 pages. Available in PDF, EPUB and Kindle. Book excerpt: An overview of recent theoretical and policy-related developments in monetary economics.

Public Finance

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Publisher : Academic Press
ISBN 13 : 0126990514
Total Pages : 976 pages
Book Rating : 4.1/5 (269 download)

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Book Synopsis Public Finance by : Richard W. Tresch

Download or read book Public Finance written by Richard W. Tresch and published by Academic Press. This book was released on 2002-05-08 with total page 976 pages. Available in PDF, EPUB and Kindle. Book excerpt: Featuring a general equilibrium framework that is both cohesive and versatile, the Second Edition of Public Finance: A Normative Theory brings new and updated information to this classic text. Through its concentration on the microeconomic theory of the public sector in the context of capitalist market economics it addresses the subjects traditionally at the heart of public sector economics, including public good theory, theory of taxation, welfare analysis, externalities, tax incidence, cost benefit analysis, and fiscal federalism. Its goal of providing a foundation, rather than attempting to present the most recent scholarship in detail, makes this Second Edition both a valuable text and a resource for professionals. * Second edition provides new and updated information * Focuses on the heart of public sector economics, including public expenditure theory and policy, tax theory and policy, cost benefit-analysis, and fiscal federalism * Features a cohesive and versatile general equilibrium framework

Macroeconomic Theory

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Author :
Publisher : Oxford University Press
ISBN 13 : 0195387716
Total Pages : 603 pages
Book Rating : 4.1/5 (953 download)

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Book Synopsis Macroeconomic Theory by : Jean-Pascal Benassy

Download or read book Macroeconomic Theory written by Jean-Pascal Benassy and published by Oxford University Press. This book was released on 2011-05-12 with total page 603 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate textbook is a "primer" in macroeconomics. It starts with essential undergraduate macroeconomics and develops in a simple and rigorous manner the central topics of modern macroeconomic theory including rational expectations, growth, business cycles, money, unemployment, government policy, and the macroeconomics of nonclearing markets. The emphasis throughout the book is on both foundations and presenting the simplest model for each topic that will deliver the relevant answers. The first two chapters recall the main workhorses of undergraduate macroeconomics: the Solow-Swan growth model, the Keynesian IS-LM model, and the Phillips curve. The next chapters present four fundamental "building blocks" of modern macroeconomics: rational expectations, intertemporal dynamic models, nonclearing markets and imperfect competition, and uncertainty. Later the book deals with growth, notably the Ramsey model, overlapping generations, and endogenous growth. Chapter 10 moves to the famous "real business cycles" (RBC), which integrate in a unified framework growth and fluctuations. The final chapters look at the issue of stabilization, how best to guard the economy from shocks, and the connections between politics and the macroeconomy. To make the book self contained, a mathematical appendix gives a number of simple technical results that are sufficient to follow the formal developments of the book.