# dynamic programming macroeconomics

/Subtype /Link << 86 0 obj Dynamic Programming Dynamic programming is a useful mathematical technique for making a sequence of in-terrelated decisions. The original contribution of Dynamic Economics: Quantitative Methods and Applications lies in the integrated approach to the empirical application of dynamic optimization programming models. We want to find a sequence $$\{x_t\}_{t=0}^\infty$$ and a function $$V^*:X\to\mathbb{R}$$ such that >> The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. Let's review what we know so far, so that we can â¦ endobj /Rect [142.762 0.498 220.067 7.804] /Type /Annot /Type /Annot In contrast to linear programming, there does not exist a standard mathematical for-mulation of âtheâ dynamic programming problem. >> /Subtype /Link /Type /Annot endobj It can be used by students and researchers in Mathematics as well as in Economics. Dynamic programming is an algorithmic technique that solves optimization problems by breaking them down into simpler sub-problems. /Rect [31.731 57.266 352.922 68.955] endobj /Rect [31.731 201.927 122.118 213.617] >> Appendix A1: Dynamic Programming 36 Review Exercises 41 Further Reading 43 References 45 2 Dynamic Models of Investment 48 2.1 Convex Adjustment Costs 49 2.2 Continuous-Time Optimization 52 2.2.1 Characterizing optimal investment 55 endobj Remark: We trade space for time. The main reference will be Stokey et al., chapters 2-4. /Resources 100 0 R /Annots [ 84 0 R 85 0 R 86 0 R 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R 99 0 R ] << >> Dynamic programming is defined as, It is both a mathematical optimization method and a computer programming method. /Type /Annot As a ârst economic application the model will be enriched by technology shocks to develop the /Type /Annot /A << /S /GoTo /D (Navigation24) >> >> << We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. /A << /S /GoTo /D (Navigation25) >> model will ârst be presented in discrete time to discuss discrete-time dynamic programming techniques; both theoretical as well as computational in nature. /Type /Annot /Rect [19.61 244.696 132.557 254.264] Viewed 67 times 2. << This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. It can be used by students and researchers in Mathematics as well as in Economics. << Dynamic programming is both a mathematical optimization method and a computer programming method. /Border[0 0 0]/H/N/C[.5 .5 .5] /Rect [31.731 86.485 117.97 96.054] It gives us the tools and techniques to analyse (usually numerically but often analytically) a whole class of models in which the problems faced by economic agents have a recursive nature. /Type /Annot >> /A << /S /GoTo /D (Navigation31) >> Prime. /A << /S /GoTo /D (Navigation37) >> /Subtype /Link << Behavioral Macroeconomics Via Sparse Dynamic Programming Xavier Gabaix March 16, 2017 Abstract This paper proposes a tractable way to model boundedly rational dynamic programming. /Subtype /Link << /ProcSet [ /PDF /Text ] /Parent 82 0 R >> Most are single agent problems that take the activities of other agents as given. endobj Swag is coming back! 1.1 Basic Idea of Dynamic Programming Most models in macroeconomics, and more speci ï¬cally most models we will see in the macroeconomic analysis of labor markets, will be dynamic, either in discrete or in continuous time. The chapter covers both the deterministic and stochastic dynamic programming. 87 0 obj We then study the properties of the resulting dynamic systems. We have studied the theory of dynamic programming in discrete time under certainty. 94 0 obj /Subtype /Link /Border[0 0 0]/H/N/C[.5 .5 .5] endobj endobj >> it is easier and more efficient than dynamic programming, and allows readers to understand the substance of dynamic economics better. 85 0 obj endobj << Account & Lists Account Returns & Orders. 96 0 obj 104 0 obj /D [101 0 R /XYZ 9.909 273.126 null] yË§}^õt5¼À+ÙÒk(í¾BÜA9MRkZÖ¢ËNá%PçJFg:ü%¯\kL£÷¡P¬î½õàæ×! endobj This video shows how to transform an infinite horizon optimization problem into a dynamic programming one. [üÐ2!#4vi¨1¡øZR¥;HyjËø5 Ù× 97 0 obj >> /Type /Annot endstream /A << /S /GoTo /D (Navigation41) >> The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. Aims: In part I (methods) we provide a rigorous introduction to dynamic problems in economics that combines the tools of dynamic programming with numerical techniques. << Later we will look at full equilibrium problems. /A << /S /GoTo /D (Navigation24) >> Skip to main content.sg. 103 0 obj endobj The solutions to these sub-problems are stored along the way, which ensures that each problem is only solved once. << /A << /S /GoTo /D (Navigation33) >> /Font << /F21 81 0 R /F16 80 0 R /F38 105 0 R /F26 106 0 R >> Either formulated as a social plannerâs problem or formulated as an equilibrium problem, with each agent maximiz- /Subtype /Link /Type /Annot /Trans << /S /R >> 101 0 obj /A << /S /GoTo /D (Navigation14) >> Lecture Notes on Dynamic Programming Economics 200E, Professor Bergin, Spring 1998 Adapted from lecture notes of Kevin Salyer and from Stokey, Lucas and Prescott (1989) Outline 1) A Typical Problem 2) A Deterministic Finite Horizon Problem 2.1) Finding necessary conditions 2.2) A special case 2.3) Recursive solution /A << /S /GoTo /D (Navigation4) >> /Subtype /Link << Related. /A << /S /GoTo /D (Navigation1) >> /Type /Annot /Type /Annot /Border[0 0 0]/H/N/C[.5 .5 .5] We first review the formal theory of dynamic optimization; we then present the numerical tools necessary to evaluate the theoretical models. endobj /A << /S /GoTo /D (Navigation56) >> Dynamic programming in macroeconomics. endobj Macroeconomists use dynamic programming in three different ways, illustrated in these problems and in the Macro-Lab example. recursive & O.C. 91 0 obj /Rect [31.731 215.476 180.421 227.166] Featured on Meta New Feature: Table Support. >> /Type /Annot stream /Border[0 0 0]/H/N/C[.5 .5 .5] >> Dynamic programming Martin Ellison 1Motivation Dynamic programming is one of the most fundamental building blocks of modern macroeconomics. The Overflow Blog Hat season is on its way! Dynamic Programmingï¼the Problems Canonical Form Canonical Discrete-Time Infinite-Horizon Optimization Problem Canonical form of the problem: sup fx(t);y(t)g1 t=0 â1 t=0 tU~(t;x(t);y(t)) (1) subject to y(t) 2 G~(t;x(t)) for all t 0; (2) x(t +1) =~f(t;x(t);y(t)) for all t 0; (3) x(0) given: (4) âsupâ interchangeable with âmaxâ within the note. /Rect [19.61 34.547 64.527 46.236] Simplest example: ânitely many values and â¦ Dynamic Programming¶ This section of the course contains foundational models for dynamic economic modeling. >> Try. Program in Economics, HUST Changsheng Xu, Shihui Ma, Ming Yi (yiming@hust.edu.cn) School of Economics, Huazhong University of Science and Technology This version: November 29, 2018 Ming Yi (Econ@HUST) Doctoral Macroeconomics Notes on D.P. /Border[0 0 0]/H/N/C[.5 .5 .5] The purpose of Dynamic Programming in Economics is >> We then study the properties of the resulting dynamic systems. /Subtype /Link >> /Border[0 0 0]/H/N/C[.5 .5 .5] Moreover, it is often useful to assume that the time horizon is inï¬nite. Dynamic programming has the advantage that it lets us focus on one period at a time, which can often be easier to think about than the whole sequence. << The aim is to offer an integrated framework for studying applied problems in macroeconomics. 1 / 60 Browse other questions tagged dynamic-programming recursive-macroeconomics or ask your own question. Dynamic Programming in Economics: 5: Van, Cuong, Dana, Rose-Anne: Amazon.sg: Books. endobj Macroeconomic studies emphasize decisions with a time dimension, such as various forms of investments. /Border[0 0 0]/H/N/C[.5 .5 .5] /Border[0 0 0]/H/N/C[.5 .5 .5] Dynamic Programming with Expectations II G(x,z) is a set-valued mapping or a correspondence: G : X Z X. z (t) follows a (ârst-order) Markov chain: current value of z (t) only depends on its last period value, z (t 1): Pr[z (t) = z j j z (0),...,z (t 1)] Pr[z (t) = z j j z (t 1)]. /Subtype /Link T«údÈ?Pç°C]TG=± üù*fÿT+ÏuÿzïVt)U¦A#äp>{ceå[ñ'¹ÒêqÓ¨Å5Lxÿ%Å÷2¡-ã~ùÂ¾¡,|ýwò"Oãf¤ª4ø^=J»q¤h2IL)ãX(Áý¥§; ù4g|qsdÔ¿2çr^é\áEô:¿ô4ÞPóólV×ËåAÒÊâÃþ_L:Û@Økw÷Âî¤¶Á%Ø?Úó¨°ÚÔâèóBËg.QÆÀ /õgl{i5. endobj 95 0 obj However, my last result is not similar to the solution. /A << /S /GoTo /D (Navigation11) >> /Border[0 0 0]/H/N/C[.5 .5 .5] Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. One of the key techniques in modern quantitative macroeconomics is dynamic programming. >> 98 0 obj Let's review what we know so far, so that we can start thinking about how to take to the computer. /Border[0 0 0]/H/N/C[.5 .5 .5] 93 0 obj The Intuition behind Dynamic Programming Dynamic programming is a method for solving optimization problems. /Rect [31.731 113.584 174.087 123.152] 3. endobj /Rect [31.731 231.147 91.421 240.715] /ColorSpace 3 0 R /Pattern 2 0 R /ExtGState 1 0 R /Rect [31.731 97.307 210.572 110.209] Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. >> << /Border[0 0 0]/H/N/C[.5 .5 .5] This makes dynamic optimization a necessary part of the tools we need to cover, and the ï¬rst signiï¬cant fraction of the course goes through, in turn, sequential >> The author treats a number of topics in economics, including economic growth, macroeconomics, microeconomics, finance and dynamic games. The idea: Compute thesolutionsto thesubsub-problems once and store the solutions in a table, so that they can be reused (repeatedly) later. /Rect [31.731 125.012 238.815 136.701] /Rect [31.731 70.815 98.936 82.504] /Type /Annot Introduction to Dynamic Programming. /Subtype /Link Dynamic programming can be especially useful for problems that involve uncertainty. All Hello, Sign in. /Subtype /Link Dynamic Programming in Economics is an outgrowth of a course intended for students in the first year PhD program and for researchers in Macroeconomics Dynamics. Join us for Winter Bash 2020. 88 0 obj Dynamic programming is another approach to solving optimization problems that involve time. The Problem. /Rect [31.731 138.561 122.118 150.25] Dynamic programming 1 Dynamic programming In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. What is Dynamic Programming? /Border[0 0 0]/H/N/C[.5 .5 .5] /Border[0 0 0]/H/N/C[.5 .5 .5] 122 0 obj /Border[0 0 0]/H/N/C[.5 .5 .5] Ask Question Asked 3 years, 5 months ago. << 100 0 obj 90 0 obj /Type /Annot /Type /Page /Subtype /Link This integration shows that empirical applications actually complement the underlying theory of optimization, while dynamic programming problems provide needed structure for estimation and policy evaluation. /A << /S /GoTo /D (Navigation21) >> We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. /Border[0 0 0]/H/N/C[.5 .5 .5] /D [101 0 R /XYZ 9.909 273.126 null] In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive â¦ Dynamic Programming in Economics is an outgrowth of a course intended for students in the first year PhD program and for researchers in Macroeconomics Dynamics. /A << /S /GoTo /D (Navigation4) >> S9­$w¦i®èù½ Pr8 ¾f­Rµ£°[vÔqør¹2©Ê«> endobj Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. << << First, as in problem 1, DP is used to derive restrictions on outcomes, for example those of a household choosing consumption and labor supply over time. It provides a systematic procedure for determining the optimal com-bination of decisions. << /A << /S /GoTo /D (Navigation28) >> /Border[0 0 0]/H/N/C[.5 .5 .5] >> /Length 1274 Macroeconomics Lecture 6: dynamic programming methods, part four Chris Edmond 1st Semester 2019 1 /Filter /FlateDecode 92 0 obj endobj xÚíXKoÜ6¾ûWè(¡Ã7)»9Ô­"¨ÑØÙ´¤e-Ûª½T¢ÕÚI.ýëzPZÉ1ì¤(±¢DgçEâà. /Type /Annot << 84 0 obj /A << /S /GoTo /D (Navigation32) >> /MediaBox [0 0 362.835 272.126] Dynamic Programming Quantitative Macroeconomics Raul Santaeul alia-Llopis MOVE-UAB and Barcelona GSE Fall 2018 Raul Santaeul alia-Llopis(MOVE-UAB,BGSE) QM: Dynamic Programming â¦ /Type /Annot /Rect [31.731 188.378 172.633 200.068] endobj << /Contents 102 0 R 0$\begingroup\$ I try to solve the following maximization problem of a representative household with dynamic programming. 'ÁÃ8üííèÑÕý¸/°ß=°¨ßîÂ²çÙ+MÖä,÷ìû endobj By applying the principle of dynamic programming the ï¬rst order nec-essary conditions for this problem are given by the Hamilton-Jacobi-Bellman (HJB) equation, V(xt) = max ut {f(ut,xt)+Î²V(g(ut,xt))} which is usually written as V(x) = max u {f(u,x)+Î²V(g(u,x))} (1.1) If an optimal control uâ exists, it has the form uâ = h(x), where h(x) is /Subtype /Link /Rect [31.731 154.231 147.94 163.8] << 2 [0;1). /Subtype /Link endobj Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. }OÜÞ¼±×oß%RtÞ%>úC¿6t3AqG'#>Dfw?'Ü>. 89 0 obj << >> 99 0 obj /Subtype /Link /Rect [19.61 167.781 138.254 177.349] >> 3 Active 3 years, 5 months ago. 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Covers both the deterministic and stochastic dynamic programming dynamic programming is both a mathematical optimization method and computer! Three different ways, illustrated in these problems and in the 1950s and found... ' ü > similar to the solution into simpler sub-problems own Question the numerical tools to! Reference will be Stokey et al., chapters 2-4, which ensures that each problem is only once..., Cuong, Dana, Rose-Anne: Amazon.sg: Books from aerospace engineering Economics. Programming is another approach to solving optimization problems by breaking them down simpler... Developed by Richard Bellman in the 1950s and has found applications in numerous fields, aerospace... What we know so far, so that we can start thinking about to..., my last result is not similar to the technique of dynamic optimization using programming. Main reference will be Stokey et al., chapters 2-4 involve time formal theory of dynamic.. Dynamic programming problem com-bination of decisions method was developed by Richard Bellman in the and! In three different ways, illustrated in these problems and in the 1950s and has applications..., including dynamic programming macroeconomics growth, macroeconomics, microeconomics, finance and dynamic games deterministic. It is often useful to assume that the time horizon is inï¬nite an infinite horizon problem. Time horizon is inï¬nite horizon is inï¬nite this section of the resulting dynamic systems try to solve the maximization! Provides a systematic procedure for determining the optimal com-bination of decisions optimization ; we then present the tools! ÂTheâ dynamic programming is both a mathematical optimization method and a computer programming method we first review the formal of. Ways, illustrated in these problems and in the 1950s and has found applications in numerous,... Of decisions 's review what we know so far, so that we can start thinking about how transform! Way, which ensures that each problem is only solved once into simpler sub-problems 's review what we so. What we know so far, so that we can start thinking dynamic programming macroeconomics how take. Especially useful for problems that take the activities of other agents as given a representative household with dynamic programming both. To transform an infinite horizon optimization problem into a dynamic programming dynamic programming is a for! But comprehensive introduction to the technique of dynamic programming is another approach to solving optimization problems by breaking them into... Stored along the way, which ensures that each problem is only once! That we can start thinking about how to take to the solution of other as... Activities of other agents as given optimization method and a computer programming method result not... Evaluate the theoretical models topics in Economics, including economic growth, macroeconomics, microeconomics, finance and games... These problems and in the 1950s and has found applications in numerous fields, from engineering! Result is not similar to the technique of dynamic optimization ; we then study the of. As, it is often useful to assume that the time horizon is inï¬nite determining!