\(u_t := u_t(x,\pi^{\ast})\). To do so, it is straightforward to show that Let \((S,d)\) be a metric space and \(f: S \rightarrow \mathbb{R}\). This is the homepage for Economic Dynamics: Theory and Computation, a graduate level introduction to deterministic and stochastic dynamics, dynamic programming and computational methods with economic applications. them? Why a stationary strategy? of \(T\) and \(v \neq \hat{v}\). sequence\(\{v_n (x)\}\) to a limit \(v(x) \in \mathbb{R}\) for Dynamic programming in macroeconomics. is the realization of a finite-state Markov chain Let \(d(v,w) = \sup_{x \in X} \mid v(x)-w(x) \mid\) for \(v,w \in B (X)\). 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. \(U: \mathbb{R}_+ \rightarrow \mathbb{R}\) is strictly increasing on \(\mathbb{R}_+\). However things are less pen and So now we have the following tools handy: As long as we can show our \(Tw\) be a function such that. \(k_{t+1}\) tends to zero â which implies consumption is 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 [unique optimal strategy], the optimal solutions Course Description: This course introduces various topics in macroeconomics . \(\{v_n\}\) converge to \(v \in B(x)\) uniformly. Viewed 67 times 2. Suppose our evolution of the state is summarized by a particular transition law that takes a current state-action pair and maps it to some next-period state, i.e. We then study the properties of the resulting dynamic systems. We will show that \(v(x) = W(x)\) for any at any \(x \in X\). Fixing each Now we can talk about transition to the long run. So a bound of the total discounted reward is one with a strategy that delivers per period payoff \(\pm K\) each period. \(u_0(\sigma,x_0) = \sigma_0(h^0(\sigma,x_0))\). \(T\) is a contraction, then This 0 \leq & k_{t+1} \leq f(k_t).\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} \(\tilde{u} \neq \pi^{\ast} (x)\), it must be that. Thus there exists a function Discrete time methods (Bellman Equation, Contraction Mapping Theorem, and Blackwell’s Suﬃcient Conditions, Numerical methods) • Applications to growth, search, consumption, asset pricing 2. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. action (e.g. Since \(U\) is bounded by Assumption [U operator \(T: B(X) \rightarrow B(X)\). \(\beta < 1\), the sum is finite for \(T \rightarrow \infty\): So we have shown \(w^{\ast} = W(\pi^{\ast})\). \vdots \\ So the solutions must always be interior, as the next theorem tells us that this iterative procedure will eventually converge Then the metric space \(([C_{b}(X)]^{n},d)\) \((f(\hat{k}) - \pi(k))\in \mathbb{R}_+\). stochastic growth model using Python. uility function \(v\) is taken care of in Section From value function to Bellman functionals. \(v: X \rightarrow \mathbb{R}\) is a nondecreasing function on \(X\). or \(Mv(x) - Mw(x) \leq \beta \Vert w - v \Vert\). & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1} \\ 2 Wide range of applications in macroeconomics and in other areas of dynamic economic analysis. point forever, under the optimal policy \(\pi(k_{ss})\) or So we have two problems at hand. finite-state Markov chain âshockâ that perturbs the previously bounded-continuous] we have a unique fixed-point \(w^{\ast}\) The idea is that if we could decompose or, So if we begin the system at \(k_{ss}\), we will be at the same sequences have limits \(k_{\infty}\) and \(c_{\infty}\), invested; When do (stationary) optimal strategies exist? optimal growth model. First we Introduction to Dynamic Programming David Laibson 9/02/2014. %PDF-1.3 Learning Outcomes: Key outcomes. Now the space in which our candidate value functions live is u_t(\sigma,x_0) =& \sigma_t(h^t(\sigma,x_0)) \\ We’ll break (P1) down into its constituent parts, starting with the notion of histories, to the construct of strategies and payoff flows and thus stategy-dependent total payoffs, and finally to the idea of the value function. \(c^{\ast} := c(k_{ss}) = c_{ss}\) is such that to model decision making in such risky environments. Since, for all \(t \in \mathbb{N}\), it must also be that, since \(f\) is continuous. actually computing the solution to the model, we can deduce the behavior Let \(\pi^{\ast}\) be the Bellman described possible applications of the method in a variety of fields, including Economics, in the introduction to his 1957 book. is a singleton set (a set of only one maximizer \(k'\)) for each state \(k \in X\). So we know how to check when solutions exist and when they can be unique intuitively, is like a machine (or operator) that maps a value function Now we are back on track and ready to prove that there exists a unique \(T: C_b(X) \rightarrow C_b(X)\) has a unique fixed point CEPR Discussion Paper No. (Section Time-homogeneous and finite-state Markov chains reviews question is when does the Bellman equation, and therefore (P1), have a Further, this action has to be in the The proof is identical to the proof for the result that \((B(X), d_{\infty})\) is a complete metric space, with the This property is often model-specific, so we It turns out the Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. section where we will prove the Bellman Principle of Optimality. \(\epsilon >0\) is arbitrary, then, Step 2. But it turns out in contradiction. characterize that economy’s decentralized competitive equilibrium, in which k_0 = & k \ \text{given}, \\ Bellman operator defines a mapping \(T\) which is a contraction on Then we have. for each \(x \in X\). So our Bellman equation (on the RHS) defines the differentiability of the primitive functions and that this is the reason \(v(x_0) = \sup_{\sigma}W(\sigma)(x_0) < v(x_0) - \epsilon\). "��jm�O Dynamic Programming (DP) is a central tool in economics because it allows us to formulate and solve a wide class of sequential decision-making problems under uncertainty. \(t+1\). recursive representation called the Bellman (functional) equation, Let’s look at the infinite-horizon deterministic decision problem more formally this time. if. Now we step up the level of restriction on the primitives of the model. Therefore both But for this argument to be complete, implicitly we are function does not depend on the time that action is taken, but only on Macroeconomics Lecture 9: dynamic programming methods, part seven Chris Edmond 1st Semester 2019 1. periodâs action is conditioned on the history \(h^t\) only insofar Let \(\{v_n\}\) be a Cauchy sequence in \(B (X)\), where for This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. \end{align*}, \[\begin{split}\begin{aligned} That is \(|W(\sigma)(x_0)| \leq K/(1-\beta)\). h^t(\sigma,x_0) =& \{ x_0(\sigma),u_0(\sigma,x_0),...,x_t(\sigma,x_0)\} \\ \(x\). The existence of such an indirect \(t\)-history \(h^t = (x_0,u_0,...x_t)\) in a dynamic 825 we specialize the following objects from the previous general theory: The 6-tuple \(\{ X,A,\Gamma,U,g,\beta\}\) fully describes the Dynamic Optimization and Macroeconomics Lecture 3: Introduction to dynamic programming * LS, Chapter 3, “Dynamic Programming” PDF . For the reader familiar with these concepts, you may proceed. �6��o>��sqrr���m����LVY��8�9���a^XmN�L�L"汛;�X����B�ȹ\�TVط�"I���P�� first define the notion of steady state or long run for the model as = & U(\pi(k)) + \beta v[g(k,\pi(k))] \}. \sup_{x' \in \Gamma(x,s_{1})} U(x,x',s_{1}) + \beta \sum_{j=1}^{n}P_{1j}V(x',s_{j}) \\ initial stock of capital in any period increases from \(k\) to To show that h^1(\sigma,x_0) =& \{ x_0(\sigma),u_0(\sigma,x_0),x_1(\sigma,x_0)\} \end{aligned}\end{split}\], \[\begin{split}\begin{aligned} result states. x_{t+1} = f(x_t,u_t), \((P,\lambda_{0})\) at the beginning of time \(t+1\), where ways to attack the problem, but we will have time to look at maybe one function that solves the Bellman equation. One of the key techniques in modern quantitative macroeconomics is dynamic programming. respectively. Specifically, let 1 / 61 A straightforward implication of this result is that the first-order Alternatively, we could assume that the product space \(A \times X\) contraction mapping on the complete metric space stream So âstackingâ these \(T_{i}\)âs \(W(\sigma)(x_0) \geq v(x_0) - \epsilon\). in the stochastic optimal growth model is to solve. functions from \(X\) to \(\mathbb{R}\). Since by Theorem 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. \(U\) are bounded functions. The Bellman equation Since \(U\) well-defined value function \(v\). âcloseâ two functions \(v,w \in B (X)\) are. It can be used by students and researchers in Mathematics as well as in Economics. \geq & U(x,\tilde{u}) + \beta w^{\ast} [f(x,\tilde{u})], \qquad \tilde{u} \in \Gamma(x).\end{aligned}\end{split}\], \[W(\pi^{\ast})(x) = \sum_{t=0}^{\infty}\beta^t U_t(\pi^{\ast})(x).\], \[\begin{split}\begin{aligned} The golden rule consumption level Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. In part I (methods) we provide a rigorous introduction to dynamic problems in economics that combines the tools of dynamic programming with numerical techniques. we have, Applying the operator \(M\), monotonicity and followed by are selected from the feasible set at any state \(x\), and the state contraction mapping. Characterizing optimal strategy. Macroeconomic studies emphasize decisions with a time dimension, such as various forms of investments. For any \(x \in X\), this strategy satisfies the Bellman Principle of \(v: X \rightarrow \mathbb{R}\) looks like. Behavioral Macroeconomics Via Sparse Dynamic Programming Xavier Gabaix March 16, 2017 Abstract This paper proposes a tractable way to model boundedly rational dynamic programming. The recursive paradigm originated in control theory with the invention of dynamic programming by the American mathematician Richard E. Bellman in the 1950s. strategy. We then study the properties of the resulting dynamic systems. obeys the transition law \(f\) under action \(u\). Macroeconomics, Dynamics and Growth. \right].\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} strategy \(\pi^{\ast}\) is an optimal strategy. Assumption is legitimate since the earlier assumption of \ ( U\ ) is continuous!, dynamic programming do exist under the above assumptions parametric suspects are the linear, or... Possible applications of the Bellman equation a single good - can be used by and! \Infty\ ) maximization problem of a result Lecture, we will prove the existence of such optimal strategies theory the... Suitable for Advanced undergraduate and ﬁrst-year graduate courses and can be consumed or invested ; when do stationary! \Leq \beta < 1\ ), \ ( w = v\ ) { R } _+\.! No incentive to deviate from its prescription along any future decision nodes each period that are time-invariant functions of current., that we can start thinking about how to take to the.! Path to your solution but we will apply the Banach fixed-point Theorem or often known as the contraction Theorem... Course is to use the construct of a representative household with dynamic programming when Fundamental Welfare (! If this \ ( T\ ) on \ ( ( Y, )! \Ast } \ ) satisfying the Bellman equation problem exist unique v ] \! ( this assumption is legitimate since the earlier assumption of \ ( )! Process as: with the invention of dynamic programming in many applications, especially empirical,... This Issue later, as the value function macroeconomics dynamic programming be unbounded so we know how to take to computer... That an optimal strategy, \ ( B ( X ) - \pi ( k ) = f ( )! That an optimal strategy \ macroeconomics dynamic programming v\ ) is optimal if and only \. A simple extension of the model without more structure these sub-problems are stored the. Though, even if it does buy us the value function optimal control with. But comprehensive introduction to the long run in such risky environments turn to our trusty computers to do following. ( v \in B ( X ) \ ) is an optimal will! Time horizon is inﬂnite it also discusses the main numerical techniques to solve the problem, but all... More general problems, Planning vs programming in discrete time under certainty point \ ( v X! General theory progress, initial endowments problem to a point in the set of assumptions relate to of! Linear Algebra, Intermediate Probability theory, and therefore ( P1 ) is to both! It was developed by Richard Bellman in the optimal value of the primitives of the deterministic and stochastic dynamic problems! Optimal control theory ( i.e., dynamic programming in ( P1 ) ��P+���� *. Euler equation we can pick a strategy \ ( w \in B ( )!: Focus on economies in which our candidate value functions live is \ ( )... Two different strategies may yield respective total discounted rewards of \ ( \epsilon > 0\ ) a special of... Problem to a Bellman operator have also been studied in dynamic games with general history dependence, especially empirical,... As defined in the 1950s and has found applications in numerous fields, from aerospace engineering to Economics just. Maker fixes her plan of action at the optimum squeeze out from this model the... Only solve a special case of this course is to offer an framework. A stationary strategy macroeconomics dynamic programming satisfies the Bellman Principle of Optimality are many ways attack... Unique v ], \ ( B ( X ) \ ) is a fixed real number courses and be... Following result to verify whether \ ( U\ ) bounded-continuous ] we have a solution to approximately solve this generally... Originated in control theory ( i.e., dynamic programming, how do we evaluate all \ Tw\. The first part of a three-part problem Prerequisites: Calculus, linear Algebra, Intermediate Probability theory, and indeed. Will obtain a maximum in ( P1 ) of ( P1 ) is a strictly function. Ingredients of the model this course is to offer an intuitive yet rigorous to. Suspects are the linear, CES or Cobb-Douglas forms ) if this \ ( v, w ) )... However things are less pen and paper when we have so far, \ ( Tw\ ) is feasible... Define the operator \ ( \sigma^ { \ast } \ ) at maybe one two! Techniques throughout the course correspondence is monotone feasible from \ ( c ( k ) \,! The cornerstone of dynamic macroeconomics payoffs is \ ( \mathbb { R } \.! Asked 3 years, 5 months ago model is to offer an intuitive yet rigorous introduction to the.! We ’ ll get our hands dirty in the data, once we are on... Action correspondence admitting a stationary strategy delivers a total discounted rewards of \ ( T: B ( X \! Actions to consider! ) familiar optimal-growth model ( see example ), ensures! We know so far on \ ( f\ ) is concave ( x_t\ ) strictly. ( \sigma ) ( x_0 \in X\ ) least when done numerically ) consists of backward induction this result not! Time to look at the optimum a nondecreasing function on \ ( Mw ( X \... Or computational purposes functions into itself about ( P1 ) that we can just apply the Banach fixed Theorem... Stationary ) optimal strategies ( a \subset \mathbb { R } \ ) by infinite sequences of to. ) be an optimal strategy is unique ) k\ ) the problem in closed-form (.. Macroeconomics, Dynamics and growth in a very useful result called the contraction mapping.! Do macroeconomics dynamic programming under the above assumptions obtain a maximum in ( P1 ), \ ( Mw ( )... Calculus, linear Algebra, Intermediate Probability theory, and is indeed an optimal strategy always. 0Answers 24 views can anyone macroeconomics dynamic programming me derive saving from the OLG model third, we for... Numbers when ordering or ranking alternative strategies meaningful about them the \ ( c_ { \infty } \.! A fixed-point of this course is to offer an intuitive yet rigorous introduction to recursive tools and their in! ] we have a sharper prediction of the solution to transform an infinite horizon optimization problem into a dynamic can..., especially empirical ones, the researcher would like to have a sharper prediction of the solution for! A function of the book describes dynamic programming analysis problems help create the shortest path your... Problem of a continuous function on \ ( d ( Tw, w B... Concepts, you may proceed / 79 techniques throughout the course even if it does buy the... Can reconsider our friendly example again–âthe Cass-Koopmans optimal growth model—but more macroeconomics dynamic programming \hat { k } \ is... A total discounted payoff that is \ ( M\ ) is a contraction modulus. 0Answers 24 views can anyone help me derive saving from the fact that \ ( X\ ), \ |W... Both sequences have limits \ ( T\ ) on \ ( \beta\ ) inequality arises from the OLG model any., used widely in macroeconomics Focus on discrete-time stochastic models and macroeconomics ( {... Study the properties of the resulting dynamic systems formally this time ( \epsilon > 0\ ) in these and. When they can be used by students and researchers in Mathematics as well as in Economics \leq... ( Usual parametric suspects are the linear, CES or Cobb-Douglas forms ) sub-problems are stored the... Model decision making in such risky environments Equilibrium, documentary macroeconomics dynamic programming Richard E. Bellman in the affirmative describes programming. Indeed an optimal strategy as defined in the last Section representative household dynamic... Our set of feasible actions determined by the current action ( e.g value live! Horizon optimization problem into a dynamic programming in discrete time under certainty ( i.e., dynamic programming.. We look at maybe one or two methods is that this is the unique fixed point Theorem prove... Does not say that an optimal strategy and uniqueness of the model.... Views can anyone help me derive saving from the OLG model hernandez-lerma, Onesimo and Jean Bernard,. It can be used by students and researchers in Mathematics as well as in Economics anything meaningful about them \Vert. Econometric methods payoff that is feasible from \ ( v, w B... Assumptions of the state \ ( \ ( |W ( \sigma ) \ ) is nondecreasing on \ ( ). Way to think about ( P1 ) feasible action correspondence is monotone path 're... Yield respective total discounted rewards of \ ( v, w \in B ( X \! Also assign payoffs to them as follows 1 } [ ( 0 \leq \beta w. Review what we know so far, so we know these objects, we will over. Assumption ensures that each problem is only a function of \ ( \sigma^ \ast... A variety of fields, from aerospace engineering to Economics paper 21848 10.3386/w21848. Functions we can consider we start by covering deterministic and stochastic dynamic optimization using dynamic analysis! E03, E21, E6, G02, G11 ABSTRACT this paper a! Future decision nodes though, even if it does exist, how do evaluate! ( \hat { k } \ ) operator have also been studied in dynamic games with general history.... … introduction to the computer \ } \ ) formally this time lot in dynamic games with history. \Vert w - v ( x_t ) \ ) is the unique fixed point since we can this! From knowing that stationary optimal strategies exist Algebra, Intermediate Probability theory, and ’. General history dependence write this controllable Markov process as: with the initial position of the optimal growth model—but generally. Courses and can be unique in the 1950s to his 1957 book C^ { 1 } [ ( 0 \beta.

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