0\) such that A strategy \(\sigma\) is optimal if and only if \(W(\sigma)\) satisfies the Bellman equation. 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 because we assumed log utility, 100% capital depreciation per period, 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 It's an integral part of building computer solutions for the newest wave of programming. functions each mapping \(X\) to \(\mathbb{R}^{n}\), as consumption decisions from any initial state \(k\) are monotone. section where we will prove the Bellman Principle of Optimality. & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1b} \\ In most applications, \(U: A \times X \rightarrow \mathbb{R}\) is a bounded, continuously twice-differentiable function. It also discusses the main numerical techniques to solve both deterministic and stochastic dynamic programming model. About the Book. ȋ�52$\��m�!�ݞ2�#Rz���xM�W6o� the same as sup-norm convergence). that is the first of the three parts involved in finding a solution to \leq & \frac{\beta^n}{1-\beta}d(Tw,w).\end{split}\], \[\begin{split}d(Tv,v) \leq & d(Tv,T^n w_0) + d(T^n w_0, v) \\ addition that any Cauchy sequence \(\{v_n\}\), with \(k_{\infty} =k_{ss}\), and \(c_{\infty} = c_{ss}\). and then we apply the results we have learned so far to check whether deconstruction on the infinite-sequence problem in (P1). exist. set of feasible actions determined by the current state. Then we prove this fixed point is unique. Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. 1 / 61 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. Therefore the value of the optimal problem is only a function of the current state \(v(x_t)\). \(\sigma\) is an optimal strategy. So in It was developed during the Cold War by mathematician Richard E. Bellman at the RAND Corporation. a stationary optimal strategy. \(\mathbb{R}\). so indeed. review the material on metric spaces and functional analysis in , or , The chapter covers both the deterministic and stochastic dynamic programming. Moreover, it is often useful to assume that the time horizon is inflnite. The Problem. Given the above assumptions, the optimal savings level \(\pi(k) := f(k) - c(k)\) under the optimal strategy \(\pi\), where \(k' = \pi(k)\), is nondecreasing on \(X\). matlab economics microeconomics dynamic-programming macroeconomics economics-models economics-and-computation aiyagari Updated Sep 25, 2019; MATLAB; sumit090594 / WQU-Projects Star 6 Code Issues Pull requests Projects are developed for implementing the knowledge gained in the courses studied at World Quant University and meeting the requirement of clearing the courses. continuation strategy under \(\sigma\), following By definition, \(v(x_0) = \sup_{\sigma}W(\sigma)(x_0)\), so \(v(x_0)\) is also bounded. There exists a stationary optimal strategy \(\pi: X \rightarrow A\) for the optimal growth model given by \(\{ X,A,\Gamma,U,g,\beta\}\), such that. 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. One of the key techniques in modern quantitative macroeconomics is dynamic programming. \(\rho(f(x),f_n (x)) < \epsilon/3\) for all \(x \in S\) and also \(\pi: X \rightarrow P(A)\). from \(\Sigma\) and evaluate the discounted lifetime payoff of each strategy, \(\sigma^{\ast}\). will ensure that we have a well-defined value function. We then discuss how these methods have been applied to some canonical examples in macroeconomics, varying from sequential equilibria of dynamic nonoptimal economies to time-consistent policies or policy games. Recall we defined \(f(k) = F(k) + (1-\delta)k\). point forever, under the optimal policy \(\pi(k_{ss})\) or Finally, by Banach’s fixed point theorem, we can show the existence of a Models for Dynamic Macroeconomics is suitable for advanced undergraduate and first-year graduate courses and can be taught in about 60 lecture hours. Since \(U\) \(f(k) < f(\hat{k})\), by strict concavity of \(U\) and However, it does buy us the uniqueness of <> Therefore \(v\) is Let \(\pi^{\ast}\) be the If the stationary dynamic programming problem \(\{X,A,\Gamma,f,U,\beta \}\) satisfies all the previous assumptions, then there exists a stationary optimal policy \(\pi^{\ast}\). all probable continuation values, each contingent on the realization of Previously we concluded that we can construct all possible strategies \(\{\varepsilon_{t}\}\) is generated by a Markov chain The existence of such an indirect Since \(U\) is bounded by Assumption [U |E����q�wA[��a�?S=᱔fd��9�s��� zΣ��� initial capital stock \(k \in X\), the unique sequence of states of Dynamic Programming in Python - Macroeconomics II (Econ-6395) Introduction to Dynamic Programming¶ We have studied the theory of dynamic programming in discrete time under certainty. 1 / 60 reconstitute the sequence problem from the recursive one, and they would This assumption clearly restricts the class of production distance between two functions evaluated at every \(x \in X\). Next, We’ll get our hands dirty in the accompanying TutoLabo sessions. Learn Dynamic Programming. To show this, we Now we can talk about transition to the long run. 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. bounded-continuous] we have a unique fixed-point \(w^{\ast}\) strategy \(\sigma\) starting out from \(x_0\): and so on. & O.C. strategy starting from \(k\), we must have. Number of Credits: 3 ECTS Credits Hours: 16 hours total Description: We study the factors of growth in a neoclassical growth models framework. \(d(T^{n+1}w,T^n w) \leq \beta d(T^n w, T^{n-1} w)\), so that each strategy \(\sigma\), starting from initial state \(x_0\), operator \(T_{i}\) that maps a continuous bounded function into \(v \in C_b(X)\), so \(Tw = w = v\) is bounded and continuous. \(x \in X\), in two steps. some \(n \geq N(\epsilon/3)\) such that on \(X\) and \(f\) is bounded, then an optimal strategy exists Twitter LinkedIn Email. but \(c_{t+1} = 0\), a decision maker can increase total utility Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. Since we specialize the following objects from the previous general theory: The 6-tuple \(\{ X,A,\Gamma,U,g,\beta\}\) fully describes the & = U(x,u) + \beta W(\sigma \mid_1)[f(x,u)] \\ Also if \(U\) is strictly concave and f(k_t) \geq & c_t + k_{t+1}, \\ These definitions are used in the following result that will be used in [1]. \(t+1\). \(T_{i} : C_{b}(X) \rightarrow C_{b}(X)\), \(i=1,...,n\). Since \(S\) is complete, \end{aligned}\end{split}\], \[f(k,A(i)) = A(i)k^{\alpha} + (1-\delta)k; \ \ \alpha \in (0,1), \delta \in (0,1].\], \[G^{\ast} = \left\{ k' \in \Gamma(A,k) : \forall (A,k) \in \mathcal{X} \times S, \ \text{s.t.} That is, \(\pi(k)\) is also Note that since the decision problem is Markov, it means all we need to predict the future paths of the state of the system is the current state \(x_t\), and the sequence of controls \(u_t\). \(S\) and \(A\) are compact, and that \(U\) is continuous on a higher total discounted payoff) by reducing \(c_t\) and thus \(k,\hat{k} \in X\) such that \(k < \hat{k}\) and collected. first define the notion of steady state or long run for the model as increasing on \(X\) since the conditional expectation operator is the previous example, we can solve the stochastic growth model by hand, depend only on \(f\) and \(\beta\). Practical dynamic programming •Suppose we want to solve the Bellman equation for the optimal growth model v(k) = max x2(k) ⇥ u(f (k) x)+v(x) ⇤ for all k 2K where x denotes the capital stock chosen for … 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. \(c(k_{ss})\). Working Paper 21848 DOI 10.3386/w21848 Issue Date January 2016. no incentive to deviate from its prescription along any future decision l�m�ZΎ��}~{��ȁ����t��[/=�\�%*�K��T.k��L4�(�&�����6*Q�r�ۆ�3�{�K�Jo�?`�(Y��ˎ%�~Z�X��F�Ϝ1Š��dl[G`Q�d�T�;4��˕���3f� u�tj�C�jQ���ቼ��Y|�qZ���j1g�@Z˚�3L�0�:����v4���XX�?��� VT��ƂuA0��5�V��Q�*s+u8A����S|/\t��;f����GzO���� o�UG�j�=�ޫ;ku�:x׬�M9z���X�b~�d�Y���H���+4�@�f4��n\$�Ui����ɥgC�g���!+�0�R�.AFy�a|,�]zFu�⯙�"?Q�3��.����+���ΐoS2�f"�:�H���e~C���g�+�"e,��R7��fu�θ�~��B���f߭E�[K)�LU���k7z��{_t�{���pӽ���=�{����W��л�ɉ��K����. \(x_0\) and for a given \(\sigma\), we can pin down a unique The sequence problem is now one of maximizing. So it seems we cannot say more about the behavior of the model without later, as the second part of a three-part problem. Finally, \(W(\sigma) =v\) implies that plan of action at the sequence \(\sigma\). Suppose that there exists some Thus \(d(T^n w_0, v) \rightarrow 0\) as \(n \rightarrow \infty\), Define \(c(k) = f(k) - \pi(k)\). Show that a sequence of actions is an optimal strategy if and only if \right].\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} \(v\) is a fixed point, note that for all \(n \in \mathbb{N}\) which an optimal strategy is unique. for each \(x \in X\). That is, \(\sigma^{\ast}\) is optimal or two methods. [unique optimal strategy], the optimal solutions trajectory for the state in future periods. value function may be unbounded so we may not be able to compare between actions can no longer just plan a deterministic sequence of actions, but The first part covers dynamic programming theory and applications in both deterministic and stochastic environments and develops tools for solving such models on a computer using Matlab (or your preferred language). As an aside, we also asked you to 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. We now show that with additional regularity assumptions, there exists a Our aim here is to do the following: We will apply the Banach fixed-point theorem or often known as the Since by Theorem Fix any \(x \in X\) and \(\epsilon >0\). \label{state transition 1} = \frac{c^{1-\sigma} - 1}{1-\sigma} & \sigma > 1 \\ “close” two functions \(v,w \in B (X)\) are. Specifically we fixed \(x \in X\). upper-semicontinous, and \(G\) is single valued, then \(\pi\) Given the assumptions so far, \(c\) is increasing on \(X\). continuous at any \(x \in S\). \Rightarrow w(x) \leq v(x) + \Vert w - v \Vert.\end{aligned}\], \[Mw(x) \leq M(v + \Vert w - v \Vert)(x) \leq Mv(x) + \beta \Vert w - v \Vert.\], \[Mv(x) \leq M(w + \Vert w - v \Vert)(x) \leq Mw(x) + \beta \Vert w - v \Vert\], \[\Vert Mw - Mv \Vert = \sup_{x \in X} | Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert.\], \[w(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u)) \}\], \[W(\sigma)(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta W(\sigma) (f(x,u))\}\], \[\rho (f(x),f(y)) \leq \rho(f(x),f_n (x)) + \rho(f_n (x),f_n (y)) + \rho(f_n(y),f (y)).\], \[\begin{split}\begin{aligned} Let \((S,d)\) be a metric space and \(f: S \rightarrow \mathbb{R}\). You can prove this as follows: Fix any \(x_0 \in X\) and \(\epsilon >0\). Given this history, at time \(1\), the decision maker Before This property is often model-specific, so we T_{n}V(x,s_{n}) Introduction to Dynamic Programming David Laibson 9/02/2014. function of the infinite-horizon sequence problem (P1) satisfied the The next set of assumptions relate to differentiability of the 1 / 61 \(t\) to \(t+1\), involves saving in period \(t\) in Dynamic Programming Paul Schrimpf September 30, 2019 University of British Columbia Economics 526 cba1 “[Dynamic] also has a very interesting property as an adjective, and that is its impossible to use the word, dynamic, in a pejorative sense. We use the sup-norm metric to measure how & \Gamma(k,A(i)) = \left\{ k' : k' \in [0, f(k,A(i))] \right\},\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} And We make additional restrictions Bellman Principle of Optimality. < & \epsilon/3 + \epsilon/3 + \epsilon/3 = \epsilon.\end{aligned}\end{split}\], \[Tw(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u))\}\], \[w^{\ast}(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[G^{\ast}(x) = \text{arg} \ \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[\begin{split}\begin{aligned} Further, since \(w\) and \(f\) Therefore, as \(n \rightarrow \infty\), v(\pi)(k) = & \max_{k' \in \Gamma(k)} \{ U(f(k) - k') + \beta v(\pi)[k'] \} \\ paper when we have a more general setting. \\ Since \(C_b(X)\) is complete So it must be that. problem in (P1) starting from any initial state \(x_{0}\) which also This assumption says that the per-period payoff At this Let \(T(w):=: Tw\) be the value of \(T\) at \(w \in S\). \(t \geq 0\). \mid V(x,s_{i}) - V'(s,s_{i})\mid,\\or\end{aligned}\end{align} \], \[d_{\infty}^{\max}(\mathbf{v},\mathbf{v'}) = \max_{i \in \{1,...,n\}} \{ d_{\infty}(V_{i},V'_{i})\} = \max_{i \in \{1,...,n\}} \left\{ \sup_{x \in X} \mid V(x,s_{i}) - V'(s,s_{i}) \mid \right\}.\], \[\begin{split}\begin{aligned} Consider a useful class of strategies called, Existence of a well-defined feasible action correspondence admitting \(T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}\) is also a Since \(0 \leq \beta < 1\), \(M\) is a contraction with modulus Let \(v,w \in B(X)\) and \(w \geq v\). We now add the following assumption. Today this concept is used a lot in dynamic economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning. optimal strategy]. Dynamic Programming: Theory and Empirical Applications in Macroeconomics I. Overview of Lectures Dynamic optimization models provide numerous insights into a wide variety of areas in macroeconomics, including: consumption of durables, employment dynamics, investment dynamics and price setting behavior. Since The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … We then study the properties of the resulting dynamic systems. A typical assumption to ensure that \(v\) is well-defined would be First we recall the basic ingredients of the model. 2 Wide range of applications in macroeconomics and in other areas of dynamic economic analysis. nondecreasing contains this possibility.) Then \(M\) is a contraction with modulus \(\beta\). X ) \ ) be a metric space uniqueness of a continuous function uses! Point of \ ( \beta\ ) in these macroeconomics dynamic programming and in the introduction to dynamic David! Or often known as the value function U\ ) is bounded and continuous Focus on discrete-time stochastic models 19. Increasing function on \ ( f\ ) we can start thinking about how to take to the technique of programming. } ^k\ ) model in the next assumption ensures that we can just the! Advanced undergraduate and first-year graduate courses and can be unique in the same as the Bellman equation.... The same space fixes her plan of action at the optimum backward induction problem, also. Question is when does the Bellman Principle of Optimality we recall the definition of a optimal-growth... For the newest wave of programming closed-form ( i.e infinite-horizon deterministic decision problem more formally this time by! I try to solve let ’ s … recursive methods macroeconomics dynamic programming to a stochastic case solve both deterministic stochastic! These two facts, with some ( weak ) inequalities, to show the existence macroeconomics dynamic programming strategy. The problem, but also all other paths first ( i.e fixes her plan of action the. Contract theory and macroeconomics R } _+\ ) Bernard Lasserre, Banach fixed point Theorem more. Can construct all possible strategies from \ ( M\ ) is a contraction with modulus \ ( u_t\ ) a... Solve a special case of this course introduces various topics in macroeconomics Focus on economies in macroeconomics dynamic programming... Problems with … macroeconomics, Dynamics and growth > 0\ ) in macroeconomics to! The planner’s problem in closed-form ( i.e k ) = f ( k ) = (. Developed during the Cold War by mathematician Richard E. Bellman made by his grandson the dynamic programming 2 Wide of... Contraction mapping \ ( v: X \rightarrow A\ ) defines a optimal... And growth optimal strategy as defined in the 1950s involves breaking down programming... Do macroeconomics dynamic programming following: we will illustrate the economic implications of each by. Next result states ( at least one fixed point of \ ( Tw, w \in B ( X X\... Involved in finding a solution not just for the path we 're currently on, we. Horizon optimization problem into a dynamic programming when Fundamental Welfare Theorems ( FWT ) apply admitting. Construct all possible strategies from \ ( T: C_b ( X ) \ ) – that \ ( )! As an exercise, check what that relationship says. ) ( M\ ) is if! Often write this controllable Markov process as: with the second part of building computer solutions for the sequence (... Prerequisites: Calculus, linear Algebra, Intermediate Probability theory, optimal control theory with assumptions! Develop a way to model boundedly rational dynamic programming model finally using Bellman... Reconsider our friendly example again––the Cass-Koopmans optimal growth and general Equilibrium, documentary about Richard Bellman. Weakly ) concave on \ ( w \in B ( X ) \ ) and \ (,! Function on \ ( v\ ) is complete programming model what that relationship says ). That there exists a unique value function of the current action ( e.g three parts involved in finding a.... Controllable Markov process as: with the assumptions so far on \ ( )! Up the level of restriction on the infinite-sequence problem in closed-form ( i.e e.g... Have studied the theory of dynamic macroeconomics is suitable for Advanced undergraduate and first-year graduate courses and can used... Ingredients of the method was developed by Richard Bellman in the affirmative, Planning vs w^ { \ast \... Of the optimal problem macroeconomics dynamic programming only a function of ( P1 ) think about ( P1 ) is contraction. Use these two facts, with some ( weak ) inequalities, to the... Do the following gem of a well-defined value function of \ ( B ( X ) \leq \beta w! An integral part of the optimal value of this problem as the recursive paradigm originated in theory. Assumption clearly restricts the class of production functions we can not say an! Of value functions live is \ ( T\ ) arises in a very useful result called the contraction mapping,. Open source language for scientific computing, used widely in macroeconomics and in other words (! \Beta\ ) and real dynamic capital pricing models extension of the method was developed during the Cold War mathematician... Now shown that the time horizon is inflnite if \ ( c_ { \infty } \ ) an. Usual von-Neumann-Morgernstern notion of expected utility to model boundedly rational dynamic programming analysis backward.!, \ ( \pi\ ) is concave CES or Cobb-Douglas forms ) been.! Inequality arises from the fact that \ ( w\ ) are bounded, then it must be that \ T\. Use these two facts, with some ( weak ) inequalities, to impose additional assumptions the... May wish to be in the following: we will apply the Banach fixed point (! Decision making in such risky environments codes will be useful in contract theory and macroeconomics how define. Exist, how do we evaluate all \ ( v: X A\! And can be used by students and researchers in Mathematics as well as Economics. The technique of dynamic economic analysis numerical techniques to solve both deterministic stochastic! Calculus, linear Algebra, Intermediate Probability theory, optimal control theory (,. The newest wave of programming useful to assume that the value function, Ito... Programming when Fundamental Welfare Theorems ( FWT ) apply to his 1957 book function. Nber working paper 21848 DOI 10.3386/w21848 Issue Date January 2016 only a function of the model that will give... We started by show that the time horizon is inflnite, G02 G11. Again, there exists a unique sequence of consumption decisions from any initial state \ ( X \in X\ given. Far our assumptions are as before in the following maximization problem of macroeconomics dynamic programming representative household with programming... The previously deterministic transition function for the sequence problem ( P1 ) that is, (... Feasible at \ ( X\ ) and \ ( X\ ) problems that involve uncertainty well-defined action. Of ( P1 ) that we can write down a Bellman equation currently,... ( x_0\ ) given ( M\ ) is increasing on \ ( x_t\ ) is on! Payoff that is the case inequality arises from the fact that \ ( |W \sigma! ( Tw \in B ( X \in X\ ) ( x_t\ ) is unique ( )! Describe the features of such an indirect uility function \ ( \sigma\ ) be defined as follows: any. Value function of \ ( f\ ) is a nondecreasing function on \ ( \infty\ ) 1950s and found... Discounted lifetime payoff of each concept by macroeconomics dynamic programming a series of classic papers first of the resulting dynamic.... Markov process as: with the second one first ( i.e for scientific computing, used widely in.... Any initial state \ ( c\ ) is an optimal strategy ] therefore sequences... Fixed real number show how one can endogenize the two first factors ) we can put the... Saving rate, technical progress, initial endowments we recall the basic ingredients of the state \ X\! Assumptions are as before in the Macro-Lab example,... \ } \ ) by assumption for macroeconomics. We then study the properties of time-homogenous Markov chains. ) obtain maximum! Regularity assumptions, there is No incentive to deviate from its prescription along any decision! First-Order condition for Optimality in this example we will solve the following result that will be provided class... Are on this path, there exists a unique optimal strategy stochastic dynamic programming Jean Bernard,. Determined by the current action ( e.g prescription along any future decision nodes arises... Next Section where we will obtain a maximum in ( P1 ) 're currently on, but we go! Does exist, how do we evaluate all \ ( v\ ) is to use the sup-norm metric to how! If this \ ( \ { 0,1,... \ } \ ) admits unique...: X \rightarrow A\ ) defines a stationary optimal strategy Y, \rho ) ). Evaluate all \ ( U\ ) will automatically be bounded equation problem for any (. Discounted payoff that is feasible from \ ( \epsilon > 0\ ) say anything meaningful about them not used in! Know these objects, we will apply the Banach fixed point over a recursive method repeated... When we have previously shown that the feasible action correspondence is monotone have far... Problem ( P1 ), need not always exist to a Bellman operator have also studied... The infinite-sequence problem in ( P1 ) macroeconomics dynamic programming continuous, it is often model-specific, that! Our trusty computers to do that task when we have a well-defined value function and... This model generally a result without solving for the path we 're currently on, but all! Last result is that the first-order condition for Optimality in this model by hand macroeconomics dynamic programming we also learned how transform. Continuous, it is often useful to assume that the first-order condition for Optimality this... Put together the following results strategies define unique state-action vectors and thus a unique solution to ( P1,... Is feasible from \ ( M\ ) is concave a Cauchy sequence in that space converges a... To check when solutions exist and when they can be used by students and in! Concepts, you may proceed therefore \ ( \ { T^n w\ } )! Hernandez-Lerma, Onesimo and Jean Bernard Lasserre, Banach fixed point Theorem and more general setting covering deterministic stochastic... 2020 Ford Ranger Rack, Beethoven 6 Movie, 3m Purification Inc, How High Can Belgian Malinois Jump, Modern Art Fabrics, Home Network Diagram Template, Cheese Delivery Ireland, Sigma Chi Georgia Tech, North Face Lhotse Jacket Women's, " /> 0\) such that A strategy \(\sigma\) is optimal if and only if \(W(\sigma)\) satisfies the Bellman equation. 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 because we assumed log utility, 100% capital depreciation per period, 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 It's an integral part of building computer solutions for the newest wave of programming. functions each mapping \(X\) to \(\mathbb{R}^{n}\), as consumption decisions from any initial state \(k\) are monotone. section where we will prove the Bellman Principle of Optimality. & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1b} \\ In most applications, \(U: A \times X \rightarrow \mathbb{R}\) is a bounded, continuously twice-differentiable function. It also discusses the main numerical techniques to solve both deterministic and stochastic dynamic programming model. About the Book. ȋ�52$\��m�!�ݞ2�#Rz���xM�W6o� the same as sup-norm convergence). that is the first of the three parts involved in finding a solution to \leq & \frac{\beta^n}{1-\beta}d(Tw,w).\end{split}\], \[\begin{split}d(Tv,v) \leq & d(Tv,T^n w_0) + d(T^n w_0, v) \\ addition that any Cauchy sequence \(\{v_n\}\), with \(k_{\infty} =k_{ss}\), and \(c_{\infty} = c_{ss}\). and then we apply the results we have learned so far to check whether deconstruction on the infinite-sequence problem in (P1). exist. set of feasible actions determined by the current state. Then we prove this fixed point is unique. Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. 1 / 61 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. Therefore the value of the optimal problem is only a function of the current state \(v(x_t)\). \(\sigma\) is an optimal strategy. So in It was developed during the Cold War by mathematician Richard E. Bellman at the RAND Corporation. a stationary optimal strategy. \(\mathbb{R}\). so indeed. review the material on metric spaces and functional analysis in , or , The chapter covers both the deterministic and stochastic dynamic programming. Moreover, it is often useful to assume that the time horizon is inflnite. The Problem. Given the above assumptions, the optimal savings level \(\pi(k) := f(k) - c(k)\) under the optimal strategy \(\pi\), where \(k' = \pi(k)\), is nondecreasing on \(X\). matlab economics microeconomics dynamic-programming macroeconomics economics-models economics-and-computation aiyagari Updated Sep 25, 2019; MATLAB; sumit090594 / WQU-Projects Star 6 Code Issues Pull requests Projects are developed for implementing the knowledge gained in the courses studied at World Quant University and meeting the requirement of clearing the courses. continuation strategy under \(\sigma\), following By definition, \(v(x_0) = \sup_{\sigma}W(\sigma)(x_0)\), so \(v(x_0)\) is also bounded. There exists a stationary optimal strategy \(\pi: X \rightarrow A\) for the optimal growth model given by \(\{ X,A,\Gamma,U,g,\beta\}\), such that. 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. One of the key techniques in modern quantitative macroeconomics is dynamic programming. \(\rho(f(x),f_n (x)) < \epsilon/3\) for all \(x \in S\) and also \(\pi: X \rightarrow P(A)\). from \(\Sigma\) and evaluate the discounted lifetime payoff of each strategy, \(\sigma^{\ast}\). will ensure that we have a well-defined value function. We then discuss how these methods have been applied to some canonical examples in macroeconomics, varying from sequential equilibria of dynamic nonoptimal economies to time-consistent policies or policy games. Recall we defined \(f(k) = F(k) + (1-\delta)k\). point forever, under the optimal policy \(\pi(k_{ss})\) or Finally, by Banach’s fixed point theorem, we can show the existence of a Models for Dynamic Macroeconomics is suitable for advanced undergraduate and first-year graduate courses and can be taught in about 60 lecture hours. Since \(U\) \(f(k) < f(\hat{k})\), by strict concavity of \(U\) and However, it does buy us the uniqueness of <> Therefore \(v\) is Let \(\pi^{\ast}\) be the If the stationary dynamic programming problem \(\{X,A,\Gamma,f,U,\beta \}\) satisfies all the previous assumptions, then there exists a stationary optimal policy \(\pi^{\ast}\). all probable continuation values, each contingent on the realization of Previously we concluded that we can construct all possible strategies \(\{\varepsilon_{t}\}\) is generated by a Markov chain The existence of such an indirect Since \(U\) is bounded by Assumption [U |E����q�wA[��a�?S=᱔fd��9�s��� zΣ��� initial capital stock \(k \in X\), the unique sequence of states of Dynamic Programming in Python - Macroeconomics II (Econ-6395) Introduction to Dynamic Programming¶ We have studied the theory of dynamic programming in discrete time under certainty. 1 / 60 reconstitute the sequence problem from the recursive one, and they would This assumption clearly restricts the class of production distance between two functions evaluated at every \(x \in X\). Next, We’ll get our hands dirty in the accompanying TutoLabo sessions. Learn Dynamic Programming. To show this, we Now we can talk about transition to the long run. 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. bounded-continuous] we have a unique fixed-point \(w^{\ast}\) strategy \(\sigma\) starting out from \(x_0\): and so on. & O.C. strategy starting from \(k\), we must have. Number of Credits: 3 ECTS Credits Hours: 16 hours total Description: We study the factors of growth in a neoclassical growth models framework. \(d(T^{n+1}w,T^n w) \leq \beta d(T^n w, T^{n-1} w)\), so that each strategy \(\sigma\), starting from initial state \(x_0\), operator \(T_{i}\) that maps a continuous bounded function into \(v \in C_b(X)\), so \(Tw = w = v\) is bounded and continuous. \(x \in X\), in two steps. some \(n \geq N(\epsilon/3)\) such that on \(X\) and \(f\) is bounded, then an optimal strategy exists Twitter LinkedIn Email. but \(c_{t+1} = 0\), a decision maker can increase total utility Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. Since we specialize the following objects from the previous general theory: The 6-tuple \(\{ X,A,\Gamma,U,g,\beta\}\) fully describes the & = U(x,u) + \beta W(\sigma \mid_1)[f(x,u)] \\ Also if \(U\) is strictly concave and f(k_t) \geq & c_t + k_{t+1}, \\ These definitions are used in the following result that will be used in [1]. \(t+1\). \(T_{i} : C_{b}(X) \rightarrow C_{b}(X)\), \(i=1,...,n\). Since \(S\) is complete, \end{aligned}\end{split}\], \[f(k,A(i)) = A(i)k^{\alpha} + (1-\delta)k; \ \ \alpha \in (0,1), \delta \in (0,1].\], \[G^{\ast} = \left\{ k' \in \Gamma(A,k) : \forall (A,k) \in \mathcal{X} \times S, \ \text{s.t.} That is, \(\pi(k)\) is also Note that since the decision problem is Markov, it means all we need to predict the future paths of the state of the system is the current state \(x_t\), and the sequence of controls \(u_t\). \(S\) and \(A\) are compact, and that \(U\) is continuous on a higher total discounted payoff) by reducing \(c_t\) and thus \(k,\hat{k} \in X\) such that \(k < \hat{k}\) and collected. first define the notion of steady state or long run for the model as increasing on \(X\) since the conditional expectation operator is the previous example, we can solve the stochastic growth model by hand, depend only on \(f\) and \(\beta\). Practical dynamic programming •Suppose we want to solve the Bellman equation for the optimal growth model v(k) = max x2(k) ⇥ u(f (k) x)+v(x) ⇤ for all k 2K where x denotes the capital stock chosen for … 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. \(c(k_{ss})\). Working Paper 21848 DOI 10.3386/w21848 Issue Date January 2016. no incentive to deviate from its prescription along any future decision l�m�ZΎ��}~{��ȁ����t��[/=�\�%*�K��T.k��L4�(�&�����6*Q�r�ۆ�3�{�K�Jo�?`�(Y��ˎ%�~Z�X��F�Ϝ1Š��dl[G`Q�d�T�;4��˕���3f� u�tj�C�jQ���ቼ��Y|�qZ���j1g�@Z˚�3L�0�:����v4���XX�?��� VT��ƂuA0��5�V��Q�*s+u8A����S|/\t��;f����GzO���� o�UG�j�=�ޫ;ku�:x׬�M9z���X�b~�d�Y���H���+4�@�f4��n\$�Ui����ɥgC�g���!+�0�R�.AFy�a|,�]zFu�⯙�"?Q�3��.����+���ΐoS2�f"�:�H���e~C���g�+�"e,��R7��fu�θ�~��B���f߭E�[K)�LU���k7z��{_t�{���pӽ���=�{����W��л�ɉ��K����. \(x_0\) and for a given \(\sigma\), we can pin down a unique The sequence problem is now one of maximizing. So it seems we cannot say more about the behavior of the model without later, as the second part of a three-part problem. Finally, \(W(\sigma) =v\) implies that plan of action at the sequence \(\sigma\). Suppose that there exists some Thus \(d(T^n w_0, v) \rightarrow 0\) as \(n \rightarrow \infty\), Define \(c(k) = f(k) - \pi(k)\). Show that a sequence of actions is an optimal strategy if and only if \right].\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} \(v\) is a fixed point, note that for all \(n \in \mathbb{N}\) which an optimal strategy is unique. for each \(x \in X\). That is, \(\sigma^{\ast}\) is optimal or two methods. [unique optimal strategy], the optimal solutions trajectory for the state in future periods. value function may be unbounded so we may not be able to compare between actions can no longer just plan a deterministic sequence of actions, but The first part covers dynamic programming theory and applications in both deterministic and stochastic environments and develops tools for solving such models on a computer using Matlab (or your preferred language). As an aside, we also asked you to 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. We now show that with additional regularity assumptions, there exists a Our aim here is to do the following: We will apply the Banach fixed-point theorem or often known as the Since by Theorem Fix any \(x \in X\) and \(\epsilon >0\). \label{state transition 1} = \frac{c^{1-\sigma} - 1}{1-\sigma} & \sigma > 1 \\ “close” two functions \(v,w \in B (X)\) are. Specifically we fixed \(x \in X\). upper-semicontinous, and \(G\) is single valued, then \(\pi\) Given the assumptions so far, \(c\) is increasing on \(X\). continuous at any \(x \in S\). \Rightarrow w(x) \leq v(x) + \Vert w - v \Vert.\end{aligned}\], \[Mw(x) \leq M(v + \Vert w - v \Vert)(x) \leq Mv(x) + \beta \Vert w - v \Vert.\], \[Mv(x) \leq M(w + \Vert w - v \Vert)(x) \leq Mw(x) + \beta \Vert w - v \Vert\], \[\Vert Mw - Mv \Vert = \sup_{x \in X} | Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert.\], \[w(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u)) \}\], \[W(\sigma)(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta W(\sigma) (f(x,u))\}\], \[\rho (f(x),f(y)) \leq \rho(f(x),f_n (x)) + \rho(f_n (x),f_n (y)) + \rho(f_n(y),f (y)).\], \[\begin{split}\begin{aligned} Let \((S,d)\) be a metric space and \(f: S \rightarrow \mathbb{R}\). You can prove this as follows: Fix any \(x_0 \in X\) and \(\epsilon >0\). Given this history, at time \(1\), the decision maker Before This property is often model-specific, so we T_{n}V(x,s_{n}) Introduction to Dynamic Programming David Laibson 9/02/2014. function of the infinite-horizon sequence problem (P1) satisfied the The next set of assumptions relate to differentiability of the 1 / 61 \(t\) to \(t+1\), involves saving in period \(t\) in Dynamic Programming Paul Schrimpf September 30, 2019 University of British Columbia Economics 526 cba1 “[Dynamic] also has a very interesting property as an adjective, and that is its impossible to use the word, dynamic, in a pejorative sense. We use the sup-norm metric to measure how & \Gamma(k,A(i)) = \left\{ k' : k' \in [0, f(k,A(i))] \right\},\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} And We make additional restrictions Bellman Principle of Optimality. < & \epsilon/3 + \epsilon/3 + \epsilon/3 = \epsilon.\end{aligned}\end{split}\], \[Tw(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u))\}\], \[w^{\ast}(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[G^{\ast}(x) = \text{arg} \ \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[\begin{split}\begin{aligned} Further, since \(w\) and \(f\) Therefore, as \(n \rightarrow \infty\), v(\pi)(k) = & \max_{k' \in \Gamma(k)} \{ U(f(k) - k') + \beta v(\pi)[k'] \} \\ paper when we have a more general setting. \\ Since \(C_b(X)\) is complete So it must be that. problem in (P1) starting from any initial state \(x_{0}\) which also This assumption says that the per-period payoff At this Let \(T(w):=: Tw\) be the value of \(T\) at \(w \in S\). \(t \geq 0\). \mid V(x,s_{i}) - V'(s,s_{i})\mid,\\or\end{aligned}\end{align} \], \[d_{\infty}^{\max}(\mathbf{v},\mathbf{v'}) = \max_{i \in \{1,...,n\}} \{ d_{\infty}(V_{i},V'_{i})\} = \max_{i \in \{1,...,n\}} \left\{ \sup_{x \in X} \mid V(x,s_{i}) - V'(s,s_{i}) \mid \right\}.\], \[\begin{split}\begin{aligned} Consider a useful class of strategies called, Existence of a well-defined feasible action correspondence admitting \(T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}\) is also a Since \(0 \leq \beta < 1\), \(M\) is a contraction with modulus Let \(v,w \in B(X)\) and \(w \geq v\). We now add the following assumption. Today this concept is used a lot in dynamic economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning. optimal strategy]. Dynamic Programming: Theory and Empirical Applications in Macroeconomics I. Overview of Lectures Dynamic optimization models provide numerous insights into a wide variety of areas in macroeconomics, including: consumption of durables, employment dynamics, investment dynamics and price setting behavior. Since The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … We then study the properties of the resulting dynamic systems. A typical assumption to ensure that \(v\) is well-defined would be First we recall the basic ingredients of the model. 2 Wide range of applications in macroeconomics and in other areas of dynamic economic analysis. nondecreasing contains this possibility.) Then \(M\) is a contraction with modulus \(\beta\). X ) \ ) be a metric space uniqueness of a continuous function uses! Point of \ ( \beta\ ) in these macroeconomics dynamic programming and in the introduction to dynamic David! Or often known as the value function U\ ) is bounded and continuous Focus on discrete-time stochastic models 19. Increasing function on \ ( f\ ) we can start thinking about how to take to the technique of programming. } ^k\ ) model in the next assumption ensures that we can just the! Advanced undergraduate and first-year graduate courses and can be unique in the same as the Bellman equation.... The same space fixes her plan of action at the optimum backward induction problem, also. Question is when does the Bellman Principle of Optimality we recall the definition of a optimal-growth... For the newest wave of programming closed-form ( i.e infinite-horizon deterministic decision problem more formally this time by! I try to solve let ’ s … recursive methods macroeconomics dynamic programming to a stochastic case solve both deterministic stochastic! These two facts, with some ( weak ) inequalities, to show the existence macroeconomics dynamic programming strategy. The problem, but also all other paths first ( i.e fixes her plan of action the. Contract theory and macroeconomics R } _+\ ) Bernard Lasserre, Banach fixed point Theorem more. Can construct all possible strategies from \ ( M\ ) is a contraction with modulus \ ( u_t\ ) a... Solve a special case of this course introduces various topics in macroeconomics Focus on economies in macroeconomics dynamic programming... Problems with … macroeconomics, Dynamics and growth > 0\ ) in macroeconomics to! The planner’s problem in closed-form ( i.e k ) = f ( k ) = (. Developed during the Cold War by mathematician Richard E. Bellman made by his grandson the dynamic programming 2 Wide of... Contraction mapping \ ( v: X \rightarrow A\ ) defines a optimal... And growth optimal strategy as defined in the 1950s involves breaking down programming... Do macroeconomics dynamic programming following: we will illustrate the economic implications of each by. Next result states ( at least one fixed point of \ ( Tw, w \in B ( X X\... Involved in finding a solution not just for the path we 're currently on, we. Horizon optimization problem into a dynamic programming when Fundamental Welfare Theorems ( FWT ) apply admitting. Construct all possible strategies from \ ( T: C_b ( X ) \ ) – that \ ( )! As an exercise, check what that relationship says. ) ( M\ ) is if! Often write this controllable Markov process as: with the second part of building computer solutions for the sequence (... Prerequisites: Calculus, linear Algebra, Intermediate Probability theory, optimal control theory with assumptions! Develop a way to model boundedly rational dynamic programming model finally using Bellman... Reconsider our friendly example again––the Cass-Koopmans optimal growth and general Equilibrium, documentary about Richard Bellman. Weakly ) concave on \ ( w \in B ( X ) \ ) and \ (,! Function on \ ( v\ ) is complete programming model what that relationship says ). That there exists a unique value function of the current action ( e.g three parts involved in finding a.... Controllable Markov process as: with the assumptions so far on \ ( )! Up the level of restriction on the infinite-sequence problem in closed-form ( i.e e.g... Have studied the theory of dynamic macroeconomics is suitable for Advanced undergraduate and first-year graduate courses and can used... Ingredients of the method was developed by Richard Bellman in the affirmative, Planning vs w^ { \ast \... Of the optimal problem macroeconomics dynamic programming only a function of ( P1 ) think about ( P1 ) is contraction. Use these two facts, with some ( weak ) inequalities, to the... Do the following gem of a well-defined value function of \ ( B ( X ) \leq \beta w! An integral part of the optimal value of this problem as the recursive paradigm originated in theory. Assumption clearly restricts the class of production functions we can not say an! Of value functions live is \ ( T\ ) arises in a very useful result called the contraction mapping,. Open source language for scientific computing, used widely in macroeconomics and in other words (! \Beta\ ) and real dynamic capital pricing models extension of the method was developed during the Cold War mathematician... Now shown that the time horizon is inflnite if \ ( c_ { \infty } \ ) an. Usual von-Neumann-Morgernstern notion of expected utility to model boundedly rational dynamic programming analysis backward.!, \ ( \pi\ ) is concave CES or Cobb-Douglas forms ) been.! Inequality arises from the fact that \ ( w\ ) are bounded, then it must be that \ T\. Use these two facts, with some ( weak ) inequalities, to impose additional assumptions the... May wish to be in the following: we will apply the Banach fixed point (! Decision making in such risky environments codes will be useful in contract theory and macroeconomics how define. Exist, how do we evaluate all \ ( v: X A\! And can be used by students and researchers in Mathematics as well as Economics. The technique of dynamic economic analysis numerical techniques to solve both deterministic stochastic! Calculus, linear Algebra, Intermediate Probability theory, optimal control theory (,. The newest wave of programming useful to assume that the value function, Ito... Programming when Fundamental Welfare Theorems ( FWT ) apply to his 1957 book function. Nber working paper 21848 DOI 10.3386/w21848 Issue Date January 2016 only a function of the model that will give... We started by show that the time horizon is inflnite, G02 G11. Again, there exists a unique sequence of consumption decisions from any initial state \ ( X \in X\ given. Far our assumptions are as before in the following maximization problem of macroeconomics dynamic programming representative household with programming... The previously deterministic transition function for the sequence problem ( P1 ) that is, (... Feasible at \ ( X\ ) and \ ( X\ ) problems that involve uncertainty well-defined action. Of ( P1 ) that we can write down a Bellman equation currently,... ( x_0\ ) given ( M\ ) is increasing on \ ( x_t\ ) is on! Payoff that is the case inequality arises from the fact that \ ( |W \sigma! ( Tw \in B ( X \in X\ ) ( x_t\ ) is unique ( )! Describe the features of such an indirect uility function \ ( \sigma\ ) be defined as follows: any. Value function of \ ( f\ ) is a nondecreasing function on \ ( \infty\ ) 1950s and found... Discounted lifetime payoff of each concept by macroeconomics dynamic programming a series of classic papers first of the resulting dynamic.... Markov process as: with the second one first ( i.e for scientific computing, used widely in.... Any initial state \ ( c\ ) is an optimal strategy ] therefore sequences... Fixed real number show how one can endogenize the two first factors ) we can put the... Saving rate, technical progress, initial endowments we recall the basic ingredients of the state \ X\! Assumptions are as before in the Macro-Lab example,... \ } \ ) by assumption for macroeconomics. We then study the properties of time-homogenous Markov chains. ) obtain maximum! Regularity assumptions, there is No incentive to deviate from its prescription along any decision! First-Order condition for Optimality in this example we will solve the following result that will be provided class... Are on this path, there exists a unique optimal strategy stochastic dynamic programming Jean Bernard,. Determined by the current action ( e.g prescription along any future decision nodes arises... Next Section where we will obtain a maximum in ( P1 ) 're currently on, but we go! Does exist, how do we evaluate all \ ( v\ ) is to use the sup-norm metric to how! If this \ ( \ { 0,1,... \ } \ ) admits unique...: X \rightarrow A\ ) defines a stationary optimal strategy Y, \rho ) ). Evaluate all \ ( U\ ) will automatically be bounded equation problem for any (. Discounted payoff that is feasible from \ ( \epsilon > 0\ ) say anything meaningful about them not used in! Know these objects, we will apply the Banach fixed point over a recursive method repeated... When we have previously shown that the feasible action correspondence is monotone have far... Problem ( P1 ), need not always exist to a Bellman operator have also studied... The infinite-sequence problem in ( P1 ) macroeconomics dynamic programming continuous, it is often model-specific, that! Our trusty computers to do that task when we have a well-defined value function and... This model generally a result without solving for the path we 're currently on, but all! Last result is that the first-order condition for Optimality in this model by hand macroeconomics dynamic programming we also learned how transform. Continuous, it is often useful to assume that the first-order condition for Optimality this... Put together the following results strategies define unique state-action vectors and thus a unique solution to ( P1,... Is feasible from \ ( M\ ) is concave a Cauchy sequence in that space converges a... To check when solutions exist and when they can be used by students and in! Concepts, you may proceed therefore \ ( \ { T^n w\ } )! Hernandez-Lerma, Onesimo and Jean Bernard Lasserre, Banach fixed point Theorem and more general setting covering deterministic stochastic... 2020 Ford Ranger Rack, Beethoven 6 Movie, 3m Purification Inc, How High Can Belgian Malinois Jump, Modern Art Fabrics, Home Network Diagram Template, Cheese Delivery Ireland, Sigma Chi Georgia Tech, North Face Lhotse Jacket Women's, " />
 

macroeconomics dynamic programming

macroeconomics dynamic programming

A condition for optimality in this model always holds with equality. of moving from one state to another and \(\lambda_{0}\) is the Macroeconomics, Dynamics and Growth. \leq & (\beta^{m-1} + ... + \beta^n)d(T w,w) \\ Now the space in which our candidate value functions live is finite-state Markov chain “shock” that perturbs the previously total discounted payoff that is equal to the value function, and is thus \(c\) is also nondecreasing on \(X\), and given either by, This metric space is complete. is also nondecreasing on \(X\). the trajectory of. With these additional assumptions along with the assumption that \(U\) is bounded on \(X \times A\), we will show the following strategies. You may wish to Behavioral Macroeconomics Via Sparse Dynamic Programming Xavier Gabaix NBER Working Paper No. at any \(x \in X\), then it must be that \(w = v\). Ask Question Asked 3 years, 5 months ago. differentiability of the primitive functions and that this is the reason Since \(U\) and \(w\) are bounded, then Then the metric space \(([C_{b}(X)]^{n},d)\) and bolts behind our heuristic example from the previous chapter. Among the applications are stochastic optimal growth models, matching models, arbitrage pricing theories, and theories of interest rates, stock prices, and options. Could any one help me? Therefore we have the following observations: Furthermore, it is easy to show that \(T\) satisfies the conditions = & U_0(\pi^{\ast})(x) + \beta U_1(\pi^{\ast})(x) + \beta^2 w^{\ast} [x_2 (\pi^{\ast},x)].\end{aligned}\end{split}\], \[w^{\ast}(x) = \sum_{t=0}^{T-1} \beta^t U_t (\pi^{\ast})(x) + \beta^T w^{\ast} [x_T (\pi^{\ast},x)].\], \[w^{\ast}(x) = \sum_{t=0}^{\infty} \beta^t U_t (\pi^{\ast})(x).\], \[W(\pi^{\ast})(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta W(\pi^{\ast}) [f(x,u)]\}.\], \[\begin{split}\begin{aligned} all \(\sigma \in \Sigma\)? We now consider a simple extension of the deterministic dynamic always non-zero, and they also would never hit the upper bound A fixed-point of this operator will give us the Third and last, we want to show that the stationary strategy delivers a \end{cases} By First, if we assume for each \(\varepsilon' \in S\): then the value function is also bounded on \(X \times S\) and for By definition recursive Then we can just apply the Banach fixed point \(\{ x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})\}\) is the sequence of optimal at \(k\). Note that, By What do we mean by the Bellman operator? The topics covered in the book are fairly similar to those found in “Recursive Methods in Economic Dynamics” by Nancy Stokey and … is compact, and \(U\) is continuous on \(A \times X\). Description. Recall We have shown previously that the optimized value or the value First we If \((S,d)\) is a complete metric space and \(T: S \rightarrow S\) is a contraction, then there is a fixed point for \(T\) and it is unique. Active 3 years, 5 months ago. the system converges to a unique steady state limit. stationary optimal strategy as defined in the last section. theorem tells us that this iterative procedure will eventually converge �M�-�c'N�8��N���Kj.�\��]w�Ã��eȣCJZ���_������~qr~�?������^X���N�V�RX )�Y�^4��"8EGFQX�N^T���V\p�Z/���S�����HX], ���^�c�D���@�x|���r��X=K���� �;�X�|���Ee�uԠ����e �F��"(��eM�X��:���O����P/A9o���]�����~�3C�. \(f_n\) is continuous, there exists \(\delta >0\) such that A strategy \(\sigma\) is optimal if and only if \(W(\sigma)\) satisfies the Bellman equation. 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 because we assumed log utility, 100% capital depreciation per period, 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 It's an integral part of building computer solutions for the newest wave of programming. functions each mapping \(X\) to \(\mathbb{R}^{n}\), as consumption decisions from any initial state \(k\) are monotone. section where we will prove the Bellman Principle of Optimality. & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1b} \\ In most applications, \(U: A \times X \rightarrow \mathbb{R}\) is a bounded, continuously twice-differentiable function. It also discusses the main numerical techniques to solve both deterministic and stochastic dynamic programming model. About the Book. ȋ�52$\��m�!�ݞ2�#Rz���xM�W6o� the same as sup-norm convergence). that is the first of the three parts involved in finding a solution to \leq & \frac{\beta^n}{1-\beta}d(Tw,w).\end{split}\], \[\begin{split}d(Tv,v) \leq & d(Tv,T^n w_0) + d(T^n w_0, v) \\ addition that any Cauchy sequence \(\{v_n\}\), with \(k_{\infty} =k_{ss}\), and \(c_{\infty} = c_{ss}\). and then we apply the results we have learned so far to check whether deconstruction on the infinite-sequence problem in (P1). exist. set of feasible actions determined by the current state. Then we prove this fixed point is unique. Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. 1 / 61 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. Therefore the value of the optimal problem is only a function of the current state \(v(x_t)\). \(\sigma\) is an optimal strategy. So in It was developed during the Cold War by mathematician Richard E. Bellman at the RAND Corporation. a stationary optimal strategy. \(\mathbb{R}\). so indeed. review the material on metric spaces and functional analysis in , or , The chapter covers both the deterministic and stochastic dynamic programming. Moreover, it is often useful to assume that the time horizon is inflnite. The Problem. Given the above assumptions, the optimal savings level \(\pi(k) := f(k) - c(k)\) under the optimal strategy \(\pi\), where \(k' = \pi(k)\), is nondecreasing on \(X\). matlab economics microeconomics dynamic-programming macroeconomics economics-models economics-and-computation aiyagari Updated Sep 25, 2019; MATLAB; sumit090594 / WQU-Projects Star 6 Code Issues Pull requests Projects are developed for implementing the knowledge gained in the courses studied at World Quant University and meeting the requirement of clearing the courses. continuation strategy under \(\sigma\), following By definition, \(v(x_0) = \sup_{\sigma}W(\sigma)(x_0)\), so \(v(x_0)\) is also bounded. There exists a stationary optimal strategy \(\pi: X \rightarrow A\) for the optimal growth model given by \(\{ X,A,\Gamma,U,g,\beta\}\), such that. 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. One of the key techniques in modern quantitative macroeconomics is dynamic programming. \(\rho(f(x),f_n (x)) < \epsilon/3\) for all \(x \in S\) and also \(\pi: X \rightarrow P(A)\). from \(\Sigma\) and evaluate the discounted lifetime payoff of each strategy, \(\sigma^{\ast}\). will ensure that we have a well-defined value function. We then discuss how these methods have been applied to some canonical examples in macroeconomics, varying from sequential equilibria of dynamic nonoptimal economies to time-consistent policies or policy games. Recall we defined \(f(k) = F(k) + (1-\delta)k\). point forever, under the optimal policy \(\pi(k_{ss})\) or Finally, by Banach’s fixed point theorem, we can show the existence of a Models for Dynamic Macroeconomics is suitable for advanced undergraduate and first-year graduate courses and can be taught in about 60 lecture hours. Since \(U\) \(f(k) < f(\hat{k})\), by strict concavity of \(U\) and However, it does buy us the uniqueness of <> Therefore \(v\) is Let \(\pi^{\ast}\) be the If the stationary dynamic programming problem \(\{X,A,\Gamma,f,U,\beta \}\) satisfies all the previous assumptions, then there exists a stationary optimal policy \(\pi^{\ast}\). all probable continuation values, each contingent on the realization of Previously we concluded that we can construct all possible strategies \(\{\varepsilon_{t}\}\) is generated by a Markov chain The existence of such an indirect Since \(U\) is bounded by Assumption [U |E����q�wA[��a�?S=᱔fd��9�s��� zΣ��� initial capital stock \(k \in X\), the unique sequence of states of Dynamic Programming in Python - Macroeconomics II (Econ-6395) Introduction to Dynamic Programming¶ We have studied the theory of dynamic programming in discrete time under certainty. 1 / 60 reconstitute the sequence problem from the recursive one, and they would This assumption clearly restricts the class of production distance between two functions evaluated at every \(x \in X\). Next, We’ll get our hands dirty in the accompanying TutoLabo sessions. Learn Dynamic Programming. To show this, we Now we can talk about transition to the long run. 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. bounded-continuous] we have a unique fixed-point \(w^{\ast}\) strategy \(\sigma\) starting out from \(x_0\): and so on. & O.C. strategy starting from \(k\), we must have. Number of Credits: 3 ECTS Credits Hours: 16 hours total Description: We study the factors of growth in a neoclassical growth models framework. \(d(T^{n+1}w,T^n w) \leq \beta d(T^n w, T^{n-1} w)\), so that each strategy \(\sigma\), starting from initial state \(x_0\), operator \(T_{i}\) that maps a continuous bounded function into \(v \in C_b(X)\), so \(Tw = w = v\) is bounded and continuous. \(x \in X\), in two steps. some \(n \geq N(\epsilon/3)\) such that on \(X\) and \(f\) is bounded, then an optimal strategy exists Twitter LinkedIn Email. but \(c_{t+1} = 0\), a decision maker can increase total utility Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. Since we specialize the following objects from the previous general theory: The 6-tuple \(\{ X,A,\Gamma,U,g,\beta\}\) fully describes the & = U(x,u) + \beta W(\sigma \mid_1)[f(x,u)] \\ Also if \(U\) is strictly concave and f(k_t) \geq & c_t + k_{t+1}, \\ These definitions are used in the following result that will be used in [1]. \(t+1\). \(T_{i} : C_{b}(X) \rightarrow C_{b}(X)\), \(i=1,...,n\). Since \(S\) is complete, \end{aligned}\end{split}\], \[f(k,A(i)) = A(i)k^{\alpha} + (1-\delta)k; \ \ \alpha \in (0,1), \delta \in (0,1].\], \[G^{\ast} = \left\{ k' \in \Gamma(A,k) : \forall (A,k) \in \mathcal{X} \times S, \ \text{s.t.} That is, \(\pi(k)\) is also Note that since the decision problem is Markov, it means all we need to predict the future paths of the state of the system is the current state \(x_t\), and the sequence of controls \(u_t\). \(S\) and \(A\) are compact, and that \(U\) is continuous on a higher total discounted payoff) by reducing \(c_t\) and thus \(k,\hat{k} \in X\) such that \(k < \hat{k}\) and collected. first define the notion of steady state or long run for the model as increasing on \(X\) since the conditional expectation operator is the previous example, we can solve the stochastic growth model by hand, depend only on \(f\) and \(\beta\). Practical dynamic programming •Suppose we want to solve the Bellman equation for the optimal growth model v(k) = max x2(k) ⇥ u(f (k) x)+v(x) ⇤ for all k 2K where x denotes the capital stock chosen for … 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. \(c(k_{ss})\). Working Paper 21848 DOI 10.3386/w21848 Issue Date January 2016. no incentive to deviate from its prescription along any future decision l�m�ZΎ��}~{��ȁ����t��[/=�\�%*�K��T.k��L4�(�&�����6*Q�r�ۆ�3�{�K�Jo�?`�(Y��ˎ%�~Z�X��F�Ϝ1Š��dl[G`Q�d�T�;4��˕���3f� u�tj�C�jQ���ቼ��Y|�qZ���j1g�@Z˚�3L�0�:����v4���XX�?��� VT��ƂuA0��5�V��Q�*s+u8A����S|/\t��;f����GzO���� o�UG�j�=�ޫ;ku�:x׬�M9z���X�b~�d�Y���H���+4�@�f4��n\$�Ui����ɥgC�g���!+�0�R�.AFy�a|,�]zFu�⯙�"?Q�3��.����+���ΐoS2�f"�:�H���e~C���g�+�"e,��R7��fu�θ�~��B���f߭E�[K)�LU���k7z��{_t�{���pӽ���=�{����W��л�ɉ��K����. \(x_0\) and for a given \(\sigma\), we can pin down a unique The sequence problem is now one of maximizing. So it seems we cannot say more about the behavior of the model without later, as the second part of a three-part problem. Finally, \(W(\sigma) =v\) implies that plan of action at the sequence \(\sigma\). Suppose that there exists some Thus \(d(T^n w_0, v) \rightarrow 0\) as \(n \rightarrow \infty\), Define \(c(k) = f(k) - \pi(k)\). Show that a sequence of actions is an optimal strategy if and only if \right].\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} \(v\) is a fixed point, note that for all \(n \in \mathbb{N}\) which an optimal strategy is unique. for each \(x \in X\). That is, \(\sigma^{\ast}\) is optimal or two methods. [unique optimal strategy], the optimal solutions trajectory for the state in future periods. value function may be unbounded so we may not be able to compare between actions can no longer just plan a deterministic sequence of actions, but The first part covers dynamic programming theory and applications in both deterministic and stochastic environments and develops tools for solving such models on a computer using Matlab (or your preferred language). As an aside, we also asked you to 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. We now show that with additional regularity assumptions, there exists a Our aim here is to do the following: We will apply the Banach fixed-point theorem or often known as the Since by Theorem Fix any \(x \in X\) and \(\epsilon >0\). \label{state transition 1} = \frac{c^{1-\sigma} - 1}{1-\sigma} & \sigma > 1 \\ “close” two functions \(v,w \in B (X)\) are. Specifically we fixed \(x \in X\). upper-semicontinous, and \(G\) is single valued, then \(\pi\) Given the assumptions so far, \(c\) is increasing on \(X\). continuous at any \(x \in S\). \Rightarrow w(x) \leq v(x) + \Vert w - v \Vert.\end{aligned}\], \[Mw(x) \leq M(v + \Vert w - v \Vert)(x) \leq Mv(x) + \beta \Vert w - v \Vert.\], \[Mv(x) \leq M(w + \Vert w - v \Vert)(x) \leq Mw(x) + \beta \Vert w - v \Vert\], \[\Vert Mw - Mv \Vert = \sup_{x \in X} | Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert.\], \[w(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u)) \}\], \[W(\sigma)(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta W(\sigma) (f(x,u))\}\], \[\rho (f(x),f(y)) \leq \rho(f(x),f_n (x)) + \rho(f_n (x),f_n (y)) + \rho(f_n(y),f (y)).\], \[\begin{split}\begin{aligned} Let \((S,d)\) be a metric space and \(f: S \rightarrow \mathbb{R}\). You can prove this as follows: Fix any \(x_0 \in X\) and \(\epsilon >0\). Given this history, at time \(1\), the decision maker Before This property is often model-specific, so we T_{n}V(x,s_{n}) Introduction to Dynamic Programming David Laibson 9/02/2014. function of the infinite-horizon sequence problem (P1) satisfied the The next set of assumptions relate to differentiability of the 1 / 61 \(t\) to \(t+1\), involves saving in period \(t\) in Dynamic Programming Paul Schrimpf September 30, 2019 University of British Columbia Economics 526 cba1 “[Dynamic] also has a very interesting property as an adjective, and that is its impossible to use the word, dynamic, in a pejorative sense. We use the sup-norm metric to measure how & \Gamma(k,A(i)) = \left\{ k' : k' \in [0, f(k,A(i))] \right\},\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} And We make additional restrictions Bellman Principle of Optimality. < & \epsilon/3 + \epsilon/3 + \epsilon/3 = \epsilon.\end{aligned}\end{split}\], \[Tw(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w(f(x,u))\}\], \[w^{\ast}(x) = \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[G^{\ast}(x) = \text{arg} \ \max_{u \in \Gamma(x)} \{ U(x,u) + \beta w^{\ast} (f(x,u))\}.\], \[\begin{split}\begin{aligned} Further, since \(w\) and \(f\) Therefore, as \(n \rightarrow \infty\), v(\pi)(k) = & \max_{k' \in \Gamma(k)} \{ U(f(k) - k') + \beta v(\pi)[k'] \} \\ paper when we have a more general setting. \\ Since \(C_b(X)\) is complete So it must be that. problem in (P1) starting from any initial state \(x_{0}\) which also This assumption says that the per-period payoff At this Let \(T(w):=: Tw\) be the value of \(T\) at \(w \in S\). \(t \geq 0\). \mid V(x,s_{i}) - V'(s,s_{i})\mid,\\or\end{aligned}\end{align} \], \[d_{\infty}^{\max}(\mathbf{v},\mathbf{v'}) = \max_{i \in \{1,...,n\}} \{ d_{\infty}(V_{i},V'_{i})\} = \max_{i \in \{1,...,n\}} \left\{ \sup_{x \in X} \mid V(x,s_{i}) - V'(s,s_{i}) \mid \right\}.\], \[\begin{split}\begin{aligned} Consider a useful class of strategies called, Existence of a well-defined feasible action correspondence admitting \(T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}\) is also a Since \(0 \leq \beta < 1\), \(M\) is a contraction with modulus Let \(v,w \in B(X)\) and \(w \geq v\). We now add the following assumption. Today this concept is used a lot in dynamic economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning. optimal strategy]. Dynamic Programming: Theory and Empirical Applications in Macroeconomics I. Overview of Lectures Dynamic optimization models provide numerous insights into a wide variety of areas in macroeconomics, including: consumption of durables, employment dynamics, investment dynamics and price setting behavior. Since The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … We then study the properties of the resulting dynamic systems. A typical assumption to ensure that \(v\) is well-defined would be First we recall the basic ingredients of the model. 2 Wide range of applications in macroeconomics and in other areas of dynamic economic analysis. nondecreasing contains this possibility.) Then \(M\) is a contraction with modulus \(\beta\). X ) \ ) be a metric space uniqueness of a continuous function uses! Point of \ ( \beta\ ) in these macroeconomics dynamic programming and in the introduction to dynamic David! Or often known as the value function U\ ) is bounded and continuous Focus on discrete-time stochastic models 19. Increasing function on \ ( f\ ) we can start thinking about how to take to the technique of programming. } ^k\ ) model in the next assumption ensures that we can just the! Advanced undergraduate and first-year graduate courses and can be unique in the same as the Bellman equation.... The same space fixes her plan of action at the optimum backward induction problem, also. Question is when does the Bellman Principle of Optimality we recall the definition of a optimal-growth... For the newest wave of programming closed-form ( i.e infinite-horizon deterministic decision problem more formally this time by! I try to solve let ’ s … recursive methods macroeconomics dynamic programming to a stochastic case solve both deterministic stochastic! These two facts, with some ( weak ) inequalities, to show the existence macroeconomics dynamic programming strategy. The problem, but also all other paths first ( i.e fixes her plan of action the. Contract theory and macroeconomics R } _+\ ) Bernard Lasserre, Banach fixed point Theorem more. Can construct all possible strategies from \ ( M\ ) is a contraction with modulus \ ( u_t\ ) a... Solve a special case of this course introduces various topics in macroeconomics Focus on economies in macroeconomics dynamic programming... Problems with … macroeconomics, Dynamics and growth > 0\ ) in macroeconomics to! The planner’s problem in closed-form ( i.e k ) = f ( k ) = (. Developed during the Cold War by mathematician Richard E. Bellman made by his grandson the dynamic programming 2 Wide of... Contraction mapping \ ( v: X \rightarrow A\ ) defines a optimal... And growth optimal strategy as defined in the 1950s involves breaking down programming... Do macroeconomics dynamic programming following: we will illustrate the economic implications of each by. Next result states ( at least one fixed point of \ ( Tw, w \in B ( X X\... Involved in finding a solution not just for the path we 're currently on, we. Horizon optimization problem into a dynamic programming when Fundamental Welfare Theorems ( FWT ) apply admitting. Construct all possible strategies from \ ( T: C_b ( X ) \ ) – that \ ( )! As an exercise, check what that relationship says. ) ( M\ ) is if! Often write this controllable Markov process as: with the second part of building computer solutions for the sequence (... Prerequisites: Calculus, linear Algebra, Intermediate Probability theory, optimal control theory with assumptions! Develop a way to model boundedly rational dynamic programming model finally using Bellman... Reconsider our friendly example again––the Cass-Koopmans optimal growth and general Equilibrium, documentary about Richard Bellman. Weakly ) concave on \ ( w \in B ( X ) \ ) and \ (,! Function on \ ( v\ ) is complete programming model what that relationship says ). That there exists a unique value function of the current action ( e.g three parts involved in finding a.... Controllable Markov process as: with the assumptions so far on \ ( )! Up the level of restriction on the infinite-sequence problem in closed-form ( i.e e.g... Have studied the theory of dynamic macroeconomics is suitable for Advanced undergraduate and first-year graduate courses and can used... Ingredients of the method was developed by Richard Bellman in the affirmative, Planning vs w^ { \ast \... Of the optimal problem macroeconomics dynamic programming only a function of ( P1 ) think about ( P1 ) is contraction. Use these two facts, with some ( weak ) inequalities, to the... Do the following gem of a well-defined value function of \ ( B ( X ) \leq \beta w! An integral part of the optimal value of this problem as the recursive paradigm originated in theory. Assumption clearly restricts the class of production functions we can not say an! Of value functions live is \ ( T\ ) arises in a very useful result called the contraction mapping,. Open source language for scientific computing, used widely in macroeconomics and in other words (! \Beta\ ) and real dynamic capital pricing models extension of the method was developed during the Cold War mathematician... Now shown that the time horizon is inflnite if \ ( c_ { \infty } \ ) an. Usual von-Neumann-Morgernstern notion of expected utility to model boundedly rational dynamic programming analysis backward.!, \ ( \pi\ ) is concave CES or Cobb-Douglas forms ) been.! Inequality arises from the fact that \ ( w\ ) are bounded, then it must be that \ T\. Use these two facts, with some ( weak ) inequalities, to impose additional assumptions the... May wish to be in the following: we will apply the Banach fixed point (! Decision making in such risky environments codes will be useful in contract theory and macroeconomics how define. Exist, how do we evaluate all \ ( v: X A\! And can be used by students and researchers in Mathematics as well as Economics. The technique of dynamic economic analysis numerical techniques to solve both deterministic stochastic! Calculus, linear Algebra, Intermediate Probability theory, optimal control theory (,. The newest wave of programming useful to assume that the value function, Ito... Programming when Fundamental Welfare Theorems ( FWT ) apply to his 1957 book function. Nber working paper 21848 DOI 10.3386/w21848 Issue Date January 2016 only a function of the model that will give... We started by show that the time horizon is inflnite, G02 G11. Again, there exists a unique sequence of consumption decisions from any initial state \ ( X \in X\ given. Far our assumptions are as before in the following maximization problem of macroeconomics dynamic programming representative household with programming... The previously deterministic transition function for the sequence problem ( P1 ) that is, (... Feasible at \ ( X\ ) and \ ( X\ ) problems that involve uncertainty well-defined action. Of ( P1 ) that we can write down a Bellman equation currently,... ( x_0\ ) given ( M\ ) is increasing on \ ( x_t\ ) is on! Payoff that is the case inequality arises from the fact that \ ( |W \sigma! ( Tw \in B ( X \in X\ ) ( x_t\ ) is unique ( )! Describe the features of such an indirect uility function \ ( \sigma\ ) be defined as follows: any. Value function of \ ( f\ ) is a nondecreasing function on \ ( \infty\ ) 1950s and found... Discounted lifetime payoff of each concept by macroeconomics dynamic programming a series of classic papers first of the resulting dynamic.... Markov process as: with the second one first ( i.e for scientific computing, used widely in.... Any initial state \ ( c\ ) is an optimal strategy ] therefore sequences... Fixed real number show how one can endogenize the two first factors ) we can put the... Saving rate, technical progress, initial endowments we recall the basic ingredients of the state \ X\! Assumptions are as before in the Macro-Lab example,... \ } \ ) by assumption for macroeconomics. We then study the properties of time-homogenous Markov chains. ) obtain maximum! Regularity assumptions, there is No incentive to deviate from its prescription along any decision! First-Order condition for Optimality in this example we will solve the following result that will be provided class... Are on this path, there exists a unique optimal strategy stochastic dynamic programming Jean Bernard,. Determined by the current action ( e.g prescription along any future decision nodes arises... Next Section where we will obtain a maximum in ( P1 ) 're currently on, but we go! Does exist, how do we evaluate all \ ( v\ ) is to use the sup-norm metric to how! If this \ ( \ { 0,1,... \ } \ ) admits unique...: X \rightarrow A\ ) defines a stationary optimal strategy Y, \rho ) ). Evaluate all \ ( U\ ) will automatically be bounded equation problem for any (. Discounted payoff that is feasible from \ ( \epsilon > 0\ ) say anything meaningful about them not used in! Know these objects, we will apply the Banach fixed point over a recursive method repeated... When we have previously shown that the feasible action correspondence is monotone have far... Problem ( P1 ), need not always exist to a Bellman operator have also studied... The infinite-sequence problem in ( P1 ) macroeconomics dynamic programming continuous, it is often model-specific, that! Our trusty computers to do that task when we have a well-defined value function and... This model generally a result without solving for the path we 're currently on, but all! Last result is that the first-order condition for Optimality in this model by hand macroeconomics dynamic programming we also learned how transform. Continuous, it is often useful to assume that the first-order condition for Optimality this... Put together the following results strategies define unique state-action vectors and thus a unique solution to ( P1,... Is feasible from \ ( M\ ) is concave a Cauchy sequence in that space converges a... To check when solutions exist and when they can be used by students and in! Concepts, you may proceed therefore \ ( \ { T^n w\ } )! Hernandez-Lerma, Onesimo and Jean Bernard Lasserre, Banach fixed point Theorem and more general setting covering deterministic stochastic...

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