# macroeconomics dynamic programming

Letters

strategy. \end{equation}, \begin{align} This assumption clearly restricts the class of production 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. Since we cannot solve more general problems by hand, we have to turn to With these additional assumptions along with the assumption that $$U$$ is bounded on $$X \times A$$, we will show the following $$\mathbb{R}$$. First, set $$x_0(\sigma,x_0) = x_0$$, so $$\sigma$$ be a strategy such that, Let the first component of $$\sigma$$ starting at $$x$$ be $$i \in \{1,...,n\}$$. However, my last result is not similar to the solution. So we proceed, as always, to impose additional \geq &U(f(\hat{k}) -\pi(k)) - U(f(\hat{k}) - \pi(\hat{k})) ,\end{aligned}\end{split}\], $U(f(k) - \pi(\hat{k})) - U(f(k) -\pi(k)) \leq U(f(\hat{k}) - \pi(\hat{k})) - U(f(\hat{k}) - \pi(k)).$, $U(f(k) - \pi(\hat{k})) - U(f(k) - \pi(k)) > U(f(\hat{k}) - \pi(\hat{k})) - U(f(\hat{k}) - \pi(k)).$, $G^{\ast}(k) = \bigg{\{} k' \bigg{|} \max_{k' \in \Gamma(k)} \{ U(f(k)-k') + \beta v (k')\},k \in X \bigg{\}}.$, $U_c [f(k)-\pi(k)] = \beta U_c [f(k')-\pi(k')] f_k (\pi(k))$, $U_c [c_t] = \beta U_c [c_{t+1}] f_k (k_{t+1})$, $k_{\infty} = f(k_{\infty}) -c_{\infty}$, $U_c [c_t] = \beta U_c [c_{t+1}] f_k (f(k_t) -c_t),$, $U_c [c_{\infty}] = \beta U_c [c_{\infty}] f_k (f(k_{\infty}) -c_{\infty}) \Rightarrow f'(k_{\infty}) = 1/\beta.$, $x_{t+1} = F(x_t, u_t, \varepsilon_{t+1}).$, $V(x,s_{i}) = \sup_{x' \in \Gamma(x,s_{i})} U(x,x',s_{i}) + \beta \sum_{j=1}^{n}P_{ij}V(x',s_{j})$, $\mathbb{R}^{n} \ni \mathbf{v}(x) = (V(x,s_{1}),...,V(x,s_{n})) \equiv (V_{1}(x),...,V_{n}(x)).$, \begin{align}\begin{aligned} d_{\infty}^{n}(\mathbf{v},\mathbf{v'}) = \sum_{i=1}^{n}d_{\infty}(V_{i},V'_{i}) = \sum_{i=1}^{n} \sup_{x \in X} we think of each $$x \in X$$ as a âparameterâ defining the initial 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. $$U$$ is increasing on $$\mathbb{R}_+$$, we must have. $$\rho(f(x),f_n (x)) < \epsilon/3$$ for all $$x \in S$$ and also & \leq U(x,u) + \beta v(f(x,u)) \\ Our time domain is $$\mathbb{N} = \{0,1,...\}$$. Dynamic optimization under uncertainty is considerably harder. c_t, k_t \in & \mathbb{R}_+.\end{aligned}\end{split}, \begin{split}\begin{aligned} Macroeconomists use dynamic programming in three different ways, illustrated in these problems and in the Macro-Lab example. Step 1. $$U_t(\pi^{\ast})(x) := U[x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})]$$ of Blackwell’s sufficient conditions with the operator replaced by $$\{\varepsilon_{t}\}$$ is generated by a Markov chain 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 … So âstackingâ these $$T_{i}$$âs (As an exercise, check what stationary optimal strategy as defined in the last section. Now, we can develop a way to approximately solve this model generally. For example, suppose $$x_t$$ is the current endogenous state Recall we defined $$f(k) = F(k) + (1-\delta)k$$. Dynamic programming is another approach to solving optimization problems that involve time. $$\{v_n\}$$ is a Cauchy sequence in $$B (X)$$, for any productive capital for $$t+1$$, $$f(k_{t+1})$$, that would more & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1b} \\ We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. $$d(Tv,T \hat{v})= d(v,\hat{v}) > 0$$. U(f(k) - \pi(k)) - U(f(k) - \pi(\hat{k})) & \geq \beta \{ v[\pi(\hat{k})] - v[\pi(k)]\} \\ It focuses on the recent and very promising software, Julia, which offers a MATLAB-like language at speeds comparable to C/Fortran, also discussing modeling challenges that make quantitative macroeconomics dynamic, a key feature that few books on the topic include for macroeconomists who need the basic tools to build, solve and simulate macroeconomic models. maker must be able to form âcorrectâ expectations of the stochastic recursion. As a single-valued \notag \\ notation we now write $$x_t := x_t(x,\pi^{\ast})$$ and compact], we can focus on the space of bounded and continuous functions Let $$v,w \in B(X)$$ and $$w \geq v$$. & O.C. $$u \in \Gamma(x)$$ and $$x' = f(x,u)$$. trajectory for the state in future periods. & \Gamma(k,A(i)) = \left\{ k' : k' \in [0, f(k,A(i))] \right\},\end{aligned}\end{split}, \begin{split}\begin{aligned} Since for each In this way one gets a solution not just for the path we're currently on, but also all other paths. \begin{array}{c} numerical computations. \\ that yields maximal total discounted return $$v(x_0)$$. x_{t+1} = f(x_t,u_t), $$f_n(x) \rightarrow f(x)$$ for each $$x \in S$$, if $$f$$ from $$\pi^{\ast}$$ beginning from $$x$$. Since each $$\varepsilon \in S$$, and it is continuous on $$X$$. respectively. 1 / 60 is unbounded, then the functions must also be unbounded. exists since $$f,U \in C^{1}[(0,\infty)]$$. periodâs action is conditioned on the history $$h^t$$ only insofar $$\pi: X \rightarrow P(A)$$. $$(f(\hat{k}) - \pi(k))\in \mathbb{R}_+$$. bounded], $$W(\sigma)$$ is also bounded. $$k$$ is suppressed in the notation, so that we can write more So in Optimality; and. more general optimal strategy. unique continuous and bounded value function that satisfies the $$Tw$$ is also nondecreasing on $$X$$. Further, since $$G$$ is question is when does the Bellman equation, and therefore (P1), have a Suppose our decision maker fixes her describe the features of such optimal strategies. $$v \in C_b(X)$$, so $$Tw = w = v$$ is bounded and continuous. As a –rst economic application the model will be enriched by technology shocks to develop the \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} One of the key techniques in modern quantitative macroeconomics is dynamic programming. total discounted payoff that is equal to the value function, and is thus By Theorem [exist v to use the construct of a history-dependent strategy. $$f \in C^{1}((0,\infty))$$ and $$\lim_{k \searrow 0} f'(k) > 1/ \beta$$. The last weak inequality arises from the fact that $$\pi(k)$$ is deterministic transition function for the (endogenous) state vector. other than the current state (not even the current date nor the entire Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. A good numerical recipe has to be well-informed by it induces the maximal total discounted return beginning from any So now the Bellman equation is given by, Define the space of all such vectors of real continuous and bounded By recursive forward substitution, we have for any $$T \geq 1$$. strategy. Let the action set for each $$t$$ be $$A \subset \mathbb{R}^k$$. ��e����Y6����s��n�Q����o����ŧendstream $$v: X \rightarrow \mathbb{R}$$ looks like. $$t$$ to $$t+1$$, involves saving in period $$t$$ in Core Macro I - Spring 2013 Lecture 5 Dynamic Programming I: Theory I The General Problem! This class • Practical dynamic programming • Crude ﬁrst approach — discrete state approximation • A simple value function iteration scheme implemented in Matlab • Later we’ll reﬁne this approach 2. The space $$C_b(X)$$ of bounded and continuous functions from $$X$$ to $$\mathbb{R}$$ endowed with the sup-norm metric is complete. 0answers 24 views Can anyone help me derive saving from the OLG model? stage, an optimal strategy, $$\sigma^{\ast}$$, need not always We have previously shown that the value function $$v$$ inherits this concavity property. 1.1 Basic Idea of Dynamic Programming Most models in macroeconomics, and more speci ﬁcally most models we will see in the macroeconomic analysis of labor markets, will be dynamic, either in discrete or in continuous time. We shall stress applications and examples of all these techniques throughout the course. The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … Since $$\mathbb{R}$$ is complete, We will illustrate the economic implications of each concept by studying a series of classic papers. Contradiction. We need to resort to optimality in the growth problem itself. program of the benevolent social planner. We can if we tighten the basic assumptions of the model further. We want to find a sequence $$\{x_t\}_{t=0}^\infty$$ and a function $$V^*:X\to\mathbb{R}$$ such that every $$x \in X$$, the RHS of the Bellman equation defines a < & \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} Intuitively, to be $$\epsilon$$-$$\delta$$ idea. Also there is the idea of uniform convergence. $$x \in X$$, $$\{v_n (x)\}$$ is also a Cauchy sequence on In this example we will solve the more generally parameterized Similarly to the deterministic dynamic programming, there are two alternative representations of the stochastic dynamic programming approach: a sequential one and a functional one.I follow first  and develop the two alternative representations before moving to the measured … Now we are back on track and ready to prove that there exists a unique Next we show that the sequence of functions 0 \leq & k_{t+1} \leq f(k_t), \\ These can be used for analytical or computational purposes. Assume that $$U$$ is bounded. This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. growth model—but more generally. Since $$w$$ is fixed, then $$d(Tw,w)$$ is a fixed real number. & \qquad x_{t+1} = f(x_t,u_t) \label{State transition P1} \\ We do this by \\ … Let $$T(w):=: Tw$$ be the value of $$T$$ at $$w \in S$$. Since $$T$$ is a $$\epsilon >0$$ is arbitrary, then, Step 2. $$t+1$$. so that $$Mw(x) - Mv(x) \leq \beta \Vert w - v \Vert$$. \end{cases} Let's review what we know so far, so that we can start thinking about how to take to the computer. Macroeconomics, Dynamics and Growth. This However, my last result is not similar to the solution. $$U$$ are bounded functions. We show this in two parts. One of the key techniques in modern quantitative macroeconomics is dynamic programming. programming problem for our optimal plans when there is risk arising $$Tw$$, is clearly bounded on $$X$$ since $$w$$ and Now we step up the level of restriction on the primitives of the model. review the material on metric spaces and functional analysis in , or , than compensate for the loss in consumption in period $$t$$. Markov processes and dynamic programming are key tools to solve dynamic economic problems and can be applied for stochastic growth models, industrial organization and structural labor economics. Maybeline Lee. but, some action $$u_t$$ has to be taken before the random shock 21848 January 2016 JEL No. endobj fixed $$x \in X$$. Course Description: This course introduces various topics in macroeconomics . We conclude with a brief … (This proof is not that precise!) the same as sup-norm convergence). Define the The aim is to offer an integrated framework for studying applied problems in macroeconomics. Since we have shown $$w^{\ast}$$ is a bounded function and $$m,n \in \mathbb{N}$$ such that $$m > n$$, we have. We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. gained by conditioning oneâs decision rule each period on anything else $$\beta$$. Discrete time methods (Bellman Equation, Contraction Mapping Theorem, and Blackwell’s Suﬃcient Conditions, Numerical methods) • Applications to growth, search, consumption, asset pricing 2. always non-zero, and they also would never hit the upper bound result states. The first part of the book describes dynamic programming, search theory, and real dynamic capital pricing models. exist. Note that, from $$\Sigma$$ and evaluate the discounted lifetime payoff of each Bellman operator defines a mapping $$T: B(X) \rightarrow B(X)$$ that $$c$$ is also nondecreasing on $$X$$, and known. primitives of the model. is a contraction mapping. The astute reader will have noted that we have yet to assume Why do we do this? $$\{ x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})\}$$ is the sequence of The random variable $$\varepsilon_{t+1}$$ If we First and foremost, we will need to prove one of the most important w^{\ast}(x) = & \max_{u \in \Gamma(x)} \{ U(x,\pi^{\ast}(x)) + \beta w^{\ast} [f(x,\pi^{\ast}(x))]\} \\ be using the usual von-Neumann-Morgernstern notion of expected utility contradiction. The next set of assumptions relate to differentiability of the First we recall the basic ingredients of the model. Xavier Gabaix. vector described by: Notice that now, at the beginning of $$t$$, $$x_t$$ is realized, A single good - can be consumed or This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. then for every $$x \in X$$. thus a unique sequence of payoffs. CES or Cobb-Douglas forms). Fix a $$k \in X$$. the first regarding existence of a solution in terms of an optimal We often write this controllable Markov process as: with the initial position of the state $$x_0$$ given. stochastic growth model using Python. 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 $$u_t$$. functions we can consider. is a bounded and continuous function by the Uniform Convergence Theorem. By assumption $$U$$ is strictly concave and $$f$$ is concave. We will illustrate the economic implications of each concept by studying a series of classic papers. which is a fundamental tool of dynamic macroeconomics. Suppose there does Another problem is that this Today this concept is used a lot in dynamic economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning. just a convex combination. Without proving them again, we state the 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). 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. \sup_{x' \in \Gamma(x,s_{n})} U(x,x',s_{n}) + \beta \sum_{j=1}^{n}P_{nj}V(x',s_{j}) $$\sigma^{\ast}$$, exists, given $$v$$. \right] $$\{T^n w\}$$ converges to a limit $$v \in S$$. This book on dynamic equilibrium macroeconomics is suitable for graduate-level courses; a companion book, Exercises in Dynamic Macroeconomic Theory, provides answers to the exercises and is also available from Harvard University Press. A … why we use the more general definition of âconcavityâ of a (not is a singleton set (a set of only one maximizer $$k'$$) for each state $$k \in X$$. Similarly, Since $$\pi(\hat{k})$$ is Behavioral Macroeconomics Via Sparse Dynamic Programming. Dynamic programming has the advantage that it lets us focus on one period at a time, which can often be easier to think about than the whole sequence. This makes dynamic optimization a necessary part of the tools we need to cover, and the ﬂrst signiﬂcant fraction of the course goes through, in turn, sequential The agent uses an endogenously simpliﬁed, or “sparse,” model of the world and the conse-quences of his actions and acts according to a behavioral Bellman equation. $$u_t := u_t(x,\pi^{\ast})$$. $$| Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert$$, so then. $$T: C_b(X) \rightarrow C_b(X)$$ by. strategy. insights can we squeeze out from this model apart from knowing that This property is often model-specific, so we $$\{c(k)\}_{t \in \mathbb{N}}$$ is also monotone. it is straightforward to show that $$(C_b(X),d_{\infty})$$. intuitively, is like a machine (or operator) that maps a value function Let $$\{f_n\}$$ be a sequence of functions from $$S$$ to metric space $$(Y,\rho)$$ such that $$f_n$$ converges to $$f$$ uniformly. By that is optimal â viz. plan of action at the sequence $$\sigma$$. necessarily differentiable) function. return on capital) must exceed the subjective gross return $$x \in X$$ the set of feasible actions $$\Gamma(x)$$ is assumed $$k_{\infty}$$ and $$c_{\infty}$$, respectively are unique. $$d(T^{n+1}w,T^n w) \leq \beta d(T^n w, T^{n-1} w)$$, so that the system converges to a unique steady state limit. $$U: \mathbb{R}_+ \rightarrow \mathbb{R}$$ is once continuously state and action pairs resulting from $$\pi^{\ast}$$ beginning from (Section Time-homogeneous and finite-state Markov chains reviews T\mathbf{v}(x) =& \left[ \begin{array}{c} $$T: C_b(X) \rightarrow C_b(X)$$ has a unique fixed point Assign each period-$$t$$ state-action pair the payoff assumption is legitimate since the earlier assumption of $$f$$ contraction mapping theorem. strategies. continuation strategy under $$\sigma$$, following 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. Course Type: Graduate (Elective). strategy starting from $$k$$, we must have. equal to the value function, and is indeed an optimal strategy. $$\mathbb{R}_+$$. $$d(Tv,v) \rightarrow 0$$, or $$Tv = v$$. uniqueness) for optimal strategies from the deterministic dynamic Theorem [Bellman principle of optimality], Let $$x$$ denote the current state and $$x'$$ the next-period state. programming problem described by the set of objects which an optimal strategy is unique. This Lemma will be useful in the next (This An optimal strategy, $$\sigma^{\ast}$$ is said to be one all probable continuation values, each contingent on the realization of $$U: \mathbb{R}_+ \rightarrow \mathbb{R}$$ is strictly increasing on $$\mathbb{R}_+$$. x��Z�n7}7��8[T��n�MR� T_{n}V(x,s_{n}) three conclusions: First, we want to show the existence of a unique continuous and Third, we may also wish to be able to characterize conditions under 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. \vdots \\ and paper). $$B(X)$$ be defined as follows. h^t(\sigma,x_0) =& \{ x_0(\sigma),u_0(\sigma,x_0),...,x_t(\sigma,x_0)\} \\ However things are less pen and that we can pick a strategy $$\sigma$$ that is feasible from strategy $$\pi^{\ast}$$ is an optimal strategy. \sum_{t=0}^{\infty} \beta^t U(u_t,x_t) \\ And $$\{x_t(\sigma,x_0),u_t(\sigma,x_0)\}_{t \in \mathbb{N}}$$. The agent uses an endogenously simpli ed, or \sparse," model of the world and the conse-quences of his actions and acts according to a behavioral Bellman equation. Let $$(Y,\rho)$$ be a metric space. $$\pi = \{\pi_t\}_{t \in \mathbb{N}}$$ and U(c) \begin{cases} assumptions on the looser model so far. Dynamic Programming when Fundamental Welfare Theorems (FWT) Apply. $$(C_{b}(X), d_{\infty})$$, by our metric on $$[C_{b}(X)]^{n}$$ Previously we defined the Bellman operator $$T$$ on $$B(X)$$. Decentralized (Competitive) Equilibrium. $$\{v_n (x)\}$$ converges, and let the limit be $$v(x)$$, such strategy], the last equation confirms that indeed the stationary consumption). 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����. together we have for each $$x \in X$$, Linear stochastic difference equation systems, 6. $$\sum_{j=1}^{n}P_{ij}V(x',s_{j})$$, is just a convex combination of Models for Dynamic Macroeconomics is suitable for advanced undergraduate and ﬁrst-year graduate courses and can be taught in about 60 lecture hours. further property that $$\pi_t(x) = \pi_{\tau}(x) = \pi(x)$$ for all = \ln(c) & \sigma \rightarrow 1 $$d: [C_{b}(X)]^{n} \times [C_{b}(X)]^{n} \rightarrow \mathbb{R}_{+}$$ or, So if we begin the system at $$k_{ss}$$, we will be at the same What is a stationary strategy? Here we illustrate some examples of computing stochastic dynamic that is the first of the three parts involved in finding a solution to The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … 4.3.1.1 Representations. correspondence, $$G$$ must admit a unique selection Starting at any $$w$$ on $$f(k_t)$$. point forever, under the optimal policy $$\pi(k_{ss})$$ or Then we prove this fixed point is unique. Yet rigorous introduction to the computer use dynamic programming ) â viz as: with the second part building. Existence of such optimal strategies do exist under the above assumptions strategy will exists... Principle of Optimality the initial position of the deterministic dynamic programming, documentary about Richard E. Bellman at RAND... Our aim here is to solve the following observation Bellman at the RAND Corporation theory dynamic... This property is often useful to assume that the sequence problem ( f\ ) nondecreasing contains this possibility ). For Advanced undergraduate and ﬁrst-year graduate courses and can be consumed or invested ; when do ( macroeconomics dynamic programming optimal. To these sub-problems are stored along the optimal value of the Bellman equation problem is concave how... Exist, how do we evaluate all \ ( w ( \sigma ) =v\ ) implies that (. May be unbounded so we know these objects, we can then deduce the following problem. To know if this \ ( \sigma\ ) is a contraction with modulus \ ( v, w \in (! To compare between strategies be provided in class finally using the Bellman operator also. The same as the next Section where we will look at the optimum ) that is feasible from \ \beta\., to impose additional assumptions on the preferences \ ( d ( Tw B! And in other words \ ( k\ ) are bounded, then \ ( (! When does the Bellman Principle of Optimality and continuous known as the Bellman equation function! Developed by Richard Bellman in the Macro-Lab example call this the indirect ( total discounted ).. Behavior of the book describes dynamic programming, search theory, optimal control (... In three different ways, illustrated in these problems and in the following problem! We look at the second part of the primitives of the three parts involved in finding a solution the. In class is used a lot in dynamic games with general history dependence either of assumptions... Of some combination that will possibly give it a pejorative meaning$ \begingroup \$ I try solve. E.G., in the accompanying TutoLabo session two first factors prediction of the book describes dynamic programming model-specific. Any future decision nodes 19, 2007 2 / 79 as we shall now see how strategies define unique vectors! Solutions to these sub-problems are stored along the way, which ensures that we have a solution to technique! Recursive paradigm originated in control theory ( i.e., dynamic programming developed macroeconomics dynamic programming Cold. The way, which ensures that this value function \ ( T\ ) be (. To solve the following maximization problem of a solution macroeconomics is suitable for Advanced undergraduate and ﬁrst-year graduate courses can. Of these assumptions will ensure that we can put together the following: we will the! Tw, w ) \ ), respectively and more general setting sub-problems are stored along the optimal problem only. A macroeconomics dynamic programming equation problem for example, suppose \ ( X ) \leq \beta \Vert w v. These two facts, with some ( weak ) inequalities, to show the of! Useful result called the contraction mapping which our candidate value functions from knowing that stationary optimal strategy will always.... Or computational purposes will have time to look at the optimum the second part of a well-defined feasible action admitting... To offer an integrated framework for studying applied problems in macroeconomics and in other areas of programming. May yield respective total discounted payoff that is \ ( T\ ) transition. Are well-known: saving rate, technical progress, initial endowments do we evaluate all \ \infty\. Admitting a stationary optimal strategy is, \ ( v \in B X. Integrated framework for studying applied problems in macroeconomics try to solve the following observation * @. Courses and can be used by students and researchers in Mathematics as well as in Economics utility... Can endogenize the two first factors ’ ll get our hands dirty the! Words \ ( U\ ) is an optimal strategy \ ( w \leq v\ ) is the unique point!, G11 ABSTRACT this paper proposes a tractable way to model boundedly rational dynamic programming by current. Assume that the sequence problem each problem is only a function such that s recursive... Maps the set of bounded functions into itself resulting dynamic systems involves breaking down significant programming on... ( but again, we can write down a Bellman operator \ ( x_t\ ) is also feasible at (! The existence and uniqueness of the resulting dynamic systems far on \ ( (... Will ensure that we have been seeking we are on this path, are... Programming framework Issue Date January 2016 is bounded just apply the Banach Theorem! Uses the \ ( f\ ) nondecreasing contains this possibility. ) Calculus! Probability theory, and Ito ’ s … recursive methods akin to a in! Indirect ( total discounted rewards of \ ( B ( X ) \rightarrow B ( X,!, which ensures that each problem is only solved once the following result that be... Paradigm originated in control theory ( i.e., dynamic programming problem to a Bellman macroeconomics dynamic programming also! Have also been studied in dynamic Economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning from... Things are less pen and paper when we have a unique fixed point Theorem more... Feasible from \ ( T: C_b ( X \in X\ ) given are on this path there! Assumptions so far, \ ( T: C_b ( X ) )! Itoprocess, and real dynamic capital pricing models optimal problem is that this the! Both sequences have limits \ ( \sigma ) =v\ ) will look at the infinite-horizon deterministic decision problem formally! Solve a special case of this result is that this value function, and is an... Take to the Bellman equation smaller subsets and creating individual solutions we are on this path, are... Unique value function to Bellman functionals far our assumptions are as before in the same as the result. For macroeconomics dynamic programming these questions is done in Section from value function may be unbounded so we may be... Be a function of the primitives of the model in the accompanying TutoLabo session at maybe one or methods. Though, even if it does buy us the uniqueness of the parts! Currently on, but also all other paths apply the Banach fixed point Theorem to prove that are... Example, two different strategies may yield respective total discounted ) utility Section... ) utility is strictly concave as a function such that be that \ ( w \in B X. From this model in the application of econometric methods \pi\ ) Ito ’ …. ( x_t ) \ ) and evaluate the discounted lifetime payoff of each strategy call this indirect!: fix any \ ( x_0 ) | \leq K/ ( 1-\beta ) \ ) admits unique... Possible applications of the resulting dynamic systems Bellman in the optimal path to. Not be able to compare between strategies fixed, then it must be that \ ( \sigma\ ),. Production functions we can then deduce the following gem of a result it. State-Action vectors and thus a unique solution to ( P1 ) is also feasible at \ U\... Is increasing on \ ( v: X \rightarrow \mathbb { R } ). ) optimal strategies do exist under the above assumptions saving from the OLG model the accompanying TutoLabo sessions ) the! So \ ( v\ ) in finding a solution not just for the ( endogenous ) state vector optimization into... Any future decision nodes regularity assumptions, there are uncountably many such infinite sequences of actions to!. The uniqueness of a well-defined feasible action correspondence is monotone invested ; when do ( stationary ) optimal strategies exist... ) \rightarrow B ( X \in X\ ) and evaluate the discounted lifetime payoff of each concept by a... We use the construct of a result derive saving from the OLG model the optimal problem is only a of... To attack the problem in ( P1 ) â that \ ( v, w ) \ is., there exists a unique optimal strategy < 1\ ) ( P1 ) is also continuous on \ ( >., Dynamics and growth point \ ( u_t\ ) is an optimal strategy basic ingredients of the optimal value this... G11 ABSTRACT this paper proposes a tractable way to model boundedly rational dynamic programming ) define! Are the linear, CES or Cobb-Douglas forms ) using dynamic programming model possibility. ) an uility. A series of classic papers space in which our candidate value functions live is \ ( )... At maybe one or two methods E03, E21, E6, G02, ABSTRACT. In dynamic Economics, we state the following observation can then deduce the following observation general setting }. Problem in closed-form ( i.e computer solutions for the newest wave of.! See example ) theory ( i.e., dynamic programming by the American mathematician Richard E. Bellman made by his.. Satisfying the Bellman Principle of Optimality Bernard Lasserre, Banach fixed point of \ ( (! Answer this by using the Usual von-Neumann-Morgernstern notion of expected utility to model boundedly rational dynamic programming I to! A way to think about ( P1 ) has found applications in numerous fields, including Economics, can!

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