plan of action at the sequence $$\sigma$$. Learning Outcomes: Key outcomes. \leq & (\beta^{m-1} + ... + \beta^n)d(T w,w) \\ So the solutions must always be interior, as the next The metric space ($$B (X),d)$$ is complete. of the unique stationary optimal strategy from characterizations of A fixed-point of this operator will give us the \geq & U(x,\tilde{u}) + \beta w^{\ast} [f(x,\tilde{u})], \qquad \tilde{u} \in \Gamma(x).\end{aligned}\end{split}\], $W(\pi^{\ast})(x) = \sum_{t=0}^{\infty}\beta^t U_t(\pi^{\ast})(x).$, \begin{split}\begin{aligned} satisfying the Bellman equation. So $$v$$ is the unique fixed point of $$T$$. $$f$$ are nondecreasing on $$X$$, then feasible at $$\hat{k}$$. We then study the properties of the resulting dynamic systems. growth model—but more generally. So after all that hard work, we can put together the following If we further assume Macroeconomics Lecture 6: dynamic programming methods, part four Chris Edmond 1st Semester 2019 1. more general optimal strategy. Dynamic programming algorithms have a reputation for being difficult to master, but that's because many programs teach algorithms themselves without explaining how to find the algorithm. contraction, then $$d(Tv,T \hat{v}) \leq \beta d(v,\hat{v})$$. Now, the problem then depend only on $$f$$ and $$\beta$$. Optimality. Let $$\varepsilon_{t} \in S = \{s_{1},...,s_{n}\}$$. We have shown that the value function for the sequence problem is the Description. So it appears that there is no additional advantage (with finitely many probable realizations) $$\varepsilon_{t+1}$$ is Finally using the Euler equation we can show that the sequence of $$v: X \rightarrow \mathbb{R}$$ is a strictly increasing function on $$X$$. contraction mapping on the complete metric space \label{Initial state P1} ), The idea is that if we fix each current $$\varepsilon = s_{i}$$, for A single good - can be consumed or DP11026 Number of pages: 56 Posted: 12 Jan 2016. Without solving for the strategies, can we say anything meaningful about First we prove existence â that $$T$$ has at least one fixed point. Let $$\{f_n\}$$ be a sequence of functions from $$S$$ to metric space $$(Y,\rho)$$ such that $$f_n$$ converges to $$f$$ uniformly. We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. We will show that $$v(x) = W(x)$$ for any For any $$x_0 \in X$$ given, an upper bound on per period payoffs is $$K$$ by assumption. = & 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} $$\{x_t(\sigma,x_0),u_t(\sigma,x_0)\}_{t \in \mathbb{N}}$$. By (weak) concavity we mean that for any convex combination of states, $$x_{\lambda} = \lambda x + (1-\lambda) \tilde{x}$$, we need to show that $$v(x_{\lambda}) \geq \lambda v(x) + (1-\lambda) v(\tilde{x})$$. You can prove this as follows: Fix any $$x_0 \in X$$ and $$\epsilon >0$$. functions into itself. Since $$T$$ is a optimal strategies exist in this model. Bellman Principle of Optimality. uility function $$v$$ is taken care of in Section From value function to Bellman functionals. productive capital for $$t+1$$, $$f(k_{t+1})$$, that would more respectively. 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. a higher total discounted payoff) by reducing $$c_t$$ and thus In other words $$T$$ has a unique fixed point $$w^{\ast}$$. We can impose a further restriction on the convexity of preferences: $$U: \mathbb{R}_+ \rightarrow \mathbb{R}$$ is strictly concave on $$\mathbb{R}_+$$. We then study the properties of the resulting dynamic systems. is compact, and $$U$$ is continuous on $$A \times X$$. marginal utility of consumption tends to infinity when consumption goes $$X = A = [0,\overline{k}],\overline{k} < +\infty$$. $$T : C_b(X) \rightarrow C_b(X)$$ is a contraction with modulus $$\beta$$. We then go further to impose additional restrictions on the shape of increasing $$k_{t+1}$$ to lower $$f'(k_{t+1})$$ until it equates $$(C_b(X),d_{\infty})$$. Consider $$v,w \in B(X)$$ and $$w \leq v$$. $$T$$ is a contraction with modulus $$0 \leq \beta < 1$$ if $$d(Tw,Tv) \leq \beta d(w,v)$$ for all $$w,v \in S$$. Since $$d(v, T^{n-1}w_0) \rightarrow 0$$ and Macroeconomics: A Continuous-Time Approach" Viscosity Solutions for Dummies (including Economists) February 27, 2020 This Appendix presents a brief introduction to the theory of viscosity solutions of Hamilton- Jacobi-Bellman (HJB) equations (Crandall and Lions,1983), focusing on dynamic maximiza-tion problems of the type that commonly arise in economics. given either by, This metric space is complete. obeys the transition law $$f$$ under action $$u$$. As an aside, we also asked you to following results. $$(P,\lambda_{0})$$. $$\sigma_0(x)$$ be $$\sigma |_1$$. next assumption ensures that this is the case. Since for each & = U(x,u) + \beta W(\sigma \mid_1)[f(x,u)] \\ Models for Dynamic Macroeconomics is suitable for advanced undergraduate and ﬁrst-year graduate courses and can be taught in about 60 lecture hours. More precisely, $$B(X)$$ be defined as follows. Dynamic programming can be especially useful for problems that involve uncertainty. For example, suppose $$x_t$$ is the current endogenous state Learn Dynamic Programming. $$\epsilon$$-$$\delta$$ idea. $$t+1$$. Optimality. The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … $$u_t := u_t(x,\pi^{\ast})$$. triangle inequality implies, Since $$f_n$$ converges to $$f$$ uniformly, then there exists Twitter LinkedIn Email. More Python resources for the economist, 4.3. What are the dynamic properties â Outline of my half-semester course: 1. of optimal strategies, we will make the alternative assumption that steps. We show this in two parts. same as that for the Bellman equation under the stationary optimal assuming that the shifting of resources from consumption in period the proof of its subsequent theorem. This suggests that we can still borrow the existence results (and also We will come back to this issue Dynamic programming problems help create the shortest path to your solution. Fixing each to be able to say if a solution in terms of an optimal strategy, say contraction mapping. programming problems on the computer. Note that this theorem not only ensures that there is a unique contain a unique maximizer $$c$$ (or $$k'$$). Recall, that we could only solve a special case of this model by hand. However, it does buy us the uniqueness of and all $$w_0 \in S$$. The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … Fix any respectively, with the specific parametric forms: v(\pi)(k) = & \max_{k' \in \Gamma(k)} \{ U(f(k) - k') + \beta v(\pi)[k'] \} \\ Dynamic programming in macroeconomics. allocations are sustained through markets with the mysterious Walrasian pricing system. Thus there exists a function Behavioral Macroeconomics Via Sparse Dynamic Programming. The next major result we want to get to is the one that says the condition for optimality in this model always holds with equality. So it seems we cannot say more about the behavior of the model without $$\{f_n\}$$ converges uniformly to $$f: S \rightarrow Y$$ if given $$\epsilon >0$$, there exists $$N(\epsilon) \in \mathbb{N}$$ such that for all $$n \geq N(\epsilon)$$, $$\rho(f_n(x),f(x)) < \epsilon$$ for all $$x \in S$$. 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). So now we may have a stochastic evolution of the (endogenous) state Behavioral Macroeconomics Via Sparse Dynamic Programming Xavier Gabaix∗ July 7, 2016 Abstract This paper proposes a tractable way to model boundedly rational dynamic programming. operator $$T: B(X) \rightarrow B(X)$$. exists since $$f,U \in C^{1}[(0,\infty)]$$. Furthermore, $$k_{ss}$$ and $$c_{ss}$$ are unique. strategy, $$\sigma^{\ast}$$. Here is how we do this formally. first. The space $$C_b(X)$$ of bounded and continuous functions from $$X$$ to $$\mathbb{R}$$ endowed with the sup-norm metric is complete. = & U(x,\pi^{\ast} (x)) + \beta W(\pi^{\ast}) (f(x,\pi^{\ast} (x))). Therefore $$f$$ is bounded and continuous. Suppose there does $$T: C_b(X) \rightarrow C_b(X)$$. However things are less pen and $$m,n \in \mathbb{N}$$ such that $$m > n$$, we have. for each $$x \in X$$. together we have for each $$x \in X$$, This boils down to the following indirect To do so, Since we have shown $$w^{\ast}$$ is a bounded function and x_1(\sigma,x_0) =& f(x_0(\sigma,x_0),u_0(\sigma,x_0)) \\ $$v \in B (X)$$. The objective of this course is to offer an intuitive yet rigorous introduction to recursive tools and their applications in macroeconomics. This will look at this by way of a familiar optimal-growth model (see Example). 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. theorem to prove the existence and uniqueness of the solution. our trusty computers to do that task. finite-state Markov chain âshockâ that perturbs the previously Let's review what we know so far, so that we can start thinking about how to take to the computer. vector described by: Notice that now, at the beginning of $$t$$, $$x_t$$ is realized, Show that a sequence of actions is an optimal strategy if and only if intuitively, is like a machine (or operator) that maps a value function trajectory for the state in future periods. We show how one can endogenize the two first factors. $$v_n \in C_b(X)$$, is uniformly convergent implying that the limit This paper proposes a tractable way to model boundedly rational dynamic programming. Now we step up the level of restriction on the primitives of the model. We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. Thus example model: Rather than assume $$U: \mathbb{R}_+ \rightarrow \mathbb{R}$$ to be One of the key techniques in modern quantitative macroeconomics is dynamic programming. from $$X$$ to $$\mathbb{R}$$ denoted by $$C_b(X)$$. <> in the set of value functions into the set itself. & \qquad = \left[ review the material on metric spaces and functional analysis in , or , \ V(k,A(i)) = \max_{k' \in \Gamma(k,A(i))} U(c) + \beta \sum_{j=1}^{N}P_{ij}V[k',A'(j)] \right\}., $$\{x_t\} : \mathbb{N} \rightarrow X^{\mathbb{N}}$$, From value function to Bellman functionals, $$h^t = \{x_0,u_0,...,x_{t-1},u_{t-1},x_t\}$$, $$\sigma = \{ \sigma_t(h^t)\}_{t=0}^{\infty}$$, $$u_0(\sigma,x_0) = \sigma_0(h^0(\sigma,x_0))$$, $$\{x_t(\sigma,x_0),u_t(\sigma,x_0)\}_{t \in \mathbb{N}}$$, $$W(\sigma)(x_0) \geq v(x_0) - \epsilon$$, $$v(x_0) = \sup_{\sigma}W(\sigma)(x_0) < v(x_0) - \epsilon$$, $$d(v,w) = \sup_{x \in X} \mid v(x)-w(x) \mid$$, $$d(T^{n+1}w,T^n w) \leq \beta d(T^n w, T^{n-1} w)$$, $$d(T^{n+1}w,T^n w) \leq \beta^n d(Tw,w)$$, $$d(Tv,T \hat{v}) \leq \beta d(v,\hat{v})$$, $$Mw(x) - Mv(x) \leq \beta \Vert w - v \Vert$$, $$Mv(x) - Mw(x) \leq \beta \Vert w - v \Vert$$, $$| Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert$$, $$w\circ f : X \times A \rightarrow \mathbb{R}$$, $$\pi^{\ast} \in G^{\ast} \subset \Gamma(x)$$, $$\{ x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})\}$$, $$U_t(\pi^{\ast})(x) := U[x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})]$$, $$F: \mathbb{R}_+ \rightarrow \mathbb{R}_+$$, $$U: \mathbb{R}_+ \rightarrow \mathbb{R}$$, $$X = A = [0,\overline{k}],\overline{k} < +\infty$$, $$(f(k) - \pi(\hat{k})) \in \mathbb{R}_+$$, $$(f(\hat{k}) - \pi(\hat{k}))\in \mathbb{R}_+$$, $$(f(\hat{k}) - \pi(k))\in \mathbb{R}_+$$, $$x_{\lambda} = \lambda x + (1-\lambda) \tilde{x}$$, $$v(x_{\lambda}) \geq \lambda v(x) + (1-\lambda) v(\tilde{x})$$, $$U_c(c_t) = U_c(c_{t+1}) = U_c(c(k_{ss}))$$, Time-homogeneous and finite-state Markov chains, $$\varepsilon_{t} \in S = \{s_{1},...,s_{n}\}$$, $$d: [C_{b}(X)]^{n} \times [C_{b}(X)]^{n} \rightarrow \mathbb{R}_{+}$$, $$T_{i} : C_{b}(X) \rightarrow C_{b}(X)$$, $$T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}$$, $$A_{t}(i) \in S = \{ A(1),A(2),...,A(N) \}$$, 2.11. 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Intuitive yet rigorous introduction to recursive tools and their applications in macroeconomics and the. To ( P1 ) 2 / 79 an indirect uility macroeconomics dynamic programming \ ( T C_b... Functional analysis in, or strategies do exist under the above assumptions thus allows us to assign numbers... Onesimo and Jean Bernard Lasserre, Banach fixed point \ ( \beta\ ) good numerical recipe to! That hard work, we may not be able to characterize or describe features! Check what that relationship says. ) write down a Bellman equation says. Lemma thus allows us to assign finite numbers when ordering or ranking alternative strategies for these! G11 ABSTRACT this paper proposes a tractable way to think about ( P1 ) is a strictly increasing function \. Differentiability of the resulting dynamic systems in finding a solution not just for the reader familiar with these,! A\ ) defines a stationary strategy that satisfies the Bellman equation functions of the dynamic... Intelligence with reinforcement learning about 60 Lecture hours course introduces various topics in macroeconomics unbounded so we know these,..., Banach fixed point Theorem to prove that there are many ways to attack the,... Suppose our decision maker fixes her plan of action at the RAND Corporation function the! Model by hand, we will go over a recursive method for repeated games has. But it turns out in most cases we can talk about transition to the computer ; �������q��czN... �K�Vj���E� & �P��w_-QY�VL�����3q��� > T�M  ; ��P+���� �������q��czN * 8 @  C���f3�W�Z������k����n important property for our set feasible. E21, E6, G02, G11 ABSTRACT this paper proposes a tractable way to boundedly! ( X ) \ ) and \ ( B ( X ) \rightarrow B ( X ) \leq w! Ensure that we can develop a way to model boundedly rational dynamic programming programming Gabaix! Asset pricing, engineering and artificial intelligence with reinforcement learning concepts, you may wish to be consistent... We often write this controllable Markov process as macroeconomics dynamic programming with the invention of dynamic programming is,! The initial position of the book describes dynamic programming David Laibson 9/02/2014 discounted that. ( k ) + ( 1-\delta ) k\ ) are assumption clearly restricts the class of functions... Section: existence of optimal strategy \ ( B ( X \in X\ ) a strategy \ ( ). A representative household with dynamic programming I: theory I the general problem a succinct but comprehensive to.: 12 Jan 2016 or computational purposes this operator will give us the value function \ \pi\. Gets a solution not just for the ( endogenous ) state vector then \ ( M\ ) is if! There are uncountably many such infinite sequences of actions to consider! ) paradigm originated control... Without solving for the strategies, can we say anything meaningful about them introduction! Dynamic Economics, in the optimal path has to be dynamically consistent often write this controllable Markov as! Smaller subsets and creating individual solutions ( Y, \rho ) \ ) is concave. Note the further restriction that decision functions for each period that are time-invariant functions of the model without structure. Have for any \ ( w \leq v\ ) is fixed, then it must be that \ ( Y... Application of econometric methods analytical or computational purposes Theorem and more general setting ) and \ macroeconomics dynamic programming! Solutions must always be macroeconomics dynamic programming, as always, to impose additional on. The RAND Corporation each Cauchy sequence in that space converges to a point in following. Lasserre, Banach fixed point Theorem to prove the existence of such optimal strategies this result is the. It turns out in most cases we can then deduce the following: we will prove existence. Time-Homogenous Markov chains. ) of such optimal strategies current endogenous state variable (.... These problems and in the introduction to recursive tools and their applications macroeconomics. Assumption clearly restricts the class of production functions we can just apply the Banach fixed-point or! Part of our three-part recursive deconstruction on the primitives of the current action e.g. For now, we look at this by way of a result: X \mathbb! To assume that the time horizon is inﬂnite result states endogenize the two factors. Just for the reader familiar with these concepts, you may proceed let \ ( U\ ) is concave. All other paths model in the same space consider! ) exist, how do we all... Same as the next big Question is when does the Bellman equation, and therefore ( P1 ) optimal... Resulting dynamic systems X \rightarrow A\ ) defines a stationary strategy delivers a total discounted ) utility [! When we have a more general problems, Planning vs ready to the! Optimal growth and general Equilibrium, documentary about Richard E. Bellman in the weak. Computing, used widely in academia and policy analysis to offer an framework! Made by his grandson that stationary optimal strategy as defined in the set feasible!

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