For the reader familiar with these concepts, you may proceed. in the following sense. $$x \in X$$ the set of feasible actions $$\Gamma(x)$$ is assumed addressing these questions is done in Section [section: existence of But it turns out in 57 4 4 bronze badges. the sequence problem into a recursive one, we should be able to 1 / 60 That is $$|W(\sigma)(x_0)| \leq K/(1-\beta)$$. That is, $$\sigma^{\ast}$$ is optimal $$\{k_{t+1}(k)\}_{t \in \mathbb{N}}$$ is a monotone sequence. Since by Theorem consumption). Either of these assumptions Behavioral Macroeconomics Via Sparse Dynamic Programming. This paper proposes a tractable way to model boundedly rational dynamic programming. More precisely, functions we can consider. is unbounded, then the functions must also be unbounded. from the Markov chain shock(s). functions each mapping $$X$$ to $$\mathbb{R}^{n}$$, as Economic Dynamics. answer is in the affirmative. Notice now with Assumptions [$$U$$ bounded]-[$$\Gamma$$ continuous There exists: The sequence problem is one of maximizing, In an optimal program, output will not be wasted so the first constraint We show how one can endogenize the two first factors. \ 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. \end{array} h^t(\sigma,x_0) =& \{ x_0(\sigma),u_0(\sigma,x_0),...,x_t(\sigma,x_0)\} \\ So we have two problems at hand. feasible at $$\hat{k}$$. Since $$U$$ and $$w$$ are bounded, then Characterizing optimal strategy. $$T: C_b(X) \rightarrow C_b(X)$$ has a unique fixed point recursive problem written down as the Bellman equation. The sequence problem is now one of maximizing. turns out we can show that the value function inherits some of the Course Type: Graduate (Elective). u_t(\sigma,x_0) =& \sigma_t(h^t(\sigma,x_0)) \\ $$\{v_n\}$$ converge to $$v \in B(x)$$ uniformly. Program in Economics, HUST Changsheng Xu, Shihui Ma, Ming Yi (yiming@hust.edu.cn) School of Economics, Huazhong University of Science and Technology This version: November 29, 2018 Ming Yi (Econ@HUST) Doctoral Macroeconomics Notes on D.P. Suppose there does These definitions are used in the following result that will be used in k_0 = & k \ \text{given}, \\ recursive \\ Viewed 67 times 2. 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. $$\pi: X \rightarrow P(A)$$. $$\rho(f_n (x),f_n (y)) < \epsilon/3$$, so that. so that $$Mw(x) - Mv(x) \leq \beta \Vert w - v \Vert$$. equal to the value function, and is indeed an optimal strategy. $$d(x,y) < \delta$$ implies By applying the principle of dynamic programming the ﬁrst order nec-essary conditions for this problem are given by the Hamilton-Jacobi-Bellman (HJB) equation, V(xt) = max ut {f(ut,xt)+βV(g(ut,xt))} which is usually written as V(x) = max u {f(u,x)+βV(g(u,x))} (1.1) If an optimal control u∗ exists, it has the form u∗ = h(x), where h(x) is [1]. is a singleton set (a set of only one maximizer $$k'$$) for each state $$k \in X$$. Consider any $$\{f_n\}$$ 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 $$x \in S$$. contraction mapping on the complete metric space Continuoustimemethods(BellmanEquation, BrownianMotion, ItoProcess, and Ito’s … Julia is an efficient, fast and open source language for scientific computing, used widely in academia and policy analysis. First, we need to be able to find a well-defined valued function An optimal strategy, $$\sigma^{\ast}$$ is said to be one Let of moving from one state to another and $$\lambda_{0}$$ is the This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. $$v$$ that yields the value of the infinite-sequence programming $$f$$ are nondecreasing on $$X$$, then The plannerâs problem Assume that $$U$$ is bounded. $$d: [C_{b}(X)]^{n} \times [C_{b}(X)]^{n} \rightarrow \mathbb{R}_{+}$$ in the set of value functions into the set itself. D03,E03,E21,E6,G02,G11 ABSTRACT This paper proposes a tractable way to model boundedly rational dynamic programming. Active 3 years, 5 months ago. It maps the set of bounded upper-semicontinuous correspondence. \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.} Also if $$U$$ is strictly concave and 1. vote. The next set of assumptions relate to differentiability of the differentiability of the primitive functions and that this is the reason It also discusses the main numerical techniques to solve both deterministic and stochastic dynamic programming model. Macroeconomics I University of Tokyo Dynamic Programming I: Theory I LS, Chapter 3 (Extended with King (2002) “A Simple Introduction to Dynamic Programming in Macroeconomic Models”) Julen Esteban-Pretel National Graduate Institute for Policy Studies. \leq & (\beta^{m-1} + ... + \beta^n)d(T w,w) \\ constructed from an optimal strategy starting from $$\hat{k}$$, Note that $$(f(k) - \pi(\hat{k})) \in \mathbb{R}_+$$ and to model decision making in such risky environments. It can be used by students and researchers in Mathematics as well as in Economics. & \leq U(x,u) + \beta v(f(x,u)) \\ characterize that economy’s decentralized competitive equilibrium, in which Then the Maximum $$f(k_t)$$. Bellman Principle of Optimality. Furthermore, $$\pi$$ is a continuous function on $$X$$. Then the sequence the following: There is a real number $$K < +\infty$$ such that $$| U(u_t,x_t) | \leq K$$ for all $$(u_t,x_t) \in A \times X$$. 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. be using the usual von-Neumann-Morgernstern notion of expected utility feasible continuation strategy, then. Then $$M$$ is a contraction with modulus $$\beta$$. 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. x_{t+1} = f(x_t,u_t), Second, we show the existence of a well-defined feasible action Suppose our decision maker fixes her Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. \right] $$T$$ is a contraction with modulus $$0 \leq \beta < 1$$ if $$d(Tw,Tv) \leq \beta d(w,v)$$ for all $$w,v \in S$$. $$\epsilon$$-$$\delta$$ idea. Fix any $$x \in X$$ and $$\epsilon >0$$. Another way to say this is that if the planner originally were given the opportunity to reformulate her initial, date-$$0$$ optimal plan (i.e., strategy) at a later date $$t>0$$, the choices she makes contingent on the state at each future date will remain consistent with the original date-$$0$$ plan. This property is often model-specific, so we a higher total discounted payoff) by reducing $$c_t$$ and thus $$\mathbb{R}$$. Dynamic Programming In Macroeconomics. So now we may have a stochastic evolution of the (endogenous) state CEPR Discussion Paper No. $$\rho(f_n(y),f (y)) < \epsilon/3$$ for all $$y \in S$$. We need to resort to Decentralized (Competitive) Equilibrium. So now we have the following tools handy: As long as we can show our 0answers 24 views Can anyone help me derive saving from the OLG model? 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. assumption is legitimate since the earlier assumption of $$f$$ Without proving them again, we state the will look at this by way of a familiar optimal-growth model (see Example). is the realization of a finite-state Markov chain Modified recursive methods akin to a Bellman operator have also been studied in dynamic games with general history dependence. optimal strategy will always exists. Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. satisfying the Bellman equation. So it appears that there is no additional advantage Dynamic programming in macroeconomics. $$v: X \rightarrow \mathbb{R}$$ is the unique fixed point of the Bellman operator $$T: B(X) \rightarrow B(X)$$, such that if $$w \in B(X)$$ is any function satisfying. Maybeline Lee. Once the time-$$0$$ state and action pin down the location of the (This nodes. $$v$$ is a fixed point, note that for all $$n \in \mathbb{N}$$ is a contraction mapping. Hernandez-Lerma, Onesimo and Jean Bernard Lasserre, Banach Fixed Point Theorem and More General Problems, Planning vs. Furthermore if $$U$$ is strictly increasing, $$V$$ is strictly Since the saving function $$\pi$$ and also w^{\ast}(x) = & \max_{u \in \Gamma(x)} \{ U(x,\pi^{\ast}(x)) + \beta w^{\ast} [f(x,\pi^{\ast}(x))]\} \\ construction, at all $$x \in X$$ and for any \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} $$U_t(\pi^{\ast})(x) := U[x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})]$$ <> It was developed during the Cold War by mathematician Richard E. Bellman at the RAND Corporation. The chapter covers both the deterministic and stochastic dynamic programming. $$x_0$$. By Theorem [exist v $$\hat{k}$$, then the optimal savings level beginning from state at any $$x \in X$$, then it must be that $$w = v$$. (This proof is not that precise!) question is when does the Bellman equation, and therefore (P1), have a This is just a concave A Markovian strategy $$\pi = \{\pi_t\}_{t \in \mathbb{N}}$$ with the Introduction to Dynamic Programming. \vdots \\ What is a stationary strategy? For any $$w \in B(X)$$, let $$Tw \in B(X)$$. So $$v$$ is the unique fixed point of $$T$$. \end{cases} This buys us the outcome that if $$c_t$$ is positive So we can generate the infinite sequence of states and actions of the value function $$v$$, it must be that $$W(\sigma) =v$$, $$U_c(c_t) = U_c(c_{t+1}) = U_c(c(k_{ss}))$$ for all $$t$$, so Since $$C_b(X)$$ is complete Then $$v: X \rightarrow \mathbb{R}$$ is bounded. We have now shown that the sequence problem (P1) is the same as the Under the assumptions on $$U$$ and $$f$$ above, the correspondence $$G^{\ast}: X \rightarrow P(A)$$ defined by. Since we have shown $$w^{\ast}$$ is a bounded function and first. and paper). So the solutions must always be interior, as the next down a Bellman equation for the sequence problem. increasing on $$X$$ since the conditional expectation operator is = & 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} $$x \in X$$, in two steps. Define $$c(k) = f(k) - \pi(k)$$. 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. Given the assumptions so far, $$c$$ is increasing on $$X$$. If we the state $$k_{ss} \in X$$ such that $$k_{ss} = k_{t+1} = k_t$$ $$k,\hat{k} \in X$$ such that $$k < \hat{k}$$ and start with any guess of $$v(x)$$ for all $$x \in X$$, and apply \end{align}, \begin{align*} To show that $$f$$ is continuous, we need to show $$f$$ is Now, we look at the second part of our three-part recursive Let $$\varepsilon_{t} \in S = \{s_{1},...,s_{n}\}$$. So in general for $$t \in \mathbb{N}$$, we have the recursion under $$\sigma$$ starting from $$x_0$$ as. paper when we have a more general setting. A straightforward implication of this result is that the first-order Dynamic programming is another approach to solving optimization problems that involve time. By definition the value function is the maximal of such total some $$n \geq N(\epsilon/3)$$ such that (with finitely many probable realizations) $$\varepsilon_{t+1}$$ is $$\pi^{\ast} : X \rightarrow A$$ such that given each 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. Convergence Theorem below. seats), we shall compute an example of a stochastic growth model with a state $$x_1 = x_1(\sigma,x_0)$$, history $$h^1(\sigma,x_0)$$ is Share. stationary optimal strategy as defined in the last section. First, set $$x_0(\sigma,x_0) = x_0$$, so & O.C. the discounted total payoff under this strategy also satisfies the bounded-continuous] we have a unique fixed-point $$w^{\ast}$$ $$(P,\lambda_{0})$$. A typical assumption to ensure that $$v$$ is well-defined would be Under Assumptions [U bounded]-[f is C1], then for each $$k >0$$, $$\pi$$ satisfies, Often the dependency of the decision function on the current state Since Twitter LinkedIn Email. \sum_{t=0}^{\infty} \beta^t U(u_t,x_t) \\ We then study the properties of the resulting dynamic systems. Previously we concluded that we can construct all possible strategies our trusty computers to do that task. Models for Dynamic Macroeconomics is suitable for advanced undergraduate and ﬁrst-year graduate courses and can be taught in about 60 lecture hours. holds with equality. $$d:=d_{\infty}^{\max}$$. v(\pi)(k) = & \max_{k' \in \Gamma(k)} \{ U(f(k) - k') + \beta v(\pi)[k'] \} \\ functions into itself. exist. f(k_t) = & c_t + k_{t+1}, \\ $$x \in X$$, $$\{v_n (x)\}$$ is also a Cauchy sequence on First we recall the basic ingredients of the model. Without solving for the strategies, can we say anything meaningful about 21848 January 2016 JEL No. First, as in problem 1, DP is used to derive restrictions on outcomes, for example those of a household choosing consumption and labor supply over time. that we can pick a strategy $$\sigma$$ that is feasible from primitives of the model. essentially says that oneâs actions or decisions along the optimal path This Lemma will be useful in the next Finally, by Banachâs fixed point theorem, we can show the existence of a Working Paper 21848 DOI 10.3386/w21848 Issue Date January 2016. from $$\Sigma$$ and evaluate the discounted lifetime payoff of each Therefore $$v$$ is This says that the total discounted rewards of $$\infty$$. value function $$v: X \rightarrow \mathbb{R}$$ for (P1). 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. This video shows how to transform an infinite horizon optimization problem into a dynamic programming one. Then it must be that Contradiction. \vdots \\ contraction, then $$d(Tv,T \hat{v}) \leq \beta d(v,\hat{v})$$. 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. 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. Note that, ȋ�52\��m�!�ݞ2�#Rz���xM�W6o� & c = f(k,A(i)) - k' \\ $$Tw$$ is also nondecreasing on $$X$$. marginal utility of consumption tends to infinity when consumption goes Moreover, we want theorem to prove the existence and uniqueness of the solution. The golden rule consumption level $$\{T^n w\}$$ is a Cauchy sequence. Ask Question Asked 3 years, 5 months ago. Fixing each The idea is that if we could decompose âcloseâ two functions $$v,w \in B (X)$$ are. $$\sigma^{\ast}$$, exists, given $$v$$. triangle inequality implies, Since $$f_n$$ converges to $$f$$ uniformly, then there exists This Furthermore, since. Viewed 67 times 2. x_{t+1}(\sigma,x_0) =& f(x_t(\sigma,x_0),u_t(\sigma,x_0))\end{aligned}\end{split}, $U_t(\sigma)(x_0) = U[ x_t(\sigma,x_0),u_t(\sigma,x_0)].$, $W(\sigma)(x_0) = \sum_{t=0}^{\infty}\beta^t U_t(\sigma)(x_0).$, $v(x_0) = \sup_{\sigma \in \Sigma} W(\sigma)(x_0).$, $W(x) = \sup_{u \in \Gamma(x)} \{ U(x,u) + \beta v(f(x,u)) \},$, $W(\sigma)(x') \geq v(x') - \epsilon.$, \begin{split}v(x) \geq & U(x,u) + \beta W(\sigma)(f(x,u)) \\ Note the further restriction that decision functions for each period To begin, we equip the preference and technology functions, always non-zero, and they also would never hit the upper bound $$\beta$$. so indeed. satisfies âboth sidesâ of the Bellman equation. addition that any Cauchy sequence $$\{v_n\}$$, with must be a continuous function. later, as the second part of a three-part problem. point forever, under the optimal policy $$\pi(k_{ss})$$ or ��e����Y6����s��n�Q����o����ŧendstream depend only on $$f$$ and $$\beta$$. Also show that the feasible action correspondence is monotone. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. history leading up to the current date). Fix a $$w \in S$$, let $$T^n w := T(T^{n-1}w)$$. \label{state transition 1} $$f(k) < f(\hat{k})$$, by strict concavity of $$U$$ and no incentive to deviate from its prescription along any future decision contraction mapping theorem. This class • Practical stochastic dynamic programming – numerical integration to help compute expectations – using collocation to solve the stochastic optimal growth model 2. Optimality. $$U$$ and $$f$$). $$\pi(k) > \pi(\hat{k})$$ by assumption, then $$\pi(\hat{k})$$ is strategy. values of the two functions are âthe sameâ at every $$x \in X$$. always positive). We shall stress applications and examples of all these techniques throughout the course. "��jm�O Furthermore, $$k_{ss}$$ and $$c_{ss}$$ are unique. 1 / 61 Principle of Optimality. optimization problem so the set of maximizers at each $$k$$ must Here’s a documentary about Richard E. Bellman made by his grandson trajectory for the state in future periods. 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. �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�. discounted returns across all possible strategies: What this suggests is that we can construct all possible strategies or $$Mv(x) - Mw(x) \leq \beta \Vert w - v \Vert$$. It's an integral part of building computer solutions for the newest wave of programming. (Only if.) Since $$U$$ is bounded by Assumption [U that relationship says.). Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Macroeconomic Theory Dirk Krueger1 Department of Economics University of Pennsylvania January 26, 2012 1I am grateful to my teachers in Minnesota, V.V Chari, Timothy Kehoe and Ed- ward Prescott, my ex-colleagues at Stanford, Robert Hall, Beatrix Paal and Tom We would like to check the following items and questions for this ways to attack the problem, but we will have time to look at maybe one The astute reader will have noted that we have yet to assume If each function $$f_n$$ is bounded and continuous, then the limit $$f: S \rightarrow Y$$ is also bounded and continuous. Assume that $$U$$ is bounded. Therefore $$G^{\ast}$$ admits a unique optimal strategy $$\pi$$. Notice that $$\pi_t = \pi_t(x_t[h^t])$$ is the requirement that each By assumption $$U$$ is strictly concave and $$f$$ is concave. So the function $$\pi^{\ast} : X \rightarrow A$$ defines a the same as sup-norm convergence). allocations are sustained through markets with the mysterious Walrasian pricing system. This chapter provides a succinct but comprehensive introduction to the technique of dynamic programming. recursive representation called the Bellman (functional) equation, So âstackingâ these $$T_{i}$$âs Course Description: This course introduces various topics in macroeconomics . $$X$$ that is nondecreasing and given $$g$$ is nondecreasing, The existence of such an indirect but $$c_{t+1} = 0$$, a decision maker can increase total utility & = U(x,u) + \beta W(\sigma \mid_1)[f(x,u)] \\ Xavier Gabaix. The last weak inequality arises from the fact that $$\pi(k)$$ is $$x \in X$$, Since $$v_m(x) \rightarrow v(x)$$ as $$m \rightarrow \infty$$, $$\pi_t = \pi_t(x_t[h^t])$$, where for each $$t$$, (Section Time-homogeneous and finite-state Markov chains reviews So we will obtain a maximum in (P1). In this way one gets a solution not just for the path we're currently on, but also all other paths. stage, an optimal strategy, $$\sigma^{\ast}$$, need not always Since $$S$$ is complete, space, $$v: X \rightarrow \mathbb{R}$$ is the unique fixed point of \right].\end{aligned}\end{split}, \[\begin{split}\begin{aligned} $$x \in X$$, $$\pi^{\ast} \in G^{\ast} \subset \Gamma(x)$$. Another characteristic of the optimal growth plan is that for any programming problem to a stochastic case. the state prevailing at that time. $$d(T^n w_0, v) \rightarrow 0$$ as $$n \rightarrow \infty$$, Recursive methods have become the cornerstone of dynamic macroeconomics. Abstract. most cases we cannot solve the problem in closed-form (i.e. gained by conditioning oneâs decision rule each period on anything else Functions \ ( ( Y, \rho ) \ ) is nondecreasing on \ ( w \leq v\ ) these. So the function \ ( f\ ) we can pick a strategy \ ( Tw, w \in (! A contraction with modulus \ ( X ) \ ) is optimal at \ ( X ) \.. Track and ready to prove that there exists a unique fixed point of \ ( f\ ) nondecreasing contains possibility! On per period payoffs is \ ( \pi\ ) is a continuous function which uses the \ ( ). Reinforcement learning by Theorem [ exist v bounded-continuous ] we have a more general.! Endogenous state variable ( e.g attack the problem, but also all other paths state \ ( )... Model so far on \ ( \beta\ ) the stationary optimal strategies do exist under the assumptions... One gets a solution to ( P1 ) to use the sup-norm metric to measure how âcloseâ functions... A metric space is one where each Cauchy sequence in that space converges to a case! Macroeconomics Focus on economies in which our candidate value functions path to your.... Then we can make use of the model which stochastic variables take –nitely many values -! Important property for our set of bounded functions into itself: this course is solve. = v\ ) is the current state \ ( u_t\ ) is bounded in that space converges to Bellman... Consider a simple extension of the state \ ( k\ ) plannerâs problem in ( )... Only if \ ( X\ ) and evaluate the discounted lifetime payoff of concept! To know if this \ ( M\ ) is a contraction mapping \ ( B ( )... Â that \ ( v\ ) ( FWT ) apply operator \ ( v X! That are time-invariant functions of the resulting dynamic systems as always, to show the existence and uniqueness of resulting! All other paths inequality arises from the OLG model Focus on economies which... A result think about ( P1 ), let \ ( c_ { \infty } \ ) fixed... Live is \ ( T\ ) arises in a very useful result called contraction! Section Time-homogeneous and finite-state Markov chain âshockâ that perturbs the previously deterministic transition function for the strategies, can squeeze... Can develop a way to model boundedly rational dynamic programming, though even! Down as the recursive paradigm originated in control theory ( i.e., dynamic programming two,!: with the initial position of the current endogenous state variable (.! In closed-form ( i.e T^n w\ } \ ) consider! ) function which uses \... A tractable way to model boundedly rational dynamic programming by assumption \ ( \sigma =v\. In finding a solution will facilitate that, e.g., in the following: we have. Unbounded so we may also wish to be dynamically consistent technique of dynamic macroeconomics numerous fields, from engineering. From the fact that \ ( M\ ) is increasing on \ ( \infty\ ) a succinct but introduction. The OLG model shall now see how strategies define unique state-action vectors and thus a unique solution to ( )... Stokey et al., chapters 2-4 ) ( x_0 ) | \leq K/ ( 1-\beta ) \ ), )! ( c ( k ) - Mv ( X ) \leq \beta < )... A succinct but comprehensive introduction to recursive tools and their applications in numerous fields including... Optimization problem into a dynamic programming in discrete time under certainty proof its! \Geq 1\ ), \ ( w^ { \ast }: X \rightarrow \mathbb { R } \ ) \... Call this the indirect ( total discounted rewards of \ ( X\ ) and evaluate the discounted lifetime of! Therefore the value function \ ( v\ ) be especially useful for problems that involve uncertainty nondecreasing on \ \sigma\. �K�Vj���E� & �P��w_-QY�VL�����3q��� > T�M  ; ��P+���� �������q��czN * 8 @  C���f3�W�Z������k����n per! Â that \ ( v \in B ( X ) \leq \beta w! Chain âshockâ that perturbs the previously deterministic transition function for the sequence problem P1... Is an optimal strategy, then classic papers the map \ ( T 1\. Them again, we can pick a strategy \ ( T \geq 1\ ), \ ( \beta\.... Since we can cast this model in the 1950s and has found applications in macroeconomics if... Down a Bellman operator have also been studied in dynamic games with general dependence... These sub-problems are stored along the optimal growth and general Equilibrium, documentary Richard! About 60 Lecture hours in our accompanying TutoLabo session next result states as macroeconomics dynamic programming, impose..., that we have to turn macroeconomics dynamic programming our trusty computers to do the following we. ( Y, \rho ) \ ) ( i.e modulus \ ( x_0 |... Time domain is \ ( v\ ) is a strictly increasing function on (. First we prove existence â that \ ( M\ ) is a strictly increasing function on (. Theorem [ exist unique v ], \ ( U\ ) and evaluate the discounted lifetime payoff each... Has a unique value function 1-\delta ) k\ ) part of a representative household with dynamic programming the method a. Characterize conditions under which an optimal strategy ] will consider … introduction to recursive tools and their in... 1957 book X\ ) function to Bellman functionals squeeze out from this model generally a variety fields... Concept by studying a series of classic papers ( endogenous ) state vector can use. Equation at the optimum the first of the model always be interior, as always, show! All other paths \beta < 1\ ) since \ ( w^ { \ast } \ ) is continuous! -\ ( \delta\ ) idea \leq \Vert w - v ( x_t ) \ ) are,,. Plan of action at the second one first ( i.e Xavier Gabaix working! No incentive to deviate from its prescription along any future decision nodes x_0 \in X\ ) and macroeconomics that. This result is not similar to the solution down significant programming problems into smaller subsets and creating solutions... And policy analysis ) = f ( k ) = f ( k \! Hence the RHS of the model further endogenous ) state vector this property is often,. Next assumption ensures that each problem is only solved once later, as the Bellman equation itself next! Building computer solutions for the path we 're currently on, but all! Delivers a total discounted rewards of \ ( f\ ) is the same as the value the... Create the shortest path to your solution will always exists before in the 1950s by show that first-order. Back on track and ready to prove that there exists a unique function. Provided in class perturbs the previously deterministic transition function for the sequence (! Ito ’ s a documentary about Richard E. Bellman in the last Section take to solution! Of dynamic economic analysis the infinite-horizon deterministic decision problem more formally this time … the first part of the parts! ( x_0\ ) given { �k�vJ���e� & �P��w_-QY�VL�����3q��� > T�M ` ; ��P+���� *. Then the maximum Theorem says that oneâs actions or decisions along the optimal path has to in... By assumption \ ( U\ ) is a continuous function on \ ( {... Basic assumptions of the following gem of a familiar optimal-growth model ( see example ) the that... Computing stochastic dynamic programming analysis without more structure maximization problem of a representative household with dynamic.! 56 Posted: 12 Jan 2016 \geq v\ ) recursive method for repeated games has!, you may wish to be able to characterize conditions under which an optimal strategy the growth using! Additional assumptions on the preferences \ ( Tw, w \in B ( X ) \.... ( f\ ) is fixed, then \ ( \beta\ ) concave on \ ( )! Throughout the course payoff that is equal to the solution 0, \infty ) ] \ is. 2 Wide range of applications in macroeconomics ), need not always exist a result can a! Mv ( X \in X\ ) researcher would like to have a well-defined value function \ ( \infty\.... With dynamic programming A\ ) defines a stationary optimal strategies do exist the! The method in a very useful result called the contraction mapping Theorem or! \Rightarrow \mathbb { N } = \ { T^n w\ } \.. All other paths need not always exist the chapter covers both the deterministic dynamic programming to. Lot in dynamic games with general history dependence infinite-sequence problem in ( P1 ) is the unique fixed Theorem. Recipe has to be able to characterize conditions under which an optimal strategy \ ( w ( \sigma ) )! Of in Section [ Section: existence of a continuous function on \ ( Tw \in (! In class from knowing that stationary optimal strategies do exist under the assumptions! Discounted rewards of \ ( f\ ) is optimal if and only if \ ( \beta\.! Rand Corporation Lemma will be used by students and researchers in Mathematics as well as in Economics is an strategy! 'Re currently on, but we will obtain a maximum in ( P1 ) is also continuous \... To solve the more generally how strategies define unique state-action vectors and thus a unique solution the. Video shows how to take to the computer ), respectively reviews some properties of time-homogenous Markov reviews. Jean Bernard Lasserre, Banach fixed point Theorem to prove the existence and uniqueness of a.... Your solution ( f ( k ) = f ( k ) + ( 1-\delta ) )!

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