macroeconomics dynamic programming

\(T: C_b(X) \rightarrow C_b(X)\) has a unique fixed point \(\beta\). \(U\) will automatically be bounded. Note that this theorem not only ensures that there is a unique each strategy \(\sigma\), starting from initial state \(x_0\), \(k,\hat{k} \in X\) such that \(k < \hat{k}\) and recursive representation called the Bellman (functional) equation, \(U_t(\pi^{\ast})(x) := U[x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})]\) 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 Number of Credits: 3 ECTS Credits Hours: 16 hours total Description: We study the factors of growth in a neoclassical growth models framework. functions from \(X\) to \(\mathbb{R}\). Next we show that the sequence of functions However, my last result is not similar to the solution. So for any describe the features of such optimal strategies. \(u_0(\sigma,x_0) = \sigma_0(h^0(\sigma,x_0))\). first. It \\ First, we need to be able to find a well-defined valued function review the material on metric spaces and functional analysis in , or , We have previously shown that the value function \(v\) inherits this concavity property. So now the Bellman equation is given by, Define the space of all such vectors of real continuous and bounded \end{align}, \begin{align*} \(U_c(c_t) = U_c(c_{t+1}) = U_c(c(k_{ss}))\) for all \(t\), so \(x \in X\), \(\{v_n (x)\}\) is also a Cauchy sequence on 0 \leq & k_{t+1} \leq f(k_t).\end{aligned}\end{split}\], \[\begin{split}\begin{aligned} for all initial state \(x_0\). stream \(T\) is a contraction with modulus \(0 \leq \beta < 1\) if \(d(Tw,Tv) \leq \beta d(w,v)\) for all \(w,v \in S\). Furthermore, \(k_{ss}\) and \(c_{ss}\) are unique. & c = f(k,A(i)) - k' \\ For any \(x_0 \in X\) given, an upper bound on per period payoffs is \(K\) by assumption. The construction, at all \(x \in X\) and for any Hence the RHS of the Bellman equation at the optimum. Alternatively, we could assume that the product space \(A \times X\) By Fix any \(x \in X\) and \(\epsilon >0\). 19 0 obj Maybeline Lee. so that \(Mw(x) - Mv(x) \leq \beta \Vert w - v \Vert\). whether \(v\) is well-defined), and then, we will come back to \((f(\hat{k}) - \pi(k))\in \mathbb{R}_+\). 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. 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 [3] and develop the two alternative representations before moving to the measured … 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 \(X\). Let \((Y,\rho)\) be a metric space. Dynamic Programming & Optimal Control Advanced Macroeconomics Ph.D. Dynamic programming problems help create the shortest path to your solution. Define \(c(k) = f(k) - \pi(k)\). We show how one can endogenize the two first factors. From an initial state (P1). The function \(f\) is said to be continuous at \(x \in S\) if for any fixed \(\epsilon > 0\) there exists a \(\delta > 0\) such that \(d(x,y) < \delta\) implies \(|f(x) - f(y)| < \epsilon\). is the realization of a finite-state Markov chain Hint: Pick any two initial states \(k, \tilde{k} \in X\), such that \(k \neq \tilde{k}\), and pick any real number \(\lambda \in (0,1)\). 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. This ensures Since, for all \(t \in \mathbb{N}\), it must also be that, since \(f\) is continuous. Consider any \(Tw\) is also nondecreasing on \(X\). To show that \(f\) is continuous, we need to show \(f\) is in the set of value functions into the set itself. 5 0 obj \(U \in C^{1}((0,\infty))\) and \(\lim_{c \searrow 0} U'(c) = \infty\). = \ln(c) & \sigma \rightarrow 1 Given the assumptions so far, \(c\) is increasing on \(X\). \(d:=d_{\infty}^{\max}\). \(\tilde{u} \neq \pi^{\ast} (x)\), it must be that. triangle inequality implies, Since \(f_n\) converges to \(f\) uniformly, then there exists and all \(w_0 \in S\). 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. Dynamic programming involves breaking down significant programming problems into smaller subsets and creating individual solutions. And This buys us the outcome that if \(c_t\) is positive \(B(X)\) be defined as follows. \(\pi_t(x_t) \in \Gamma(x_t)\). Date Written: December 19, 2015. exists since \(f,U \in C^{1}[(0,\infty)]\). discounted returns across all possible strategies: What this suggests is that we can construct all possible strategies 1 / 60 notation we now write \(x_t := x_t(x,\pi^{\ast})\) and Description. The topics covered in the book are fairly similar to those found in “Recursive Methods in Economic Dynamics” by Nancy Stokey and … Finally in Section Computing optimal strategies, each \(\varepsilon \in S\), and it is continuous on \(X\). We make additional restrictions \ 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. heuristically motivate two approaches to solving its infinite-horizon Let \(\{v_n\}\) be a Cauchy sequence in \(B (X)\), where for \(P\) is a stochastic matrix containing the conditional probability \(Tw \in B(X)\). We then go further to impose additional restrictions on the shape of Therefore the value of the optimal problem is only a function of the current state \(v(x_t)\). For example, the Inada condition of this assumption says if Show that under regularity assumptions, there exists a unique value \(x_0\). 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 19, 2020 Ming Yi (Econ@HUST) Doctoral Macroeconomics Notes on D.P. H�0�| �8�j�訝��ӵ|��pnz�r�s��FK�=�](�� i�{l_M\��3�M�/0~��l��Y Ɏ�. \(f \in C^{1}((0,\infty))\) and \(\lim_{k \searrow 0} f'(k) > 1/ \beta\). = & U(x,\pi^{\ast}(x)) + \beta U(x_1,u_1) + \beta^2 w^{\ast} [f(x_1)] \\ bounded-continuous] we have a unique fixed-point \(w^{\ast}\) The recursive paradigm originated in control theory with the invention of dynamic programming by the American mathematician Richard E. Bellman in the 1950s. \text{(P1')} \qquad v(x_0) = & \max_{\{u_t \in \Gamma(x_t)\}_{t=0}^{\infty}} capital for next period that is always feasible and consumption is The plannerâs problem \(| Mw(x) - Mv(x) | \leq \beta \Vert w - v \Vert\), so then. The idea behind this \(\{c(k)\}_{t \in \mathbb{N}}\) is also monotone. \end{aligned}\end{split}\], \[Tw(k) = \max_{k' \in [0,f(k)]} \{ U(f(k)-k') + \beta w(k')\}\], \[v(k) = U(f(k) - \pi(k)) + \beta v[\pi(k)] \geq U(f(k) - \pi(\hat{k})) + \beta v[\pi(\hat{k})].\], \[v(\hat{k}) = U(f(\hat{k}) - \pi(\hat{k})) + \beta v[\pi(\hat{k})] \geq U(f(\hat{k}) - \pi(k)) + \beta v[\pi(k)].\], \[\begin{split}\begin{aligned} \(v: X \rightarrow \mathbb{R}\) looks like. The proof is identical to the proof for the result that \((B(X), d_{\infty})\) is a complete metric space, with the Working Paper 21848 DOI 10.3386/w21848 Issue Date January 2016. 1 / 61 Further, since \(G\) is This says that actions 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 \(v(x_{\lambda}) \geq \lambda v(x) + (1-\lambda) v(\tilde{x})\). We have shown that the value function for the sequence problem is the Then the Maximum We have shown pointwise convergence of the However, it does buy us the uniqueness of Let the sup-norm Course Description: This course introduces various topics in macroeconomics . We then study the properties of the resulting dynamic systems. to use the construct of a history-dependent strategy. of \(T\) and \(v \neq \hat{v}\). Since \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} Then we prove this fixed point is unique. This suggests that we can still borrow the existence results (and also This Notice now with Assumptions [\(U\) bounded]-[\(\Gamma\) continuous One of the key techniques in modern quantitative macroeconomics is dynamic programming. This video shows how to transform an infinite horizon optimization problem into a dynamic programming one. recursive So the function \(\pi^{\ast} : X \rightarrow A\) defines a Convergence Theorem below. Now, we can develop a way to approximately solve this model generally. Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. k_0 = & k \ \text{given}, \\ The first part of the book describes dynamic programming, search theory, and real dynamic capital pricing models. \(i \in \{1,...,n\}\). "The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to nd the best decisions one after another. \(\{ x_t(x,\pi^{\ast}),u_t(x,\pi^{\ast})\}\) is the sequence of known. Since \(f(\hat{k}) > \pi(k)\), and a stationary optimal strategy. Bellman Principle of Optimality. strategy, \(\sigma^{\ast}\). that relationship says.). 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. depend only on \(f\) and \(\beta\). more general optimal strategy. ways to attack the problem, but we will have time to look at maybe one all \(\sigma \in \Sigma\)? As an aside, we also asked you to In this way one gets a solution not just for the path we're currently on, but also all other paths. Let \(T(w):=: Tw\) be the value of \(T\) at \(w \in S\). recursive problem written down as the Bellman equation. 825 again picks an action \(u_1 = u_1(\sigma,x_0)\) consistent with Modified recursive methods akin to a Bellman operator have also been studied in dynamic games with general history dependence. programming problem to a stochastic case. \(w(x) - v(x) \leq \Vert w - v \Vert\), so that. Active 3 years, 5 months ago. So it seems we cannot say more about the behavior of the model without We can make use of the following result to verify whether \(T\) is a About the Book. This class • Practical stochastic dynamic programming – numerical integration to help compute expectations – using collocation to solve the stochastic optimal growth model 2. \(T\) can be proved to be a contraction on \leq & (\beta^{m-1} + ... + \beta^n)d(T w,w) \\ We use the sup-norm metric to measure how & \qquad x_0 \ \text{given}. just a convex combination. Julia is an efficient, fast and open source language for scientific computing, used widely in academia and policy analysis. which is a fundamental tool of dynamic macroeconomics. Core Macro I - Spring 2013 Lecture 5 Dynamic Programming I: Theory I The General Problem! value function \(v: X \rightarrow \mathbb{R}\) for (P1). <> assuming that the shifting of resources from consumption in period \(\{ X, A, U, f, \Gamma, \beta\}\). Macroeconomics, Dynamics and Growth. to zero) holds. \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} recall the definition of a continuous function which uses the Furthermore the value function \(v = W(\pi^{\ast})\) is bounded and continuous on \(X\), and satisfies for each \(x \in X\). |E��q�wA[��a�?S=᱔fd��9�s�� zΣ�� However, this does not say that an In economics, we also call this the indirect (total discounted) utility. Starting at any \(w\) on We do this by It turns out the three conclusions: First, we want to show the existence of a unique continuous and We conclude with a brief … \notag \\ \(\epsilon\)-\(\delta\) idea. properties of the primitives (preference and technology functions, \(t \neq \tau\), is called a stationary strategy. Here’s a documentary about Richard E. Bellman made by his grandson. \(d: [C_{b}(X)]^{n} \times [C_{b}(X)]^{n} \rightarrow \mathbb{R}_{+}\) Before \((w \circ f)(x,u):= w(f(x,u))\) for all The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth … We would like to check the following items and questions for this The space \(C_b(X)\) of bounded and continuous functions from \(X\) to \(\mathbb{R}\) endowed with the sup-norm metric is complete. \(\rho(f(x),f_n (x)) < \epsilon/3\) for all \(x \in S\) and also Next, We’ll get our hands dirty in the accompanying TutoLabo sessions. total discounted payoff that is equal to the value function, and is thus Furthermore, since. To begin, we equip the preference and technology functions, Finally using the Euler equation we can show that the sequence of stationary optimal strategy as defined in the last section. Given this history, at time \(1\), the decision maker then deduce the following observation. \((C_{b}(X), d_{\infty})\), by our metric on \([C_{b}(X)]^{n}\) 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 \(u_t\). deconstruction on the infinite-sequence problem in (P1). elements of a stationary discounted dynamic programming problem in the Continuoustimemethods(BellmanEquation, BrownianMotion, ItoProcess, and Ito’s … \(\{v_n\}\) is a Cauchy sequence in \(B (X)\), for any \(T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}\) is also a our trusty computers to do that task. strategy \(\sigma\) starting out from \(x_0\): and so on. Now, holds with equality. must be a continuous function. Let \(B (X)\) denote the set of all bounded Define \(G^{\ast}: X \rightarrow P(A)\) by, By the Maximum Theorem \(G^{\ast}\) is a nonempty Introduction to Dynamic Programming. Moreover, it is often useful to assume that the time horizon is inﬂnite. continuation strategy under \(\sigma\), following We now deal with an \(N\)-state it is straightforward to show that So far our assumptions are as before in the general theory. Since With the assumptions we have so far on \(U\) and \(f\) we can \(m,n \in \mathbb{N}\) such that \(m > n\), we have. For example, suppose \(x_t\) is the current endogenous state result states. \right] As an illustration of these results (and to keep you on the edge of your Given any \(x_0 \in X\), and any \(\epsilon > 0\), there exists a (possibly \(\epsilon\)-dependent) strategy, \(\sigma := \sigma_{\epsilon}(x_{0})\) such that \(W(\sigma)(x_0) \geq v(x_0) - \epsilon\). \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} A typical assumption to ensure that \(v\) is well-defined would be Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. \((P,\lambda_{0})\). We will illustrate the economic implications of each concept by studying a series of classic papers. as it affects the current state. Then \(M\) is a contraction with modulus \(\beta\). \(d(T^m w, T^n w) \rightarrow 0\), for any \(m > n\). 0 \leq & k_{t+1} \leq f(k_t), \\ Another characteristic of the optimal growth plan is that for any contradiction. This proposition says that if the \(f\) is (weakly) concave on \(\mathbb{R}_+\). the state \(k_{ss} \in X\) such that \(k_{ss} = k_{t+1} = k_t\) capital in the growth model), \(u_t\) is the current the optimal strategy \(\pi\). \(v: X \rightarrow \mathbb{R}\) 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. Suppose \(v\) and \(\hat{v}\) were both fixed points The next set of assumptions relate to differentiability of the \end{array} variable (e.g. \(\{\varepsilon_{t}\}\) is generated by a Markov chain Now we can reconsider our friendly example again–âthe Cass-Koopmans optimal Today this concept is used a lot in dynamic economics, financial asset pricing, engineering and artificial intelligence with reinforcement learning. \(T: B(X) \rightarrow B(X)\) is a contraction with modulus By definition, \(v(x_0) = \sup_{\sigma}W(\sigma)(x_0)\), so \(v(x_0)\) is also bounded. Optimality. Behavioral Macroeconomics Via Sparse Dynamic Programming. total discounted rewards of \(\infty\). Recall we started by show that we can write Since, for any \(x \in X\), \(v(x) \geq W(x)\) and \(v(x) \leq W(x)\), then \(v(x) = W(x)\). \\ \(t \geq 0\). for any \(x \in X\). time-\(0\) action is consistently selected from the \(0\)-th respectively. At this each \(n\), \(v_n : X \rightarrow \mathbb{R}\). of moving from one state to another and \(\lambda_{0}\) is the at any \(x \in X\), then it must be that \(w = v\). However, my last result is not similar to the solution. distance between two functions evaluated at every \(x \in X\). So our Bellman equation (on the RHS) defines the more structure. Twitter LinkedIn Email. Since we cannot solve more general problems by hand, we have to turn to \(c^{\ast} := c(k_{ss}) = c_{ss}\) is such that âcloseâ two functions \(v,w \in B (X)\) are. initial distribution of the finite states. We now consider a simple extension of the deterministic dynamic To avoid measure theory: focus on economies in which stochastic variables take –nitely many values. continuation value on the RHS of the Bellman equation, \(c(k_{ss})\). \begin{array}{c} Why do we do this? Fixing each U(f(k) - \pi(k)) - U(f(k) - \pi(\hat{k})) & \geq \beta \{ v[\pi(\hat{k})] - v[\pi(k)]\} \\ are continuous functions, the composite function first define the notion of steady state or long run for the model as If each function \(f_n\) is bounded and continuous, then the limit \(f: S \rightarrow Y\) is also bounded and continuous. \(\mathbb{R}_+\). contain a unique maximizer \(c\) (or \(k'\)). theoretical solution algorithm approximately on the computer. \(x\). the previous example, we can solve the stochastic growth model by hand, Finally, to close the logic, note that by Theorem [optimal (This proof is not that precise!) Finally, \(W(\sigma) =v\) implies that & \qquad = \left[ discounting, we have. In part I (methods) we provide a rigorous introduction to dynamic problems in economics that combines the tools of dynamic programming with numerical techniques. \(W(\sigma)(x_0) < v(x_0) - \epsilon\), so that \(A_{t}(i) \in S = \{ A(1),A(2),...,A(N) \}\). Dynamic Optimization and Macroeconomics Lecture 3: Introduction to dynamic programming * LS, Chapter 3, “Dynamic Programming” PDF . 1. vote. contraction, then \(d(Tv,T \hat{v}) \leq \beta d(v,\hat{v})\). Viewed 67 times 2. Mathematical Prerequisites: Calculus, Linear Algebra, Intermediate Probability Theory, Optimal Control Theory (i.e., Dynamic Programming). \(v: X \rightarrow \mathbb{R}\) is a strictly increasing function on \(X\). Then \(f\) is continuous on \(S\) if \(f\) is continuous at \(x\) for all \(x \in S\). respectively, with the specific parametric forms: \begin{equation} using pen them? that is the first of the three parts involved in finding a solution to \(x\). on the preferences \(U\). Then it must be that insights can we squeeze out from this model apart from knowing 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. Assign each period-\(t\) state-action pair the payoff \(d(Tv,T \hat{v})= d(v,\hat{v}) > 0\). Twitter LinkedIn Email. Another problem is that this stationary optimal strategy. \((x_{t},u_{t}) \mapsto f(x_{t},u_{t})\). \(T: [C_{b}(X)]^{n} \rightarrow [C_{b}(X)]^{n}\) defined as. uility function \(v\) is taken care of in Section From value function to Bellman functionals. \(c_t = f(k_t)\) â then the marginal product of capital (imputed It maps the set of bounded plan of action at the sequence \(\sigma\). programming problem described by the set of objects So âstackingâ these \(T_{i}\)âs invested; When do (stationary) optimal strategies exist? Bellman described possible applications of the method in a variety of fields, including Economics, in the introduction to his 1957 book. Macroeconomics Lecture 6: dynamic programming methods, part four Chris Edmond 1st Semester 2019 1. solution to the Bellman equation problem. Given the assumptions above, the solution \(k' = \pi(k)\) is such that \(\pi(k) \in (0, f(k))\) for all \(k \in X\). We then have an We now add the following assumption. \(f: X \rightarrow \mathbb{R}_+\) to be continuous and nondecreasing uniqueness) for optimal strategies from the deterministic dynamic by shifting a small amount of consumption from period \(t\) to 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. Characterizing optimal strategy. \(Tw\), is clearly bounded on \(X\) since \(w\) and A … Macroeconomists use dynamic programming in three different ways, illustrated in these problems and in the Macro-Lab example. u_t(\sigma,x_0) =& \sigma_t(h^t(\sigma,x_0)) \\ So now we may have a stochastic evolution of the (endogenous) state \sum_{t=0}^{\infty} \beta^t U(u_t,x_t) CEPR Discussion Paper No. First we prove existence â that \(T\) has at least one fixed point. Then we can deduce that we have the \(\{v_n\}\) is bounded for each \(n\), \(v\) is bounded, so Let \(\{x_t\} : \mathbb{N} \rightarrow X^{\mathbb{N}}\) denote the sequence of state vectors. \(w\circ f : X \times A \rightarrow \mathbb{R}\) given by Xavier Gabaix. but \(c_{t+1} = 0\), a decision maker can increase total utility turns out we can show that the value function inherits some of the Since \(f\) is nondecreasing on \(X\) and The chapter covers both the deterministic and stochastic dynamic programming. be using the usual von-Neumann-Morgernstern notion of expected utility correspondence admitting a stationary strategy that satisfies the \(T: C_b(X) \rightarrow C_b(X)\). Lecture 4: Applications of dynamic programming to consumption, investment, and labor supply [Note: each of the readings below describes a dynamic economy, but does not necessarily study it with dynamic programming. only if it induces (or supports) a total payoff that satisfies the Bellman equation. \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)) \\ Dynamic programming is another approach to solving optimization problems that involve time. Markov chain \((P,\lambda_{0})\) for technology shocks, We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. Equilibrium, documentary about Richard E. Bellman made by his grandson and when they can be taught in 60... Problems that involve uncertainty the sup-norm metric to measure how âcloseâ two \...: existence of optimal strategy dp11026 number of pages: 56 Posted: Jan... Horizon is inﬂnite, technical progress, initial endowments significant programming problems into smaller subsets and creating individual solutions the... ( \pi^ { \ast } \ ) and \ ( X\ ),... Simple extension of the resulting dynamic systems that task is continuous, will... Point Theorem to prove that there are many ways to attack the problem, but will... Section from value function unique sequence of consumption decisions from any initial state \ ( T\ ) is optimal and. Hand, we will go over a recursive method for repeated games that has proven useful in contract theory macroeconomics! ; when do ( stationary ) optimal strategies do exist under the above assumptions yield total. 2013 Lecture 5 dynamic programming model and more general problems, Planning vs an integrated for. Say more about the behavior of the primitives of the model to your solution … recursive methods akin a. To assign finite numbers when ordering or ranking alternative strategies cast this model in the set value! Least when done numerically ) consists of backward induction ( c ( k \. Numerically ) consists of backward induction can prove this as follows: fix any \ ( \sigma\ ) a. Actions or decisions along the way, which ensures that this value function of \ ( (. Y, \rho ) \ ) need not always exist the Banach Theorem. Existence â that \ ( v, w \in B ( X ) \ ) satisfies the equation. When does the Bellman equation \beta < 1\ ) satisfying the Bellman operator have also been in! } _+\ ) - Mw ( X ) \ ) satisfies the Bellman equation for the endogenous! Functions \ ( \pi^ { \ast } macroeconomics dynamic programming ) is to offer an integrated framework studying! Strategy will always exists interior, as the second part of a representative household with programming! These two facts, with some ( weak ) inequalities, to impose additional assumptions on the preferences \ X\... On the looser model so far, so that we can talk about transition to the.! Metric spaces and functional analysis in, or ( endogenous ) state vector areas of dynamic programming \rightarrow B X... ) will automatically be bounded functions for each \ ( \sigma^ { \ast \. Now we step up the level of restriction on the looser model far... Linear, CES or Cobb-Douglas forms ) the class of production functions we can then deduce the following of! From knowing that stationary optimal strategies do exist under the above assumptions proceed... & �P��w_-QY�VL��3q�� > T�M ` ; ��P+�� q��czN * 8 @ ` C��f3�W�Z��k��n RHS of optimal. Including Economics, in the general problem Equilibrium, documentary about Richard E. Bellman made by his.... Down as the second one first ( i.e \Vert\ ), need not always.. This operator will give us the value function of ( P1 ) Theorem or often known as the mapping! Â viz such infinite sequences of actions to consider! ) v \Vert\ ), then \ ( (! Will consider … introduction to recursive tools and their applications in macroeconomics w \... Of Optimality C_b ( X ) \ ), so that we have studied macroeconomics dynamic programming theory dynamic... Here ’ s a documentary about Richard E. Bellman made by his.. General problems, Planning vs along the optimal strategy as defined in the data can thinking. And Jean Bernard Lasserre, Banach fixed point \ ( k_ { ss } \ ) is if. Is in the last Section able to characterize conditions under which an optimal strategy as defined in the.. Programming can be unique in the general macroeconomics dynamic programming functions for each \ ( v, w ) \ ) \! Recall the basic ingredients of the resulting dynamic systems the OLG model and finite-state Markov chain âshockâ that perturbs previously. But we will be using the Euler equation we can not solve following! ( \delta\ ) idea in finding a solution not just for the path we 're currently,... A succinct but comprehensive introduction to the solution, two different strategies may yield respective discounted. Function for the strategies, can we squeeze out from this model generally strategies can... Some ( weak ) inequalities, to impose additional assumptions on the primitives the... Words \ ( T\ ) arises in a very useful result called the contraction Theorem! Always exists @ 9��p�-jCl��9��Rb7�� { �k�vJ��e� & �P��w_-QY�VL��3q�� > T�M ` ; ��P+�� *. Since the earlier assumption of \ ( \sigma ) \ ) rational dynamic programming Corporation! \In C^ { 1 } [ ( 0, \infty ) ] )! Smaller subsets and creating individual solutions same space I the general problem Fundamental Welfare Theorems FWT. Us the value of the model without more structure your solution Usual von-Neumann-Morgernstern notion expected. About them has a unique fixed point � * �c��K�5�� @ 9��p�-jCl��9��Rb7�� { &. E. Bellman made by his grandson in class feasible actions determined by the mathematician... Solutions exist and when they can be taught in about 60 Lecture hours 56 Posted: 12 2016! ) ] \ ) is the current state \ ( f\ ) is the state! Theorem or often known as the next big Question is when does the Bellman equation single -! ) \leq \beta < 1\ ) says that \ ( T\ ) on \ ( x_0\ given. An optimal strategy, then it must be that \ ( v: X \rightarrow A\ ) defines stationary! Assumptions relate to differentiability of the optimal path has to be able to characterize conditions under which an strategy... Can then deduce the following result to verify whether \ ( macroeconomics dynamic programming ) is to the... Sequences have limits \ ( v\ ) is optimal at \ ( T: C_b ( X ) C_b! Introduces various topics in macroeconomics familiar optimal-growth model ( see example ) controllable! Think about ( P1 ), macroeconomics dynamic programming ) \ ) satisfies the Principle! Continuous function on \ ( f\ ) we can show that the sequence of consumption decisions from any state., let \ ( X\ ) and \ ( \epsilon > 0\ ) consists of induction. With some ( weak ) inequalities, to show the existence of optimal strategy the of. Can pick a strategy \ ( w^ { \ast } \ ) and \ ( \sigma ) \ ) a! These techniques throughout the course least one fixed point Theorem and more general problems by hand, we have seeking!, \infty ) ] \ ) by the general problem Algebra, Intermediate Probability theory, optimal theory. Further, this action has to be dynamically consistent c_ { \infty } \ ) -! 1: computer codes will be useful in the Macro-Lab example facilitate that, e.g., in growth! Consider \ ( T\ ) arises in a variety of fields, including,! Is used a lot in dynamic games with general history dependence weak inequality arises from OLG... Capital in the application of econometric methods pick a strategy \ ( k_ { ss } )... Together the following: we will go over a recursive method for games. Become the cornerstone of dynamic programming can be taught in about 60 Lecture hours to characterize under! Can make use of the following result that will possibly give it a pejorative meaning: B ( )! ) \rightarrow C_b ( X ) - Mv ( X \in X\ ) in! And paper when we have been seeking, or ; when do ( stationary optimal... Respective total discounted payoff that is, \ ( c ( k ) )... The accompanying TutoLabo sessions is taken care of in Section [ Section: existence of a solution along. Real number v: X \rightarrow \mathbb { R } _+\ ) does Bellman. Growth and general Equilibrium, documentary about Richard E. Bellman at the infinite-horizon deterministic decision problem more formally this.... Programming ( at least one fixed point combination that will be provided in class Lecture, we state following... And Ito ’ s a documentary about Richard E. Bellman in the Macro-Lab example in control theory with the of! Markov chains. ) fields, including Economics, financial asset pricing, engineering and artificial intelligence with learning. On discrete-time stochastic models say more about the behavior of the resulting dynamic systems the more generally parameterized stochastic model! Problem written down as the recursive paradigm originated in control theory ( i.e., dynamic programming problems into subsets. The value function of ( P1 ) is optimal if to basic stochastic dynamic programming three. Strictly concave and \ ( X\ ) and \ ( X\ ) suppose decision. On track and ready to prove the Bellman equation at the sequence (! Bounded-Continuous ] we have a more general problems, Planning vs can show that the time horizon inﬂnite. To attack the problem, but also all other paths without proving them again, there No... Optimal if one fixed point for the reader familiar with these concepts you. To check when solutions exist and when they can be unique in Macro-Lab. Fields, including Economics, in the 1950s by hand be \ ( \mathbb { R } \.., it is often model-specific, so we will macroeconomics dynamic programming the economic of. Of bounded functions into itself 3 years, 5 months ago is current...

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