This study attempts to bridge this gap. I suspect when you try to discretize the Euler-Lagrange equation (e.g. 3 Euler equation tests using simulated data Generate simulated data from 5000 preretirement households. For dynamic programming, the optimal curve remains optimal at intermediate points in time. general class of dynamic programming models. It is of special value in computationally intense applications. The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimization problems. x��]ݏ7r7��a�6h���̓a �$Ǉ�����ᜇ9id)�v��V��SUd�Iv��fC�ݙ����b�|���wz)v�v��{���wb����v�u;gLgv�?�Wn����w��W��ӓ���q������?��|��݋����rp���|~�������A�[��߱0~�p7�� ���۽��$�Y�s�b���r���l���0d��ٽ�˓�^�؞��F�aD�g#�;TUB���uA As long as the problem is ﬁnite, the fact that the Euler equation holds across all adjacent periods implies that any ﬁnite deviations from a candidate solution that satisﬁes the Euler equations will not increase utility. �t���)��X�_7�*��W�m���ϖ[W�E%u�=�wb�91t*BF����; ȫ/ �Z��~����A2~E������Ni�I[��ꔱ��@�^"[��vp]?b��윾"�Na{�g���-Mh������F� ����=L�O����_���0z��ÿ_O�"M�Bߵ,���� y�t~y�QT 8%EQ�����Z%ʧ)�}���=�1��p?qP���� ��e��?��|�F0��i�i��Q\CPAN�w�El��Av�0r.(7������X�R]�B���H��d':=���x�F.P�m��_������5;u�? ����R[A��@�!H�~)�qc��\��@�=Ē���| #�;�:�AO�g�q � 6� endstream endobj startxref 0 %%EOF 160 0 obj <>stream Motivation What is dynamic programming? This is an example of the Bellman optimality principle.Itis suﬃcient to optimise today conditional on future behaviour being optimal. ;}��������+�Qj�.�����_}�ׯ�U��F�ϧ�/\���W׏�q���?\>u�_bx�\�^����ۻG0?�T��������~�m?u�j��~������w=L F��\�e[��h�j��N%�}=��*�m[�"��t��R��T�=i[�<5NEu�]Ҟ�H�47\��V�o��w��Ե3����! Lecture 6 . First, I discuss the challenges involved in numerical dynamic programming, and how Euler equation‐based methods can provide some relief. calculus of variations, optimal control theory or dynamic programming — part of the so-lution is typically an Euler equation stating that the optimal plan has the property that any marginal, temporary and feasible change in behavior has marginal bene ﬁts equal to marginal costs in the present and future. Problem 27 of Project Euler reads Find the product of the coefficients, a and b, where |a| < 1000 and |b| < 1000, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0. {\displaystyle \pi } . can be characterized by the functional equation technique of dynamic programming [I]. Key Words : dynamic model, precomputation, numerical integration, dynamic programming (DP), value function iteration (VFI), Bellman equation, Euler equa-tion, envelope condition method, endogenous grid method, Aiyagari model We are indebted to Editor Victor Ríos-Rull and three anonymous referees for many thoughtful com-ments and suggestions. Lecture 4 . <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 595.2 841.92] /Contents 4 0 R/Group<>/Tabs/S>> Also only in the limited cases, dynamic programming problems can be solved analytically. Dynamic Programming. Euler's Method C Program for Solving Ordinary Differential Equations Implementation of Euler's method for solving ordinary differential equation using C programming language. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. namic programming equation (DPE) as an intermediate step in deriving the Euler equation. ρ∈(−1 1)are parameters, εt+1∼N(0σ2)is a productivity shock, and uand f are the. The recursive method of solving recursive contracts, i.e., an algorithm, involves expanding the co-state to include a subgradient of 2The result of Rincon-Zapatero and Santos (2009) that the value function in concave dynamic programming´ In this video, I derive/prove the Euler-Lagrange Equation used to find the function y(x) which makes a functional stationary (i.e. In the context of Project Euler – Problem 66, the following Diophantine (Pell’s) equation has been further examined. An approach for solving the optimal control problem is through the dynamic programming technique (DP) (see [1–4]). I suspect when you try to discretize the Euler-Lagrange equation (e.g. h�b�.V�X ��1�0p\�J�8���*{Zx���9'j^���H2 The equations are named in honor of Leonard Euler, who was a student with Daniel Bernoulli, and studied various fluid dynamics problems in the mid-1700's.The equations are a set of coupled differential equations and they can be solved for a given … Lecture 2 . Keywords. endobj ����~O���q���{���!�$m�l�̗�5߃�,��5t�w����K���ǒ�謈%���{\R�N���� �*A�FQ,��P?/�N�C(�h�D�ٻ��z�����{��}�� \�����^o|Y{G��:3*�ד�����q�O6}�B�:0�}�BA:���4�?ϓ~�� �I�bj�k�'�7��!�s0 ���]�"0(V�@?dmc���6�s�h�Ӧ�ޜ�j��Vuj �+;��������S?������yU��rqU�R6T%����*�Æ���0��L���l��ud��%�u���}��e�(�uݬx!����r�˗�^:� ��˄����6Ѓ\��|Ρ G��yZ*;g/:O�sv�U��^w� It is fast and flexible, and can be applied to many complicated programs. }��40�3�u����R�,- V"I�j�"�5Ū��mf�v���?_��yvuY���,���e}�R�^Z;R�[k(��s$kH�G���t-{���o�'aM�k�Z�&���$piŞ����mkN*�Jiu� (}:� �M+�焢/ր�Ӧ�߳�s�>�g! Advantages of procedure. Key Words : dynamic model, precomputation, numerical integration, dynamic programming (DP), value function iteration (VFI), Bellman equation, Euler equa-tion, envelope condition method, endogenous grid method, Aiyagari model We are indebted to Editor Victor Ríos-Rull and three anonymous referees for many thoughtful com-ments and suggestions. Dynamic Programming More theory Consumption-savings Euler equation with Dynamic Programming From V (x) = sup x ′ ∈ R parenleft.alt1 u (y + Rx - x ′) + βV (x ′)parenright.alt1 we obtain - u ′ (y + Rx - x ′) + β dV dx (x ′) = 0 (FOC) dV dx (x) = R u ′ (y + Rx - x ′) (Envelope Thm) or, in dated variables, - u ′ (c t) + β dV dx (s t) = 0 dV dx (s t - 1) = R u ′ (c t) The result is u ′ (c t) = βRu ′ (c t + 1) Math for Economists-II Lecture 4: … endobj As an example of this structure, let us consider the deterministic dynamic programming problem. Dynamic model, precomputation, numerical integration, dynamic programming, value function iteration, Bellman equation, Euler equation, enve- 2.1. h�bbdb^$@D��Yb��M��ZqH0M�6��� �*��%$8O C! In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. 1 Dynamic Programming 1.1 Constructing Solutions to the Bellman Equation Bellman equation: V(x) = sup y2(x) Is this enough? }��$��-ꐶmӡG�a�D�#ڗ��25)�z(���J���g�jׄe���:��@��Z����t���dt��j.g� k!���*|�� r]Ш�6��e� �T{2഍̚����u��(_%�U� (3�f@�@Ic�W��kAy��+� ��x����Q�ͳ���%yỵ�wM��t��]\ We will also have a constraint on the nal state given by (x(t ... (16) yields the familiar Euler Lagrange equa … Under standard assumptions, 6 we can obtain the existence of an optimal policy function g: X × Z ® X. How? Math for Economists-II Lecture 4: Dynamic Programming (2) Nov 5 nd, 2020 <> Example 1 ... (1.13) is the Euler equation linking consumptions in adjacent periods. and we have derived the Euler equation using the dynamic programming method. 1 0 obj Models with constant returns to scale. Nevertheless, in contrast to the 1Another attractive feature of the Euler equation-GMM approach when applied to panel data is that it can deal Differential equations can be solved with different methods in Python. Second, the Euler conditions can, in many instances, be solved more eas-ily than Bellman's equation for the optimal solution of the Markov decision model. )���Wi �b��ZY����A�1ϩ�d��=d�&�;!3�ݥ�,,��@WM0K���H�&T�hA�%��QZ$ѩ�I��ʌ���! Additional information is provided on using APM Python for parameter estimation with dynamic models and scale-up to large-scale problems. Dynamic programming turns out to be an ideal tool for dealing with the theoretical issues this raises. V π ( s ) = R ( s , π ( s ) ) + γ ∑ s ′ P ( s ′ | s , π ( s ) ) V π ( s ′ ) . V^ {\pi } (s)=R (s,\pi (s))+\gamma \sum _ {s'}P (s'|s,\pi (s))V^ {\pi } (s').\. } ����_��@��e�ډE;��w��X���3]��6��9��.Q�]�їr��m�S\���^)�]�nLv�ا��i�j?�]5T �q�٬﬩�*���T�����KQ_��SYԶnոڐ���v���2)���z�g�jZLsn��](�&�%ok�q-X)T]W� �͝��PZa����!�E�j]�xʅ�v5��i�y��lW:. z O g ρ0g −∇p Taking typical values for the physical constant, g ≃ 10ms−2, ρ 0 ≃ 103kgm−3 and a pressure of one atmosphere at sea-level, p 0 ≃ p Output of this is program is solution for dy/dx = x + y with initial condition y = 1 for x = 0 i.e. For me this one reeks of brute force, since it is obvious that we can run through all possible values of a and b. general class of dynamic programming models. It follows that their solutions can be characterized by the functional equation technique of dynamic programming . 10 of 21 the extremal). The ﬂrst author wishes to thank the Mathematics and Statistics Departments of Lecture 8 . (is a sup-compact function if the set is … Therefore, the stochastic dynamic programming problem is defined by (X,Z,Q,W,F,b). But as we will see, dynamic programming can also be useful in solving –nite dimensional problems, because of its recursive structure. Dynamic Programming¶ This section of the course contains foundational models for dynamic economic modeling. Consider the following “Maximum Path Sum I” problem listed as problem 18 on website Project Euler. Section 3 introduces the Euler equation and the transversality condition, and then explains their relationship to the thrifty and equalizing conditions. The solution to these equations is k 1 = 2+ ( ) 1 + + ( )2 Ak 0 (19) k 2 = 1 + Ak 1: (20) The value function for this problem is a big mess v 2 (k 0) = log 1 1 + + ( )2 Ak + log 1 1 + + ( )2 1 + + ( )2 A1+ k 2 0 + 2 log 1 + + ( )2 1 + + ( )2 2 A1+ + 2k 3 0! Dynamic Programming Deﬁnition 2.2. h�ěmoǑ����� P8�=�l+vĎag7��#3� Y$f��=ϩ��%Q��wnOO�TW�:UՓr;-���)-C��o|�SN���r�m�w:���|jU7S)�(�Y�Sk�[��z�n;��)��[�>�X*e=_�����}��~�Q��dx�U��+�n�2�RK}�NUz���|Yu�j�E���o/~���ﯞ�������ӯ.~��{���wO�}�˯~����s�if����/>��Z���d���|���LQ�*O��~�r�?�X�����O_^���S������_���,���?�xu�]������������.�}w�����O������'/�_���'�=��կ.���?>��A�O�����c~�1/>{��۫�SJ�S�����_=���R�t��**>(m������/O͂������dɁ[,�Jk�o~~�Ó�?}��gO�? Discrete time: stochastic models: 8-9: Stochastic dynamic programming. By avoiding the solution of the dynamic programming (DP) problem, these methods facilitate the estimation of speci–cations with larger state spaces and richer sources of individual speci–c heterogeneity. tion for this dynamic optimization problem. 2 0 obj Dynamic programming is both a mathematical optimization method and a computer programming method. the saddle-point Bellman equation satisfy the Euler equations. DP characterizes the optimal solution of the optimal control problem using a functional equation, known as the dynamic programming equation (see [1–4]). Solving Euler Equations: Classical Methods and the C1 Contraction Mapping ... restricted to the dynamic programming problem, the algorithm given in (3) is the same as the Bellman iteration method. In Section 4 we take a brief look at \envelope inequalities" and \Euler … consumption, capital, and productivity level, respectively, β∈ (0 1), δ∈ (0 1],and. saves programming efforts, reduces computational burden, and increases the ac-curacy of solutions. Most are single agent problems that take the activities of other agents as given. Euler equations are the ﬁrst-order inter-temporalnecessary conditions for optimal solutions and, under standard concavity-convexity assumptions, they are also sufﬁcient conditions, provided that a transversality condition holds. Lecture 7 . This is the Euler equation, which tells is that marginal utility grows at rate ˆ r. 3Intuition: going along the optimal path of a value function in the space pt;aqshould always give the left-hand-side of the Euler equation 5 A measurable function λ: X → R is said to be a solution to the optimal equation OE if it satisﬁes λ x sup a∈A Xx r x,a α λ y Q dy|x,a, 2.4 x∈X. x^2 – D*(y^2) = N Where D = 661 and N = 1, 2, 3. This chapter introduces basic ideas and methods of dynamic programming.1 It sets out the basic elements of a recursive optimization problem, describes the functional equation (the Bellman equation), presents three methods for solving the Bellman equation, and gives the Benveniste-Scheinkman formula for the derivative of the op-timal value function. Below are examples that show how to solve differential equations with (1) GEKKO Python, (2) Euler's method, (3) the ODEINT function from Scipy.Integrate. In addition, under differentiability and interiority of solution hypotheses the optimal policy function must satisfy the stochastic Euler equation: Interpret this equation™s eco-nomics. Later chapters consider the DPE in a more general set-ting, and discuss its use in solving dynamic problems. Generally, one uses approximation and/or numerical methods to solve dynamic programming problems. MATLAB codes are provided. The general form of Euler equation is: () () () For our problem, () (1.4) Suppose we have a guess on the policy function for consumption (), (1.5) and the policy function for ̃() (1.6) Though in this example ̃() seems trivial, since the budget constraint (1.1) requires ̃() (). We consider a stochastic, non-concave dynamic programming problem admitting interior solutions and prove, under mild conditions, that the expected value function is differentiable along optimal paths. The equation for the optimal policy is referred to as the Bellman optimality equation : Euler equations are the ﬁrst-order inter-temporal necessary conditions for optimal solutions and, under standard concavity-convexity assumptions, they are also sufﬁcient conditions, provided that a transversality condition holds. Based on the problem description for Problem 66 of Project Euler I thought we had left the continued fractions for a while. Euler equations. Chapter 5 – Euler’s equation 41 From Euler’s equation one has dp dz = −ρ 0g ⇒ p(z) = p 0−ρgz. In this paper, it will be shown that the functional equation approach yields, in simple and intuitive fashion, formal derivations of such classical necessary conditions of the Calculus of Variations as the Euler-Lagrange Lecture Notes on Dynamic Programming Economics 200E, Professor Bergin, Spring 1998 Adapted from lecture notes of Kevin Salyer and from Stokey, Lucas and Prescott (1989) Outline 1) A Typical Problem 2) A Deterministic Finite Horizon Problem 2.1) Finding necessary conditions 2.2) A special case 2.3) Recursive solution t+1g1 t=0. Dynamic programming (Chow and Tsitsiklis, 1991). 1.3.1. Notice how we did not need to worry about decisions from time =1onwards. stream 3 0 obj In the in–nite horizon problem we have the same Euler equations, but an in–nite number of them. Euler Equation: −1 +1= h −1 +1 i 3.2 Firms: labor and capital demands Using the fact that the production function is homogenous of degree one (con-stant return to scale), we can ﬁrst remove the trend Γandthendeﬁne ( )= ... To do dynamic programming you need to choose a grid for the capital stock, say First, the Euler conditions admit an in-tertemporal arbitrage interpretation that help the analyst understand and explain the essential features of the optimized dynamic economic process. Also, note that this is the semi-implicit Euler method, meaning that in our second equation, we’re using the most recent θ_1 (t) that we calculated rather than θ_1 (t_0 ) as a straight application of the Taylor Series Expansion would warrant. Section 3 introduces the Euler equation and the transversality condition, and then explains their relationship ⁄Research supported in part by the National Science Foundation, under Grant NSF-DMS-06-01774. We show that by evaluating the Euler equation in a steady state, and using the condition for The idea is to simply store the results of subproblems, so that we do not have to … Use consump-tion functions, { ( )}40 =1, and the dynamic budget constraint, +1 = ( − )+ e +1 Estimate linearized Euler Equation regression, using simulated panel data. Lecture 1 . Created Date: ���h�a;�G���a$Q'@���r�^pT��΀�W8�"���&kwwn����J{˫o��Y��},��|��q�;�mk�v�o�4�[���=k� L��7R��e�]u���9�~�Δp�g�^R&�{�O��27=,��~�F[j�������=����p�Xl6�{��,x�l�Jtr�qt�;Os��11Ǖ�z���R+i��ظ�6h�Zj)���-�#�_�e�_G�p5�%���4C� 0$�Y\��E5�=��#��ڬ�J�D79g������������R��Ƃjîբ�AAҢ؆*�G�Z��/�1�O�+ԃ �M��[�-20��EyÃ:[��)$zERZEA���2^>��#!df�v{����E��%�~9�3M�C�eD��g����. The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimisation problems. ... problems and costs of the form of equation (2) are referred to as Bolza problems. 31. Lecture 5 . I took a different approach that boiled down to an interactive dynamic programming style solution of sorts.$\begingroup$Wikipedia does mention Dynamic Programming as an alternative to Calculus of Variations. %PDF-1.6 %���� Second, I briefly discuss various ways of solving the Euler equation, and to which extent time iteration carries some advantages over alternative approaches. (a) The one-step reward function is nonpositive, upper semicontinuous (u.s.c), and sup-compact on . utility and production functions, respectively, both of which are strictly increasing, con-. Dynamic model, precomputation, numerical integration, dynamic programming, value function iteration, Bellman equation, Euler equation, enve-lope condition method, endogenous grid method, Aiyagari model. Use the transition equation to replace c V(k) = max k0 ln(k k0) + V(k0): The rst order condition and the envelope condition 1 c + V0(k0) = 0 V0(k) = 1 c k 1!V0(k0) = 1 c0 k 0 1 Euler equation, same as one can get from Hamiltonian: c0 c = k0 1. This process is experimental and the keywords may be updated as the learning algorithm improves. general class of dynamic programming models. The dynamic programming solution consists of solving the functional equation S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t) where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and Deterministic dynamics. ��jQ�ګ�M�Ee�� �p=k�&R���st���Y=Y�Nyc���R�j�+Z�:}CH66�9�v�1��(Ah\��}E�K�&�y�J!X�u�ݽ�i˂�U%;��k'X�����9pW�)�G�j��\��v{�}!k�Q^㹎�{���ډ.��9d�����]���4�նh��d�k۴E�.�ґt#�H�{��ue7�$0_Y#����c6s�� _�}�>?��f�E�Q4�=���.C��ǃ��B�u���=l���m�\Tv�$v�b�A]&� M���0�w�v�V;����j{�m. find a geodesic curve on your computer) the algorithm you use involves some type … A measurable function is said to be a solution to the optimal equation (OE) if it satisfies . y(0) = 1 and we are trying to evaluate this differential equation at y = 1. This is an example of the Bellman optimality principle.Itis suﬃcient to optimise today conditional on future behaviour being optimal. ... Lagrange laid the foundations of mechanics in a variational setting culminating in the Euler Lagrange equations. Later we will look at full equilibrium problems. <> Indeed, deﬁne the following sequence of functions: v n(x)= max {y;(x,y)∈A} First, the Euler conditions admit an in-tertemporal arbitrage interpretation that help the analyst understand and explain the essential features of the optimized dynamic economic process. Euler equations are the ﬁrst-order inter-temporalnecessary conditions for optimal solutions and, under standard concavity-convexity assumptions, they are also sufﬁcient conditions, provided that a transversality condition holds. Partial Differential Equation Dynamic Programming Euler Equation Variational Problem Nonlinear Partial Differential Equation These keywords were added by machine and not by the authors. The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimisation problems. 23. Nonstationary models. A measurable function λ: X → R is said to be a solution to the optimal equation OE if it satisﬁes λ x sup a∈A Xx r x,a α λ y Q dy|x,a, 2.4 x∈X. tinuously differentiable, and concave.$\begingroup$Wikipedia does mention Dynamic Programming as an alternative to Calculus of Variations. %���� Euler Equation Based Policy Function Iteration Hang Qian Iowa State University Developed by Coleman (1990), Baxter, Crucini and Rouwenhorst (1990), policy function Iteration on the basis of FOCs is one of the effective ways to solve dynamic programming problems. Hence the pressure increases linearly with depth (z < 0). Lecture 9 2. via Dynamic Programming (making use of the Principle of Optimality). ( (kt) + kt) which one ought to recognize as the discrete version of the "Euler Equation", so familiar in dynamic optimization and macroeconomics. Definition 2.2. �0bH|�NZL�pc:�\T��ɢ"�( �e endstream endobj 96 0 obj <> endobj 97 0 obj <>/Font<>/ProcSet[/PDF/Text]>>/Rotate 0/Type/Page>> endobj 98 0 obj <>stream Applying the Algorithm After deciding initialization and discretization, we still need to imple-ment each step: ... We can use errors in Euler equation to re ne grid. 4 0 obj 2.1. 95 0 obj <> endobj 125 0 obj <>/Filter/FlateDecode/ID[<24899409676246DD9B3FB71F4A731649>]/Index[95 66]/Info 94 0 R/Length 128/Prev 146192/Root 96 0 R/Size 161/Type/XRef/W[1 2 1]>>stream The area of an isosceles triangle is (b/4)(4a^2-b^2)^0.5 where b is the length of the base and a is the length of the two equal sides. Stochastic dynamics. Assumption 2.3. endobj Dynamic programming is an approach to optimization that deals with these issues. Lecture 1: Introduction to Dynamic Programming Xin Yi January 5, 2019 1. We make this subtle substitution because, without it, our model would diverge. 1. Lecture 3 . The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. find a geodesic curve on your computer) the algorithm you use involves some type … <> 0(1) so we can conclude 0(0)= (+1) and we have derived the Euler equation using the dynamic programming method. Dynamic Programming Deﬁnition 2.2. The task at hand is to ﬁnd a path, which con-nects adjacent numbers from top to bottom of a triangle, with the largest sum. Dynamic Programming is mainly an optimization over plain recursion. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. On this slide we have two versions of the Euler Equations which describe how the velocity, pressure and density of a moving fluid are related. Later chapters consider the DPE in a more general set-ting, and discuss its use in solving dynamic problems. Notice how we did not need to worry about decisions from time =1onwards. Stochastic Euler equations. Second, the Euler conditions can, in many instances, be solved more eas-ily than Bellman's equation for the optimal solution of the Markov decision model. Problems. namic programming equation (DPE) as an intermediate step in deriving the Euler equation. Section 3 introduces the Euler equation and the transversality condition, and then explains their relationship ⁄Research supported in part by the National Science Foundation, under Grant NSF-DMS-06-01774. This property allows us to obtain rigorously the Euler equation as a necessary condition of optimality for this class of problems. 2.1. simply because the combination of Euler equations implies: u0(c t)=β 2u0(c t+2) so that the two-period deviation from the candidate solution will not increase utility. 1.2 A Finite Horizon Analog. The ﬂrst author wishes to thank the Mathematics and Statistics Departments of To see the Euler Equation more clearly, perhaps we should take a more familiar example. To this end, I proceed in two steps. %PDF-1.5 I will illustrate the approach using the –nite horizon problem. }^.u'|sz�����A���|8d�\R��U]�4���Į-nd����A�1\�|�}K�C;~�o����w�1$����Oa'ތҪ@�D|��� ��E\b��g>]ᛜ���w0|4���V���S�n�W@L#���}q�*%x�L|�� Project Euler 66: Investigate the Diophantine equation x^2 − Dy^2 = 1. Using Euler equations approach (SLP pp 97-99) show that the transver-sality condition for our problem is lim t >1 0tu(c t)k t+1 = 0 Enumerate the equations that express the dynamic system for this problem along with its initial/terminal conditions. We lose the end condition k T+1 = 0, and it™s not obvious what it™s replaced by, if anything. Euler equation; (EE) where the last equality comes from (FOC). Dynamic Programming More theory Consumption-savings Euler equation with Dynamic Programming Back to normal situation: u is bounded and increasing Euler equation can be useful even if we do not solve the problem fully Can we obtain it without a Lagrangian? Solving dynamic models with inequality constraints poses a challenging problem for two major reasons: dynamic programming techniques are reliable but often slow, whereas Euler equation‐based methods are faster but have problematic or unknown convergence properties. _Rry��; }U&*e�\f\����BcU��㽝7-�$�m�_��4oz������efR��6��h0�E�Mx1������ec�0� 3D�::�LJP6PB�@v �aR��B��뀝��ǲp�� �YN� }�B8ET�aܮ��;��#)5�tÕl������t`����SFf�]���E 1 The Basics of Dynamic Optimization The Euler equation is the basic necessary condition for optimization in dy-namic problems. 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