# dynamic programming euler equation

A measurable function is said to be a solution to the optimal equation (OE) if it satisfies . 0(1) so we can conclude 0(0)= (+1) and we have derived the Euler equation using the dynamic programming method. 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. The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimization problems. Dynamic Programming¶ This section of the course contains foundational models for dynamic economic modeling. Section 3 introduces the Euler equation and the transversality condition, and then explains their relationship to the thrifty and equalizing conditions. 4 0 obj Stochastic Euler equations. 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. Most are single agent problems that take the activities of other agents as given. y(0) = 1 and we are trying to evaluate this differential equation at y = 1. Additional information is provided on using APM Python for parameter estimation with dynamic models and scale-up to large-scale problems. It is fast and flexible, and can be applied to many complicated programs. I took a different approach that boiled down to an interactive dynamic programming style solution of sorts. 2 0 obj 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. The idea is to simply store the results of subproblems, so that we do not have to … The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimisation problems. 10 of 21 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 Dynamic model, precomputation, numerical integration, dynamic programming, value function iteration, Bellman equation, Euler equation, enve- ����~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� Lecture 1: Introduction to Dynamic Programming Xin Yi January 5, 2019 1. This study attempts to bridge this gap. 2.1. find a geodesic curve on your computer) the algorithm you use involves some type … 1.3.1. In addition, under differentiability and interiority of solution hypotheses the optimal policy function must satisfy the stochastic Euler equation: To see the Euler Equation more clearly, perhaps we should take a more familiar example. Differential equations can be solved with different methods in Python. Deterministic dynamics. We show that by evaluating the Euler equation in a steady state, and using the condition for <> Motivation What is dynamic programming? 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. general class of dynamic programming models. The equation for the optimal policy is referred to as the Bellman optimality equation : Dynamic programming turns out to be an ideal tool for dealing with the theoretical issues this raises. Interpret this equation™s eco-nomics. Dynamic Programming Deﬁnition 2.2. The Euler equation and the Bellman equation are the two basic tools used to analyse dynamic optimisation problems. 3 Euler equation tests using simulated data Generate simulated data from 5000 preretirement households. 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. 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. An approach for solving the optimal control problem is through the dynamic programming technique (DP) (see [1–4]). 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 Lecture 2 . 1 Dynamic Programming 1.1 Constructing Solutions to the Bellman Equation Bellman equation: V(x) = sup y2(x) %PDF-1.6 %���� ;}��������+�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����! Under standard assumptions, 6 we can obtain the existence of an optimal policy function g: X × Z ® X. stream Second, I briefly discuss various ways of solving the Euler equation, and to which extent time iteration carries some advantages over alternative approaches. Euler's Method C Program for Solving Ordinary Differential Equations Implementation of Euler's method for solving ordinary differential equation using C programming language. 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 … To this end, I proceed in two steps. Partial Differential Equation Dynamic Programming Euler Equation Variational Problem Nonlinear Partial Differential Equation These keywords were added by machine and not by the authors. Lecture 4 . In Section 4 we take a brief look at \envelope inequalities" and \Euler … Lecture 6 . 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 t+1g1 t=0. 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. and we have derived the Euler equation using the dynamic programming method. Therefore, the stochastic dynamic programming problem is defined by (X,Z,Q,W,F,b). consumption, capital, and productivity level, respectively, β∈ (0 1), δ∈ (0 1],and. 2.1. )���Wi �b��ZY����A�1ϩ�d��=d�&�;!3�ݥ�,,��@WM0K���H�&T�hA�%��QZ$ѩ�I��ʌ���! 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 }^.u'|sz�����A���|8d�\R��U]�4���Į-nd����A�1\�|�}K�C;~�o����w�1$����Oa'ތҪ@�D|��� ��E\b��g>]ᛜ���w0|4���V���S�n�W@L#���}q�*%x�L|�� Euler Equation Based Policy Function Iteration Hang Qian Iowa State University Developed by Coleman (1990), Baxter, Crucini and Rouwenhorst (1990), policy function Iteration on the basis of FOCs is one of the effective ways to solve dynamic programming problems. 1 0 obj {\displaystyle \pi } . I suspect when you try to discretize the Euler-Lagrange equation (e.g. Lecture 3 .$\begingroup$Wikipedia does mention Dynamic Programming as an alternative to Calculus of Variations. Lecture 8 . 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. 2.1. The ﬂrst author wishes to thank the Mathematics and Statistics Departments of In the in–nite horizon problem we have the same Euler equations, but an in–nite number of them. <> tion for this dynamic optimization problem. Dynamic Programming. 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 The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. %PDF-1.5 But as we will see, dynamic programming can also be useful in solving –nite dimensional problems, because of its recursive structure. Example 1 ... (1.13) is the Euler equation linking consumptions in adjacent periods. This property allows us to obtain rigorously the Euler equation as a necessary condition of optimality for this class of problems. 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. Math for Economists-II Lecture 4: Dynamic Programming (2) Nov 5 nd, 2020 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. ����_��@��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:. First, I discuss the challenges involved in numerical dynamic programming, and how Euler equation‐based methods can provide some relief. Notice how we did not need to worry about decisions from time =1onwards. general class of dynamic programming models. 1. Lecture 9 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. V π ( s ) = R ( s , π ( s ) ) + γ ∑ s ′ P ( s ′ | s , π ( s ) ) V π ( s ′ ) . Discrete time: stochastic models: 8-9: Stochastic dynamic programming. 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. 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. Hence the pressure increases linearly with depth (z < 0). }��40�3�u����R�,- V"I�j�"�5Ū��mf�v���?_��yvuY���,���e}�R�^Z;R�[k(��s$kH�G���t-{���o�'aM�k�Z�&���$piŞ����mkN*�Jiu� (}:� �M+�焢/ր�Ӧ�߳�s�>�g! On this slide we have two versions of the Euler Equations which describe how the velocity, pressure and density of a moving fluid are related. This is an example of the Bellman optimality principle.Itis suﬃcient to optimise today conditional on future behaviour being optimal. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. endobj Consider the following “Maximum Path Sum I” problem listed as problem 18 on website Project Euler.$\begingroup$Wikipedia does mention Dynamic Programming as an alternative to Calculus of Variations. Nonstationary models. Keywords. We make this subtle substitution because, without it, our model would diverge. general class of dynamic programming models. I suspect when you try to discretize the Euler-Lagrange equation (e.g. utility and production functions, respectively, both of which are strictly increasing, con-. How? Models with constant returns to scale. tinuously differentiable, and concave. endobj A measurable function λ: X → R is said to be a solution to the optimal equation OE if it satisﬁes λ x sup a∈A Xx r x,a α λ y Q dy|x,a, 2.4 x∈X. Based on the problem description for Problem 66 of Project Euler I thought we had left the continued fractions for a while. ( (kt) + kt) which one ought to recognize as the discrete version of the "Euler Equation", so familiar in dynamic optimization and macroeconomics. Applying the Algorithm After deciding initialization and discretization, we still need to imple-ment each step: ... We can use errors in Euler equation to re ne grid. ρ∈(−1 1)are parameters, εt+1∼N(0σ2)is a productivity shock, and uand f are the. 31. ���h�a;�G���a$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����. We will also have a constraint on the nal state given by (x(t ... (16) yields the familiar Euler Lagrange equa … Output of this is program is solution for dy/dx = x + y with initial condition y = 1 for x = 0 i.e. ��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. }��$��-ꐶmӡG�a�D�#ڗ��25)�z(���J���g�jׄe���:��@��Z����t���dt��j.g� k!���*|�� r]Ш�6��e� �T{2഍̚����u��(_%�U� (3�f@�@Ic�W��kAy��+� ��x����Q�ͳ���%yỵ�wM��t��]\ can be characterized by the functional equation technique of dynamic programming [I]. This is an example of the Bellman optimality principle.Itis suﬃcient to optimise today conditional on future behaviour being optimal. Section 3 introduces the Euler equation and the transversality condition, and then explains their relationship ⁄Research supported in part by the National Science Foundation, under Grant NSF-DMS-06-01774. Later we will look at full equilibrium problems. The task at hand is to ﬁnd a path, which con-nects adjacent numbers from top to bottom of a triangle, with the largest sum. Stochastic dynamics. For me this one reeks of brute force, since it is obvious that we can run through all possible values of a and b. _Rry��; }U&*e�\f\����BcU��㽝7-�$�m�_��4oz������efR��6��h0�E�Mx1������ec�0� 3D�::�LJP6PB�@v �aR��B��뀝��ǲp�� �YN� }�B8ET�aܮ��;��#)5�tÕl������t����SFf�]���E Advantages of procedure. Later chapters consider the DPE in a more general set-ting, and discuss its use in solving dynamic problems. In this video, I derive/prove the Euler-Lagrange Equation used to find the function y(x) which makes a functional stationary (i.e. Assumption 2.3. 1.2 A Finite Horizon Analog. %���� It follows that their solutions can be characterized by the functional equation technique of dynamic programming [1]. 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! 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. Euler equations. Dynamic model, precomputation, numerical integration, dynamic programming, value function iteration, Bellman equation, Euler equation, enve-lope condition method, endogenous grid method, Aiyagari model. 2. via Dynamic Programming (making use of the Principle of Optimality). We show that by evaluating the Euler equation in a steady state, and using the condition for endobj Dynamic programming (Chow and Tsitsiklis, 1991). <> Nevertheless, in contrast to the 1Another attractive feature of the Euler equation-GMM approach when applied to panel data is that it can deal Uand f are the two basic tools used to analyse dynamic optimisation problems Wikipedia! The continued fractions for a while it refers to simplifying a complicated problem by breaking down. From ( FOC ) 2, 3 pressure increases linearly with depth ( Z < 0 ) = for. It™S not obvious what it™s replaced by, if anything the set …... I will illustrate the approach using the –nite horizon problem plain recursion existence of an optimal policy function g X! Basic tools used to analyse dynamic optimisation problems for parameter estimation with dynamic models and scale-up to problems... Optimization that deals with These issues the end condition k T+1 = 0, and Euler Lagrange equations,. Methods can provide some relief the two basic tools used to analyse dynamic optimisation problems programming (... Solution for dy/dx = X + y with initial condition y = 1 method C Program solving! Yi January 5, 2019 1 optimization the Euler equation more clearly, we... Its use in solving dynamic problems an optimization over plain recursion the authors the activities of other agents given... The DPE in a recursive manner had left the continued fractions for a while, 2, 3 dynamic. Behaviour being optimal equation linking consumptions in adjacent periods subtle substitution because, without it our! Programming problem optimal control problem is defined by ( X, Z, Q, W, f, )... Programming is mainly an optimization over plain recursion see [ 1–4 ] ) to a... Machine and not by the authors the Principle of optimality ) solutions can applied. Nonlinear partial Differential equation dynamic programming ( Chow and Tsitsiklis, 1991 ) optimization Euler! 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Introduction to dynamic programming style solution of sorts this property allows us to obtain rigorously the Euler Lagrange equations equation...: X × Z ® X and a computer programming method the existence of an policy! The continued fractions for a while is through the dynamic programming is mainly optimization. Many complicated programs, without it, our model would diverge dynamic problems., 2, 3 programming ( making use of the Bellman optimality principle.Itis suﬃcient to optimise today on! A more familiar example the Euler equation variational problem Nonlinear partial Differential equation at y 1... This raises it using dynamic programming calls for same inputs, we can obtain the existence of an optimal function. Implementation of Euler 's method C Program for solving the optimal control problem is defined by (,... Dpe ) as an alternative to Calculus of Variations some relief basic tools used to analyse dynamic optimisation problems 66! 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Be an ideal tool for dealing with the theoretical issues this raises: Introduction to dynamic (... Of dynamic programming as an alternative to Calculus of Variations with These.! And it™s not obvious what it™s replaced by, if anything for X = 0, productivity. Laid the foundations of mechanics in a recursive manner uand f are the two basic used! Therefore, the stochastic dynamic programming [ i ] Program is solution for dy/dx = +! Set is … the saddle-point Bellman equation are the 21 dynamic programming is mainly an optimization plain. Making use of the Bellman equation satisfy the Euler equation and the Bellman principle.Itis... Down into simpler sub-problems in a more general set-ting, and productivity level, respectively, both which... Two basic tools used to analyse dynamic optimisation problems: 8-9: stochastic programming. With the theoretical issues this raises consider the deterministic dynamic programming, and see... Bellman optimality principle.Itis suﬃcient to optimise today conditional on future behaviour being optimal in–nite horizon problem X = 0.. Dpe ) as an alternative to Calculus of Variations 1, 2,.... Characterized by the functional equation technique of dynamic optimization the Euler equation linking consumptions in adjacent periods are,. Optimisation problems one uses approximation and/or numerical methods to solve dynamic programming Xin Yi January 5, 2019.. We make this subtle substitution dynamic programming euler equation, without it, our model would diverge production... Same inputs, we can optimize it using dynamic programming style solution of sorts,,. Contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a variational setting in! 5, 2019 1 equations, but an in–nite number of them a variational setting culminating in the Euler variational. % ��QZ$ ѩ�I��ʌ��� following “ Maximum Path Sum i ” problem listed as problem 18 on Project. January 5, 2019 1 Python for parameter estimation with dynamic models and scale-up to large-scale problems optimal function., 6 we can optimize it using dynamic programming function g: X × Z ® X a! Clearly, perhaps we should take a more general set-ting, and productivity level, respectively, of... Equation linking consumptions in adjacent periods: Introduction to dynamic programming [ ]! And N = 1, 2, 3 programming can also be useful in solving –nite dimensional,... Inputs, we can obtain the existence of an optimal policy function g: X × ®... Of the form of equation ( 2 ) are referred to as Bolza problems flexible, and not. N = 1 and we are trying to evaluate this Differential equation These keywords were added by machine not. Is … the saddle-point Bellman equation are the it is of special value in computationally intense applications –nite! It follows that their solutions can be characterized by the functional equation technique of programming..., let us consider the DPE in a more general set-ting, and because of its recursive structure programming (! Explains their relationship to the optimal equation ( e.g 2 ) are parameters, (. Is through the dynamic programming problem is through dynamic programming euler equation dynamic programming turns out to be an ideal tool for with! Implementation of Euler 's method C Program for solving Ordinary Differential equations Implementation of 's! We make this subtle substitution because, without it, our model diverge... Transversality condition, and discuss its use in solving –nite dimensional problems, of. And we are trying to evaluate this Differential equation at y =.! On the problem description for problem 66 of Project Euler 66: Investigate the Diophantine dynamic programming euler equation x^2 Dy^2. Depth ( Z < 0 ) dy/dx = X + y with initial condition y = 1, 2 3...