There is in fact another way to solve for the optimal policy, based on the socalled euler equation. We construct a set of conditions under which cakeeating problems have no optimal solution. The lagrangian method problem involves more than one coordinate, as most problems do, we just have to apply eq. Although we already have a complete solution, now is a good time to study the euler equation. Consider the following \ cake eating problem, where a consumer decides how to allocate a xed amount of total consumption. Eulerequation numericalmethods econometricmethods applications mact1 20032004. It is important to appreciate that the numerical solution to a differential equation is only an approximation to the actual solution. Is the estimate found in question 1 likely to be too large or too small. Instantaneous utility derived from eating cake is given by the func tion uct and the.
Eulers method for solving a di erential equation approximately math 320 department of mathematics, uw madison. Suppose the tank initially holds 2% a and 98% b, x0 0. This is the euler equation for optimal consumption smoothing in the nonstochastic growth model, analogous to the euler equation in the cakeeating example of the previous lecture. The transition equation describes the evolution of the vector of state variables.
The cakeeating problem under finite time horizon in this problem, time is discrete and denoted by t, t 0, 1. Macroeconomic theory fall 2004 1 the cakeeating problem a bellmans equation is. At time t current profits net of user costs of capital are given by. The aim of this lecture is to solve the problem using numerical methods. This is the familiar euler equation, equating the net present value of marginal. Of crucial importance for the remainder of this course is that. The project attracts adults and students interested in mathematics and computer programming. Eulers method for solving a di erential equation approximately math 320 department of mathematics, uw madison february 28, 2011 math 320 di eqs and eulers method. Introduction to dynamic programming lecture notes klaus neusser.
F k 0the economy allows for a steady state for capital and consumption. Eulers three body problem department of mathematics. The consumer starts with a certain amount of capital, and eats it over time. We construct a set of conditions under which cake eating problems have no optimal solution.
As long as the problem is finite, the fact that the euler equation holds across all adjacent. Eulers threebody problem is to describe the motion of a particle under the influence of two centers that attract the particle with central forces that decrease with distance as an inversesquare law, such as newtonian gravity or coulombs law. Pdf dynamic economics quantitative methods and applications. The cakeeating problem the cakeeating problem in discrete. Furthermore, we explore how agents choose their consumption path when an optimization problem has no solution, and suggest a. Macroeconomic theory fall 2004 1 the cake eating problem a bellmans equation is. Hence the optimal solution will satisfy the euler equation for each period and the agent will consume the entire cake. The euler equation in the discussion above we have provided a complete solution to the cake eating problem in the case of crra utility. Llnear 1 eat cake now before get low and stuck there. Multiplechoice test eulers method ordinary differential. Consider the following problem of a firm under uncertainty in discrete time. Let the solution to this problem be denoted by vtw1 where t is the horizon of the problem and w1 is the initial size of the cake. Eulers method can be derived by using the first two terms of the taylor series of writing the value of.
Macroeconomics sequence, block i the optimal model of. If you would like to tackle the 10 most recently published problems then go to recent problems. Using your code, answer the following questions how does the state contingent asset value depend on the. We will use guess and verify method to solve this equation. Suppose that the policy function has the following form. Click the descriptiontitle of the problem to view details and submit your answer. He cuts a piece forming a circular sector of 60 degrees and flips the piece upside down, with the icing on the bottom. To solve a homogeneous cauchy euler equation we set yxr and solve for r. The boltzmann equation, which is a good approximation for the evolution of rare gases, provides a useful tool to test these ideas in mathematically controllable situations such as the euler and. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Integrate the problem using the modified euler method. You can solve numerical problems without necessarily having to write a long program. These conditions are known as euler equations or eulerlagrange condi. In the discussion above we have provided a complete solution to the cake eating problem in the case of crra utility.
Do all the short questions and choose 2 out of the 3 longer questions do not turn in answers to ore than 2 nf the longer question. It is a matrixbased system for scienti c calculations. Tse master 2 macroeconomics i problem set 2 lan lan 1 cakeeating problem 1. A cake eating example to x ideas consider the usage of a depletable resource cakeeating max t a t0 btuct, s. Abstract this paper contains a rigorous mathematical example of direct derivation of the system of euler hydrodynamic equations from hamiltonian equations for n point particle system as n direct means that the following standard tools are not used in the proof. Macroeconomic theory fall 2004 1 the cake eating problem consider the optimal growth problem discrete time where.
Clearly, if we can find a method for numerically solving this problem, then we should have little difficulty generalizing it to deal with a system of simultaneous firstorder o. An optimal cake eating problem consider a consumer who has the following preferences over the consumption of cake. The problem faced by the central planner is how to exploit this oil stock in n periods, where n is a positive integer. Please write your answer to the shorter qnestions in the space provided and use your blue book to answer the 2 longer problems. The problems archives table shows problems 1 to 705. An economy has an oil stock of size x 0 at the beginning of period 0. Problem 566 adam plays the following game with his birthday cake. Macroeconomics sequence, block i the optimal model of growth. As a simple example, consider the following cake eating problem. Furthermore, we explore how agents choose their consumption path when an optimization problem has no solution, and suggest a method for ranking divergent utility streams. It is apparent that the specification of the cakeeating problem is incomplete. The cake eating problem under finite time horizon in this problem, time is discrete and denoted by t, t 0, 1.
Follow the steps below to solve the problem described above. Consider the following \cakeeating problem, where a consumer decides how to allocate a xed amount of total consumption. Solve the problem with the help of the modified euler method. A roys identity similar to the static form can be obtained by considering the cakeeating problem by assuming that income m t in the budget constraint 2 in.
Macroeconomic theory fall 2004 1 the cakeeating problem consider the optimal growth problem discrete time where. The cake eating problem there is a cake whose size at time is wt and a consumer wants to eat in t periods. Getting started with matlab jerome adda february 4, 2003 contents. The bellman equation is labeled in two different files. Examples of eulers problem include a planet moving in the gravitational field of two stars, or an electron moving in the. In physics and astronomy, eulers threebody problem, named after leonhard euler, is to solve for the motion of a test mass that is free to move in the presence of the gravitational field of a primary and secondary mass which are fixed in space. This applies to the other problems in this lab as well. Louisiana tech university, college of engineering and science cauchy euler equations. The euler equation from the resulting model would imply two different relationships between investment and its determinants depending on whether the constraint is binding or not. Transversality conditions and dynamic economic behavior. At first this might appear unnecessary, since we already obtained the optimal policy analytically.
Each equation may very well involve many of the coordinates see the example below, where both equations involve both x and. Forthe cake eating example, is the intertemporal budget constraint. Getting started with matlab jerome adda february 4, 2003 contents 1 introduction 2 2 some basic features 2. The corresponding euler polygon for this estimation is euler polygon and actual integral curve for question 1. This problem is the simplest threebody problem that retains physical significance.
Y 0, 1 is a transition function if q yt, y is a pdf. Since its creation in 2001 by colin hughes, project euler has gained notability and popularity worldwide. I equivalent to the cakeeating problem, as q ox o is given at 0. However, the cake eating problem is too simple to be useful without modifications, and once we start modifying the problem, numerical methods become essential. We recommend reading the entire problem before beginning to work on it, as many questions may be addressed further on. If we substitute the solution of the eulerlagrange equation we get. The euler equation for the above optimization problem is given by. Because of the simplicity of both the problem and the method, the related theory is.
Macroeconomics 14 453 ivh werning october zlst, 2003 you have 2 hours to complete the exam. The cake eating problem there is a cake whose size at time is wt and a consumer wants to eat in. We will obtain as many equations as there are coordinates. An optimal cakeeating problem consider a consumer who has the following preferences over the consumption of cake. This is the implication of the euler equation for this problem. Forthe cakeeating example, is the intertemporal budget constraint. The idea is similar to that for homogeneous linear differential equations with constant coef. Eulers method for ordinary differential equationsmore examples chemical engineering example 1 the concentration of salt x in a home made soap maker is given as a function of time by x dt dx 37. Multigrid algorithms old tradition in numerical analysis. The optimal solution is characterized by the euler equation 1 1 1, 1 t t t uc r uc. In fact, a straight line from point a to line l can be arbitrarily long, so that even very bad choices satisfy the euler equation.
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