Forthe cake eating example, is the intertemporal budget constraint. C h a p t e r 10 analytical hamiltonjacobibellman su. In each period the agent decides to eat the entire cake and receive utility uc or wait. Macroeconomic theory fall 2004 1 the cake eating problem a bellman s equation is. Consider the following \cakeeating problem, where a consumer decides how to allocate a xed amount of total consumption. Using this solution, explain the time paths of c t and k t starting from the given initial condition k 0. Y 0, 1 is a transition function if q yt, y is a pdf. Assuming that no corner solution appears derivation by the control. I when we iterate once more, now tomorrow is the last day on earth.
An optimal cake eating problem consider a consumer who has the following preferences over the consumption of cake. Macroeconomic theory fall 2004 1 the cakeeating problem a bellmans equation is. By our inada conditions, we know these will never bind. A cake eating example here we will look at a very simple dynamic optimization problem. However, the cake eating problem is too simple to be useful without modifications, and once we start modifying the problem, numerical methods become essential. Begin with equation of motion of the state variable. The key to this formulation is that we decide what to do in period t 1 with the assumption that our actions in the remaining periods will be optimal. There is in fact another way to solve for the optimal policy, based on the socalled euler equation. Intuition for vfi i in the iteration period, all future states are the same. It writes the value of a decision problem at a certain point in time in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices. Of crucial importance for the remainder of this course is that. Having bwon the lhs frees the rhs continuation aluev function wfrom being the same function. Richard bellman was an american mathematician in the 20th century who invented dynamic programming in in. The cake eating problem there is a cake whose size at time is wt and a consumer wants to eat in.
Computer code for deterministic cake eating problem clear. The cakeeating problem under infinite time horizon 1. Bellman optimality equation for q the relevant backup diagram. Results are illustrated by an example of a gametheoretic model of. Setup and explain the bellman equation that determines these functions.
The problem faced by the central planner is how to exploit this oil stock in n periods, where n is a positive integer. Ive been looking at the cake eating problem over a finite horizion and have been trying to figure out if we can derive a policy function for such a problem. To begin, we consider yet another variation of the cakeeating problem already analyzed in various guises in chapter 4 see, especially, example 4. Although we already have a complete solution, now is a good time to study the euler equation. Hence if supbwx 6 wx, then keep getting a better closer function to the xed point where. The bellman equation for v has a unique solution corresponding to the.
What does the contraction property imply about lim nv. Write out the bellman equation the above problem can be reexpressed as follows. Gambling game martin branda kpms mff uk 20180518 2 34. I in a \cakeeating example, this means eat everything. Optimal control theory and the linear bellman equation. Using itos lemma, derive continuous time bellman equation. At each point of time, t 1,2,3,t you can consume some of the cake and thus save the remainder. The taste shock, z, may take on only two values, 0 bellman s principle of optimality that is used to solve these problems recursively note. Introduction to dynamic programming lecture notes klaus neusser. Transforming an infinite horizon problem into a dynamic programming one duration. View homework help the cakeeating problem under infinite time horizon from eco 4145 at university of ottawa. At first this might appear unnecessary, since we already obtained the optimal policy analytically. As a simple example, consider the following cake eating problem. The consumer starts with a certain amount of capital, and eats it over time.
We obtain the following euler equation from the two first order conditions 2 1c0. Let us consider a speci c example from economics called the cake eating problem. Empirical implications eitm summer institute 2014 dynamic optimization. Twostage transportation problem content 1 twostage transportation problem 2 dynamic programming and bellman principle 3 example. For convenience, rewrite with constraint substituted into objective function. The bellman equation is labeled in two different files. The main tool we will use to solve the cake eating problem is dynamic programming. Getting started with matlab jerome adda february 4, 2003 contents 1 introduction 2 2 some basic features 2. This principle is at the heart of the dynamic programming technique and is intimately related to the idea of time consistency see kydland and prescott, 1977.
An introduction to dynamic programming in discrete time. An optimal cakeeating problem consider a consumer who has the following preferences over the consumption of cake. 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. The envelope theorem can be derived for the restricted optimization problem. The optimal consumption path satisfies the socalled euler equation. Note that this equation is not linear in s, and therefore solving the equation is nontrivial but well understood. In order to solve this problem, we need certain conditions to be true.
We construct v as a nx2 matrix, containing the value function. Dynamic programming and bellmans principle piermarco. Reinforcement learning derivation from bellman equation. Macroeconomic theory fall 2004 1 the cakeeating problem consider the optimal growth problem discrete time where. Bellman equation, in order to see how the value and policy functions at period 0 for the given inital. In the discussion above we have provided a complete solution to the cake eating problem in the case of crra utility. A bellman equation, named after its discoverer, richard bellman, also known as a dynamic programming equation, is a necessary condition for optimality associated with. The cakeeating problem under finite time horizon in this problem, time is discrete and denoted by t, t 0, 1. Dynamic economics quantitative methods and applications to. In the absence of noise, the optimal control problem in continuous time can be solved in two ways. The transition equation describes the evolution of the vector of state variables. Markov decision processes and bellman equations emo todorov. We assume now that the cake must be eaten in its entirety in one period. Optimal feedback synthesis glossary bibliography biographical sketch summary dynamic programming is a method that provides an optimal feedback synthesis for a control problem by solving a nonlinear partial differential equation, known as the.
Here we will look at a very simple dynamic optimization problem. Forthe cakeeating example, is the intertemporal budget constraint. Use the method of undetermined coefficients to show that the value function takes the linear form vx a b x. An economy has an oil stock of size x 0 at the beginning of period 0. A hamiltonjacobibellmantype equation is derived for finding optimal solutions in differential games with random duration. The cakeeating problem under infinite time horizon the. Outline dynamic optimization 2 university of houston. To verify that this stochastic update equation gives a solution, look at its xed point. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Use of envelope condition and repeated substitution we go back to euler equation 1. The aim of this lecture is to solve the problem using numerical methods. Using blackwells conditions, show that this bellman operator is a contraction mapping. Chapter 1 introduction we will study the two workhorses of modern macro and.
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