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