TAILIEUCHUNG - Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 4

Chapter 4 Practical Dynamic Programming . The curse of dimensionality We often encounter problems where it is impossible to attain closed forms for iterating on the Bellman equation. Then we have to adopt some numerical approximations. This chapter describes two popular methods for obtaining numerical approximations. | Chapter 4 Practical Dynamic Programming . The curse of dimensionality We often encounter problems where it is impossible to attain closed forms for iterating on the Bellman equation. Then we have to adopt some numerical approximations. This chapter describes two popular methods for obtaining numerical approximations. The first method replaces the original problem with another problem by forcing the state vector to live on a finite and discrete grid of points then applies discrete-state dynamic programming to this problem. The curse of dimensionality impels us to keep the number of points in the discrete state space small. The second approach uses polynomials to approximate the value function. Judd 1998 is a comprehensive reference about numerical analysis of dynamic economic models and contains many insights about ways to compute dynamic models. . Discretization of state space We introduce the method of discretization of the state space in the context of a particular discrete-state version of an optimal saving problem. An infinitely lived household likes to consume one good which it can acquire by using labor income or accumulated savings. The household has an endowment of labor at time t st that evolves according to an m-state Markov chain with transition matrix P. If the realization of the process at t is S then at time t the household receives labor income of amount wsi. The wage w is fixed over time. We shall sometimes assume that m is 2 and that st takes on value 0 in an unemployed state and 1 in an employed state. In this case w has the interpretation of being the wage of employed workers. The household can choose to hold a single asset in discrete amount at G A where A is a grid ai a2 . an . How the model builder chooses the 93 94 Practical Dynamic Programming end points of the grid A is important as we describe in detail in chapter 17 on incomplete market models. The asset bears a gross rate of return r that is fixed over time. The household s maximum

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