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According to my textbook, the analytical technique for solving a Bellman's Equation is as follows:

  1. Guess a form for $V_0(x)$

  2. Solve the maximization problem with respect to the control and obtain a policy function x′=h0(x)

  3. Update the guess by plugging the policy function such that $V1(x)=F(x,h0(x))+\beta V0(h0(x))$

  4. Repeat the above until $V_{i+1}=V_i$ At this point, the equation is solved.

My question is, why does this technique, specifically the step, $V_{i+1}=V_i $, solve the Bellman equation? Thanks!

michael_fortunato
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  • Your description isn't clear, e.g. you don't define $F, V1, V0$ or $\beta$. But it is an iterative method and $V_{i+1} = V_i$ is the condition for convergence, i.e. if $V$ in the current iteration is the same as $V$ in the previous iteration $V$ will not change in any future iterations. – BDN Jun 15 '18 at 18:23

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