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I can't decide between using value function iteration or projection on chebyshev polynomials.

I'm inclined to use projection, however I need to compute (for welfare analysis) the value of the value function at some specific age.

Using the decision rules computed by projection and derived from FOCS is it possible to compute the expected lifetime discounted utility?

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You could solve for the value function ex-post from the Bellman equation. For simplicity, consider a deterministic dynamic programming problem (in OLG model, you'd have age as another state variable, but I'll abstract from that). If $x$ is state and its next-period value $x'$ is control, $U(x,x')$ is one-period utility and you have somehow obtained decision rule $x' = g(x)$, it must be the case that the value function satisfies

$$ V(x) = U(x,g(x)) + \beta V(g(x)) $$

This is a functional equation in $V()$ that could also be solved by projection method. Alternatively, you could iterate on the Bellman equation (on some grid over state space) just like in VFI, but skipping the optimization step (since you already know the optimal choice), and eventually the value function should converge.

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