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docs: v_t defined as function of v_t ? #235

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sbenthall opened this issue Apr 5, 2023 · 4 comments
Open

docs: v_t defined as function of v_t ? #235

sbenthall opened this issue Apr 5, 2023 · 4 comments

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@sbenthall
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sbenthall commented Apr 5, 2023

In the documentation of the (non-separable) value function, the signature is given as:

$v_t = w(s_t,x_t,v_t,s_{t+1},x_{t+1},v_{t+1})$

Note that $v_t$ is repeated on both sides of the equation.
Is that intentional?

See:

https://github.com/EconForge/dolo.py/blob/master/docs/model_specification.md#value
https://dolo.readthedocs.io/en/latest/model_specification.html#value

@albop
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albop commented Apr 5, 2023 via email

@sbenthall
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I'm sorry if this is a dense question, but isn't $v_t = v_t$ always, so if $v_t$ is an argument to $w$, the function is trivial?

@albop
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albop commented Apr 13, 2023

Actually the way I did Epstein-Zin a while ago was by iterating on two variables:

V_t <- U(c_t) + \beta (V_{t+1}/EV_t^{1/alpha})^{-\gamma}
EV_t <- (V_{t+1})^\alpha

In equilibrium, EV_t is of course the same on the left and on the right.

I admit it is a little bit of a corner case ....

In other applications, I used arbitrage equations for the same purpose (treating left and right hand side in the same way) but that is also not super satisfying.

@sbenthall
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Oh, I see. So with some solution algorithms, you might run this to convergence to solve for a single period, before incrementing t.

So there's two update/convergence loops?

Is the solution algorithm for this in the repository?

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