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I am working in economics and I am trying to build a model that take into account the fact that indivudal can take a decision once in their life time that changes the value of a parameter R.

To be more clearer: Time t is discrete and goes from 0 to T I have a vector of parameters Xt (for example, temperature, population etc.) to whose probablistic law is known (let say that probability of X(t+1) given X(t) is known, but we can imagine other ideas).

The individual is maximizing a value function which is the sum over t of V(Rt, Xt)*a(t) where a decreasing function acting like a discount factor. The maximization is done over R, which you can only change once at some time t < T.

I'd like to find a good way to derive the optimal time and the optimal new value of R.

If someone has seen this kind of problem somewhere I will be more than glad to be redirected to the appropriate literature.

Thanks a lot for your helps

Best

Tochoka

Tochoka
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1 Answers1

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You have a 2D nonlinear minimization problem. The two parameters are the time of change $t$ and the new value of $R$. These problems are cast as minimization, so you want to minimize the negative of your value function. Routines are available in any numerical analysis text. I like the discussion in chapter 10 of Numerical Recipes. You should probably create a number of datasets for the $Xt$ and use the same datasets at each $t,R$ point

Ross Millikan
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  • Thanks, but I really want to go as far as possible without using the numerical tool. Even if this means I have to make som simplifying assumptions on the value function and the parameters evolution. My biggest problem is for t because I can always derive the first order derivative of the value function for R (I can assume without loss of generality that V is concave in R my case, and not monotonic). – Tochoka Jan 24 '16 at 10:57
  • I think that will depend on your functions. Without more information I don't know what to suggest. Maybe you could pick a few versions of $V$ with a simple form (probably with no random variation), do the derivatives analytically, and see some patterns. – Ross Millikan Jan 24 '16 at 15:09