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Comments on: Infinite time and economics http://www.tvhe.co.nz/2007/09/26/infinite-time-and-economics/ The Visible Hand in Economics Thu, 27 Sep 2007 06:48:42 +0000 hourly 1 https://wordpress.org/?v=6.9.4 By: Kimble http://www.tvhe.co.nz/2007/09/26/infinite-time-and-economics/#comment-365 Thu, 27 Sep 2007 06:48:42 +0000 http://tvhe.wordpress.com/2007/09/26/infinite-time-and-economics/#comment-365 The feel good comment of the year.

]]> By: rauparaha http://www.tvhe.co.nz/2007/09/26/infinite-time-and-economics/#comment-364 Thu, 27 Sep 2007 05:30:44 +0000 http://tvhe.wordpress.com/2007/09/26/infinite-time-and-economics/#comment-364 I asked Jacek and he said that you’re right. A game with a probability of dying in each period can be rewritten as an infinite horizon game with a discount rate incorporating the probability of death. Given that, it would be able to support trigger strategies although, as you say, there would be a reduced number of available punishment strategies with a higher discount rate.

]]> By: Matt Nolan http://www.tvhe.co.nz/2007/09/26/infinite-time-and-economics/#comment-363 Thu, 27 Sep 2007 00:29:51 +0000 http://tvhe.wordpress.com/2007/09/26/infinite-time-and-economics/#comment-363 Interesting. I’d like to ask a few question, as you know dynamics is definitely not my strong point 😉

Now the multiplicity of trigger strategies depends on the discount factor right? So the more you discount the future, then there will be a smaller number of credible trigger strategies in the game. I never defined what the probability of dieing looked like, however can’t we just say that the associated probability of dieing in different periods is like having a time dependent discount factor (note that for this to be an infinite horizon game there must be some non-negative probability of the individual never dieing, that is why I said it is more appropriate for a firm than an individual).

For example suppose the probability of dieing is history independent and constant. In fact say that in each period there is a 10% chance of dieing. Now if the payoff when you die is zero, then even if you do not discount the value of the future based on interest rates or a preference for current consumption, you will still value next periods game only 90% as much as you value this periods game. When you set up your bellman value function, isn’t this the same as having a discount factor of 0.9 out in front of the recursive term?

Now I’m not saying that this is what the probability of dieing does look like, it will differ depending on the object and depending on beliefs and historical knowledge. However, changing the functional form of the probability of dieing is the same as changing the functional form of the discount factor.

So as a result, the probability of dieing will only influence trigger strategies in the same way as a discount factor. In fact, the probability of dieing should be one of the factors that defines the value of the discount factor.

Finally, I know that the statement that an infinite horizon game is an unbounded one is completely not controversial. However, I still thought it was an important point because many people get confused. I’ve heard plenty of people say infinite horizon games are stupid because nothing goes forever, I just wanted to clear up the fact that the game doesn’t have to go forever for the ex ante choice to be made under the guise of an infinite horizon framework.

]]> By: rauparaha http://www.tvhe.co.nz/2007/09/26/infinite-time-and-economics/#comment-362 Wed, 26 Sep 2007 06:42:47 +0000 http://tvhe.wordpress.com/2007/09/26/infinite-time-and-economics/#comment-362 Woohoo, I get to nit-pick back again 😉 I slightly disagree with you about an infinite horizon being the same as uncertainty about when games will end. Having an infinite horizon allows an infinite number of trigger strategy equilibria to exist. Even a small probability of the game ending will cause some of those equilibria to collapse since no punishment strategy will be able to support them. As the probability of the game ending rises, fewer of the equilibria will be sustainable. So the solution set of an infinite game is likely to be different from the solution set for a game where there is some probability of the game ending in each period. I think that conflating the two situations is misleading.

I do agree that infinite games are ones in which the time frame of the game is unbounded though; although, since that really just draws on the definition of infinity it’s a pretty uncontroversial statement.

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