A growth model with regime switch

I have a very general question. I am reading this paper :

http://www.webmeets.com/files/papers/eaere/2015/177/Discounting-HelsinkiBlind.pdf

There is a catastrophic event probability and after the catastrophic event the consumption level is reduced to zero. However, authors are making a steady state analysis prior to the catastrophic event. The catastrophic event probability is $h\left(X\right)$ (let s say that it follows a Poisson process.) $X$ stands for the pollution for example.

Let $T$ be the event occurrence time and denote $F\left(t\right)=Pr\left\{ T\leq t\right\}$ and $f\left(t\right)=F^{'}\left(t\right)$ as the corresponding probability distribution and density functions respectively.

$$h\left(S\left(t\right)\right)\Delta=\frac{f\left(t\right)\Delta}{1-F\left(t\right)}=-\frac{d\left[ln\left(1-F\left(t\right)\right)\right]}{dt}$$

where $\Delta$ is an infinitesimal time interval. The term $h\left(S\left(t\right)\right)\Delta$ specifies the conditional probability that an abrupt event will occur between $\left[t,t+\Delta\right]$.

My question is that : Due to this specification, the probability distribution function will be equal to 1 when $t$ tends to $\infty$. Then, in this case, for sure that catastrophic event will occur at the long run.

So, how is it possible to discuss about a steady state with catastrophic events ? At some moment in time, the economy will change regime with the catastrophic event and this steady state will not be a "permanent" one. How can it be possible to justify this ?

• Infinity is always larger than any "some point in time". Still, I agree that after sufficiently many time periods, the probability of the catastrophic event will be very high. But if these "sufficiently many" time periods are "very many", then a steady-state analysis is reasonable - after all, eventually the earth will be destroyed by the sun for sure. Dec 13, 2017 at 20:44
• @AlecosPapadopoulos I think I found a technical explanation for that. In fact, the objective function in this kind of models is expressed by expectation terms, which means that the model is a stochastic model. However, by taking the hazard function as another state variable, it is possible to express the "deterministic equivalent" of the model. (this is firstly introduced by Kamien and Schwarz, 1971). Then, catastrophe probability takes the role of a discount factor which is $\rho + h(S)$ (lets say adjusted discount). Indeed, it is the same idea with endogenous discount models. Dec 14, 2017 at 11:07
• And if it is possible to express the model as a deterministic problem, then it is legitimate to make a steady state analysis Dec 14, 2017 at 11:08

1 Answer

Continuing on the comments exchange, the conclusion that the overall effect is to change the discount factor, is similar to the one reached in the original Overlapping Generations Model of Blanchard, where individuals are faced with a "probability of death" (surely a "catastrophic event" I believe).

See the Blanchard & Fischer book "Lectures in Macroeconomics", p. 117. The authors note that the result has been originally obtained by Cass and Yaari (1967) "Individual Saving, Aggregate Capital Accumulation, and Efficient Growth", a paper that is part of the book Essays on The Theory of of Optimal Economic Growth

• Thanks Alecos ! I also thought of it after making the comment. There is a same kind of model in continuous time OLG model of Calvo and Obstfeld (1988, Econometrica) Dec 14, 2017 at 14:22
• @optimalcontrol You're welcome. Dec 14, 2017 at 14:23