Note on positive lower bound of capital in the stochastic growth model

Sustained positive consumption in a model of stochastic growth: The role of risk aversion

“…Further, the requirement that the worst possible realization of the production technology is infinitely productive at zero rules out economies where productivity may not be high under bad realizations of the technology shock. In a more recent contribution, Chatterjee and Shukayev (2008) weaken the productivity requirement by requiring that the lowest possible marginal productivity at zero exceed the discount rate; they show that if, in addition, the utility function is bounded below, then the economy is bounded away from zero with probability one (though the economy may not necessarily grow near zero under the worst realization of the productivity shock and, in particular, the almost sure lower bound on long run consumption may depend on initial condition). The restriction that the utility function is bounded below rules out a large class of CES utility functions that are widely used in the macroeconomic growth literature 7 (and for some of which, optimal paths have been independently shown to be bounded away from zero for all δ ∈ (0, 1) and for particular choices of the production technology).…”

“…As for the second question relating to sufficient conditions for sustaining positive consumption in the long run, existing models of stochastic growth make strong assumptions to ensure that the limiting distribution of capital is bounded away from zero. 5 Brock and Mirman (1972) and Mirman and Zilcha (1975) impose two conditions that ensure expansion of capital and consumption near zero even under the worst realization of the stochastic technology. The two conditions are as follows: marginal productivity at zero is infinite for all realizations of the random shock and there is a strictly positive probability mass on the "worst" realization of the technology 6 .…”

“…4 See, Mirman and Zilcha (1975). 5 A number of papers impose conditions on endogenously determined elements such as the optimal policy function, or the stochastic process of optimal capital stocks. For example, Boylan (1979) imposes conditions that include a requirement that for each realization of the shock, the optimal capital stock next period is concave in the current stock.…”

On Lipschitz continuity of the iterated function system in a stochastic optimal growth model

“…This, however, still does not ensure that there is a compact interval, that excludes the zero stock, in which the support of the invariant distribution of output must lie. Further, given y > 0, Chatterjee and Shukayev (2008) do not provide a lower bound y , dependent on y and the primitives of the model.…”

“…In a recent paper, Chatterjee and Shukayev (2008) allow for the general framework of Brock-Mirman, and show that if the utility function is bounded (below), then given any initial output level, y > 0, there exists y ∈ (0, y), such that the Markov process generated by the IFS, which starts from y will stay in (y , ∞) for all time. In other words, the possibility of a map of the IFS lying entirely below the 45 • line (for all positive output levels), for some realization of the random shock, can be ruled out.…”