Monotonicity and continuity of the critical capital stock in the Dechert-Nishimura model
The authors show that the critical capital stock of the Dechert and Nishimura model [\textit{D. Dechert} and \textit{K. Nishimura}, J. Econ. Theory 31, 332--354 (1983; Zbl 0531.90018)] is a decreasing and continuous function of the discount factor (\(\beta\)). The following characterization of the behavior of the critical capital stock (\(x^c\)) is presented: (i) For high discounting, there is no critical capital stock (\(x^c\)) or nonzero steady state. Hence, all optimal paths converge to zero. (ii) For middle range discounting, \(x^c\) varies continuously and it is strictly decreasing with respect to \(\beta\). (iii) For mild discounting, \(x^c\) is equal to zero implying that every optimal path starting from a nonzero initial state converges to the nonzero steady state. Case (iii) exists only if the boundary value (\(\beta_0\)), given in terms of the primitives of the model, is less than one. If \(\beta_0\) is not smaller than one, then case (ii) covers up until \(\beta=1\). Concerning middle range discounting, it is also shown that \(x^c\) merges to the nonzero steady state when \(\beta\) decreases to a certain boundary value and that \(x^c\) converges to the minimum sustainable capital stock when \(\beta\) increases to \(\min\{\beta_0,1\}\).
scientific article; zbMATH DE number 6000196
scientific article