Discounting and long-run behavior: Global bifurcation analysis of a family of dynamical systems (Q5931251)

In the paper the relationship between the discount rate and nature of asymptotic behavior of dynamic optimization models is studied. The theory is developed under the conditions of independence from starting state, and unique switching.NEWLINENEWLINENEWLINEDynamical system is the pair \((X,g)\), where \(X= [0,1]\) and \(g\) is a map from \(X\) to \(X\). The model is described by a triple \((\Omega,u,\delta)\), where \(\Omega= X\times X\) is transition possibility set; \(u\) is the utility function, supposed to be continuous, concave and monotone; \(\delta\) is the discount factor, \(0<\delta< 1\). An optimal program, value function \(V\) and policy function \(h(x)\) are defined. The dynamical system \((X,h)\) generated by optimization model \((\Omega,u,\delta)\) is studied.NEWLINENEWLINENEWLINEThe family of examples is considered where the theory applied to these examples yields a global bifurcation diagram from which it follows that there is a critical discount factor \(\widehat\delta\), such that for \(\delta>\widehat\delta\) optimal programs exhibit global asymptotic stability to the stationary optimal stock (fixed point of the policy function); and for \(\delta> \widehat\delta\) optimal programs converge to the two-period cycle, that is investigated as well.NEWLINENEWLINENEWLINEThe authors write that the paper ``is best viewed as an exercise in trying to understand the relationship btween a dynamic optimization model\dots and optimal policy function generated by it''.