Equilibrium cycling with small discounting (Q757217)

For a so-called Cobb-Douglas three sector technology \[ y_ i=b_ i\prod^{2}_{j=0}k_{ji}^{\alpha_{ji}},\quad \sum^{2}_{j=0}\alpha_{ji}=1,\quad i=0,1,2, \] under the natural assumptions \(\sum^{2}_{i=0}k_{0i}=1\) and \(\sum^{2}_{i=0}k_{ij}=k_ j\) the optimal growth problem \[ (1)\quad Max\quad \int^{\infty}_{0}T(y_ 1,y_ 2,k_ 1,k_ 2)e^{-(r-g)t}dt,\quad \dot k_ i=y_ i-gk_ i,\quad i=1,2 \] is considered. Here r-g is the discount rate and T defines the efficiency frontier. Using the Pontryagin maximum principle the problem (1) is reduced to a system \[ (2)\quad \dot k=y(k,p)-gk,\quad k=(k_ 1,k_ 2),\quad \dot p=-w(k,p)+rp,\quad y=(y_ 1,y_ 2). \] Then with the help of the Hopf bifurcation theorem one can show that for any \(r>g\) there exists a Cobb-Douglas technology such that the equilibrium cycles in the system (2). This result refutes the opinion of an impossibility of persistent cyclic behavior in optimal growth models with small but positive discount rates.