Nonlinear and evolutionary phenomena in deterministic growing economies

Abstract

We discuss the implications of nonlinearity in competitive models of optimal endogenous growth. Departing from a simple representative agent setup with convex risk premium and investment adjustment costs, we define an open economy dynamic optimization problem and show that the optimal control solution is given by an autonomous nonlinear vector field in <3 with multiple equilibria and no optimal stable solutions. We give a thorough analytical and numerical analysis of this system qualitative dynamics and show the existence of local singularities, such as fold (saddle-node), Hopf and Fold-Hopf bifurcations of equilibria. Finally, we discuss the policy implications of global nonlinear phenomena. We focus on dynamic scenarios arising in the vicinity of Fold-Hopf bifurcations and demonstrate the existence of global dynamic phenomena arising from the complex organization of the invariant manifolds of this system. We then consider this setup in a non-cooperative differential game environment, where asymmetric players choose open loop no feedback strategies and dynamics are coupled by an aggregate risk premium mechanism. When only convex risk premium is considered, we show that these games have a specific state-separability property, where players have optimal, but naive, beliefs about the evolution of the state of the game. We argue that the existence of optimal beliefs in this fashion, provides a unique framework to study the implications of the self-confirming equilibrium (SCE) hypothesis in a dynamic game setup. We propose to answer the following question. Are players able to concur on a SCE, where their expectations are self-fulfilling? To evaluate this hypothesis we consider a simple conjecture. If beliefs bound the state-space of the game asymptotically and strategies are Lipschitz continuous, then it is possible to describe SCE solutions and evaluate the qualitative properties of equilibrium. If strategies are not smooth, which is likely in environments where belief-based solutions require players to learn a SCE, then asymptotic dynamics can be evaluated numerically as a Hidden Markov Model (HMM). We discuss this topic for a class of games where players lack the relevant information to pursue their optimal strategies and have to base their decisions on subjective beliefs. We set up one of the games proposed as a multi-objective optimization problem under uncertainty and evaluate its asymptotic solution as a multi-criteria HMM.We show that under a simple linear learning regime there is convergence to a SCE and portray strong emergence phenomena as a result of persistent uncertainty