The choice of a growth path under a linear quadratic approximation

Orlando Gomes

doi:10.14195/2183-203x_22_4

Abstract

There is a recent strand of literature which suggests that second order approximations of linear quadratic objective functions in the steady state vicinity, namely when assuming stochastic scenarios, lead to very interesting and useful results. For example, applications in monetary policy resort to such technique. In this paper we find that, for a specific optimal control problem under a purely deterministic setup, a second-order approximation of the objective function may lead to inaccurate results, particularly when one considers exogenous variables as arguments of the objective function. These results are related to the stability conditions, which in the present case can be written as constraints to a discount rate associated with future outcomes. We designate the proposed model as an ‘optimal growth control’ model, from which we compute general conditions about stability and analyse the application of such a framework to a fertility - human capital problem.

Highlights

Existe uma corrente recente de literatura que aponta para a relevância do método de aproximação linear quadrática da função objectivo na vizinhança do ponto de equilíbrio, nomeadamente quando se assume um cenário estocástico
There is a recent strand of literature which suggests that second order approximations of linear quadratic objective functions in the steady state vicinity, namely when assuming stochastic scenarios, lead to very interesting and useful results
Recent work in economic policy has focused on second-order approximations to the objective function in order to evaluate the dynamics of the models under discussion

Summary

Introduction1

Recent work in economic policy has focused on second-order (linear quadratic) approximations to the objective function in order to evaluate the dynamics of the models under discussion. Under the continuous time-infinite horizon framework as considered here, we will be able to see that the difference between a second-order approach and a linear treatment of the assumed specific model is relevant only if one considers exogenous variables in the objective function. This purely deterministic continuous time framework is not the ideal scenario for applying a linear quadratic approximation.

The OGC Model

Exogenous Disturbances

The Introduction of an Exogenous Variable

Illustration with a Fertility - Human Capital Case