General existence of competitive equilibrium in the growth model with an endogenous labor–leisure choice

We prove the existence of competitive equilibrium in the canonical optimal growth model with elastic labor supply under general conditions. In this model, strong conditions to rule out corner solutions are often not well justified. We show using a separation argument that there exist Lagrange multipliers that can be viewed as a system of competitive prices. Neither Inada conditions, nor strict concavity, nor homogeneity, nor differentiability are required for existence of a competitive equilibrium. Thus, we cover important specifications used in the macroeconomics literature for which existence of a competitive equilibrium is not well understood. We give examples to illustrate the violation of the conditions used in earlier existence results but where a competitive equilibrium can be shown to exist following the approach in this paper.