General existence of competitive equilibrium in the growth model with an endogenous labor–leisure choice (original) (raw)
2020, Journal of Mathematical Economics
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.
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The inclusion of a labour#leisure choice in endogenous growth models has interesting and somewhat counter-intuitive e#ects. In existing one sector models a condition for indeterminacy is that labour demand is upwardsloping, which is di#cult to reconcile with evidence. In this paper we give conditions for indeterminacy in a one sector model with decreasing returns to labour. We show that this requires that consumption and leisure are both highly intertemporally substitutable while the factors of production are highly complementary. # We wish to thank G. Cazzavillan, R. Farmer, M. Salmon and participants in seminars in Florence, Istanbul and Warwick for helpful comments. All remaining errors are our own. 1 Introduction In this paper we study whether the market outcome can be indeterminate in a one sector model of endogenous growth, where agents elastically supply labour and there are decreasing returns to labour. Recently there has been a renewed interest in the problem of ind...
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