A Two-Sector model of Optimal Growth in Which Labour is Employed only in the Industry of Consumption Goods: A Complete Characterization of Equilibrium Paths

In this paper, I develop a frictionless two-sector optimal growth model with endogenous labour supply in which the non-reproducible factor can be employed only in the production of consumption goods. From a theoretical point of view, I show that in this setting the optimal allocation of labour is constant over time, whereas its actual utilization relies on the availability of a convex technology. In parallel, a convex technology in the sector of investment goods is instead sufficient for the determinacy of meaningful equilibrium paths. Furthermore, calibrating the model to the US economy, I show that such a two-sector economy smooths the complementarity between the marginal propensities to consume and to save, and it also able to replicate the countercyclical (procyclical) pattern of the relative price of capital goods (real wages).