A Two-Sector Model of Optimal Growth in which Labour is Employed only in the Industry of Investment Goods: A Complete Characterization of Equilibrium Paths

In this paper, I develop a two-sector growth model with endogenous labour supply in which labour is employed only in the production of investment goods. From a theoretical point of view, I show that no matter the shape of the technology to produce consumption goods, a convex technology in the sector of investment goods is necessary and sufficient for a meaningful stationary solution and the determinacy of equilibrium trajectories. In addition, from a numerical perspective, I show that a calibrated version of the model relaxes the complementarity between the propensity to consume and to save and it is also able to provide a rationale for the procyclical patterns of the relative price of capital goods and the real wage.