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 the non-reproducible factor is employed uniquely in the production of investment goods. Relying on Cobb-Douglas technologies, logarithmic utility of consumption and separate CRRA disutility for labour provision, I show that a centralized solution characterized by a meaningful steady state and the determinacy of equilibrium trajectories requires constant returns to scale only in the production of new equipment. Moreover, I show that a calibrated version of the model tailored on the US economy leads to a marginal rate of transformation between consumption and investment goods lower than one in absolute value, and it is also able to provide a rationale for the procyclical patterns of the relative price of capital goods and the real wage.