Optimal long-run money growth rate in a cash-in-advance economy with labor-market frictions

We revisit the Friedman rule in a labor search model and extend Heer (2003), Cooley and Quadrini (2004), and Wang and Xie (2013) to one that allows for endogenous growth. We show that, even without a liquidity effect or a CIA constraint on firms' wage payment, our model offers a different channel for moderate money growth to increase welfare. Intuitively, in a one-sector endogenous growth economy, the technology is of constant returns with respect to capital. When the labor market is frictional, a moderate increase in money growth induces an expansion in vacancy and employment. Labor and capital are complements in production. With an increase in employment, when the technology is neoclassical, the decreasing return in capital leads to a lower marginal product of labor. However, in an endogenous growth framework wherein the technology exhibits socially constant returns in capital, the marginal product of labor is constant. Due to a constant marginal product of labor, modest inflation raises employment, enlarges economic growth, and increases welfare. Moreover, the optimal long-run inflation rate departs from the Friedman rule, even when the Hosios rule holds. Finally, we find that our model with sustainable growth fits the data better than that without sustainable growth.