A Mathematical Formulation of the Dual Nature of Unproductive Labor

In this paper, we present a novel model that explicitly formalizes the dual nature of unproductive labor and derives the optimal size of the unproductive sector. Two sets of results emerge regarding the relationship between the unproductive sector and the profitability and accumulation of capital. First, when the unproductive sector does not make any contribution to improving the efficiency of productive labor’s value creation process, the unproductive sector will cease to exist. In this case, when the labor supply is flexible, profit continues to be produced although it is smaller than otherwise, as long as the exploitation rate is positive; in contrast, if the labor supply is fixed, the absence of the unproductive sector necessarily squeezes the exploitation rate to zero, consequently yielding zero profit. Second, when the labor supply is flexible, an economy can always rely on improving the efficiency of the unproductive sector to prevent economic stagnation and to enhance growth performance; however, when the labor supply is fixed, an economy can do so only when the wage rate is sufficiently high relative to the labor productivity conditions in the productive sector.