Two Simple Formulas Relating the Growth and Profit Rates

Abstract

This article compares the magnitudes of the growth and profit rates within an economic model that represents two successive periods of production. Its main result is a pair of simple formulas determining respectively the (growth rate)/(profit rate) ratio and the (profit rate) – (growth rate) difference. The first formula permits to show that the ratio is a decreasing function of the capital/income ratio. The second formula allows us to point out that the difference, as a general rule, is an increasing function of the capital/income ratio although, under certain conditions, it may be otherwise. Both formulas consider the simplest case, when savings equals capital increase and, in addition, the capital/income ratio is constant. The general case in which these conditions are not necessarily satisfied is also considered. In the simplest case, the ratio between the two rates is equal to that between the savings rate and the capital share while the difference is equal to that between these same variables divided by the capital/income ratio. In the general case, the results just indicated are modified by a quotient having in both formulas the same numerator: the capital increase not due to savings (as a fraction of income) minus the increase in the capital/income ratio. In the first formula, the quotient is preceded by a positive sign and its denominator is the capital share while in the second one it is preceded by a negative sign and its denominator is the capital/income ratio. As illustrated by means of an empirical application, these results help to explain the inequality between the growth and profit rates whose importance for income distribution is underlined by Piketty.