The Genuine Savings Criterion and the Value of Population

Arrow, Dasgupta and Maler demonstrate thatin any dynamic model of the economy with changing population, population should properly be one of the state variables of the system. It enters both in the maxim and, at least under total utilitarianism, and into the production function in one way or another. If population growth is exponential and there are constant returns to scale, then a simple transformation to per capita variables can be used to eliminate one state variable. However, this simple transformation cannot be made if growth is not exponential, as it obviously is not and cannot be. If the growth of population is exogenous, then introducing it into the system does not affect the optimal policy. However, if one asks whether the system is sustainable, in the sense of at least maintaining total welfare (integral of discounted utilities), then the criterion is that the value of the rates of change of the state variables is non-negative, so that the shadow price of population becomes relevant. In this paper, we derive explicit formulas in a simple model, showing that the rate of growth of per capita capital is not the correct formula but must have other terms added to it. We also study the question under an alternative criterion of long-run average utilitarianism.