# Simultaneous Optimization of Population and Capital

# Simultaneous Optimization of Population and Capital

The problem of optimum savings has been discussed by Ramsey on the assumption of a constant population and later by a number of economists on the more general assumption that the labour force expands at a constant exogenously fixed rate; different rates of population growth lead to different solutions; i.e. the path of optimum capital accumulation is relative to the population growth. In contrast, Meade and others have been concerned with the problem of optimum population, assuming among other things that at any given time the economy is provided with a given rate of savings as well as a given stock of capital equipment to be used; it follows that the path of optimum population is relative to capital accumulation. It is evident that these two partial optimization procedures should be synthesized so as to give a genuine supreme path, which is optimum with respect to both capital and population. This final chapter generalizes the Ramsey–Meade problem in that direction and shows that two kinds of long‐run paths—efficient and optimum paths—will under some conditions converge to the Golden Growth path when the time horizon of the paths becomes infinite; the two long‐run tendencies that are derived may be regarded as extensions of those discussed in the chapters entitled First and Second Turnpike Theorems. The different sections of the chapter discuss: the generalized Ramsey–Meade problem; the finding that the Golden Equilibrium rate of growth is greater than the Silvery Equilibrium rate; the Average Final State Turnpike Theorem; the strong superadditivity of processes—a sufficient condition for strong convergence; the tendency towards the ‘top facet’ as the general rule; cyclic phenomena; the Average Consumption Turnpike Theorem and its proof; and aversion to fluctuation in consumption.

*Keywords:*
Average Consumption Turnpike Theorem, Average Final State Turnpike Theorem, capital accumulation, constant population, consumption, convergence, cyclic phenomena, economic growth, efficiency, Golden Equilibrium, Golden Growth, growth rate, labour force, J. E. Meade, optimum capital accumulation, optimum population, optimum savings, population growth, F. P. Ramsey, Silvery Equilibrium, superadditivity, Turnpike Theorems

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