A GOLDEN GOLDEN‐RULE FOR WELFARE‐MAXIMIZATION IN AN ECONOMY WITH A VARYING POPULATION GROWTH RATE

Abstract

It is demonstrated mathematically that for an economy with a variable rate of population growth over a large finite period of time th e Samuelson-Diamond criterion for social welfare is not optimized at the Swan-Phelps Golden Rule level for constant population growth but rather at a lower level which is called the Golden Golden-Rule. The Swan-Phel ps Rule states that for constant population growth the most efficient golden-age path equates the interest rate to the rate of population growth. However the optimum golden-age state in an economy in which the rate of population growth varies will be at a lower ratio than the Swan-Phelps Golden-Rule State when the population growth rate is an incr easing function of the capital-labor ratio and at a higher capital-labor ratio when the rate is a decreasing function. The derivation of the golden golden-age path is the same whether the rate of population growth is an increasing or decreasing function of the capital-labor ratio. Basically a planner following the Golden Golden-Rule will tell society to consume at a higher rate in the beginning thus lowering per capita wealth and hence the rate of population increase (assuming the representative man likes children and will produce more as his income permits and fewer as his income does not permit) until the economy is operating at desired or turnpike level. Society should continue to consume at that level until nearly the end of the time period when it would move away from the turnpike level back toward the preassigned old level of operation. In this manner per capita consumption is maximized. If a planner should prefer the Bentham-Lerner criterion for maximizing the total utility of all people who will ever live the turnpike solution depends upon the utility origi n. However it is both stated and demonstrated mathematically that ther e is an upper bound on the optimal capital-labor ratio which is always b elow the Schumpeter captial-saturation level. It is noted that there is nothing sacred about the Golden Golden-Rule and that a change in one of its basic assumptions would invalidate it.