Exact Measures of Income in Two Capital-Resource-Time Economies

Exact optimal paths are calculated for two closed economies, each with an accumulable capital, a non-renewable resource and exogenous technical progress. The first economy has hyperbolic discounting and (possibly) hyperbolic technical progress. On its optimal path, generally, welfare-equivalent income > wealth-equivalent income > Sefton-Weale income > NNP; and sustainable income = NNP only if consumption is constant. These results support the view that there can be no single definition of income. The Solow (1974) constant consumption solution is a special case; and for low enough discounting, growth is optimal even when technical progress is zero. The second economy has a non-linear frontier between consumption and investment goods. In it, Weitzman’s (1997) technical progress premium formula works only if an upwards correction factor is applied to the rate of progress in production, to convert it to a rate of progress in NNP.