THE THEORY OF ECONOMIC GROWTH more rapidly than the labor supply. The second system is so unpro- ductive that the full employment path leads only to forever diminish- ing income per capita. Since net investment is always positive and labor supply is increasing, aggregate income can only rise The basic conclusion of this analysis is that, when production takes place under the usual neoclassical conditions of variable pro- portions and constant returns to scale, no simple opposition between natural and warranted rates of growth is possible. There may not be -in fact in the case of the Cobb-Douglas function there never can be-any knife-edge The system can adjust to any given rate of growth of the labor force, and eventually approach a state of steady proportional expansion IV, EXAMPLES In this section I propose very briefly to work out three three simple choices of the shape of the production function for whic it is possible to solve the basic differential equation( 6)explicitly. Example 1: Fixed Proportions. This is the Harrod-Domar case It takes a units of capital to produce a unit of output and b units of labor. Thus a is an acceleration coefficient. Of course, a unit of output can be produced with more capital and / or labor than this (the isoquants are right-angled corners); the first bottleneck to be eached limits the rate of output. This can be expressed in the form (2)by saying Y= F(K, L)= min K L where"min(.. ""means the smaller of the numbers in parentheses. The basic differential equation(6)becomes =mi(1 Evidently for very small r we must have<-, so that in this range a2b,1.e,r 2 b, the equa- tion becomes r=h-nr. It is easier to see how this works graphi ally. In Figure Iv the function s min(r, a)is epresented by a