§9. Impatience Principle B (n(m - 1) Equations)

To express algebraically Impatience Principle B22 we are compelled to recognize for each year a separate rate Edition: current; Page: [295] of interest. The rate of interest connecting the first year with the second will be called i', that connecting the second year with the third, i'', and so on to i(m-1). Under this principle, the rates of time preference for all the different individuals in the community for each year will be reduced to a level equal to the rate of interest. This condition, algebraically expressed, is contained in several continuous equations, of which the first is:

i' = f1'= f2' =... = fn'.

This expresses the fact that the rate of time preference of the first year's income compared with next is the same for all the individuals, and is equal to the rate of interest between the first year and the next. A similar continuous equation may be written with reference to the time preferences and the rate of interest as between the second year's income and the next, namely:

i'' = f1'' = f2'' =... = fn'.

Since the element of risk is supposed to be absent, it does not matter whether we consider these second-year rates of interest and time preference as the ones which are expected, or those which will actually obtain, for, under our assumed conditions of no risk, there is no discrepancy between expectations and realizations.

A similar set of continuous equations applies to time-exchange between each succeeding year and the next, up to that connecting year (m - 1) with year m. There will therefore be m - 1 continuous equations of the Edition: current; Page: [296] above type. Since each such continuous equation is evidently made up of n constituent equations, there are in all n (m - 1) equations in the second set of equations.