The Maximization of Utility Over Time

SUPPOSE provisionally the existence of a utility function F. Let us assume further that the individual in question consumes only three goods x, y, and z. The argument can easily be extended later to any number of commodities. The individual is at the point in time 0 and plans for the period of time 1, 2, , n. Consumption takes place at these n discontinuous points in time. We denote by xi the quantity of the commodity x which the individual plans to consume at the point in time j; similarly, y2 and zj. He does not accumulate any commodity stocks. The utility function F will depend on those quantities which the individual expects to consume: