BUSF_A01.qxd

APPENDIX

Appendix: Formal derivation of the separation theorem

Figure 2.1
The productive
investment
opportunities
available to an
individual whose
wealth totals W 0

W 0 could all be invested, in which case it would yield wealth totalling W 1 , after one year. These

investment opportunities consist of a number of individual opportunities, like the shaded one.

Here an investment of amount amade immediately would generate amount bafter one year.

In Figure 2.1, 0W 0 represents the total amount of wealth available for present con-

sumption and/or investment by an individual. The curved line W 0 W 1 represents the

investments available to our investor. All of the present wealth (0W 0 ) may be invested,

thus yielding 0W 1 after a year, or all of the present wealth (0W 0 ) could be consumed,

leaving no wealth at the end of the year. A third (and perhaps more likely) possibility

is that the investor might choose to consume part of the present wealth and invest part

• after all, there is a need to eat, both now and next year. If our investor decided to
  invest C 0 W 0 of the present wealth (and consume 0C 0 ), this investment would result in
  0C 1 at the year end.

Formal derivation of the separation theorem

The nature of investment

Investment is in essence laying out cash in order to give rise to future cash receipts.

Usually most or all of the outlay occurs before the inflows. Thus investment has a time

dimension.

Let us now consider the position of an investor who has an amount of wealth which

may either be spent (consumed) or be invested in productive assets (plant, machinery,

trading inventories (stock) and so forth). At this stage, let us restrict ourselves to con-

sidering only investment horizons of one period (say a year). Thus the individual may

invest part or all of the current wealth for one year and consume the proceeds of the

investment next year, or consume it all immediately.