BUSF_A01.qxd

Appendix: Formal derivation of the separation theorem

now, it would, with interest at rate r, become W 0 +rW 0 after a year. Thus the slope of

the borrowing/lending line is (W 0 +rW 0 )/W 0 = 1 +r.

The existence of borrowing/lending opportunities enables levels of investment to

be undertaken in excess of the amount of wealth left after a desired level of consump-

tion has been undertaken. This is achieved by borrowing the required amount now

and repaying it (with interest) next year. Similarly, not all of the present wealth left

after consuming the desired amount need be invested in production; some of it can be

lent, to be received back (again with interest) next year. This broadening of the pos-

sibilities may well enable the investor to achieve higher levels of satisfaction (utility).

Figure 2.5 combines Figures 2.3 and 2.4, that is, it introduces the borrowing/lend-

ing line into the investment/consumption configuration. Figure 2.5 does look rather

complicated, but it is basically only what we have already met.

Figure 2.5
Utility of
consumption/
investment of some
individual with the
borrowing/lending
opportunity

This indicates that most individuals can reach a greater level of utility through a combination

of productive investment andborrowing or lending than could be achieved without the

borrowing/lending option. It also shows that individuals will be unanimous about the level

of productive investment that will maximise their utility, despite having widely differing

attitudes to present and future consumption.

Without the borrowing/lending opportunity, point P represented the optimum

consumption/investment combination for our investor, achieving utility curve U 1 U 1.

With the introduction of borrowing/lending, a higher level of satisfaction can be

achieved, moving up to utility curve U 2 U 2. Our individual can now invest amount