The Mathematics of Arbitrage

40 3 Utility Maximisation on Finite Probability Spaces

proportionality relation fails to hold. Then one immediately deduces from a
marginal variation argument that the investment of the agent cannot be opti-
mal. Indeed, by investing a little more in the more favourable Arrow asset and
a little less in the less favourable one, the economic agent can strictly increase
her expected utility under the same budget constraint. Hence for the optimal
investment the proportionality must hold true. The above result also identifies
the proportionality factor asy=u′(x), wherexis the initial endowment of
the investor. This marginal utility of the indirect utility functionu(x)also
allows for a straightforward economic interpretation.

Theorem 3.1.3 indicates an easy way to solve the utility maximisation
problem at hand: calculatev(y) using (3.19), which reduces to a simple one-
dimensional computation. Once we knowv(y), the theorem provides easy
formulae to calculate all the other quantities of interest, e.g.,X̂T(x),u(x),
u′(x)etc.
Another message of the previous theorem is that the value functionx→
u(x) may be viewed as a utility function as well, sharing all the qualitative
features of the original utility functionU. This makes sense economically, as
the “indirect utility” functionu(x) denotes the expected utility of an agent
with initial endowmentx, when optimally investing in the financial marketS.

Let us now give an economic interpretation of the formulae foru′(x)initem
(iii) along these lines: suppose the initial endowmentxis varied tox+h,for
some small real numberh. The economic agent may use the additional endow-
menthto finance, in addition to the optimal pay-off functionX̂T(x),hunits
of the num ́eraire asset, thus ending up with the pay-off functionX̂T(x)+hat
timeT. Comparing this investment strategy to the optimal one corresponding
to the initial endowmentx+h,whichisX̂T(x+h), we obtain

lim

h→ 0

u(x+h)−u(x)

h

= lim

h→ 0

E[U(X̂T(x+h))−U(X̂T(x))]

h

(3.23)

≥lim

h→ 0

E[U(X̂T(x)+h)−U(X̂T(x))]

h

(3.24)

=E[U′(X̂T(x))].

Using the fact thatuis differentiable and thathmay be positive as well
as negative, we must have equality in (3.24) and have therefore found another
proof of formula (3.21) foru′(x); the economic interpretation of this proof
is that the economic agent, who is optimally investing, is indifferent of first
order towards a (small) additional investment into the num ́eraire asset.
Playing the same game as above, but using the additional endowment
h∈Rto finance an additional investment into the optimal portfolioX̂T(x)
(assuming, for simplicity,x= 0), we arrive at the pay-off functionx+xhX̂T(x).

Comparing this investment withX̂T(x+h), an analogous calculation as in
(3.23) leads to the formula foru′(x) displayed in (3.22). The interpretation