Economics 202A Lecture Outline, October
30November 1, 2007 (version 2.0)
Maurice Obstfeld
Optimal Consumption in a Frictionless World: Complete Markets
To understand consumption under uncertainty, we start with the benchmark
case of
complete markets.
Complete asset markets e/ectively allow consumers
to buy insurance against any contingency (or to sell insurance). This is possi
ble because there exist assets with returns di/erentiated across every state of
nature, and, subject to an overall budget constraint, individuals can purchase
any (positive or negative) amount of such assets.
This is not realistic ° although one way to read the proliferation of exotic
derivative products in recent years is as an evolution of realworld markets
toward the ideal of completeness.
Why, then, consider this case? Because the availability of this benchmark
°
like the hypothetical "frictionless plane" in physics °
allows us to get a
handle on more complex problems.
For example, Newton±s law
F
=
ma
is
counterintuitive until one learns to abstract from the force exerted by friction.
Assumptions.
Let±s start with a pure endowment model (no investment or
production). There are two periods. On date 1, individual
i
±s endowment is
y
i
.
From the perspective of date 1, however, the date 2 endowment is a random
variable. There are also only two possible states of nature on date 2. In state 1
the endowment is
y
i
(1)
, in state 2 it is
y
1
(2)
:
Let
c
i
denote the individual±s date 1 consumption,
c
i
(1)
and
c
i
(2)
the indi
vidual±s
contingency plans
for consumption on date 2. The plans are contingent
on the state that actually occurs on date 2. The probability that state
s
occurs
is
°
(
s
)
, where, summing over all states
s
,
°
s
°
(
s
) = 1
:
A key hypothesis is that the individual chooses the consumption plan that
maximizes average lifetime utility,
U
i
=
°
(1)
°
u
(
c
i
) +
±u
±
c
i
(1)
²³
+
°
(2)
°
u
(
c
i
) +
±u
±
c
i
(2)
²³
=
u
(
c
i
) +
±
°
°
(1)
u
±
c
i
(1)
²
+
°
(2)
u
±
c
i
(2)
²³
=
u
(
c
i
) +
±
E
u
±
c
i
(
s
)
²
;
where
c
(
s
)
denotes consumption in state
s
. This is the von NeumannMorgenstern
expected utility criterion and, being linear in probabilities, it is somewhat spe
cial. One of its consequences (as we shall see) is that it forces the intertemporal
substitution elasticity to equal the (inverse) coe¢ cient of absolute risk aversion
for isoelastic utility. We shall de²ne the risk aversion coe¢ cient later.
A basic
ArrowDebreu security
for state
s
pays its owner 1 unit of output
on date 2 if state
s
occurs and nothing otherwise. (In contrast, a riskless
bond
pays its owner the same amount of output in every state.)
Let
r
be the rate of interest on a bond. We de²ne
r
by the de²nition that
1
=
(1 +
r
)
is the price (all prices are in terms of date 1 consumption) of a bond
1
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paying its owner 1 unit of output on date 2 regardless of the state of nature.
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 Fall '07
 AKERLOF
 Economics, Utility, Euler equation

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