Figure4.12gives the main intuition on the aggregation problem. At the equilibrium allocation asset prices are determined by the trade of two agents, however, they could also be thought of as being derived from a single utility function that is maximized over the budget set based on aggregate endowments (the upper right corner of the Edgeworth Box).
Actually, the answer to our first question is even simpler since it does not need any information on the individual’s utility functions. Any asset price vector that is arbitrage-free can also be generated by a single utility function. Hence, in a sense, anything goes!
ciz
cjs
cjz
cis
Equilibrium allocation q
Rep. Agent q
Fig. 4.12 Aggregating individual decision problems into one representative agent
Theorem 4.11 (Anything Goes Theorem) Letqbe an arbitrage-free asset price vector for the market structureA. Then there exists an economy with a representative consumer maximizing an expected utility function such thatq is the equilibrium price vector of this economy.
Proof Sinceq is arbitrage-free there exists some risk neutral probability 0 such thatq0D0A. Choose then
UR.c0; : : : ;cs/WDc0C XS sD1
scs:
At the pricesqthe representative agent will consume aggregate endowments, which can be seen immediately from the first order condition.33 ut The argument made in this proof is the reason why the state price measure is also called the “risk neutral measure”. It could be thought of as being derived in a risk neutral world, i.e., in an economy in which a single risk-neutral representative agent determines asset prices. Note, however, that every real agent in the economy might
33Obviously, one could also find a representative consumer with strictly concave utilities since one only needs to satisfy that his marginal rates of substitution at aggregate endowments coincide with the state prices.
190 4 Two-Period Model: State-Preference Approach be risk neutral, so that somehow the representative agent does not reallyrepresent the agents. This is the point of our question 2: “Is it possible to find an aggregate utility function that has the same properties as the individual utility functions?”. For the answer of question 2, allocational efficiency will be quite useful, as has first been noticed by Constantinides [Con82].
Note first that Pareto-efficiency is equivalent to maximizing some welfare function. In other words, any Pareto efficient allocation can be obtained from the maximization of some welfare function in which the weights are chosen appropriately and the maximization of a welfare function results in a Pareto-efficient allocation. A welfare functionassigns a social utility to each allocation. It is in a certain sense the analog of a utility function on consumption bundles in classical economies. We define the welfare function as an aggregate of individual utilities.
Let i > 0be the weight of agentiin the social welfare functionPI
iD1 iUi.ci/. The next argument shows that choosing the welfare weights i > 0equal to the reciprocal of the agents’ marginal utility of consumption in period 0 attained in the financial market equilibrium,
iD 1
@0Ui.ci/;
one can generate the equilibrium consumption allocation from the social welfare function: recall that under differentiability and boundary assumptions Pareto- efficiency implies
r1U1.c1/
@0U1.c1/ D: : :D r1UI.cI/
@0UI.cI/ DW: Define
UR.W/WD sup
c1;:::;cI
( I X
iD1
iUi.ci/ˇˇ ˇˇˇ
XI iD1
ciDW )
where i D 1=.@0Ui.ci//. The first order condition for this maximization problem is 1rU1.c1/D: : :D IrUI.cI/DWand34rUR.W/D, hence:
r1UR.W/D r1Ui.ci/
@0Ui.ci/ and @0UR.W/D1:
34This last claim is the so called envelope theorem.
Consider
max UR.cR/ such thatcRW q
A
:
The first order condition is
q0 D r1UR.cR/0
@0UR.cR/AD r1Ui.ci/0
@0Ui.ci/AD0A: Note thatcR DW, the aggregate wealth of the economy.
Hence, we have found a “technique” to replace the individual utility functions by some aggregate utility function. In particular, we see that concavity of the individual utility functions is inherited by the aggregate utility function. Hence, as we argued above the likelihood ratio process should be decreasing. That is to say, postulating some utility function of the representative agent we can now test whether asset prices are in line with optimization by referring to aggregate consumption data.35 But when does the aggregate utility function really represent the individuals? We now give a first result in this direction (others can be found in the exercise book):
if all individual utility functions are of the expected utility type with common time preference and common beliefs, then the representative agent is also an expected utility maximizer with the same time preference and the same beliefs. Hence, our result shows that any heterogeneous set of risk aversions can be aggregated into one aggregate risk aversion. More precisely:
Proposition 4.12 Assume that for all iD1; : : : ;I the utility functions uiagree and that the time discountingıis also independent of i. Moreover assume that the beliefs ps, sD1; : : : ;S, are homogeneous, i.e., let Uibe given by
Ui.ci/Dui.ci0/Cˇ XS sD1
psui.cis/ for iD1; : : : ;I:
Then UR.cR/DuR.cR/CˇPS
sD1psuR.cRs/, for some function uRWR!R. Proof We use the definition ofUR:
UR.W/D sup
c1;:::;cI
( I X
iD1
iUi.ci/ˇˇ ˇˇˇ
XI iD1
ciDW )
35See Sect.4.6.3for empirical studies along this line.
192 4 Two-Period Model: State-Preference Approach
where iD1=.@0Ui.ci//gives
UR.W/D sup
c1;:::;cI
( I X
iD1 i
ui.ci0/Cˇ XS sD1
psui.cis/ ˇˇ ˇˇˇ
XI iD1
ciDW )
D sup
c1;:::;cI
(PI
iD1 iui.ci0/
„ ƒ‚ …
uR.W0/
Cˇ XS sD1
ps
PI
iD1 iui.cis/
„ ƒ‚ …
uR.Ws/
ˇˇˇˇ ˇ
XI iD1
ciDW )
DuR.W0R/Cˇ XS sD1
psuR.WsR/: ut
Similar results are possible, for example for Prospect Theory preferences. Note that in the case of Prospect Theory the representative agent may not need to be risk loving over losses since this non-concavity of the utility gets smoothed out by the maximization as Fig.4.13suggests.36
This looks like wonderful news: taking the representative agent perspective one can even forget about non-concavities in the individual utility functions.
This observation was first made in an article titled “Prospect Theory: Much Ado About Nothing!” [LL03]. So can we really forget about Prospect Theory just by aggregating the preferences of single agents? Well we should not get too enthusiastic since the representative agent technique has a natural limitation: it is generally not useful to tell us anything about asset prices that we do not know yet. More precisely, it is not useful for comparative statics, or “out of sample predictions”. Indeed, as shown in the exercise book, basing one’s investment decisions on the representative agents technique may result in severe losses, since asset prices would be predicted to go in the wrong direction.
Fig. 4.13 Smoothing out individual non-concavities on the aggregate
u1 u2
aggregate utility
36For an in-depth treatment of this smoothing aggregation in general see [DDT80] and, for the case of Cumulative Prospect Theory, De Giorgi, Hens and Rieger [DGHR10].
This leads us to the final question of this section: “Is it possible to use the aggregate decision problem to determine asset prices ‘out of sample’, i.e., after some change, e.g., of the dividend payoffs?” If this is possible, some authors37 say one gets “demand aggregation”. This means that not only at the equilibrium point the representative agent demand function coincides with the sum of the individual demands, but it coincides for any prices. Demand aggregation is possible, however under quite restrictive assumptions. In Hens and Pilgrim [HP03] we find the following cases in which a positive answer to our third question is possible:
1. Identical utility functions and identical endowments 2. Quasi-linearity:Ui.ci0; : : : ;ciS/Dci0Cui.ci1; : : : ;cis/ 3. Expected utility with common beliefsand
(a) no-aggregate riskPK
kD1Aks DPK
kD1Akzfor alls;z or (b) complete marketsand
• CRRA and collinear endowmentsor
• identical CRRAor
• Quadratic utility functions
Some of these results have been extended to incomplete markets, see [HP03].
We conclude this section by giving some example how in the representative agent utility the heterogeneous preferences get aggregated:
• Expected utility with common beliefs and no-aggregate risk:
Ui.ci0; : : : ;ciS/WDui.ci0/Cˇi XS sD1
psui.cis/; iD1; : : : ;I;
aggregates to
UR.W0; : : : ;WS/DuR.W0/CˇR XS sD1
psuR.Ws/
for any concaveuR.
• Expected utility with common beliefs and common time preference and quasi- linear quadratic preferences:
Ui.ci0; : : : ;ciS/WDci0Cˇ XS sD1
ps
cis12 i.cis/2
; iD1; : : : ;I;
37For example, Rubinstein [Rub74] or Constantinindes [Con82].
194 4 Two-Period Model: State-Preference Approach
aggregates to
UR.cR0; : : : ;cRS/DcR0 Cˇ XS sD1
ps
cRs 12 R.cRs/2
where
R D
XI iD1
1
i
!1
:
• Expected logarithmic utility with common time preference and collinear endow- ments
Ui.ci0; : : : ;ciS/WDln.ci0/Cˇ XS sD1
pisln.cis/; iD1; : : : ;I;
andwiDıiW,iD1; : : : ;I, aggregates to
UR.cR0; : : : ;cRS/Dln.cR0/Cˇ XS sD1
pRs ln.cRs/
wherepRs DPI iD1ıipis.
• Mean-Variance utilities with heterogeneous expectations on the means and common beliefs on the covariances,
Vi.i; /WD˛i
22; iD1; : : : ;I;
aggregates to
VR.R; /DR˛R 2 2; whereR DPI
iD1aii;withaiD Pri=˛i iri=˛i.
In each of these examples, the representative agent is of the same type as the individual agents and, moreover, he generates a mapping from the individual agents’
characteristics into asset prices that also can be used for asset price predictions, i.e., after some change of the asset payoffs, for example.