We use the two-period model as outlined in Chap.3 and first give the definition of financial markets equilibria in economic terms, i.e., in terms of asset prices and quantities of assets bought and sold. As before, the periods are enumeratedtD0; 1.
In the second periodt D 1 a finite number of states of the world,s D 1; 2; : : : ;S can occur (compare Fig.4.3).
As before, we denote the assets byk D 0; 1; 2; : : : ;K. The first asset,k D 0, is the risk-free asset delivering the certain payoff 1 in all second period states. The assets’ payoffs are denoted byAks. The time 0 price of assetkis denoted byqk. Recall the states-asset-payoff matrix,
AD.Aks/D 0 B@
A01 AK1 ::: :::
A0S AKS 1 CAD
A0 AK D
0 B@
A1 :::
AS
1 CA;
which gathers the essence of the asset structure.
Each investoriD1; : : : ;Iis described by his exogenous wealth in all states of the worldwiD.wi0; : : : ;wiS/0. Given these exogenous entities and given the asset prices qD.q0; : : : ;qK/0he can finance his consumptionci D.ci0; : : : ;ciS/0by trading the assets. We denote byiD.i;0; : : : ; i;K/0the vector of asset trade of agenti. Note
170 4 Two-Period Model: State-Preference Approach thati;kcan be positive or negative, i.e., agents can buy or sell assets. In these terms, the agent’s decision problem is:
imax2RKC1Ui.ci/ such that ci0C XK kD0
qki;kDwi0
and cisD XK kD0
Aksi;kCwis0; sD1; : : : ;S;
which, considering that some parts of the wealth may be given in terms of assets,18 can be written as:
Omax
i2RKC1Ui.ci/ such that ci0C XK kD0
qkOi;kD XK kD0
qkAi;kCwi0
and cisD XK kD0
AksOi;kCwi?s; sD1; : : : ;S:
Afinancial markets equilibrium is a system of asset prices and an allocation of assets such that every agent optimizes his decision problem and markets clear, formally:
Definition 4.7 A financial markets equilibrium is a list of portfolio strategiesOopt;i, iD1; : : : ;I, and a price systemqk,kD0; : : : ;K, such that for alliD1; : : : ;I,
Oopt;iDarg max
Oi2RKC1
Ui.ci/ such that ci0C XK kD0
qkOi;kD XK kD0
qkAi;kCwi0
and cisD XK kD0
AksOi;kCwi?s; sD1; : : : ;S;
and markets clear:
XI iD1
Oopt;i;kD XI
iD1
Ai;k; kD0; : : : ;K:
Note that we only required asset markets to clear. What about markets for consumption? Are we sure that they are also in equilibrium? Formally, can we show that also the sum of the consumption is equal to the sum of the available resources,
18See Chap.4.1.3for this transformation of the decision problem.
i.e.,
XI iD1
ci0D XI
iD1
wi0 and XI
iD1
cisD XI
iD1
wis; sD1; : : : ;S‹
Noting that wis D PK
kD0AksAi;k C wi?s, this follows from the agents’ budget restrictions:
XI iD1
ci0C XK kD0
qkOopt;i;k
! D
XI iD1
wi0C XK kD0
qkAi;k
!
and
XI iD1
cisD XI
iD1
XK kD0
AksOopt;i;kCwi?s
!
; sD1; : : : ;S;
because asset markets clear: PI
iD1Oopt;i;k D PI
iD1Ai;k, k D 0; : : : ;K. Hence, nothing is missing in the Definition4.7.
It is immediate to see that in a financial market equilibrium there cannot be arbitrage opportunities. This is true, because otherwise the agents would not be able to solve their maximization problem since any portfolio they consider could still be improved by adding the arbitrage portfolio. Hence, deriving asset prices from an equilibrium model automatically leads to arbitrage-free prices.
As mentioned before, a financial markets equilibrium can be illustrated by an Edgeworth Box (Fig.4.8). At the equilibrium allocation both agents have optimized their consumption by means of asset trade given their budget constraint and markets clear.
initial allocation
i= 2
i= 1 ciz
cjs
cjz
cis
q equilibrium allocation
Fig. 4.8 A financial markets equilibrium in an Edgeworth Box
172 4 Two-Period Model: State-Preference Approach The geometry of the Edgeworth Box suggests that asset prices should be related to the agents’ marginal rates of substitution. And indeed, on investigating the first order conditions for solving their optimization problems we see that the marginal rates of substitution are one candidate for state prices. The first order condition for any agent is:
qkD XS sD1
@csUi.ci0; : : : ;ciS/
@c0Ui.ci0; : : : ;ciS/
„ ƒ‚ …
si
Aks; kD0; : : : ;K:
In particular, for the case of expected utility
Ui.ci0; : : : ;ciS/Dui.ci0/Cıi XS sD1
probisui.cis/ we get:
qkD XS sD1
probisıi@csui.cis/
@c0ui.ci0/
„ ƒ‚ …
si
Aks; kD0; : : : ;K:
Hence, we get a nice theory of state prices that links them to the agents’ time pref- erences, their beliefs, their risk aversion and their consumption. The consumption is hereby dependent on the aggregate availability of resources.
We recall how to express a financial markets equilibrium in finance terms:
imax2KC2Ui.ci/ such that ci0Dwi0.1c/ XK kD0
Oi;kwi0
and cisD XK kD1
RksOi;k
!
wi0;finCwi?s; sD1; : : : ;S:
This puts us in a position to define a financial markets equilibrium in finance terms:
Definition 4.8 A financial markets equilibrium is a list of portfolio strategiesi, iD1; : : : ;I, and a system of returnsRk,kD0; : : : ;K, such that for alliD1; : : : ;I, opt;iDarg max
i2KC2
Ui.ci/ such that ci0Dwi0 XK kD0
Oi;kwi0;fin
and cisD XK kD1
RksOi;k
!
wi0;finCwi?s; sD1; : : : ;S;
and markets clear:
XI iD1
opt;i;kriDM;k; kD0; : : : ;K;
whereriWDwi0;fin=.P
iwi0;fin/andM;kis the relative market capitalization of assetk.
The market clearing condition in Definition4.8may look a bit unusual because it is not often stated explicitly in finance models.19So let us make sure it is indeed equivalent to the equality of demand and supply of assets:
Multiplying each market clearing condition for assets, XI
iD1
Oopt;i;kD XI
iD1
Ai;k; kD0; : : : ;K;
by the price of that asset and extending the expressions by the financial wealth of the agents,wi;finDPK
kD0qkAi;k, yields the equivalence:
XI iD1
i;k;optriD XI
iD1
qkOopt;i;k wi0;fin
wi0;fin P
iwi0;fin D qkP
iAi;k
PK kD0.qkP
iAi;k/ DM;k: Before passing on to the next section we should once more mention that everything can also be expressed in terms of factors. A financial markets equilibrium is then a system of factor returns such that all agents take the factor risk that suits best their consumption plans and markets clear. This is further discussed in the exercise book.
19Most finance models work right away with a representative investor being in equilibrium with himself. Hence, the market clearing condition is not stated explicitly.
174 4 Two-Period Model: State-Preference Approach