To find the equilibrium in a system that runs over more than two-periods, it is necessary to define first the uncertainty associated with time and information.
We follow the approach of Lucas [LJ78] and define a model over discrete time, i.e., tD0; 1; 2; : : : ;T, by a tree-like extension of our two-period model (see also [Con82]). The information structure of this “Lucas tree model” is given by a finite set ofstates!t2˝tin eacht. A path of state realizations over time is denoted by the vector!t D .!0; !1; : : : ; !t/. The uncertainty with respect to information decreases with the time since the paths are not recombining. The time-uncertainty can be described graphically by anevent tree1consisting of an initial date (tD 0)
1The mathematical term for an event tree is a filtration,F0;F1; : : : ;FT, i.e. a sequence of partitions of a setf1; : : : ;sgsuch thatF0 D ff1; : : : ;sg;¿g,FTD ff1g;f2g; : : : ;fsggand for allet 2Ft there existset12Ft1such thatetet1.
© Springer-Verlag Berlin Heidelberg 2016
T. Hens, M.O. Rieger,Financial Economics, Springer Texts in Business and Economics, DOI 10.1007/978-3-662-49688-6_5
211
Fig. 5.1 Generating paths by drawing states!t2˝t
0
2 2
1
1 1
2 Ω0 Ω1 Ω2
Fig. 5.2 Associated event
tree (0,11)
(0,12) (0,21) (0,22)
(0,11) (0,12) (0,21) (0,22)
(0,1,1) (0,1,2) (0,2,1) (0,2,2)
and a set of paths!tat timet. In any intermediate time periodtthe event tree consists of a partition of the set of paths so that two paths that cannot yet be distinguished attbelong to the same subset. See Fig.5.1for an example.
The probability measure determining the occurrence of the states is denoted byP.
We will definePover the set of paths. We callPthephysical measure, since it is exogenously given by nature, and use it to model the exogenous dividends process.
If the realizations are independent over time,Pcan be calculated as a product of the probabilities associated with the realizations building the vector!t. For example, the probability of getting two times “head” by throwing a fair coin is equal to the probability of getting “head” once (equal to 0.5) multiplied with the probability of getting “head” in the second run (equal to 0.5) (Fig.5.2).
In the Lucas [LJ78] model the payoffs are determined by the dividend payments and capital gains in every period. Let i D 1; : : : ;I denote the investors and k D 1; : : : ;K some long-lived assets in unit supply that enable wealth transfers over different periods of time. In addition, there is a consumption good. This good is perishable, i.e., it cannot be used to transfer wealth over time. All assets pay off in terms of the consumption good. This clear distinction between means to transfer wealth over time and means to consume is one of the important modeling assumptions of Lucas [LJ78].
Ultimately, we are of course interested in the evolution of payoffs of the assets, but not every node in the tree has to result in a payoff. It could well be that other events than payoffs trigger trades and, therefore, have to be represented in the tree.
As an example, think of a company paying out dividends only once a year, but having quarterly earning reports: obviously, the earning reports will lead to changes in the probability distribution of the dividend payoffs, thus we have to include them into our tree model although there is no payment connected with them. Moreover, even events that are unrelated to the company’s dividend payoff have sometimes to be considered. An example would be substantial changes in the investor population or preferences: they will result in trades that can change the stock prices of the company.
5.1 The General Equilibrium Model 213 Following Lucas [LJ78] once more, we assumeperfect foresight, which means thatconditionally on the events, all investors agree on the prices. Although this seems to be a strong assumption, our model is still flexible enough to accommodate different opinions: we just have to split states into sub-states whenever some investors disagree about the prices in the original state. Then, we assume the same price expectations in each state and allow agents to hold different probabilities of the occurrence of the states.
In acompetitive equilibriumwith perfect foresight, every investor decides about his portfolio strategy according to his consumption preferences2over time. Investors may disagree on the probability distribution of the states, but by construction, they agree on the prices conditionally on the states. This leads to the following definition of an equilibrium extending Definition4.8to the multi-period case.3
Definition 5.1 A competitive equilibrium with perfect foresight is a list of portfolio strategies ti and a sequence of pricesqkt, t D 0; 1; : : : ;T, such that for all i D 1; : : : ;I
.0i; : : : ; Ti/2arg max
t2RKC1 tD0;1;:::;T
Ui.cons/ s.t. ti,consC XK kD1
qktkDŠ Wti; ti,cons0;
with WtiWD XK kD1
Dkt Cqkt
ti;k1Cwit; for alltD0; 1; : : : ;T;
whereDkt are the total dividend payments of assetk,witis the endowment of investor iin periodtand markets clear4:
XI iD1
ti;k
DŠ 1 for allkand allt:
2Note that investors’ preferences are defined over consumption and not over the depot value.
The utility function representing the investors’ time preferences and risk attitude determines the consumption, which is smoothed over the realized states.
3Note that in contrast to Chap.4,kD0does not denote the risk-free asset but consumption. This is because long-lived assets that are re-traded are rarely risk-free since their prices might fluctuate.
4In principle, all quantities will be dependent on the entire history/path!t – or at least on the realized state!t – and we should write, e.g.,qkt.!t/, as there might be a different path!0t for whichqkt.!0t/is a different value. Thus, in writingqkt above we not only name a function where instead its value is meant, but we are not precise on which value we actually mean, either. Doing so is therefore – if not stated differently – understood just as an abbreviation for ease of reading.
Please observe that we cannot writeqk.t/, because there is not “one” functionqorqkwhich gets evaluated at two different points in time:qtandqtC1are two different functions, as they are defined on two different domains (˝tWD˝0 : : : ˝tand˝tC1D˝t ˝tC1respectively).
Note thatti;k.!t/2Ris the amount of assetk, respectively forkD0the amount of the consumption good, that agentihas in periodtgiven the path!tandtiis the portfolio strategy along the set of paths.tiis accordingly this amount of all assets for all paths!tandiis the list of portfolio strategies over time. Thus, the objective functionUiis a function on the space of all consumption paths. Note also that we normalized the price of the consumption good to one and that we used Walras law to exclude the market clearing condition for the consumption good.
To be more concrete, investigate the decision problem of an expected utility maximizer in the situation/node!t restricted to the decision variables in that and the adjacentnodes:
max
ti;k;ti;cons;tiC1;cons
ui.ti;cons/Cıti
X
!tC12˝tC1
prob.!tC1/ui
tCi;cons1 .!t!tC1/
;
where ui is the utility function of investor i (compare Sect.4.1.3). The budget constraints are:
ti;consC XK kD1
qktti;k
D XK kD1
.Dkt Cqkt/ti;k1Cwit; ti;cons0
and for allwtC1 2˝tC1
tCi;cons1 .!t!tC1/C XK kD1
qktC1.!t!tC1/ tCi;k1.!t!tC1/
D XK kD1
.DktC1.!t!tC1/CqktC1.!t!tC1// ti;kCwitC1.!t!tC1/ ; ti;cons0:
We start the economy with some initial endowment of assets i1 such that P
ii1 D 1. Assets start paying dividends int D 0, i.e., the budget constraint at the beginning is
0i;consC XK kD1
qk00i;kD XK kD1
.Dk0Cqk0/i;k1Cwi0:
We can think oftD0as the starting point of our analysis, i.e.,i1can be interpreted as the allocation of assets that we inherit from a previous period (“the past”). Hence, in a sense the economy can be thought of as restarted attD0.
5.1 The General Equilibrium Model 215 Instead of using theamountof assetkheld in the portfolio of investoriin timet, the investors’ demand can be expressed in terms of the asset allocation or percentage of the budget value. We defineit;k D .qktti;k/=Wti (in analogy to the two-period model definition from Sect.4.1.3). Thereby we getti;k D it;kWti=qkt. Equalizing demand with supply, i.e.,
XI iD1
it;kWti qkt
DŠ 1 for allkand allt,
gives the following result:
Proposition 5.2 The price of asset k is the average wealth of the traders’ asset allocation for asset k, i.e.,
qkt D XI iD1
it;kWti:
This pricing rule is equivalent to the simple equilibrium condition that demand is equal to supply. We have made no other assumptions to derive this result!
We are interested in optimal asset allocation strategies, thus it is more useful to formulate the investments in terms of percentage of wealth rather than in terms of the absolute number of assets. Therefore, we rewrite the model in terms ofit;k(i.e., strategy as a percentage of wealth) instead ofti;k (i.e., strategy in terms of asset units) which leads to the following reformulation of Definition5.1:
Definition 5.3 Acompetitive equilibriumwith perfect foresight is a list of portfolio strategiesit, and a sequence of pricesqkt for allt D 0; 1; : : : ;T, such that for all iD1; : : : ;I
i2arg max
it2KC1 tD0;1;:::;T
Ui.consWi/ s.t. WtiD XK kD1
Dkt Cqkt qt1 it;k1
!
Wti 1Cwit;
for alltD0; 1; : : : ;T and markets clear:
X
i
it;kWtiDqkt; for allkand allt.
Again, for simplicity we have omitted the dependence of the decisions on the paths. In other words, in a competitive equilibrium all investors choose anasset allocationitD0:::T that maximizes their utility over time under the restriction of a
budget constraint with a stochastic compound interest rate.5As in Definition5.1, the compatibility of these decision problems today and in all later periods and events is assured by the assumption of perfect foresight. This equilibrium is therefore also called equilibrium in plans and price expectations.