So far we have restricted our attention to financial market equilibria. We did not give any argument how such an equilibrium is reached, starting from a non-equilibrium state. One may argue that the economy is always subject to exogenous “shocks”
that will disturb the current equilibrium. Hence, if there were no forces that drive the economy back to the equilibrium then it is very unreasonable to assume that real
202 4 Two-Period Model: State-Preference Approach Fig. 4.16 Stability of an
equilibrium in a simple one dimensional setting. The figure shows the excess demand of a single asset as a function of its price
q∗
iθi(q)−θ¯i
life phenomena can be described by an equilibrium. Everything we learned so far would not have any justification! Exogenous shocks could, for example, be changes in the economic fundamentals like the payoffs of the assets or the exogenous wealth of the agents. Also they could result from changes in agents’ beliefs about the states of the world (induced by natural disasters or sudden unforeseen political events).
In this book we will consider three types of dynamics that can be distinguished by their speed of adjustment. We start with the fastest type, the short-run dynamics.
In the short run we look at the intraday adjustment of market prices due to excess demand or excess supply of assets. We postulate that prices move in the direction of excess demand, i.e., when demand exceeds supply, prices will increase and when supply exceeds demand, prices will decrease. This was the original idea of Adam Smith on which he based the conjecture that competitive equilibria will always be reached. While this argument is compelling for the stability of one market (see Fig.4.16for an illustration), it is not obvious at all if markets are linked to each other because the demand of any asset does also depend on the price of any other asset. Note that these cross-price effects naturally arise when agents have portfolio considerations like diversification. In this section we will show two results on the stability of financial market equilibria in a CAPM economy.
With simple mean-variance preferences of the form Ui.ci1; : : : ;ciS/WDi.ci1; : : : ;ciS/ i
22;i.ci1; : : : ;ciS/;
we prove global stability of the unique financial market equilibrium. This case is obtained, for example, if utility functions are of the CARA-type and returns are log-normally distributed. If, however, mean-variance preferences are obtained from normally distributed returns and Prospect Theory preferences then the mean- variance utility looks more complicated than that, see [DGP08]. As an effect, CAPM equilibria may be unstable, for example because due to exogenous shocks they may jump from one possible equilibrium to another equilibrium. Intraday crashes
08-1987 09-1987 11-1987 01-1988 1500
2000 2500 3000
Fig. 4.17 Black Monday of October 19, 1987 in which the DJIA lost 20 % of its value in a single day!
that occur for no obvious reason like the Black Monday of October 19, 1987 (see Fig.4.17) have been explained by this switching from one equilibrium to another.38 Before we conclude by proving the stability for simple mean-variance utilities, we will first give the geometric intuition for these crashes when the utilities are more complicated. Figure4.18shows the phenomenon of multiple equilibria in the Edgeworth Box. For a given set of endowments and preferences two different market clearing prices are obtained. One may think of one equilibrium as the “optimistic”
one and one as the “pessimistic” one. In the optimistic equilibrium, the asset price is high because everybody believes the asset is attractive and, hence, prices are driven up. In the pessimistic equilibrium the reverse holds true.
Translating Fig.4.18into an excess demand diagram, we get a situation like that displayed in Fig.4.19. Note that if there are multiple equilibria we actually need to have at least three of them, two of which are stable and one is unstable.
So how can it happen that on small changes of the exogenous characteristics we get drastic changes of the endogenous entities? Well, this happens if the economy is initially in one equilibrium that disappears or becomes unstable due to the exogenous changes, as the sequence in Fig.4.20shows.
A more compact way of showing the same phenomenon is to comprise the three parts of Fig.4.20 into one. This can be done by looking at a mapping from the exogenous characteristics, e.g., the asset payoffs, to the endogenous asset prices (Fig.4.21).
38The classical reference is Leland and Genotte [GL90].
204 4 Two-Period Model: State-Preference Approach
i= 2
i= 1 ciz
cjs
cjz
cis
Equilibrium 2
Equilibrium 1
q
q
Fig. 4.18 Multiple equilibria in an Edgeworth Box
iθi(q)−θ¯i
q∗
Fig. 4.19 Multiple equilibria in the excess demand diagram
We will finish this section by giving the formal argument that for simple mean- variance utilities financial market equilibria are stable. Before we do so let us mention that in the next chapter we will analyze two more adjustment processes:
The medium-term adjustment in which price expectations adjust on the basis of the realized returns (see Chap.5) and the long-run wealth adjustment in which the unsuccessful agents leave the market. The latter is also called the market selection dynamics, and is studied in evolutionary finance (see Sect.5.7.1).
Now we prove the short-term stability of equilibria if prices are continuously adjusted in the direction of excess demand. This adjustment process is called the
“Law of Demand and Supply”:
Proposition 4.14 Assume that agents have mean-variance preferences of the form Ui.ci1; : : : ;ciS/Di.ci1; : : : ;ciS/ i
22;i.ci1; : : : ;ciS/: (4.5) Then there is a unique globally stable market equilibrium.
Fig. 4.20 Multiple equilibria in the excess demand diagram
q∗
iθi(q)−θ¯i
iθi(q)−θ¯i
q∗
q∗
iθi(q)−θ¯i
Fig. 4.21 A market crash resulting from small changes in the asset payoffs
q∗
A
206 4 Two-Period Model: State-Preference Approach
Proof NotecisDRsiwi0and recall the budget constraintPK
kD0i;kD1. Substitute cisusing (4.5), then for any portfoliowe get:
Ui.i;1; : : : ; i;K/Dwi0
i.R0i/ iwi0
2 2;i.R0i/
:
The solution is, as before, 0D1
XK kD1
k; iD.COVi.R//1i.R/Rf1
iwi0 :
Written in economic terms (recallRkD Aqkk) this reads:
iD 1
iwi0.q/
COVi.A/1
i.A/Rfq
;
and finally, sincei;kD i;qkkwi, we get the asset demand function i;kD i;k.1ic/wi0
qk D 1
i
COVi.A/1
i.A/Rfq
; as in Sect.4.4.
The continuous time dynamics (Law of Demand and Supply) guarantee that prices adjust in a way that reduces the difference between demand and supply.
Mathematically, this can be expressed by P
qtDa XI iD1
i;k.qt/ XI
iD1
Ai;k
!
;
wherea> 0is the speed of adjustment. Note that the market demand is monotone, i.e.
.qOt Qqt/ XI
iD1
i;k.qOt/ XI
iD1
i;k.qQt/
!
< 0; for allqOt;qQt; (4.6)
because @qti.qt/ D Rfi.COVi.A//1, which is a negative definite matrix if there are no redundant assets. To prove stability, we define a so called “Lyapunov function”L.t/WD kqtqk2and show thatL.t/is decreasing int. We compute
L.t/P D@tL.t/D2.qtq/qPt
D2.qtq/a XI
iD1
i;k.qOt/ XI
iD1
i;k.qQt/
!
;
hence we can apply the monotonicity (4.6) and arrive at L.t/P D2.qtq/a
XI iD1
i;k.qt/i;k.q/
!
< 0:
This proves that the market is in fact globally stable. ut