Download Projekt 21: 21 Strategic Behavior on Financial Markets

? Projekt 21:
Strategic Behavior on Financial Markets
?
1
Projekt 21:
21
.
Strategic Behavior on Financial Markets
→ Deutsche Version Seite 3 / German version page 3
Jointly with Yakar Kannai, Faculty of Mathematics and Computer
Science, Weizmann Institute, Rehovot, Israel.
We exhibit the model of an exchange economy with money and credit as a
non-cooperative game.
This model can be characterized as follows.
1. There exists money, serving both as a mean of exchange and as a store
of value;
2. agents are price makers (and not just price takers);
3. there exists a central bank who issues money, accepts deposits, and
lends;
4. bankruptcy is not ruled out, but is penalized;
5. yet the bank tries to keep the number of bankrupt firms as small as
possible.
Our model includes elements of the above items. There is a finite set of agents
involved in trade of a Shapley-Shubik type ([3], [4]), along with a central bank
able to issue money, distribute it as loans, and accept deposits. The central
bank has the authority to determine the various interest rates. Agents would
derive a negative utility from being bankrupt, whereas positive cash holdings
at the end of the period have positive utility, the latter presumably deriving
from subsequent use of money at a later period.
Each agent is endowed with positive amounts of a consumer nondurable commodity and money. Agents issue bids in terms of money towards purchasing
a quantity of the consumption good. (Agents cannot consume directly their
commodity endowment in whole or parts.) Agents may exceed their endowment (and thus take a loan from the bank), or else they may bid less than
? Projekt 21:
Strategic Behavior on Financial Markets
?
2
their endowment, their money surplus going to the bank as a deposit. There
is a central bank in the market which controls the interest rates for deposits
and loans and increases the total amount of money, if the books cannot be
balanced otherwise. As soon as bids are announced, the price of the commodity is given by the ratio of the aggregate bid to the aggregate supply of
the good. Each agent then receives for consumption the good bought by his
bid and the money proceeds of the selling of his commodity endowment. In
addition, our agent receives returns from her bank deposit or has to pay the
loan (with interest).
At the end of the day, each agent has 1) consumed an amount of the commodity (deriving from it a positive amount of utility), 2) is unable to repay
his loan with the prescribed interest, so that he is bankrupt and derives a
negative utility from this fact, or else 3) has a positive amount of cash left,
from which she derives positive utility.
The bank announces a policy concerning interest rates on deposits and loans.
Formally, this policy is a (vector-valued) function of the agents’ bids. The
agents, in turn, may take into account the bank’s policy. In this manner
a well defined game (the financial market game) is specified. Hence, bids
play the role of strategies. The bank may try to achieve certain objectives
through its policies. E.g., the wish to eliminate unnecessary bankruptcies
the desire to combat inflation could be such objectives. We exhibit a policy
which leads to certain desirable outcomes.
We establish the existence of a Nash equilibrium for the financial market
game. (An essential element of the proof is the construction of a compact set
of strategies which is mapped into itself by the best response correspondence.)
Under certain regularity conditions we demonstrate the existence of a (Nash)
equilibrium in mixed strategies. For a specific policy we prove existence of
an equilibrium in pure strategies.
Our goal is to put forward a multi-period model where the utility for holding
cash reserves at the end of the j-th period is derived from the utility of having
this reserve as an endowment for the j + 1-st period, and obtaining (using
backward induction) a subgame-perfect equilibrium in pure strategies. We
plan to achieve this goal in a sequel to this paper.
See page 4 for some literature.
? Projekt 21: Finanzmärkte ?
3
Projekt 21:
21
Strategisches Verhalten auf Finanzmärkten
Gemwinsames Projekt mit Yakar Kannai, Faculty of Mathematics and
Computer Science, Weizmann Institute, Rehovot, Israel.
Im Projekt “Strategic Behavior on Financial Markets” wird der Versuch
gemacht, strategisches Verhalten auf Finanzmärkten einzuführen. Die derzeit
dominierende Ideologie setzt Preisnehmerverhalten seitens der auf einem Finanzmarkt agierenden Agenten voraus. Damit wird eine infinitesimal schnelle
Anpassung des Marktgeschehens über Arbitrage an das Gleichgewicht angenommen. Offensichtlich ist aber strategisches Verhalten auf Finanzmärkten zu
beobachten. Die Börsenentwicklung der letzten Zeit zeigt gerade, daß Anlagemöglichkeiten in Finanzgütern, bei denen eine bekannte Verteilung der
stochastischen Entwicklung vorausgesetzt wird, durch strategisches Verhalten der Anleger beinträchtigt werden.
In diesem Projekt wird ein Modell konstruiert in dem die Zentralbank Zinssätze
in Abhängigkeit vom Verhalten der Agenten auf den Finanzmärkten festsetzt.
Diese Politik wird bekannt gegeben. Es ist die Aufgabe der Zentralbank einerseits inflationäre Tendenzen durch Kontrolle der Geldmenge zu steuern
und andererseits Unternehmensschließungen mangels Liquidität zu verhindern. Zinspolitiken sollen nach Möglichkeit beide Ziele im Auge haben – und
diese Zielsetzung kann modellmäßig sehr wohl präzisiert werden.
Bei vorgegebener Zinspolitik ziehen die Agenten Nutzen aus Kapitaleinsatz wie auch aus Anlageverhalten. Preise und Zinsen werden von diesem
(“strategischen”) Verhalten der Agenten beeinflußt und die Auszahlungen
schwanken daher mit den Geboten sämtlicher Agenten. Auf diese Weise
entsteht ein n–Personenspiel, dessen Nash–Gleichgewichtspunkte untersucht
werden können.
In einer neu erschienen Arbeit [2] wird ein Gleichgewicht im statischen Modell
beschreiben. Dynamische Modelle sollen gegebenenfalls mit Hilfe iterierte
Spiele später behandelt werden.
? REFERENCES ?
4
Einige Literatur:
References
[1] Karatzas M. Shubik M. Geanakoplos, J. and W. Sudderth, A strategic
market game with active bankruptcy, JME 34 (2000), 359–396.
[2] Y. Kannai and J. Rosenmüller, Strategic behavior on financial markets,
Working Paper 351, Institute of Mathematical Economics, University of
Bielefeld, Bielefeld, Germany, July 23, 2003, 24 pp.
[3] L. S. Shapley, Non-cooperative general exchange, in: Theory and measurement of externalities (S.A.Y. Lin, ed.), Academic Press, New York,
1976, pp. 155 – 177.
[4] L.S. Shapley and M. Shubik, Trading using one commodity as means of
payment, Journal of Political Economy 85 (1977), 937 – 968.