From Disequilibrium Markets to Equilibrium
Abstract
The modeling of financial markets as disequilibrium models by ordinary differential equations has become a popular modeling tool. One famous example of such a model is the BejaGoldman model [beja1980dynamic] which we consider in this paper. We study the passage from disequilibrium dynamics to equilibrium. Mathematically, this limit corresponds to an asymptotic limit also known as a TikhonovFenichel reduction. Furthermore, we analyze the stability of the reduced equilibrium model and discuss the economic implications. We conduct several numerical examples to visualize and support our analysis.
Keywords: BejaGoldmann Model, disequilibrium, equilibrium, rational market, asymptotic limit, TikhnovFenichel reduction, high frequency trader
1 Introduction
In the past decades the interest in rational markets built on the general equilibrium theory has shifted to irrational markets also known as disequilibrium models.
General equilibrium theory dates back to the early works of Walras [walras2013elements] and has been further developed in the last century by McKenzie, Arrow and Debreu [walker2006walrasian].
Heuristically speaking, an equilibrium price is reached when supply matches demand. This equilibrium theory in particular assumes that we have a rational market meaning that
there are no transaction costs and perfect informations. Many major contributions in finance such as the portfolio theory by Markowitz [markowitz1952portfolio] and Merton [merton1969lifetime] or the capital asset pricing model by Sharpe [sharpe1964capital] and Lintner [lintner1965security] are built on the rational market hypothesis.
Mainly based on the restricted nature of the assumptions of a rational markets, the general equilibrium theory has been critizised [beja1977orders, beja1980dynamic, heertje2002recent, ackerman2002still]. This has lead to the theory of disequilibrium markets [beja1980dynamic, chiarella1986perfect, he2011dynamic, day1990bulls] probably first introduced by Beja and Goldman [beja1980dynamic]. Major contributions in the field of agentbased models are build on the idea of disquilibrium markets [frankel1990chartists, day1990bulls, chiarella1992dynamics, chiarella1992developments, kirman1993ants, lux1995herd, hommes2006heterogeneous]. Before we are able to present the general idea of disequlibrium markets we have to introduce the notion of aggregated excess demand [mantel1974characterization, debreu1974excess, sonnenschein1972market].
The excess demand denotes the aggregated supply and demand of all financial agents. More precisely the excess demand is defined as the sum of agents’ supply subtracted from agent’s demand.
For an equilibrium price
(1) 
has to hold. The price denotes the logarithmic stock price of an asset. More generally one can even consider for a nonlinear function , zero at the origin as suggested by several authors [campbell1997econometrics, kempf1999market, cont2000herd]. Nevertheless, the linear form (1) can be seen as linearization of the nonlinear form as argued in [beja1980dynamic, SABCEMM]. The general form of a disequilibrium market model is then given by
(2) 
The constant denotes the market depth or the speed of price adjustment [kempf1999market]. In comparison to the equlibirum market model (1) the market depth is finite.
Mathematically speaking, the disequilibirum model (2) is a relaxation of the algebraic relation (1) with the relaxation parameter . In this paper we study the asymptotic limit for disequilibrium models which consist of price equations of the type (2) coupled to an additional ODE. Thus, we study the limit from an disequilibirum model to an equilibirum model.
Such asymptotic limits are also known in the case of a singular perturbation problem as TikhonovFenichel reductions [fenichel1979geometric, tikhonov1952systems, goeke2014constructive, Hoppensteadt]. This theory for dynamical systems has been mainly applied to chemical reaction kinetics [heineken1967mathematical, lax2018singular, frank2018quasi].
The advantage of this asymptotic limit is to obtain a possibly much simpler reduced form of the original dynamical system. Such a reduced system can be conveniently analyzed by classical tools and is a good approximation of the original dynamics for small .
In this paper we consider two different asymptotic limits of the BejaGoldman model. The BejaGoldman model is a two dimensional dynamical system. The price equation is of the form (2) and is coupled
to the time evolution of the chartist estimate. More precisely, the aggregated excess demand is given by the excess demand of two representative agents, chartist and fundamentalists.
First we study the so called liquid market limit which corresponds to an infinite large market depth. The reduced model can be seen as the equilibrium market version of the BejaGoldman model.
Secondly we study the liquid chartist limit which can be seen as the limit of an infinite fast reaction speed of chartists. One may characterize this limit as a disequilibrium market model with highfrequency trader.
The resulting reduced models are one dimensional ordinary differential equations coupled to an algebraic equation, which we can easily analyze.
The outline of the paper is as follows. In the next section we introduce the BejaGoldman model in detail and present the dynamical behavior. In section 3 we give an introduction to singular perturbation problems and the TikonovFenichel reduction. Then we derive the reduced BejaGoldman model in the liquid market and liquid chartist limit, present numerical experiments and analyze the obtained reduced models.
In section 4 we give an economic interpretation of the results and a small conclusion of this work.
2 The BejaGoldman Model
We introduce the BejaGoldman model [beja1980dynamic] and present the dynamical behavior of the model, as studied in [beja1980dynamic]. The BejaGoldman model with parameters and scaling parameter (or relaxation parameter) is given by
(3a)  
(3b) 
Here, denotes the logarithmic stock price and so called chartist’ price estimate. The parameter denotes the bond return and the fundamental price. The aggregated excess demand is defined as the sum of the fundamental () and chartists’ () demands:
The parameter respectively denote the market power of fundamentalists respectively chartists. This modeling approach of chartist and fundamental demand dates back to the work by Zeeman [zeeman1974unstable] and has been frequently used in agent based modeling [levy2000microscopic, chiarella2006asset, brock1997rational, brock1998heterogeneous, chiarella2006asset, franke2012structural]. The scaling parameter denotes the inverse market depth. The chartists’ estimate is defined as an relaxation of the price change. The parameter is the inverse reaction speed of chartists.
We can rewrite the BejaGoldman model (3) as follows:
(4a)  
(4b) 
The matrix form of our system (4) reads.
with
Mathematically, our model is an inhomogeneous linear differential system. The eigenvalues of are given by
and the eigenvectors are:
Proposition 1.
The solution basis of the ODE
is given by
in the case of real eigenvalues . In the case of complex eigenvalues the solution basis reads:
We define the fundamental matrix of the BejaGoldman model by . Then, the solution of the BejaGoldman model is given by
We want to recap the stability results of the BejaGoldman model as discussed in [beja1980dynamic].
Proposition 2.
We aim to visualize the different limit behavior of the price and chartist estimate as analyzed in Proposition 2. For sufficiently large market power of the fundamentalists it is always possible to obtain stable dynamics. More precisely stable dynamics lead to convergence of the chartist estimate to zero and to the equilibrium price . This behavior can be obtained for the regions , as defined in Figure 1, see Figure 2 and Figure 3. Increasing the market power of chartists leads to oscillatory stable behavior (region ), then to oscillatory unstable behavior (region ) and finally to a blow up (region ). This dynamic is depicted in the Figures 3 and 4. Furthermore, we have an example of the dynamics at the stability border (see Figure 4).
3 Asymptotic Limits
In this section we study the asymptotic limits of the BejaGoldman model (4). First, we define the precise form of a singular perturbation problem and then introduce the main tools in order to study asymptotic limits.
Definition 1.
Given an open set and an analytic function . Then the set
is called zero set of .
Definition 2.
We call a parameter dependent system of autonomous ordinary differential equations of the form
(5) 
with analytical function a singular perturbation problem, provided that the zero set of does not solely consists of isolated points.
By rescaling the time we can rewrite (5) as:
(6) 
The goal is to determine the limit system as . By analyzing the limit system, which is of lower dimension and thus in general easier to analyze, one expects to gain information on the original system (5) for small . In particular, the limit system is (for small ) a good approximation to the original system. Examples of such asymptotic reductions are given by [grad1963asymptotic, frank2018quasi, heineken1967mathematical]. To our knowledge such asymptotic limits have not been applied rigorously to any economic model. Formally (i.e. without any proof of convergence), singular perturbation models can be studied by an asymptotic expansion also known as Hilbert expansion [mckean1967chapman]. The ansatz for the solution of the system (5) reads:
(7) 
As a next step one inserts the asymptotic expansion (7) into the original dynamical system. Then one can deduce the reduced system by comparing different orders of . As one would expect this methodology does not always lead to the correct reduction as no convergence result is provided. Therefore, in this paper we consider a rigorous reduction approach based on the early works of Tikhonov and Fenichel [fenichel1979geometric, tikhonov1952systems] as well as Hoppensteadt [Hoppensteadt]. More precisely, we follow the constructive appraoch introduced by Goeke, Noethen and Walcher [goeke2014constructive, noethen2011tikhonov] and use Lax, Selinger, Walcher [LaxHoppensteadt] for an optimized convergence result. We will employ the following theorem as presented in [goeke2014constructive].
Theorem 1.
Given a system of the form (6) for analytical . We assume there exist a point in the zero set of , such that is maximal in a neighborhood of . Then there exists a neighborhood of such that is a dimensional submanifold. If a sum decomposition of the form
exists, then the following results hold:

There exist a product decomposition
with analytic functions
such that holds. Furthermore, the zero set of satisfies .

Define
Then the system
(8) is well defined on .

If there exists a such that
is fulfilled (i.e. the real part of every nonzero eigenvalue of the Jacobian at point is uniformly bounded from above by a negative constant), then there exists a and a neighborhood of such that solutions of (6) with initial conditions in for all on uniformly converge to the solution of the reduced system (8) on as .
Remark 1.

is often called the slow manifold, which we also will do in the following.

One should emphasize that the convergence result stated above is only valid on an finite interval. After leaving the Interval (i.e. ) the solution may start to oscilate, blowup or show any other behaviour; see e.g. [kruff2019rosenzweig] Section 4.1.

In [LaxHoppensteadt], conditions are presented such that the convergence holds  loosely spoken  for all positive times. In particular, Proposition 2.10 therein implies that the existence of exactly one stationary point which is exponentially stable (i.e. the linearization of the reduces system at has only one negative eigenvalue) in tie with a technical condition is sufficient for the desired convergence for all positive times.
In the rest of the chapter we show that the BejaGoldman model is a singular perturbation problem in the liquid market limit and liquid chartist limit. Then we derive the corresponding reductions by use of Theorem 1. Finally, we analyze the reduced systems and present several numerical tests.
3.1 Liquid Market Limit
In this section we study the liquid market limit which corresponds to an infinite market depth. We treat the parameter as a given parameter.
Proposition 3.
The BejaGoldman model as defined in (4) is a singular perturbation model in the liquid market regime.
Proof.
The slow manifold is given by the zero set of:
Thus we have , which does not solely consist of isolated points. ∎
As a next step we derive the reduced system.
Proposition 4.
Proof.
We utilize Theorem 1 and define:
The Jacobian of is given by:
and hence the conditions of Theorem 1 are satisfies as the nonzero eigenvalue is simple and negative for . Next, we define
In particular, holds and the Jacobian of reads:
Then and we can define as follows:
Hence, we get
Then the reduced system reads
where is given by
or alternatively by the previously computed ODE. The convergence for all follows by Proposition 2.10 of [LaxHoppensteadt], using that there exists only one stationary point of the limit system, which is exponentially stable. (We omit a discussion of the additional technical condition that needs to be satisfied. But it can easily be proven that the condition holds using the stability properties of the original system (4)).
Computing the explicit solution is then basic analysis.
∎
Corollary 1.
Proof.
We insert a Hilbert expansion:
Then we can insert the previous expansion in our model and obtain for different orders:
Price equation:  
Chartist equation:  
We use the price equation of order to rewrite the chartist equation of order as follows:
As next step we utilize the equality
which is obtained by differentiating the price equation of order . We get
and thus the statement is shown. ∎

If holds then the slow manifold still exists and is still invariant, but now is repelling, i.e. every solution not starting on the slow manifold will never reach the slow manifold. Solutions starting on the slow manifold will now diverge to as , which corresponds to the unstable case for system (4).
Finally, we present some numerical results. We have computed the solutions of the original BejaGoldman model with the Matlab solver ode15s which is especially designed for stiff ordinary differential equations. From Figure 7 we can deduce that the original model asymptotically converges to the reduced model if . This asymptotic behavior of both models for different parameters can be seen in Figure 9. Thus, our numerical test validate our analysis that in this case the BejaGoldman model converges to the reduced model (9) as . In addition we study the case if no reduction exists (). In fact the slow manifold still exists but is repelling. In Figure 8 we show the cases of initial values laying on the slow manifold or not laying on the slow manifold. One can observe that the original system is always repelling but diverges slower in the case of initial values on the slow manifold.
3.2 Liquid Chartist Limit
In this section we study the limit which corresponds to a infinite fast reaction speed of chartists. The parameter is treated as a constant and is fixed.
Proposition 5.
The BejaGoldman model as defined in (4) is a singular perturbation model in the liquid chartist regime.
Proof.
The slow manifold is given by the zero set of:
Then the zero set does not solely consist of isolated points. ∎
Proposition 6.
Let the BejaGoldman model be as defined in (4) and assume . Then the reduced system in the liquid chartist limit reads
(10)  
(11) 
and the convergence holds for all . is given by . The unique solution of the stock price equation is given by:
Proof.
The result for finite time follows directly by Tikhonov’s Theorem [tikhonov1952systems]. One also can use Theorem 1 with
as well as
A straightforward computation shows
which gives (10), where . Inserting yields (11). The convergence on infinite intervals follows again by [LaxHoppensteadt] Proposition 2.10 (in the same manner as in the proof of Proposition 4). Solving the reduced equation is again a simple computation. ∎
Corollary 2.
Proof.
As before, we insert a Hilbert expansion:
Then comparing the different orders, we get:
Price equation:  
Chartist equation: 
∎

If holds then the slow manifold still exists and is still invariant, but now is repelling. Solutions starting on the slow manifold will now diverge to as , which corresponds to the unstable case for system (4).

The case is degenerate: There exists no slow manifold in this case. As shown previously in the liquid market regime it is possible to study the original system for different small values .
We aim to complete this section with some numerical tests for the liquid chartist limit. The asymptotic behavior of the solution of the original model and the reduced model is depicted in Figure 10. The distance between both models with respect to is shown in Figure 12. Both numerical tests support our analysis that for the BejaGoldman model converges to the reduced model (10). Furthermore, in the case no reduction exists and Figure 11 depicts the repelling slow manifold. Here, we see that the solution of the formally reduced system remains on the slow manifold while the solution of the original model oscillates with increasing amplitude.
4 Economic Implications and Conclusion
Before we discuss the liquid market and liquid chartist limit we want to quickly summarize the behavior of the original BejaGoldman model in the standard parameter regime.
The stability behavior depends on an interplay of the parameters . In particular, the system is always stable if the strength of chartist is below the inverse market depth i.e. .
If no chartists are present the system is always stable. Increasing the impact of chartist leads after a certain threshold to stable oscillatory behavior, then to unstable oscillatory behavior and to a blow up for sufficiently large . In fact, not only the impact of chartists, but especially the speed of the adjustment of their estimate heavily influences the price behavior.
In case of a valid reduction we expect that the behavior of the original system is well approximated by the corresponding reduction. Thus, in summary the model characteristics of the reduced model in the
liquid market limit and liquid chartist limit are given by:

Liquid Market Limit In the equilibrium market case, which corresponds to the limit of an infinite market depth, the reduced system only depends on the parameter . The system is stable and is a valid reducton of the original system if the strength of fundamentalists is larger than the strength of chartists times the speed of the adjustment of the estimate i.e. . Thus, the price converges to the value and the chartist estimate to zero.

Liquid Chartist Limit The behavior of the reduced system in the liquid chartist case is very similar to the liquid market case. The reduced model depends on the parameter . The reduced system exists and is stable if the inverse market depth is larger that the strength of chartists i.e. . The asymptotic equilibrium price is given by and the chartist estimate converges to zero.
The previously described market regimes are only present in extreme situations i.e. in case of an infinite market depth or in case of infinitely fast high frequency trader. Unfortunately, the reduced models are only a valid approximation of the original model for special sets of parameters. As discussed previously, the reduction in the liquid market limit (liquid chartist limit) is only a reduction of the original model for (). Therefore one may ask critically what the benefits of TikhonovFenichel reductions are. In general, we habe 3 main advantages:

The reductions replicate original model behavior.

The reductions are simpler to analyze than the original systems (as the dimension is always lower).

The reductions make it possible to study the original system even if no explicit solution of the original model exists.
In cases where the original model can be solved explicitly, TikhonovFenichel reductions still can be applied but one does not obtain novel insights of the original model.
Therefore this study has to be seen as a novel example to utilize the asymptotic theory, although we do not gain new information on the BejaGoldman model.
This work rather shows the novel interpretation of a financial market model as a singular perturbation problem and exemplified we study the TikhonovFenichel reductions.
Thus, we have derived the reduced system of the BejaGoldman model for two asymptotic limits, the liquid market limit and the liquid chartist limit.
Furthermore, we have analyzed the behavior of these reduced systems and have verified our analysis by several numerical test.
Especially the simple analysis of the reduced systems which perfectly replicate the asymptotic behavior of the original model reveals the importance of TikhonovFenichel reductions.
Finally, we want to point out that such an analysis can be extended to more general disequilibirum models provided they are in singular perturbation form.
This is of major importance in the case that the original model is more complicated (i.e. of higher dimension) and the asymptotic behavior cannot be studied directly.
Acknowledgement
T. Trimborn was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC2023 Internet of Production – 390621612.
T. Trimborn gratefully acknowledges support by the HansBöcklerStiftung and the RWTH Aachen University StartUp grant.
T. Trimborn acknowledges the support by the ERS Prep Fund  Simulation and Data Science.
The work was partially funded by the Excellence Initiative of the German federal and state governments.