Steady states, stability and bifurcations in multi-asset market models

As we will see in the sequel, one attractive feature of exponential replicator dynamics is that it ensures a steady state in which the price of the risky asset mirrors its fundamental value, provided that investors have sufficient funds. See Dindo and Tuinstra (2011) for an insightful discussion of exponential replicator dynamics and Bischi et al. (2015) and Schmitt et al. (2017) for recent applications. However, Agliari et al. (2018) study a related stock market participation model on the basis of the discrete choice approach by Brock and Hommes (1998). To improve our understanding of multi-asset market dynamics, future work may also consider the transition probability approach by Lux (1995).

With an abuse of notation, \(r_{l}\) denotes the ‘feasibility threshold’ for both the 2D and 4D model. A comparison of \(r_l=\frac{D}{\alpha N}\) (2D model) and \(r_l=\frac{D_1}{\alpha _1 N}+\frac{D_2}{\alpha _2 N}\) (4D model) reveals, among others, that for \(D=D_1=D_2\) and \(\alpha =\alpha _1=\alpha _2\) (symmetric markets) twice as many investors are needed to make the FSS feasible.

We have also observed numerically the same Neimark–Sacker bifurcation value \( \beta _{u}^{(II)}\) with a three-risky-asset extension of the present model.

Simulation results reveal that interesting associated phenomena occur also when threshold \(r_{l}\) is crossed with parameters that do not satisfy condition (17). In this case, the motion of investor shares, previously confined to subset \(x_{1,t}+x_{2,t}=1\), takes place in the inner part of the phase space, with investors switching across all assets (including the safe asset) and prices fluctuating below their NFSS values \( \hat{P}_{i}=D_{i}/r_{l}\), \(i=1,2\).

Elementary row and column operations performed on matrix \(\mathbf {J}(\mathbf { s}^{*})-v\mathbf {I}_{4}\) lead to a simplified matrix \(\mathbf {Q=Q(}\nu \mathbf {)}\) with additional zero entries. Then, \(Det(\mathbf {Q})=Det\left( \mathbf {J}(\mathbf {s}^{*})-v\mathbf {I}_{4}\right) \) can easily be computed by cofactor expansion, yielding a tractable expression for the characteristic polynomial (23).

Note that the lower-dimensional set defined by \(x_{1}+x_{2}=1\) is invariant for the map that governs the dynamical system. In fact, from \( x_{2,t}=1-x_{1,t}\) we obtain \(x_{2,t+1}=1-x_{1,t+1}\), as can be checked directly from Eq. (11).

At the transcritical bifurcation value, where \(r=r_{l}\), stability condition (31) becomes identical to the stability condition (27) of the FSS.