Interethnic relations, informal trading networks, and social integration: imitation, habits, and social evolution

Abstract

The ethnically homogeneous middleman groups (EHMGs), which are informal trading networks, are ubiquitous in less-developed economies where the legal infrastructure for contract enforcement is not well developed. This paper develops a formal model of social interaction among members of the EHMG as well as in more general situations in a multi-ethnic or multi-cultural society consisting of identifiable ethnic or linguistic groups. Behavioral patterns are transmitted between generations and altered via imitation in social contacts. The model demonstrates how different discriminatory behavioral patterns can evolve and persist over time. One result is that the trust between such groups can increase due to a higher frequency of inter-group contacts. In concluding the paper, we speculate about how the model can be expanded to include changes in legal structures, especially contract law. This could lead to an increase in the trust between the different groups.

Appendices

Appendix A: The dynamics in a phase diagram

In this appendix we analyze how r and $\pi _{1}$ influence the equilibrium values of the p’s. We find the dynamics in the $p_{2}-p_{3}$-dimensions of the simplex $p_{1}+p_{2}+p_{3}=1$.

1.1 A.1 The dynamics of $p_{3}$

The stationary points of $p_{3}$ are found by solving the equation $\dot{p}_{3}=0$. It is clear from the equations of motion, 3.7, that $p_{3}=0$ defines a set of stationary points for $p_{3}$. It is also readily seen that these points are locally attracting, by evaluating $\dot{p}_{3}/p_{3}$ at $p_{3}=0$:

$$\begin{aligned} \frac{\dot{p}_{3}}{p_{3}}=-r\left[ n\left( 1\right) c_{b}-n\left( 0\right) c_{w}\right] -\pi _{1}\left( 1-r\right) n\left( 1-p_{2}\right) c_{b} \end{aligned}$$

(A.1)

Under Assumption 3.11, this is negative and, hence, $p_{3}=0$ is a stable value. If $p_{3}=1$, the equation of motion reduces to:

$$\begin{aligned} \dot{p}_{3}=-\pi _{1}\left( 1-r\right) n\left( 0\right) c_{b} \end{aligned}$$

(A.2)

This is negative, unless $\pi _{1}=0$. The stable equilibrium $p_{3}=1$ in the case of no interaction is thus altered if in the outside world there is a positive probability of meeting an individual that can be trusted ($\pi _{1}>0$), which is stated in proposition 4.2.

If there are other stationary points of $p_{3}$ than $p_{3}=0$, then they have to solve equation:

$$\begin{aligned} \frac{1-p_{3}}{n\left( 1-p_{2}-p_{3}\right) }\left[ n\left( p_{3}\right) c_{w}-n\left( 1-p_{3}\right) c_{b}\right] =\pi _{1}\frac{1-r}{r}c_{b} \end{aligned}$$

(A.3)

For a given value of $p_{2}$ and a sufficiently small $\pi _{1}$, this equation has at least two roots under Assumptions 3.8 and 3.11. For a sufficiently large $\pi _{1}$ and a sufficiently small r, it might have no root, and $\dot{p}_{3}<0$ everywhere. The case with two roots has been pictured in the phase diagram, Fig. 2. The lines $\left( A_{3}^{*},A_{3}^{o}\right)$ and $\left( A_{3}^{**},A_{3}^{oo}\right)$ are the loci of stationary points for $p_{3}$. The first one is a repellent and the second one an attractor. The $p_{3}$ coordinates of points $A_{3}^{*}$ and $A_{3}^{**}$, $p_{3}^{*}$ and $p_{3}^{**}$, solve equation:

$$\begin{aligned} \frac{1-p_{3}}{n\left( 1-p_{3}\right) }\left[ n\left( p_{3}\right) c_{w}-n\left( 1-p_{3}\right) c_{b}\right] =\pi _{1}\frac{1-r}{r}c_{b} \end{aligned}$$

(A.4)

The $p_{2}$ coordinate is, of course, zero. The $p_{3}$ coordinates of $A_{3}^{o}$ and $A_{3}^{oo}$, $p_{3}^{o}$ and $p_{3}^{oo}$, solve equation:

$$\begin{aligned} \frac{1-p_{3}}{n\left( 0\right) }\left[ n\left( p_{3}\right) c_{w}-n\left( 1-p_{3}\right) c_{b}\right] =\pi _{1}\frac{1-r}{r}c_{b} \end{aligned}$$

(A.5)

The $p_{2}$ coordinates are $1-p_{3}^{o}$ and $1-p_{3}^{oo}$, respectively.

The behavior of these stationary points can be visualized in simple diagrams. Define $f^{*}(p_{3})$, $f^{o}(p_{3})$, and h as:

$$\begin{aligned} f^{*}(p_{3})&:=\frac{1-p_{3}}{n\left( 1-p_{3}\right) }\left[ n\left( p_{3}\right) c_{w}-n\left( 1-p_{3}\right) c_{b}\right] \nonumber \\ f^{o}(p_{3})&:=\frac{1-p_{3}}{n\left( 0\right) }\left[ n\left( p_{3}\right) c_{w}-n\left( 1-p_{3}\right) c_{b}\right] \nonumber \\ h&:= \pi _{1}\frac{1-r}{r}c_{b} \end{aligned}$$

(A.6)

We note that:

$$\begin{aligned} f^{o}(p_{3})=\frac{n(1-p_{3})}{n(0)}f^{*}(p_{3}) \end{aligned}$$

(A.7)

That is, $\vert f^{o}(p_{3})\vert \ge \vert f^{*}(p_{3})\vert$ and they are equal if $p_{3}=1$ or if the expression in the square brackets is zero. In Figs. 3, 4, and 6 the stationary points of $p_{3}$ are indicated for different values of h. Figure 3 corresponds to the situation in the phase diagram in Fig. 2 with three distinct regions of relevance for the dynamics of $p_{3}$, one with an increasing and two with a decreasing $p_{3}$.

In Fig. 4 the situation is pictured when $\pi _{1}$ approaches zero or r approaches one. Now, $p_{3}^{**}$ and $p_{3}^{oo}$ approach one and $p_{3}^{*}$ and $p_{3}^{o}$ become identical, in fact, the curve ($A_{3}^{*}$, $A_{3}^{o}$) becomes a straight line and the curve ($A_{3}^{**}$, $A_{3}^{oo}$) degenerates to the point $p_{3}=1$. The equilibrium in dominant strategies in a world without interaction is a limiting case of an equilibrium in a world with vanishing interaction. The resulting phase diagram with two regions for the dynamics of $p_{3}$ is pictured in Fig. 5.

In Fig. 6 $p_{3}^{*}$ and $p_{3}^{**}$ disappear and eqilibrium D disappears, too, Fig. 7, or becomes globally very unstable, Fig. 8, depending on the size of $\pi _{1}$.

1.2 A.2 The dynamics of $p_{2}$

Fig. 2

Phase diagram for $\dot{p}_{2}$ and $\dot{p}_{3}$

Fig. 3

Stationary points of $p_{3}$ for some values of $\pi _{1}$ and r, $0<\pi _{1}<1$ and $0<r<1$

Fig. 4

Stationary points of $p_{3}$ as $\pi _{1}\rightarrow 0$ or $r\rightarrow 1$

Fig. 5

Phase diagram for $\dot{p}_{2}$ and $\dot{p}_{3}$ as $\pi _{1}\rightarrow 0$ and $0<r<1$

Fig. 6

Stationary points of $p_{3}$ for sufficiently large values of $\pi _{1}$ or sufficiently small values of r, $0<\pi _{1}<1$ and $0<r<1$

Fig. 7

Phase diagram for $\dot{p}_{2}$ and $\dot{p}_{3}$ as $\pi _{1}\rightarrow 1$ and r is sufficiently small

Fig. 8

Phase diagram for $\dot{p}_{2}$ and $\dot{p}_{3}$ for $0<\pi _{1}<1$ and r sufficiently small

The equation of motion for $p_{2}$ is:

$$\begin{aligned} \dot{p}_{2} =&\left( 1-p_{2}-p_{3}\right) \left( 1-\pi _{1}\right) \left( 1-r\right) n\left( p_{2}+p_{3}\right) c_{w} \nonumber \\&-p_{2}\pi _{1}\left( 1-r\right) n\left( 1-p_{2}-p_{3}\right) c_{b} \nonumber \\&+p_{2}p_{3}r\left[ n\left( 1-p_{3}\right) c_{b}-n\left( p_{3}\right) c_{w}\right] +\left( 1-p_{2}-p_{3}\right) p_{3}rn\left( 1-p_{3}\right) c_{b} \end{aligned}$$

(A.8)

If $p_{2}=0$, it reduces to:

$$\begin{aligned} \dot{p}_{2}=\left( 1-p_{3}\right) \left[ \left( 1-\pi _{1}\right) \left( 1-r\right) n\left( p_{3}\right) c_{w}+p_{3}rn\left( 1-p_{3}\right) c_{b}\right] \end{aligned}$$

(A.9)

This is positive if $p_{3}<1$. For $p_{2}=1$ we similarly find:

$$\begin{aligned} \dot{p}_{2}=-\pi _{1}\left( 1-r\right) n\left( 0\right) c_{b} \end{aligned}$$

(A.10)

This is negative, unless $\pi _{1}=0$. Hence, due to continuity, there is a region in the phase diagram for sufficiently small values of $p_{2}$, where $p_{2}$ is growing and a region for sufficiently high values of $p_{2}$, where $p_{2}$ is getting smaller. Along the edges of the phase diagram, for $p_{3}=0$ and $p_{2}+p_{3}=1$, we can find expressions for the limits of these regions. If $p_{3}=0$, we find:

$$\begin{aligned} \dot{p}_{2}=\left( 1-p_{2}\right) \left( 1-\pi _{1}\right) \left( 1-r\right) n\left( p_{2}\right) c_{w}-p_{2}\pi _{1}\left( 1-r\right) n\left( 1-p_{2}\right) c_{b} \end{aligned}$$

(A.11)

and for $p_{2}+p_{3}=1$:

$$\begin{aligned} \dot{p}_{2}=-p_{2}\pi _{1}\left( 1-r\right) n\left( 0\right) c_{b}+p_{2}p_{3}r\left[ n\left( 1-p_{3}\right) c_{b}-n\left( p_{3}\right) c_{w}\right] \end{aligned}$$

(A.12)

The stationary points on the $p_{3}=0$ axis are given by the solution of:

$$\begin{aligned} \frac{1-p_{2}}{n\left( 1-p_{2}\right) }\frac{n\left( p_{2}\right) }{p_{2}}=\frac{\pi _{1}}{1-\pi _{1}}\frac{c_{b}}{c_{w}} \end{aligned}$$

(A.13)

Under Assumption 3.10, this equation has a unique solution, $p_{2}^{*}$, which does not depend on r and which is monotone in $\pi _{1}$ and equal to one if $\pi _{1}$ is zero and zero if $\pi _{1}$ is one. The stationary points for $p_{2}+p_{3}=1$ are at $p_{2}=0$ or given as the solutions to the equation:

$$\begin{aligned} \pi _{1}\frac{1-r}{r}c_{b}=\frac{p_{3}}{n\left( 0\right) }\left[ n\left( 1-p_{3}\right) c_{b}-n\left( p_{3}\right) c_{w}\right] \end{aligned}$$

(A.14)

This can be rewritten as:

$$\begin{aligned} h=-\frac{p_{3}}{1-p_{3}}f^{o}(p_{3}) \end{aligned}$$

(A.15)

If $\pi _{1}$ is sufficiently small and r sufficiently big, this equation will have at least two roots, which are the $p_{3}$ coordinates of the points $A_{2}^{o}$ and $A_{2}^{oo}$. It is readily seen that these points will lie to the left of the points $A_{3}^{o}$ and $A_{3}^{oo}$ in the phase diagram in Fig. 2, since the expression in the square brackets will have to have the opposite sign in the expressions for the stationarity of $p_{2}$ and $p_{3}$.

In Figs. 9 and 10 we have indicated, how the stationary points can be found. Figure 9 corresponds to the phase diagram in Fig. 2. Similarly, in Fig. 10 the limiting situation with $h\rightarrow 0$, corresponding to the phase diagram in Fig. 5 is illustrated.

1.3 A.3 Global stability

Fig. 9

Stationary points of $p_{2}$ (and $p_{3}$) for some values of $\pi _{1}$ and r, $0<\pi _{1}<1$ and $0<r<1$

Fig. 10

Stationary points of $p_{2}$ (and $p_{3}$) as $\pi _{1}\rightarrow 0$ or $r\rightarrow 1$

Fig. 11

Phase diagram for $\dot{p}_{2}$ and $\dot{p}_{3}$ for sufficiently small r

An inspection of the various phase diagrams clearly indicates the smaller is r the more attractive globally becomes equilibrium G. For sufficiently small values of r the D equilibrium even disappears. Furthermore, for sufficiently large values of h (small r and/or large $\pi _{1}$), the phase diagram consists of only two regions and G is the only equilibrium, the location of which is determined by the value of $\pi _{1}$, see the phase diagram in Fig. 11. Hence, contact feeds trust.

Appendix B: Feasible interaction equilibria

We will show the existence of fixed points of the mappings in Sect. 4.2.1. We show several lemmata.

Lemma 5.1

The mapping $g(\gamma (p))$ has at least three fixed points, $p=0$, $p=1$, and $p=\hat{p}\in (0,1)$.

Proof

First two are trivial. For the third one, we note that the function g is monotone and implicitly defined by

$$\begin{aligned} \frac{g(\pi _{1})}{1-g(\pi _{1})}=\frac{\pi _{1}}{1-\pi _{1}}\frac{n\left( g(\pi _{1})\right) c_{b}}{n\left( 1-g(\pi _{1})\right) c_{w}} \end{aligned}$$

(B.1)

and $\gamma$ by

$$\begin{aligned} \frac{\gamma (p_{1})}{1-\gamma (p_{1})}=\frac{p_{1}}{1-p_{1}}\frac{\nu \left( \gamma (p_{1})\right) \kappa _{b}}{\nu \left( 1-\gamma (p_{1})\right) \kappa _{w}}\text {.} \end{aligned}$$

(B.2)

The function g has a fixed point $\check{p}$ given by $n(\check{p} )c_{b}=n\left( 1-\check{p}\right) c_{w}$. In view of Assumption , this point exists, and it is unique if n(p) is strictly monotone. By the same argument we find the fixed point $\check{\pi }$ of $\gamma$. If $1>p>\check{p}$, then $g(p)<p$, and if $0<p<\check{p}$, then $g(p)>p$. The corresponding result holds for $\gamma$ and $\check{\pi }$. Combining these results, we find that if $1>p>\max \{\check{p},\check{\pi }\}$ , then $g(\gamma (p))<p$, and if $0<p<\min \{\check{p},\check{\pi }\}$, then $g(\gamma (p))>p$. By continuity, the fixed point $\hat{p}=g\left( \hat{\pi } \right)$ exists and $\min \{\check{p},\check{\pi }\}\le \hat{p}=g\left( \hat{\pi }\right) \le \max \{\check{p},\check{\pi }\}$, and by the same token $\hat{\pi }=\gamma (\hat{p})$ also satisfies $\{\check{p},\check{\pi }\}\le \hat{\pi }=\gamma (\hat{p})\le \max \{\check{p},\check{\pi }\}$. $\square$

Remark B.1

If the two groups are identical, $\hat{p}$ is unique and equal to $\check{p}= \check{\pi }$.

Lemma B.2

The mapping $g(\delta _{1}(p))$ has at least one fixed point, $p=0$.

Proof

Trivial. $\square$

Lemma B.3

The mapping $d_{1}(\delta _{1}(p))$ has at least one fixed point, $p=0$.

Proof

Trivial. $\square$

There exist then at least the six feasible equilibria listed in Sect. 4.2.1.

Appendix C: Proof of proposition 4.4

We linearize the system 4.2 at the various equilibria and calculate the eigenvalues of the determinants of the matrices of partial derivatives of the equations of motion. The matrices of partial derivatives are given in Appendix D. The eigenvalues associated with the various equilibria are:

Equilibrium (i):

$$\begin{aligned} \left( \begin{array}{c} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \end{array} \right) =\left( \begin{array}{c} -a-\frac{r}{1-r}\left( a-b\right)<0 \\ -\alpha -\frac{\rho }{1-\rho }\left( \alpha -\beta \right)<0 \\ \frac{1}{2}\left( -a-\alpha +\sqrt{\left( a+\alpha \right) ^{2}-4\left( a\alpha -\beta b\right) }\right)<0 \\ \frac{1}{2}\left( -a-\alpha -\sqrt{\left( a+\alpha \right) ^{2}-4\left( a\alpha -\beta b\right) }\right) <0 \end{array} \right) \text {,} \end{aligned}$$

(C.1)

where $a=a\left( 1\right)$, $\alpha =\alpha \left( 1\right)$,$\ b=b\left( 0\right)$,$\ \beta =\beta \left( 0\right)$. Since $a\left( 1\right)>b\left( 0\right) >0$ and $\alpha \left( 1\right)>\beta \left( 0\right) >0$ , all four eigenvalues are real and negative. Hence, equilibrium (1) is locally stable.

Equilibrium (iii):

$$\begin{aligned} \left( \begin{array}{c} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \end{array} \right) =\left( \begin{array}{c} -e\left( 1,0\right)<0 \\ -\epsilon \left( 1,0\right)<0 \\ \frac{1}{2}\left( -\beta -b+\sqrt{\left( \beta +b\right) ^{2}-4\left( \beta b-\alpha a\right) }\right)<0 \\ \frac{1}{2}\left( -\beta -b-\sqrt{\left( \beta +b\right) ^{2}-4\left( \beta b-\alpha a\right) }\right) <0 \end{array} \right) \text {,} \end{aligned}$$

(C.2)

where $a=a\left( 0\right)$, $\alpha =\alpha \left( 0\right)$,$\ b=b\left( 1\right)$,$\ \beta =\beta \left( 1\right)$. Since $b\left( 1\right)>a\left( 0\right) >0$ and $\beta \left( 1\right)>\alpha \left( 0\right) >0$ , all four eigenvalues are real and negative. Hence, equilibrium (3) is locally stable.

Equilibrium (iv):

$$\begin{aligned} \left( \begin{array}{c} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \end{array} \right) =\left( \begin{array}{c} -e\left( 1,0\right)<0 \\ \varepsilon \left( 0,1\right)<0 \\ \frac{1}{2\left( 1-\rho \right) }\left( -w+\sqrt{w^{2}-4\left( 1-\rho \right) \left[ b\beta -\alpha a\left( 1-\rho \right) \right] }\right)<0 \\ \frac{1}{2\left( 1-\rho \right) }\left( -w-\sqrt{ w^{2}-4\left( 1-\rho \right) \left[ b\beta -\alpha a\left( 1-\rho \right) \right] }\right) <0 \end{array} \right) \text {,} \end{aligned}$$

(C.3)

where $a=a\left( 0\right)$, $\alpha =\alpha \left( 0\right)$,$\ b=b\left( 1\right)$,$\ \beta =\beta \left( 1\right)$, and $w:=b\left( 1-\rho \right) +\beta$. Since $b\left( 1\right)>a\left( 0\right) >0$ and $\beta \left( 1\right)>\alpha \left( 0\right) >0$, all four eigenvalues are real and negative. Hence, equilibrium (4)—and, by symmetry, equilibrium (5)—is locally stable.

Equilibrium (vi):

$$\begin{aligned} \left( \begin{array}{c} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \end{array} \right) =\left( \begin{array}{c} e\left( 0,1\right)<0 \\ \varepsilon \left( 0,1\right)<0 \\ \frac{1}{2\omega }\left( -w+\sqrt{w^{2}-4\omega \left[ b\beta -\alpha a\omega \right] }\right)<0 \\ \frac{1}{2\omega }\left( -w-\sqrt{w^{2}-4\omega \left[ b\beta -\alpha a\omega \right] }\right) <0 \end{array} \right) \text {,} \end{aligned}$$

(C.4)

where $a=a\left( 0\right)$, $\alpha =\alpha \left( 0\right)$,$\ b=b\left( 1\right)$,$\ \beta =\beta \left( 1\right)$, $\omega =\left( 1-r\right) \left( 1-\rho \right)$, and $w:=b\left( 1-\rho \right) +\beta \left( 1-r\right)$. Since $b\left( 1\right)>a\left( 0\right) >0$ and $\beta \left( 1\right)>\alpha \left( 0\right) >0$, all four eigenvalues are real and negative. Hence, equilibrium (6) is locally stable.

Equilibrium (ii):

Here, in order to make the problem tractable, we have to make the assumption that both groups have identical functions describing their behavior. That is, $a(x)=\alpha (x)$, $b(x)=\beta (x)$, etc., which, of course, implies that $\hat{p}=\hat{\pi }$. Then the eigenvalues are:

$$\begin{aligned} \left( \begin{array}{c} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \end{array} \right) =\left( \begin{array}{c} d>0 \\ -\hat{p}a(\hat{p})-\frac{r}{1-r}\left[ a\left( 1\right) -b\left( 0\right) \right]<0 \\ -\hat{p}a(\hat{p})-\frac{r}{1-r}\left[ a\left( 1\right) -b\left( 0\right) \right]<0 \\ -2a(\hat{p})+d<0 \end{array} \right) , \end{aligned}$$

(C.5)

where $d:=\hat{p}\left( 1-\hat{p}\right) \left( 1-r\right) \left[ \frac{ \partial n\left( 1-\hat{p}\right) }{\partial \left( 1-p\right) }c_{w}+\frac{ \partial n\left( \hat{p}\right) }{\partial p}c_{b}\right]$. The last inequality follows from the fact that, by Assumption 3.10, $a( \hat{p})>d$.

The positive eigenvalue d is associated with the eigenvector

$$\begin{aligned} \left( \begin{array}{c} e_{p_{1}} \\ e_{p_{2}} \\ e_{\pi _{1}} \\ e_{\pi _{2}} \end{array} \right) =\left( \begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \end{array} \right) \text {.} \end{aligned}$$

(C.6)

Hence, a perturbation increasing (or decreasing) $p_{1}$ and $\pi _{1}$ by a small amount, holding $p_{1}+p_{2}$ and $\pi _{1}+\pi _{2}$ constant (which is a movement in the two simplices) will cause a movement away from the equilibrium (2). Equilibrium (2) is thus unstable.

Appendix D: The matrices of partial derivatives of the equations of motion

In this appendix, we give the matrices of the partial derivatives of the equations of motion, Eq. 4.2. For the definitions of the various parameters, we refer to the discussion in the main text.

For equilibrium (i), (1, 0, 0; 1, 0, 0), we have:

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial \dot{p}}{\partial p} &{} \frac{\partial \dot{p}}{\partial \pi } \\ \frac{\partial \dot{\pi }}{\partial p} &{} \frac{\partial \dot{\pi }}{\partial \pi } \end{array} \right) =\left( \begin{array}{cccc} \frac{r}{1-r}b-a &{}\quad \frac{r}{1-r}b &{}\quad b &{}\quad 0 \\ -\frac{r}{1-r}a &{}\quad -\frac{1}{1-r}a &{}\quad -b &{}\quad 0 \\ \beta &{}\quad 0 &{}\quad \frac{\rho }{1-\rho }\beta -\alpha &{}\quad \frac{\rho }{1-\rho }\beta \\ -\beta &{} 0 &{} -\frac{\rho }{1-\rho }\alpha &{} -\frac{1}{1-\rho }\alpha \end{array} \right) \text {,} \end{aligned}$$

(D.1)

for equilibrium (ii), $(\hat{p},1-\hat{p},0;\hat{\pi },1-\hat{\pi },0)$:

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial \dot{p}}{\partial p} &{} \frac{\partial \dot{p}}{\partial \pi } \\ \frac{\partial \dot{\pi }}{\partial p} &{} \frac{\partial \dot{\pi }}{\partial \pi } \end{array} \right) =\left( \begin{array}{cc} \Psi &{} \begin{array}{cc} a(\hat{p}) &{}\quad 0 \\ -a(\hat{p}) &{}\quad 0 \end{array} \\ \begin{array}{cc} a(\hat{p}) &{}\quad 0 \\ -a(\hat{p}) &{}\quad 0 \end{array}&\Psi \end{array} \right) , \end{aligned}$$

(D.2)

where

$$\begin{aligned} \Psi :=\left( \begin{array}{cc} -a(\hat{p})+d+\hat{p}\frac{r}{1-r}b\left( 0\right) &{} \hat{p}\frac{r}{1-r} b\left( 0\right) \\ \begin{array}{c} \left[ (1-\hat{p})a(\hat{p})-\frac{r}{1-r}a\left( 1\right) \right. \\ \left. +(1-\hat{p})\frac{r}{1-r}b\left( 0\right) -d\right] \end{array} &{} \begin{array}{c} \left[ -\hat{p}a(\hat{p})-\frac{r}{1-r}a\left( 1\right) \right. \\ \left. +(1-\hat{p})b\left( 0\right) \right] \end{array} \end{array} \right) \text {,} \end{aligned}$$

(D.3)

for equilibrium (iii), (0, 1, 0; 0, 1, 0):

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial \dot{p}}{\partial p} &{} \frac{\partial \dot{p}}{\partial \pi } \\ \frac{\partial \dot{\pi }}{\partial p} &{} \frac{\partial \dot{\pi }}{\partial \pi } \end{array} \right) =\left( \begin{array}{cccc} -b &{}\quad 0 &{}\quad a &{}\quad 0 \\ b-e &{}\quad -e &{}\quad -a &{} \quad 0 \\ \alpha &{}\quad 0 &{}\quad -\beta &{}\quad 0 \\ -\alpha &{}\quad 0 &{}\quad \beta -\epsilon &{}\quad -\epsilon \end{array} \right) , \end{aligned}$$

(D.4)

for equilibrium (iv), (0, 1, 0; 0, 0, 1):

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial \dot{p}}{\partial p} &{} \frac{\partial \dot{p}}{\partial \pi } \\ \frac{\partial \dot{\pi }}{\partial p} &{} \frac{\partial \dot{\pi }}{\partial \pi } \end{array} \right) =\left( \begin{array}{cccc} -b &{}\quad 0 &{}\quad a &{}\quad 0 \\ b-e &{}\quad -e &{}\quad -a &{}\quad 0 \\ \alpha &{}\quad 0 &{}\quad -\frac{\beta }{1-\rho } &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \beta +\frac{\rho }{1-\rho }\alpha &{}\quad -\delta \end{array} \right) , \end{aligned}$$

(D.5)

and, finally, for equilibrium (vi), (0, 0, 1; 0, 0, 1):

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial \dot{p}}{\partial p} &{} \frac{\partial \dot{p}}{\partial \pi } \\ \frac{\partial \dot{\pi }}{\partial p} &{} \frac{\partial \dot{\pi }}{\partial \pi } \end{array} \right) =\left( \begin{array}{cccc} -\frac{b}{1-r} &{}\quad 0 &{}\quad a &{}\quad 0 \\ b+\frac{r}{1-r}a &{}\quad -d &{}\quad 0 &{}\quad 0 \\ \alpha &{}\quad 0 &{}\quad -\frac{\beta }{1-\rho } &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \beta +\frac{\rho }{1-\rho }\alpha &{}\quad -\delta \end{array} \right) \text {.} \end{aligned}$$

(D.6)