Heterogeneity in social values and capital accumulation in a changing world

Abstract

In a society characterized by a multitude of heterogeneous agents and a large number of possibly immaterial goods, each one having distinct social and personal values, we study the impact of these relative values on intergenerational capital accumulation, as a function of economic and social parameters such as capital mobility, productivity and personal and social values discrepancies. Each agent is modelled by a one-period production function and a two-period intertemporal utility. Agents live, produce and consume over one period, but optimize over two periods, so providing a remaining stock of goods for the next generation. This creates a dynamics in capital accumulation depending on social and individual values. A threshold appears in capital stock accumulation that depends on personal and social values’ volatilities, and below which the initial stock will be depleted. Whereas volatility in social values increases the threshold, impairing capital accumulation, adverse shocks in goods’ values may reverse the dynamics of the accumulation process. Finally, capital mobility specifically favors forerunners, but capital accumulation in one or several sectors may shift social values in their direction, at the expense of other sectors.

Appendix

The optimization problem:

$$\begin{aligned}&\left( 1-\rho +\delta \rho V_{t}^{t}\left( \phi _{t}+Y_{t}\right) \right) V_{t}+\gamma \phi _{t}+\varepsilon \rho \left( \left( V_{t}-\chi _{t}\right) ^{t}\left( \phi _{t}+Y_{t}\right) \right) \left( V_{t}-\chi _{t}\right) \\&\quad =\left( \alpha +\delta \rho \right) V_{t}\left( \left( V_{t}\right) ^{t}C_{t}\right) +\beta \left( V_{t}-\chi _{t}\right) \left( \left( V_{t}-\chi _{t}\right) ^{t}C_{t}\right) \\&\qquad +\,\gamma C_{t}+\varepsilon \rho \left( V_{t}-\chi _{t}\right) \left( \left( V_{t}-\chi _{t}\right) ^{t}C_{t}\right) \end{aligned}$$

can be written in a matricial form:

$$\begin{aligned} W_{t}=\left( \gamma +N\right) C_{t} \end{aligned}$$

(10)

where we have defined:

$$\begin{aligned} W_{t}= & {} \left( 1-\rho +\delta \rho V_{t}^{t}\left( \phi _{t}+Y_{t}\right) \right) V_{t}+\gamma \phi _{t}+\varepsilon \rho \left( \left( V_{t}-\chi _{t}\right) ^{t}\left( \phi _{t}+Y_{t}\right) \right) \left( V_{t}-\chi _{t}\right) \\ N_{ij}= & {} \left( \alpha +\delta \rho \right) \left( V_{t}\right) _{i}\left( V_{t}\right) _{j}+\left( \beta +\varepsilon \rho \right) \left( V_{t}-\chi _{t}\right) _{i}\left( V_{t}-\chi _{t}\right) _{j} \end{aligned}$$

Finding $C_{t}$ and then $Y_{t}$ to solve the optimization problem amounts to invert the matrix $\left( \gamma +N\right)$. To do so, we decompose N in several terms. First, we set:

$$\begin{aligned} X_{t}= & {} \frac{\sqrt{\alpha }V_{t}}{\sqrt{\gamma }} \nonumber \\ Z_{t}= & {} \frac{\sqrt{\beta +\varepsilon \rho }\left( V_{t}-\chi _{t}\right) }{\sqrt{\gamma }} \end{aligned}$$

(11)

and introduce the normalization $\left( V_{t}\right) ^{t}\left( V_{t}\right) =\left( \chi _{t}\right) ^{t}\left( \chi _{t}\right) =\left( E_{t}\chi _{t+1}\right) ^{t}\left( E_{t}\chi _{t+1}\right) =1$. We call $H_{t}$ the vector space spanned by $X_{t}$ and $Z_{t}$. We also introduce the compact notation (for $i=1,2$):

$$\begin{aligned} X^{\left( 1\right) }=X_{t}\text {, }X^{\left( 2\right) }=Z_{t} \end{aligned}$$

Omitting temporarily the time index t, we then write N as the sum:

$$\begin{aligned} N=\sum _{i=1,2}M^{\left( i\right) } \end{aligned}$$

where $M^{\left( i\right) }=X^{\left( i\right) }\left( X^{\left( i\right) }\right) ^{t}$. We will find $\left( \gamma +N\right) ^{-1}$ by an expansion in powers of N. We thus need to compute $\left( \sum _{i}M^{\left( i\right) }\right) ^{k}$. It is obtained by using the following relations:

$$\begin{aligned} \left( \sum _{i}M^{\left( i\right) }\right) ^{k}=\sum _{i,l}c_{k}^{\left( il\right) }P^{\left( il\right) } \end{aligned}$$

with $c_{k}^{\left( il\right) }$ satisfying the following recursive relation:

$$\begin{aligned} c_{k+1}^{\left( il\right) }=\sum _{m}c_{k}^{\left( im\right) }\gamma ^{\left( ml\right) } \end{aligned}$$

(12)

and where we defined:

$$\begin{aligned} P^{\left( il\right) }= & {} X^{\left( i\right) }\left( X^{\left( l\right) }\right) ^{t} \\ \gamma ^{\left( il\right) }= & {} \left( X^{\left( i\right) }\right) ^{t}X^{\left( l\right) } \end{aligned}$$

Gathering the coefficients $c_{k}^{\left( im\right) }$ and $\gamma ^{\left( ml\right) }$ in a matricial form:

$$\begin{aligned} C_{k+1}= & {} \left( c_{k}^{\left( im\right) }\right) ,G=\left( \gamma ^{\left( ml\right) }\right) \\ C_{k}= & {} C_{1}G^{k-1} \end{aligned}$$

the previous relation (12) rewrites:

$$\begin{aligned} C_{k+1}=C_{k}G \end{aligned}$$

and one can check that:

$$\begin{aligned} C_{1}= & {} \left( \delta _{il}\right) \\ C_{k}= & {} G^{k-1} \end{aligned}$$

This formula allows to obtain $\left( \sum _{i}M^{\left( i\right) }\right) ^{k}$ as a trace over indices $i=1,2$:

$$\begin{aligned} \left( \sum _{i}M^{\left( i\right) }\right) ^{k}=Tr\left( G^{k-1}P^{t}\right) \end{aligned}$$

so that ultimately:

$$\begin{aligned} \left( 1+\sum _{i}M^{\left( i\right) }\right) ^{-1}= & {} 1+\sum _{k=1}^{\infty }\left( -\sum _{i}M^{\left( i\right) }\right) ^{k}=1+\sum _{k=1}^{\infty }Tr\left( \left( -1\right) ^{k}\left( G\right) ^{k-1}P^{t}\right) \\= & {} 1-Tr\left( \left( 1+G\right) ^{-1}P^{t}\right) \end{aligned}$$

and as a consequence:

$$\begin{aligned} \left( \gamma +N\right) ^{-1}=\gamma ^{-1}\left( 1+\frac{N}{\gamma }\right) ^{-1}=\gamma ^{-1}\left( 1-Tr\left( \left( 1+G\right) ^{-1}P^{t}\right) \right) \end{aligned}$$

(13)

$\left( \gamma +N\right) ^{-1}$ can thus be found straightforwardly by writing $1+G$ and P as:

$$\begin{aligned}&1+G=\left( \begin{array}{cc} 1+X^{t}X &{} X^{t}Z \\ X^{t}Z &{} 1+Z^{t}Z \end{array} \right) \\&P^{t}=\left( \begin{array}{cc} XX^{t} &{} ZX^{t} \\ XZ^{t} &{} ZZ^{t} \end{array} \right) \end{aligned}$$

and by using the following identity on the space $H_{t}$:

$$\begin{aligned} \frac{1}{\left( Z^{t}Z\right) \left( X^{t}X\right) -\left( X^{t}Z\right) ^{2} }Tr\left( \begin{array}{cc} Z^{t}Z &{} -X^{t}Z \\ -X^{t}Z &{} X^{t}X \end{array} \right) \left( \begin{array}{cc} XX^{t} &{} ZX^{t} \\ XZ^{t} &{} ZZ^{t} \end{array} \right) =1 \end{aligned}$$

Actually, this last relation yields on $H_{t}$:

$$\begin{aligned}&\frac{1}{\left( 1+Z^{t}Z\right) \left( 1+X^{t}X\right) -\left( X^{t}Z\right) ^{2}}Tr\left( \begin{array}{cc} 1+Z^{t}Z &{} -X^{t}Z \\ -X^{t}Z &{} 1+X^{t}X \end{array} \right) \left( \begin{array}{cc} XX^{t} &{} ZX^{t} \\ XZ^{t} &{} ZZ^{t} \end{array} \right) \\&\quad =\frac{\left( Z^{t}Z\right) \left( X^{t}X\right) -\left( X^{t}Z\right) ^{2}+XX^{t}+ZZ^{t}}{\left( 1+Z^{t}Z\right) \left( 1+X^{t}X\right) -\left( X^{t}Z\right) ^{2}} \end{aligned}$$

and we are thus led to the following expression for the restriction $\left( \gamma +N\right) _H^{-1}$ of $\left( \gamma +N\right) ^{-1}$ on $H_{t}$:

$$\begin{aligned} \left( \gamma +N\right) _H^{-1}= & {} \gamma ^{-1}\left( 1-Tr\left( \left( 1+G\right) ^{-1}P^{t}\right) \right) \\= & {} \gamma ^{-1}\frac{1+Z^{t}Z+X^{t}X-XX^{t}-ZZ^{t}}{\left( 1+Z^{t}Z\right) \left( 1+X^{t}X\right) -\left( X^{t}Z\right) ^{2}} \end{aligned}$$

In the sequel, all dynamic quantities will be projected on $H_{t}$, through their scalar product with $V_{t}$ or $\chi _{t}$. As a consequence, for the sake of clarity, we can identify $\left( \gamma +N\right) _H^{-1}$ and $\left( \gamma +N\right) ^{-1}$. Ultimately, using (11), and reintroducing the time dependency, we can rewrite $\left( \gamma +N\right) ^{-1}$ as a function of $V_{t}$ and $\left( V_{t}-\chi _{t}\right)$:

$$\begin{aligned}&\left( \gamma +N\right) ^{-1} =\gamma ^{-1}\left( 1-Tr\left( \left( 1+G\right) ^{-1}P^{t}\right) \right) \\&\quad =\frac{\gamma +\left( \alpha +\delta \rho \right) +\left( \beta +\varepsilon \rho \right) \left( V_{t}-\chi _{t}\right) ^{t} \left( V_{t}-\chi _{t}\right) -\left( \alpha +\delta \rho \right) V_{t}\left( V_{t}\right) ^{t}-\left( \beta +\varepsilon \rho \right) \left( V_{t}-\chi _{t}\right) \left( V_{t}-\chi _{t}\right) ^{t}}{\left( \gamma +\left( \beta +\varepsilon \rho \right) \left( V_{t}-\chi _{t}\right) ^{t}\left( V_{t}-\chi _{t}\right) \right) \left( \gamma +\left( \alpha +\delta \rho \right) \right) -\left( \alpha +\delta \rho \right) \left( \beta +\varepsilon \rho \right) \left( \left( V_{t}\right) ^{t}\left( V_{t}-\chi _{t}\right) \right) ^{2}} \end{aligned}$$

Now, if we let:

$$\begin{aligned} z=\sin ^{2}x \end{aligned}$$

where

$$\begin{aligned} x=\frac{1}{2}\arg (V_{t},\chi _{t}) \end{aligned}$$

the scalar products $\left( V_{t}-\chi _{t}\right) ^{t}\left( V_{t}-\chi _{t}\right)$, $\left( V_{t}\right) ^{t}\left( V_{t}-\chi _{t}\right)$ and $\left( V_{t}\right) ^{t}V_{t}$ can be computed and give the following alternate expression for $\left( \gamma +N\right) ^{-1}$:

$$\begin{aligned} \left( \gamma +N\right) ^{-1}=\frac{\gamma +4\left( \beta +\varepsilon \rho \right) z+\left( \alpha +\delta \rho \right) -\left( \alpha +\delta \rho \right) V_{t}\left( V_{t}\right) ^{t}-\left( \beta +\varepsilon \rho \right) \left( V_{t}-\chi _{t}\right) \left( V_{t}-\chi _{t}\right) ^{t}}{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\left( \alpha +\delta \rho \right) \right) -4\left( \alpha +\delta \rho \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}} \end{aligned}$$

We are interested in the dynamics for $\phi _{t}.\chi _{t}$, the social value of a certain stock of capital goods. Recall the form of the production function:

$$\begin{aligned} Y_{t}= & {} \left( \eta \left( \phi _{t}.\chi _{t}\right) \right) ^{1-a}\left( \frac{K_{2}\left( \xi ,\left( \phi _{t}.V_{t}\right) \right) \xi V_{t}+K_{1}\left( 1-\xi ,\left( \phi _{t}.\chi _{t}\right) \right) \left( 1-\xi \right) \chi _{t}}{\sqrt{1-4\xi \left( 1-\xi \right) z^{2}}}\right) -\eta \left( \phi _{t}\right) \\\simeq & {} \eta \left( \phi _{t}.\chi _{t}\right) \left( K_{2}\xi V_{t}+K_{1}\left( 1-\xi \right) \chi _{t}\right) -\eta \left( \phi _{t}\right) \end{aligned}$$

where $K_{2}$ and $K_{1}$ are the values of $K_{2}\left( \xi ,\left( \phi _{t}.V_{t}\right) \right)$ and $K_{1}\left( 1-\xi ,\left( \phi _{t}.\chi _{t}\right) \right)$ at the optimal value of $\xi$. We define the following quantity:

$$\begin{aligned} {\tilde{\phi }}_{t}=\phi _{t}+Y_{t}=\left( 1-\eta \right) \phi _{t}+\eta \left( \phi _{t}.\chi _{t}\right) \left( K_{2}\xi V_{t}+K_{1}\left( 1-\xi \right) \chi _{t}\right) \end{aligned}$$

and rewrite the vector $W_{t}$ involved in the consumption as:

$$\begin{aligned} W_{t}= & {} uV_{t}+v\left( V_{t}-\chi _{t}\right) +\gamma \phi _{t}\\ W_{t}= & {} \left( 1-\rho +\delta \rho V_{t}^{t}\left( \phi _{t}+Y_{t}\right) \right) V_{t}+\gamma \phi _{t}+\varepsilon \left( \left( V_{t}-\chi _{t}\right) ^{t}\left( \phi _{t}+Y_{t}\right) \right) \left( V_{t}-\chi _{t}\right) \\ u= & {} 1-\rho +\delta \rho V_{t}^{t}\left( \phi _{t}+Y_{t}\right) \\= & {} 1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}.V_{t}+\eta \left( \phi _{t}.\chi _{t}\right) \left( K_{2}\xi +\left( 1-2z\right) K_{1}\left( 1-\xi \right) \right) \right) \\ v= & {} \varepsilon \rho \left( \left( V_{t}-\chi _{t}\right) ^{t}\left( \phi _{t}+Y_{t}\right) \right) \\= & {} \varepsilon \rho \left( \left( 1-\eta \right) \phi _{t}.\left( V_{t}-\chi _{t}\right) +2z\eta \left( \phi _{t}.\chi _{t}\right) \left( K_{2}\xi -K_{1}\left( 1-\xi \right) \right) \right) \end{aligned}$$

The consumption is thus given by (10):

$$\begin{aligned} C_{t}= & {} \left( \gamma +N\right) ^{-1}W_{t}=\frac{\left( \left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) u-2\left( \alpha +\delta \rho \right) zv-\left( \alpha +\delta \rho \right) \gamma V_{t}.\phi _{t}\right) }{ \left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\left( \alpha +\delta \rho \right) \right) -4\left( \alpha +\delta \rho \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}V_{t} \\&+\,\frac{\left( \left( \gamma +\left( \alpha +\delta \rho \right) \right) v-2\left( \beta +\varepsilon \rho \right) zu-\left( \beta +\varepsilon \rho \right) \gamma \left( V_{t}-\chi _{t}\right) .\phi _{t}\right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\left( \alpha +\delta \rho \right) \right) -4\left( \alpha +\delta \rho \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\left( V_{t}-\chi _{t}\right) \\&+\,\gamma \frac{\gamma +4\left( \beta +\varepsilon \rho \right) z+\left( \alpha +\delta \rho \right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\left( \alpha +\delta \rho \right) \right) -4\left( \alpha +\delta \rho \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\phi _{t} \end{aligned}$$

and the dynamics for $\phi _{t}$ is deduced from the intertemporal constraint equation:

$$\begin{aligned} \phi _{t+1}= & {} \phi _{t}+Y_{t}-C_{t} \nonumber \\= & {} \left( \eta \left( \phi _{t}.\chi _{t}\right) \left( K_{2}\xi +K_{1}\left( 1-\xi \right) \right) \right. \nonumber \\&\left. -\,\frac{\left( \left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) u-2\alpha zv-\alpha \gamma V_{t}.\phi _{t}\right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha \right) -4\alpha \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\right) V_{t} \nonumber \\&-\,\left( \eta \left( \phi _{t}.\chi _{t}\right) K_{1}\left( 1-\xi \right) \right. \nonumber \\&\left. +\, \frac{\left( \left( \gamma +\alpha \right) v-2\left( \beta +\varepsilon \rho \right) zu-\left( \beta +\varepsilon \rho \right) \gamma \left( V_{t}-\chi _{t}\right) .\phi _{t}\right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha \right) -4\alpha \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\right) \left( V_{t}-\chi _{t}\right) \nonumber \\&+\,\left( \left( 1-\eta \right) -\gamma \frac{\gamma +4\left( \beta +\varepsilon \rho \right) z+\alpha }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha \right) -4\alpha \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\right) \phi _{t}\nonumber \\ \end{aligned}$$

(14)

We also choose, as in Sect. 4.2:

$$\begin{aligned} K_{1}\equiv K_{1}\left( \xi ,\left( \phi _{t}.\chi _{t}\right) \right)= & {} {{{\bar{K}}}}_{1}+\left( \phi _{t}.\chi _{t}\right) {\hat{K}}_{1}\\ K_{2}\equiv K_{2}\left( \xi ,\left( \phi _{t}.V_{t}\right) \right)= & {} {{{\bar{K}}}}_{2}+\left( \phi _{t}.V_{t}\right) {\hat{K}}_{2} \end{aligned}$$

In the expression $K_{1}\left( \phi _{t}.\chi _{t}\right) \xi +K_{2}\left( \phi _{t}.\chi _{t}\right) \left( 1-\xi \right)$, note that $\xi$ depends on $V_{t}$. However, due to the envelope theorem, the production functions do not depend, to the first order, on the derivatives of $\xi$ with respect to the other parameters. As a consequence we can, in first approximation, consider $\xi$ as a constant. Its dependence in the other parameters, $V_{t}$ and $\chi _{t}$, is of second order only, that is $\xi =\frac{1+\varphi z}{2}$ as second order approximation, with $\varphi$ a constant parameter.

1.1 Case 1

We start to solve the benchmark case $V_{t}=\chi _{t}$ so that $z=0$. Since the personal and social values are identical, we can assume that $K_{2}=K_{1}={{{\bar{K}}}}_{1}+\left( \phi _{t}.\chi _{t}\right) {\hat{K}}_{1}$.The dynamics equation (14) then reduces to:

$$\begin{aligned} \phi _{t+1}= & {} \phi _{t}+Y_{t}-C_{t} \nonumber \\= & {} \left( \eta \left( \phi _{t}.\chi _{t}\right) \left( K_{1}\xi +K_{2}\left( 1-\xi \right) \right) -\frac{\left( u-\alpha \chi _{t}.\phi _{t}\right) }{\gamma +\left( \alpha +\delta \rho \right) }\right) \chi _{t}-\eta \phi _{t} \end{aligned}$$

(15)

with

$$\begin{aligned} u =1-\rho +\delta \rho \left( \left( 1-\eta \right) +\eta \left( K_{1}\xi +\left( 1-2z\right) K_{2}\left( 1-\xi \right) \right) \right) \left( \phi _{t}.\chi _{t}\right) \end{aligned}$$

To find the evolution of the system, let us consider the exogenous dynamics for $\chi _{t}$ that was considered in the text:

$$\begin{aligned} \chi _{t+1}=\chi _{t}+\delta _{t+1} \end{aligned}$$

Normalizing the vector of values, $1=\chi _{t+1}^{2}=1+2\chi _{t}.\delta _{t+1}+\delta _{t+1}^{2}$, leads to the relation:

$$\begin{aligned} \chi _{t}.\delta _{t+1}=-\frac{1}{2}\delta _{t+1}^{2} \end{aligned}$$

so that $\delta _{t+1}$ is not orthogonal to $\chi _{t}$. Statistically, we decompose $\chi _{t+1}$ as $\chi _{t+1}=\chi _{t}+\delta _{t+1}=\chi _{t}+x\chi _{t}+\varepsilon _{t+1}$ with $\left\langle \varepsilon _{t+1}\right\rangle =0$, $\left\langle \varepsilon _{t+1}^{2}\right\rangle =\sigma ^{2}$, $\left\langle \chi _{t}.\varepsilon _{t+1}\right\rangle =0$. The condition on the norm becomes:

$$\begin{aligned} 1=\left\langle \left( \chi _{t}+x\chi _{t}+\varepsilon _{t+1}\right) ^{2}\right\rangle =1+2x+x^{2}+\sigma ^{2} \end{aligned}$$

so that $x=-1+\sqrt{1-\sigma ^{2}}$ and:

$$\begin{aligned} \chi _{t+1}= & {} \sqrt{1-\sigma ^{2}}\chi _{t}+\varepsilon _{t+1} \\ \left\langle \chi _{t+1}.\chi _{t}\right\rangle= & {} \sqrt{1-\sigma ^{2}} =\left\langle 1-\frac{1}{2}\delta _{t+1}^{2}\right\rangle \end{aligned}$$

Now, multiply the dynamics equation (15) by $\chi _{t+1}$. In average, it writes:

$$\begin{aligned} \frac{\left\langle \phi _{t+1}.\chi _{t+1}\right\rangle }{\sqrt{1-\sigma ^{2} }}= & {} \left( \eta \left( {{{\bar{K}}}}_{1}-1\right) +\frac{\alpha -\delta \rho \eta \left( {{{\bar{K}}}}_{1}-1\right) }{\gamma +\alpha +\delta \rho }\right) \left\langle \phi _{t}.\chi _{t}\right\rangle -\frac{1-\rho }{\left( \gamma +\alpha +\delta \rho \right) }\\&+\,\eta \left( \frac{\gamma +\alpha }{\gamma +\alpha +\delta \rho }\right) {\hat{K}}_{1}\left\langle \left( \phi _{t}.\chi _{t}\right) ^{2}\right\rangle \end{aligned}$$

Since the expectation is taken on $\chi _{t+1}$ at time t, $\left\langle \left( \phi _{t}.\chi _{t}\right) ^{2}\right\rangle =\left\langle \phi _{t}.\chi _{t}\right\rangle ^{2}$, one is led to:

$$\begin{aligned} \frac{\left\langle \phi _{t+1}.\chi _{t+1}\right\rangle }{\sqrt{1-\sigma ^{2} }}= & {} \frac{\left( \alpha +\eta \left( \alpha +\gamma \right) \left( {{{\bar{K}}}}_{1}-1\right) \right) }{\alpha +\gamma +\delta \rho }\left\langle \phi _{t}.\chi _{t}\right\rangle \\&-\,\frac{1-\rho }{\left( \gamma +\alpha +\delta \rho \right) }+\eta \frac{\gamma +\alpha }{\gamma +\alpha +\delta \rho }\hat{ K}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle ^{2} \end{aligned}$$

and this equation can be written in a compact form as:

$$\begin{aligned} \phi _{t+1}.\chi _{t+1}=a\phi _{t}.\chi _{t}-b+c\left( \phi _{t}.\chi _{t}\right) ^{2} \end{aligned}$$

where

$$\begin{aligned} a= & {} \left( \frac{\left( \alpha +\eta \left( \alpha +\gamma \right) \left( {{{\bar{K}}}}_{1}-1\right) \right) }{\alpha +\gamma +\delta \rho }\right) \sqrt{1-\sigma ^{2}} \\ b= & {} \frac{1-\rho }{\gamma +\alpha +\delta \rho }\sqrt{1-\sigma ^{2}} \\ c= & {} \eta \frac{\gamma +\alpha }{\gamma +\alpha +\delta \rho }\sqrt{1-\sigma ^{2}}{\hat{K}}_{1} \end{aligned}$$

Such an equation can be solved by a continuous time approximation:

$$\begin{aligned} y^{\prime }=\left( a-1\right) y-b+cy^{2} \end{aligned}$$

whose solution is:

$$\begin{aligned} \phi \left( t\right) .\chi \left( t\right) =\frac{\phi ^{+}-\phi ^{-}\frac{ \phi _{0}.\chi _{0}-\phi ^{+}}{\phi _{0}.\chi _{0}-\phi ^{-}}\exp \left( \Delta t\right) }{1-\frac{\phi _{0}.\chi _{0}-\phi ^{+}}{\phi _{0}.\chi _{0}-\phi ^{-}}\exp \left( \Delta t\right) } \end{aligned}$$

with

$$\begin{aligned} \Delta= & {} \sqrt{\left( a-1\right) ^{2}+4bc} \\ \phi ^{\pm }= & {} \frac{-\left( a-1\right) \pm \Delta }{2c} \\ \phi _{0}.\chi _{0}= & {} \phi \left( 0\right) .\chi \left( 0\right) \end{aligned}$$

As explained in the text, the dynamics presents a threshold pattern. Since $\phi ^{-}<0$, the following cases arise:

• If $\phi _{0}.\chi _{0}<\phi ^{+}$, then the dynamics converges, $\phi \left( t\right) .\chi \left( t\right) \rightarrow 0$.

• If $\phi ^{+}<\phi _{0}.\chi _{0}$, the dynamics diverges for a finite time $t_{0}$, defined by: $t_{0}=\frac{1}{\Delta }\ln \left( \frac{\phi ^{-}-\phi _{0}.\chi _{0}}{\phi ^{+}-\phi _{0}.\chi _{0}} \right)$.

• If the initial social value of the stock is low, then there is a decrease in stocks. On the contrary, for an initial value that is large enough, there is an exponential accumulation.

1.2 Case 2

To take into account the heterogeneity of agents with respect to the entire society, we will now consider the corrections due to $\left( V_{t}-\chi _{t}\right)$. We consider that $V_{t}$ differs from $\chi _{t}$ in the following way:

$$\begin{aligned} V_{t}=\sqrt{1-{\tilde{\sigma }}^{2}}\chi _{t}+{\tilde{\varepsilon }}_{t} \end{aligned}$$

where ${\tilde{\varepsilon }}_{t}$ is a random gaussian noise, with mean zero and variance ${\tilde{\sigma }}^{2}$ independent from $\varepsilon _{t}$.

We first compute some expectations that are relevant to derive the dynamics for the social value of the capital stock:

$$\begin{aligned} \left\langle \left( V_{t}-\chi _{t}\right) .\chi _{t+1}\right\rangle= & {} \left\langle \left( V_{t}-\chi _{t}\right) .\left( \sqrt{1-\sigma ^{2}} \chi _{t}+\varepsilon _{t}\right) \right\rangle \\= & {} \left\langle \left( \left( \sqrt{1-{\tilde{\sigma }}^{2}}-1\right) \chi _{t}+ {\tilde{\varepsilon }}_{t}\right) .\left( \sqrt{1-\sigma ^{2}}\chi _{t}+\varepsilon _{t}\right) \right\rangle \\= & {} \left( \sqrt{1-{\tilde{\sigma }}^{2}}-1\right) \sqrt{1-\sigma ^{2}} \\ \left\langle V_{t}.\chi _{t+1}\right\rangle= & {} \left\langle V_{t}.\left( \sqrt{1-\sigma ^{2}}\chi _{t}+\varepsilon _{t}\right) \right\rangle \\= & {} \left\langle \left( \left( \sqrt{1-{\tilde{\sigma }}^{2}}\right) \chi _{t}+ {\tilde{\varepsilon }}_{t}\right) .\left( \sqrt{1-\sigma ^{2}}\chi _{t}+\varepsilon _{t}\right) \right\rangle \\= & {} \sqrt{1-{\tilde{\sigma }}^{2}}\sqrt{1-\sigma ^{2}} \end{aligned}$$

and:

$$\begin{aligned} \left\langle z\right\rangle= & {} \left\langle \frac{1-V_{t}.\chi _{t}}{2} \right\rangle =\frac{1-\left\langle V_{t}.\chi _{t}\right\rangle }{2} \\= & {} \frac{1-\left\langle \left( \sqrt{1-{\tilde{\sigma }}^{2}}\chi _{t}+\tilde{ \varepsilon }_{t}\right) .\chi _{t}\right\rangle }{2} \\= & {} \frac{1-\sqrt{\left( 1-{\tilde{\sigma }}^{2}\right) }}{2} \end{aligned}$$

We will also need the expectations of some squared terms, such as $\left\langle \left( \phi _{t}.V_{t}\right) ^{2}\right\rangle$, $\left\langle \left( \phi _{t}.V_{t}\right) \left( \chi _{t}.V_{t}\right) \right\rangle$, $\left\langle \left( \chi _{t}.V_{t}\right) \left( \chi _{t}.V_{t}\right) \right\rangle$ and $\left\langle \left( \phi _{t}.V_{t}\right) \left( \phi _{t}.\chi _{t+1}\right) \right\rangle$. Note that $\left\langle \left( \phi _{t}.V_{t}\right) ^{2}\right\rangle =\left\langle \left( \phi _{t}.V_{t}\right) \left( \phi _{t}.V_{t}\right) \right\rangle$. All these terms involve contributions of the type: $\sum \Big \langle \left( \phi _{t}\right) _{i}.\left( {\tilde{\varepsilon }}_{t}\right) _{i}\left( \phi _{t}\right) _{i}.\left( {\tilde{\varepsilon }}_{t}\right) _{j}\Big \rangle$ or $\sum \left\langle \left( \phi _{t}\right) _{i}.\left( {\tilde{\varepsilon }} _{t}\right) _{i}\left( \phi _{t}\right) _{i}.\left( \varepsilon _{t+1}\right) _{j}\right\rangle$. The second term is null, given that ${\tilde{\varepsilon }}_{t}$ and $\varepsilon _{t+1}$ are independent. The other terms involve contributions like $\left\langle \left( \phi _{t}.{\tilde{\varepsilon }}_{t}\right) ^{2}\right\rangle$, $\left\langle \left( \phi _{t}. {\tilde{\varepsilon }}_{t}\right) \left( \chi _{t}.{\tilde{\varepsilon }} _{t}\right) \right\rangle$, $\left\langle \left( \chi _{t}.\tilde{ \varepsilon }_{t}\right) ^{2}\right\rangle$.

Let us consider the first term only, $\left\langle \left( \phi _{t}.\tilde{ \varepsilon }_{t}\right) ^{2}\right\rangle$. The reasoning will be similar for the other terms.

Since we are working with vectors with large number of components, and given that the random terms $\left( {\tilde{\varepsilon }}_{t}\right)$ and $\left( \varepsilon _{t+1}\right)$ are gaussian and isotropic, the expectations can be computed and yield $\sum \left\langle \left( \phi _{t}\right) _{i}\left( {\tilde{\varepsilon }}_{t}\right) _{i}\left( \phi _{t}\right) _{i}\left( {\tilde{\varepsilon }}_{t}\right) _{j}\right\rangle =\sum \left( \phi _{t}\right) _{i}\left( \phi _{t}\right) _{i}\left\langle \left( {\tilde{\varepsilon }} _{t}\right) _{i}\left( {\tilde{\varepsilon }}_{t}\right) _{j}\right\rangle =\sum \left( \phi _{t}\right) _{i}\left( \phi _{t}\right) _{i}\frac{{\tilde{\sigma }}^{2}}{N}=\frac{\phi _{t}.\phi _{t}}{N}{\tilde{\sigma }}^{2}$.

Therefore, all variance terms arising in the mean of squared values are of order $\frac{1}{N}$ and are negligible.

We will thus write $\left( \phi _{t}.V_{t}\right) ^{2}=\left\langle \phi _{t}.V_{t}\right\rangle ^{2}$ and the same for other quantities.

For the same reasons, and for the sake of simplicity, in the sequel we will implicitly understand z, everywhere it appears, as its mean value, that is $\frac{1-\sqrt{\left( 1-{\tilde{\sigma }}^{2}\right) }}{2}$.

We will ultimately assume, as explained in the text, that ${\bar{K}}_{2}=\tau {{{\bar{K}}}}_{1}$, ${\hat{K}}_{2}=\tau {\hat{K}}_{1}$. We also write $\xi =\frac{1}{2}\left( 1+\varphi z\right)$ and $1-\xi =\frac{1}{2}\left( 1-\varphi z\right)$. The optimal choice is to produce toward $V_{t}$, since the agent has a comparative advantage in this direction. This choice of parametrization is justified by the envelope theorem ensuring that $\xi -\frac{1}{2}$ should be of second order in $V_{t}-\chi _{t}$. As shown before, the correction is actually proportional to z, and in first approximation its value depends on the derivatives of the productivity that are included in the factor $\varphi$.

These considerations lead us to the following dynamics for $\left\langle \phi _{t}.\chi _{t}\right\rangle$ :

$$\begin{aligned}&\left\langle \phi _{t+1}.\chi _{t+1}\right\rangle =\eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left\langle K_{2}\xi +K_{1}\left( 1-\xi \right) \right\rangle \left\langle V_{t}.\chi _{t+1}\right\rangle \\&\quad -\frac{\left( \left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left\langle u\right\rangle -2\left( \alpha +\rho \delta \right) z\left\langle v\right\rangle -\left( \alpha +\rho \delta \right) \gamma \left\langle V_{t}.\phi _{t}\right\rangle \right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha +\rho \delta \right) -4\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\left\langle V_{t}.\chi _{t+1}\right\rangle \\&\quad -\eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left\langle K_{1}\left( 1-\xi \right) \right\rangle \left\langle \left( V_{t}-\chi _{t}\right) .\chi _{t+1}\right\rangle \\&\quad -\frac{\left( \left( \gamma +\alpha +\rho \delta \right) \left\langle v\right\rangle -2\left( \beta +\varepsilon \rho \right) z\left\langle u\right\rangle -\left( \beta +\varepsilon \rho \right) \gamma \left\langle \left( V_{t}-\chi _{t}\right) .\phi _{t}\right\rangle \right) }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha +\rho \delta \right) -4\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\\&\qquad \left\langle \left( V_{t}-\chi _{t}\right) .\chi _{t+1}\right\rangle \\&\quad +\left( \left( 1-\eta \right) -\gamma \frac{\gamma +4\left( \beta +\varepsilon \rho \right) z+\alpha +\rho \delta }{\left( \gamma +4\left( \beta +\varepsilon \rho \right) z\right) \left( \gamma +\alpha +\rho \delta \right) -4\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\right) \\&\qquad \left\langle \phi _{t}.\chi _{t+1}\right\rangle \end{aligned}$$

In the sequel, we will consider the simplification $\gamma =0$. It makes the computation more tractable and will not impair the arguments. The previous equation reduces to:

$$\begin{aligned} \left\langle \phi _{t+1}.\chi _{t+1}\right\rangle= & {} \left( \eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left\langle K_{2}\xi +K_{1}\left( 1-\xi \right) \right\rangle \right. \nonumber \\&\left. -\,\frac{4\left( \beta +\varepsilon \rho \right) z\left\langle u\right\rangle -2\left( \alpha +\rho \delta \right) z\left\langle v\right\rangle }{\left( 4\left( \beta +\varepsilon \rho \right) z\right) \left( \alpha +\rho \delta \right) -4\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}}\right) \left\langle V_{t}.\chi _{t+1}\right\rangle \nonumber \\&-\,\left( \eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left\langle K_{1}\left( 1-\xi \right) \right\rangle \right. \nonumber \\&\left. +\,\frac{\left( \left( \alpha +\rho \delta \right) \left\langle v\right\rangle -2\left( \beta +\varepsilon \rho \right) z\left\langle u\right\rangle \right) }{4\left( \beta +\varepsilon \rho \right) z\left( \alpha +\rho \delta \right) -4\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( z\right) ^{2}} \right) \nonumber \\&\quad \left\langle \left( V_{t}-\chi _{t}\right) .\chi _{t+1}\right\rangle +\left( 1-\eta \right) \left\langle \phi _{t}.\chi _{t+1}\right\rangle \end{aligned}$$

(16)

Now, compute some quantities that are relevant to the dynamics (16), given the chosen assumptions:

$$\begin{aligned} \left\langle K_{1}\left( 1-\xi \right) \right\rangle= & {} \frac{1-\varphi z}{2}\left( {{{\bar{K}}}}_{1}+ {\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle \right) \\ \left\langle K_{2}\xi \right\rangle= & {} \frac{1+\varphi z}{2}\tau \left( {{{\bar{K}}}}_{1}+{\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle \sqrt{1-{\tilde{\sigma }}^{2}}\right) \\ \left\langle \sqrt{1-{\tilde{\sigma }}^{2}} K_{2}\xi +K_{1}\left( 1-\xi \right) \right\rangle= & {} \frac{\sqrt{1-{\tilde{\sigma }}^{2}}\left\langle 1+\varphi z\right\rangle \tau +\left( 1-\varphi z\right) }{2}{\bar{K}}_{1}\\&+\,\frac{\left( 1-{\tilde{\sigma }} ^{2}\right) \left( 1+\varphi z\right) \tau +\left( 1-\varphi z\right) }{2} {\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle \\= & {} \frac{\left( 1-2z\right) \left( 1+\varphi z\right) \tau +\left( 1-\varphi z\right) }{2}{{{\bar{K}}}}_{1}\\&+\,\frac{\left( 1-2z\right) ^{2}\left( 1+\varphi z\right) \tau +\left( 1-\varphi z\right) }{2}{\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle \\ \left\langle \left( V_{t}-\chi _{t}\right) .\chi _{t+1}\right\rangle= & {} \left( \sqrt{1-{\tilde{\sigma }}^{2}}-1\right) \sqrt{1-\sigma ^{2}}\\ \left\langle V_{t}.\chi _{t+1}\right\rangle= & {} \sqrt{1-{\tilde{\sigma }}^{2}} \sqrt{1-\sigma ^{2}}\\ \left\langle \phi _{t}.\chi _{t+1}\right\rangle= & {} \left\langle \phi _{t}.\chi _{t}\right\rangle \sqrt{1-\sigma ^{2}} \end{aligned}$$

which allows to rewrite:

$$\begin{aligned} \frac{\left\langle \phi _{t+1}.\chi _{t+1}\right\rangle }{\sqrt{1-\sigma ^{2} }}= & {} \eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left\langle \sqrt{1-{\tilde{\sigma }}^{2}}K_{2}\xi +K_{1}\left( 1-\xi \right) \right\rangle \nonumber \\&-\,\frac{2z\left( \beta +\varepsilon \rho \right) \left( 1+\sqrt{ 1-{\tilde{\sigma }}^{2}}\right) \left\langle u\right\rangle +\left( \left( 1-2z\right) \sqrt{1-{\tilde{\sigma }}^{2}}-1\right) \alpha \left\langle v\right\rangle }{4z\alpha \left( \beta +\varepsilon \rho \right) \left( 1-z\right) } \nonumber \\&+\,\left( 1-\eta \right) \left\langle \phi _{t}.\chi _{t}\right\rangle \end{aligned}$$

(17)

with the relevant quantities $\left\langle u\right\rangle$ and $\left\langle v\right\rangle$ expanded as:

$$\begin{aligned} \left\langle u\right\rangle= & {} 1-\rho \nonumber \\&+\rho \delta \left\langle \left( 1-\eta \right) \left( 1-2z\right) \right. \nonumber \\&\left. +\,\eta \left( \tau \frac{1+\varphi z}{2}\left( {\bar{K}}_{1}+{\hat{K}} _{1}\left( \phi _{t}.\chi _{t}\right) \left( 1-2z\right) \right) \right. \right. \nonumber \\&\left. \left. +\,\left( 1-2z\right) \frac{1-\varphi z}{2}\left( {{{\bar{K}}}}_{1}+{\hat{K}}_{1}\left( \phi _{t}.\chi _{t}\right) \right) \right) \right\rangle \left\langle \phi _{t}.\chi _{t}\right\rangle \nonumber \\= & {} 1-\rho \nonumber \\&+\rho \delta \left( 1-\eta \right) \left( 1-2z\right) \left\langle \phi _{t}.\chi _{t}\right\rangle \nonumber \\&+\,\eta \rho \delta \left\langle \left( {\bar{K}}_{1}\left( \tau \frac{ 1+\varphi z}{2}+\left( 1-2z\right) \frac{1-\varphi z}{2}\right) +\left( 1-2z\right) \left( \tau \frac{1+\varphi z}{2}\right. \right. \right. \nonumber \\&\left. \left. \left. +\,\frac{1-\varphi z}{ 2}\right) {\hat{K}}_{1}\left( \phi _{t}.\chi _{t}\right) \right) \left( \phi _{t}.\chi _{t}\right) \right\rangle \nonumber \\= & {} 1-\rho \nonumber \\&+\,\rho \delta \left( \left( 1-\eta \right) \left( 1-2z\right) +\eta {{{\bar{K}}}}_{1}\left( \tau \frac{1+\varphi z}{2}+\left( 1-2z\right) \frac{1-\varphi z}{ 2}\right) \right) \left\langle \phi _{t}.\chi _{t}\right\rangle \nonumber \\&+\,\rho \delta \eta \left( \left( 1-2z\right) \left( \tau \frac{ 1+\varphi z}{2}+\frac{1-\varphi z}{2}\right) {\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle ^{2}\right) \end{aligned}$$

(18)

$$\begin{aligned} \left\langle v\right\rangle= & {} \varepsilon \left( \left( 1-\eta \right) \left\langle \phi _{t}.\left( V_{t}-\chi _{t}\right) \right\rangle +2z\eta \left\langle \phi _{t}.\chi _{t}\right\rangle \left( K_{2}\xi -K_{1}\left( 1-\xi \right) \right) \right) \nonumber \\= & {} \varepsilon \rho 2z\left\langle -\left( 1-\eta \right) +\eta \left( \frac{ 1+\varphi z}{2}\tau \left( {{{\bar{K}}}}_{1}+\left( 1-2z\right) {\hat{K}}_{1}\left( \phi _{t}.\chi _{t}\right) \right) \right. \right. \nonumber \\&\left. \left. -\frac{1-\varphi z}{2}\left( {{{\bar{K}}}}_{1}+{\hat{K}} _{1}\left( \phi _{t}.\chi _{t}\right) \right) \right) \right\rangle \left\langle \phi _{t}.\chi _{t}\right\rangle \end{aligned}$$

(19)

We will need to compute the second term in the right hand side of (17):

$$\begin{aligned}&-\,\frac{2z\left( \beta +\varepsilon \rho \right) \left( 1+\sqrt{ 1-{\tilde{\sigma }}^{2}}\right) \left\langle u\right\rangle +\left( \left( 1-2z\right) \sqrt{1-{\tilde{\sigma }}^{2}}-1\right) \left( \alpha +\rho \delta \right) \left\langle v\right\rangle }{4z\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( 1-z\right) } \\= & {} -\frac{4z\left( \beta +\varepsilon \rho \right) \left( 1-z\right) \left\langle u\right\rangle +\left( \left( 1-2z\right) ^{2}-1\right) \left( \alpha +\rho \delta \right) \left\langle v\right\rangle }{4z\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( 1-z\right) } \\= & {} -\frac{4z\left( \beta +\varepsilon \rho \right) \left( 1-z\right) \left\langle u\right\rangle -4z\left( 1-z\right) \left( \alpha +\rho \delta \right) \left\langle v\right\rangle }{4z\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) \left( 1-z\right) } \\= & {} -\frac{\left( \beta +\varepsilon \rho \right) \left\langle u\right\rangle -\left( \alpha +\rho \delta \right) \left\langle v\right\rangle }{\left( \alpha +\rho \delta \right) \left( \beta +\varepsilon \rho \right) } \\= & {} -\frac{1-\rho }{\alpha +\rho \delta } \\&-\frac{\rho \delta }{\alpha +\rho \delta }\left( \left( 1-\eta \right) \left( 1-2z\right) +\eta {{{\bar{K}}}}_{1}\left( \tau \frac{1+\varphi z}{2}+\left( 1-2z\right) \frac{1-\varphi z}{2}\right) \right) \left\langle \phi _{t}.\chi _{t}\right\rangle \\&+\,\frac{2\varepsilon \rho z}{\beta +\varepsilon \rho }\left( -\left( 1-\eta \right) +\eta {{{\bar{K}}}}_{1}\left( \tau \frac{1+\varphi z}{2}+\frac{1-\varphi z}{2} \right) \right) \left\langle \phi _{t}.\chi _{t}\right\rangle \\&+\,\eta \left( \frac{\varepsilon z}{\beta +\varepsilon \rho }\left( \left( 1+\varphi z\right) \tau \left( 1-2z\right) -\left( 1-\varphi z\right) \right) \right. \\&\left. -\,\frac{\rho \delta }{\alpha +\rho \delta }\left( 1-2z\right) \left( \tau \frac{1+\varphi z}{2}+\frac{1-\varphi z}{2}\right) \right) {\hat{K}} _{1}\left\langle \phi _{t}.\chi _{t}\right\rangle ^{2} \end{aligned}$$

so that the dynamics (17) in its expanded form is thus:

$$\begin{aligned} \frac{\left\langle \phi _{t+1}.\chi _{t+1}\right\rangle }{\eta \sqrt{ 1-\sigma ^{2}}}= & {} \left( \frac{\left( 1-2z\right) \tau \left( 1+\varphi z\right) +\left( 1-\varphi z\right) }{2}{{{\bar{K}}}}_{1}\right. \\&+\,\frac{\left( 1-\eta \right) }{\eta }-\frac{\rho \delta \left( \frac{ \left( 1-\eta \right) \left( 1-2z\right) }{\eta }+{\bar{K}}_{1}\left( \tau \frac{ 1+\varphi z}{2}+\left( 1-2z\right) \frac{1-\varphi z}{2}\right) \right) }{ \alpha +\rho \delta } \\&+\,\left. \frac{2\varepsilon \rho z}{\beta +\varepsilon \rho }\left( -\frac{ \left( 1-\eta \right) }{\eta }+{{{\bar{K}}}}_{1}\left( \tau \frac{1+\varphi z}{2}+\frac{ 1-\varphi z}{2}\right) \right) \right) \left\langle \phi _{t}.\chi _{t}\right\rangle \\&+\,\left( \frac{\left( 1-2z\right) ^{2}\tau \left( 1+\varphi z\right) +\left( 1-\varphi z\right) }{2}\right. \\&\left. -\,\frac{\rho \delta \left( 1-2z\right) }{ \alpha +\rho \delta }\left( \tau \frac{1+\varphi z}{2}+\frac{1-\varphi z}{2} \right) \right. \\&+\,\left. \frac{2\varepsilon \rho z}{\beta +\varepsilon \rho }\left( \frac{ \left( 1+\varphi z\right) \tau }{2}\left( 1-2z\right) -\frac{\left( 1-\varphi z\right) }{2}\right) \right) {\hat{K}}_{1}\left\langle \phi _{t}.\chi _{t}\right\rangle ^{2} \\&-\,\frac{\left( 1-\rho \right) }{\left( \alpha +\delta \rho \right) \eta } \end{aligned}$$

Reordering the various terms and applying the same resolution method as for case 1 leads directly to the results presented in the text.

1.3 Case 3

The dynamical system describing the evolution of $V_{t}$ and $\chi _{t}$ is now:

$$\begin{aligned} V_{t}= & {} \sqrt{1-{\tilde{\sigma }}^{2}}\chi _{t}+{\tilde{\varepsilon }}_{t} \\ \chi _{t+1}= & {} \sqrt{1-\sigma ^{2}}V_{t}+\varepsilon _{t} \end{aligned}$$

As explained in the text, it describes the evolution in social values by a group of precursor agents. Note that the second equation can be rewritten as:

$$\begin{aligned} \chi _{t+1}=\sqrt{\left( 1-\sigma ^{2}\right) \left( 1-{\tilde{\sigma }} ^{2}\right) }\chi _{t}+\varepsilon _{t}+\sqrt{1-\sigma ^{2}}{\tilde{\varepsilon }}_{t} \end{aligned}$$

Using the techniques described in the previous case, we find the dynamics for the stock of capital goods when $\gamma =\beta =\varepsilon =0$.

$$\begin{aligned} \phi _{t+1}= & {} \eta \left( \phi _{t}.\chi _{t}\right) K_{2}\xi \left( 1- \frac{\rho \delta }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) } \right) V_{t} \\&-\left( \frac{1-\rho +\rho \delta \left( 1-\eta \right) \phi _{t}.V_{t}}{ 2\left( 1-z\right) \left( \alpha +\rho \delta \right) }+\frac{\rho \delta \eta \left( \phi _{t}.\chi _{t}\right) \left( \left( 1-2z\right) K_{1}\left( 1-\xi \right) \right) }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) V_{t} \\&+\,\eta \left( \phi _{t}.\chi _{t}\right) K_{1}\left( 1-\xi \right) \left( 1- \frac{\rho \delta \left( 1-2z\right) }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) \chi _{t} \\&-\,\left( \frac{1-\rho +\rho \delta \left( 1-\eta \right) \phi _{t}.V_{t}}{ 2\left( 1-z\right) \left( \alpha +\rho \delta \right) }+\frac{\rho \delta \eta \left( \phi _{t}.\chi _{t}\right) K_{2}\xi }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) \chi _{t}+\left( 1-\eta \right) \phi _{t} \\= & {} \eta \left( \phi _{t}.\chi _{t}\right) \left( \frac{1+\varphi z}{2} \right) \left( {{{\bar{K}}}}_{2}+{\hat{K}}_{2}\phi _{t}.V_{t}\right) \left( 1-\frac{\rho \delta }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) V_{t} \\&-\,\left( \frac{1-\rho +\rho \delta \left( 1-\eta \right) \phi _{t}.V_{t}}{ 2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right. \\&\left. +\,\frac{\rho \delta \eta \left( \phi _{t}.\chi _{t}\right) \left( \left( 1-2z\right) \left( \frac{1-\varphi z}{2}\right) \left( {\bar{K}}_{1}+{\hat{K}}_{1}\phi _{t}.\chi _{t}\right) \right) }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) } \right) V_{t} \\&+\,\eta \left( \phi _{t}.\chi _{t}\right) \left( \frac{1-\varphi z}{2} \right) \left( {{{\bar{K}}}}_{1}+{\hat{K}}_{1}\phi _{t}.\chi _{t}\right) \left( 1-\frac{ \rho \delta \left( 1-2z\right) }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) \chi _{t} \\&-\left( \frac{1-\rho +\rho \delta \left( 1-\eta \right) \phi _{t}.V_{t}}{ 2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right. \\&\left. +\,\frac{\rho \delta \eta \left( \phi _{t}.\chi _{t}\right) \left( \frac{1+\varphi z}{2}\right) \left( {{{\bar{K}}}}_{2}+{\hat{K}}_{2}\phi _{t}.V_{t}\right) }{2\left( 1-z\right) \left( \alpha +\rho \delta \right) }\right) \chi _{t}+\left( 1-\eta \right) \phi _{t} \end{aligned}$$

We define $y_{t}=\left\langle \phi _{t}.\chi _{t}\right\rangle$ and set $\tau =1$, so that we will focus only on the precursor effect and discard the influence of an advantage in productivity along $V_{t}$. We also normalize the coefficients to $\alpha =1$, $\delta =1$, ${{{\bar{K}}}}_{1}={\bar{K}}_{2}=1$, ${\hat{K}}_{1}={\hat{K}}_{2}$. One is led to:

$$\begin{aligned} \frac{y_{t+1}}{\sqrt{1-\sigma ^{2}}}=-\frac{1-\rho }{\rho +1}+\frac{z\eta -2z+z^{2}\eta \varphi +1}{\rho +1}y_{t}+\frac{\left( 1-2z\right) \eta {\hat{K}}_{1}}{\rho +1}y_{t}^{2} \end{aligned}$$

as stated in the text. Once again, the dynamics presents a threshold pattern, with:

$$\begin{aligned} \phi _{+}=\frac{-\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) - \frac{1+\rho }{\sqrt{1-\sigma ^{2}}}\right) +\sqrt{\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}} \right) ^{2}+4\left( 1-\rho \right) \left( 1-2z\right) \eta {\hat{K}}_{1}}}{ 2\left( 1-2z\right) \eta {\hat{K}}_{1}} \end{aligned}$$

The variations of the threshold with respect to some of the parameters are computed straightforwardly:

$$\begin{aligned}&\frac{\partial \phi _{+}}{\partial \sigma ^{2}} >0 \\&\frac{\partial \phi _{+}}{\partial {\hat{K}}_{1}}<0 \\&\frac{\partial \phi _{+}}{\partial \left( 1+z\varphi \right) } <0 \\&\frac{\partial \phi _{+}}{\partial \eta } =-\frac{z\left( 1+z\varphi \right) }{2}\\&\quad \left( 1-2z\right) \eta {\hat{K}}_{1}\left( 1-\frac{\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}}\right) }{\sqrt{\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}}\right) ^{2}+4\left( 1-\rho \right) \left( 1-2z\right) \eta {\hat{K}}_{1}}}\right) \\&\quad +\,\frac{{\hat{K}}_{1}}{\eta }\frac{\partial \phi _{+}}{\partial {\hat{K}}_{1}} \end{aligned}$$

and thus $\frac{\partial \phi _{+}}{\partial \eta }<0$. Concerning $\frac{\partial \phi _{+}}{\partial \rho }$, remark that:

$$\begin{aligned}&2\left( 1-2z\right) \eta {\hat{K}}_{1}\frac{\partial \phi _{+}}{\partial \rho } =\frac{1}{\sqrt{1-\sigma ^{2}}}\\&\quad +\,\frac{-\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}}\right) -2\sqrt{ 1-\sigma ^{2}}\left( 1-2z\right) \eta {\hat{K}}_{1}}{\sqrt{1-\sigma ^{2}}\sqrt{ \left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{ \sqrt{1-\sigma ^{2}}}\right) ^{2}+4\left( 1-\rho \right) \left( 1-2z\right) \eta {\hat{K}}_{1}}} \end{aligned}$$

As a consequence, $\frac{\partial \phi _{+}}{\partial \rho }>0$ if:

$$\begin{aligned} 0< & {} \sqrt{\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{ 1+\rho }{\sqrt{1-\sigma ^{2}}}\right) ^{2}+4\left( 1-\rho \right) \left( 1-2z\right) \eta {\hat{K}}_{1}} \\&-\,\left( \left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{ 1+\rho }{\sqrt{1-\sigma ^{2}}}\right) +2\sqrt{1-\sigma ^{2}}\left( 1-2z\right) \eta {\hat{K}}_{1}\right) \end{aligned}$$

Now, given that:

$$\begin{aligned}&\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{ \sqrt{1-\sigma ^{2}}}\right) ^{2}+4\left( 1-\rho \right) \left( 1-2z\right) \eta {\hat{K}}_{1} \\&\quad -\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}}+2\sqrt{1-\sigma ^{2}}\left( 1-2z\right) \eta {\hat{K}} _{1}\right) ^{2} \\&\qquad =\left( 1-\rho \right) -\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{1-\sigma ^{2}}}\right) \\&\quad \qquad \; \sqrt{1-\sigma ^{2} }-\left( 1-\sigma ^{2}\right) \left( 1-2z\right) \eta {\hat{K}}_{1} \\&\qquad =2-\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) \right) \sqrt{1-\sigma ^{2}}-\left( 1-\sigma ^{2}\right) \left( 1-2z\right) \eta {\hat{K}}_{1} \end{aligned}$$

one obtains:

$$\begin{aligned} \frac{\partial \phi _{+}}{\partial \rho }< & {} 0\text { if }\eta {\hat{K}}_{1}> \frac{2-\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) \right) \sqrt{1-\sigma ^{2}}}{1-\sigma ^{2}} \\ \frac{\partial \phi _{+}}{\partial \rho }> & {} 0\text { otherwise.} \end{aligned}$$

For the last parameter, z, let us call:

$$\begin{aligned} a=\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{ \sqrt{1-\sigma ^{2}}}\right) \end{aligned}$$

and note that:

$$\begin{aligned} \frac{\partial a}{\partial z}=\eta \left( 1+z\varphi \right) -2<0 \end{aligned}$$

We can thus write:

$$\begin{aligned} \frac{\partial \phi _{+}}{\partial z}= & {} \frac{1}{2\left( 1-2z\right) \eta {\hat{K}}_{1}}\frac{\partial a}{\partial z}\\&\times \!\left( \! -1+\frac{a}{\sqrt{\left( \left( z\eta \left( 1+z\varphi \right) +1-2z\right) -\frac{1+\rho }{\sqrt{ 1-\sigma ^{2}}}\right) ^{2}+4\left( 1-\rho \right) \left( 1\!-\!2z\right) \eta {\hat{K}}_{1}}}\!\right) \\&-2\frac{{\hat{K}}_{1}}{\left( 1-2z\right) }\frac{\partial \phi _{+}}{ \partial {\hat{K}}_{1}} \end{aligned}$$

so that:

$$\begin{aligned} \frac{\partial \phi _{+}}{\partial z}>0 \end{aligned}$$

1.4 Case 4

In that case, we consider again $\gamma =\beta =\varepsilon =0$, as well as the normalizations $\alpha =1$ and $\delta =1$. We also assume that each agent produces only its own good, that is, his production is proportional to $V_{t}$. In other words, we assume that the productivity of agent i is $K_{i}={\bar{K}}_{i}+{\hat{K}}_{i}\left( \phi _{t}.V_{t}\right)$. We also assume the normalization ${{{\bar{K}}}}_{i}=1$, that is $K_{i}=1+{\hat{K}}_{i}\left( \phi _{t}.V_{t}\right)$, and ${\hat{K}}_{1}={\hat{K}}_{2}$.

Given these assumptions, the dynamics for $\phi _{t}$ of any sector is then:

$$\begin{aligned} \phi _{t+1}= & {} \left( \eta \left( \phi _{t}.\chi _{t}\right) \left( 1+{\hat{K}} _{1}\left( \phi _{t}.V_{t}\right) \right) \right) V_{t} \\&-\left( \frac{1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}.V_{t}+\eta \left( \phi _{t}.\chi _{t}\right) \left( 1+{\hat{K}}_{1}\left( \phi _{t}.V_{t}\right) \right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}\right) }\right) \frac{\chi _{t}+V_{t}}{2} \\&+\left( 1-\eta \right) \phi _{t} \end{aligned}$$

or, setting $y_{t}=\left( \phi _{t}.\chi _{t}\right)$:

$$\begin{aligned} \phi _{t+1}= & {} \left( \eta y_{t}\left( 1+{\hat{K}}_{1}\left( \phi _{t}.V_{t}\right) \right) \right) V_{t}\\&-\left( \frac{1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}.V_{t}+\eta y_{t}\left( 1+{\hat{K}} _{1}\left( \phi _{t}.V_{t}\right) \right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}\right) }\right) \frac{\chi _{t}+V_{t}}{2}\\&+\left( 1-\eta \right) \phi _{t} \end{aligned}$$

Let us now use the hypotheses described in the text to derive the dynamics of the system. The values $V_{t}^{\left( i\right) }$ are assumed, for the sake of simplicity, to be constant and orthogonal.

$$\begin{aligned} V_{t}^{\left( 1\right) }= & {} V_{0}^{\left( 1\right) } \\ V_{t}^{\left( 2\right) }= & {} V_{0}^{\left( 2\right) } \\ V_{t}^{\left( 1\right) }.V_{t}^{\left( 2\right) }= & {} V_{0}^{\left( 1\right) }.V_{0}^{\left( 2\right) }=0 \end{aligned}$$

We will assume that the social value at time $t+1$ is a weighted sum of the two groups’ values, the weight being given by the relative social value of the stock of each group at time t. In other words, the social value evolves endogenously according to the relative evolution of the two groups:

$$\begin{aligned} \chi _{t+1}= & {} \frac{\left( \phi _{t}^{\left( 1\right) }.\chi _{t}\right) V_{t}^{\left( 1\right) }+\left( \phi _{t}^{\left( 2\right) }.\chi _{t}\right) V_{t}^{\left( 2\right) }}{\sqrt{\left( \phi _{t}^{\left( 1\right) }.\chi _{t}\right) ^{2}+\left( \phi _{t}^{\left( 2\right) }.\chi _{t}\right) ^{2}+2\left( \phi _{t}^{\left( 1\right) }.\chi _{t}\right) \left( \phi _{t}^{\left( 2\right) }.\chi _{t}\right) V_{t}^{\left( 1\right) }.V_{t}^{\left( 2\right) }}} \nonumber \\= & {} \frac{y_{t}^{\left( 1\right) }V_{0}^{\left( 1\right) }+y_{t}^{\left( 2\right) }V_{0}^{\left( 2\right) }}{\sqrt{\left( y_{t}^{\left( 1\right) }\right) ^{2}+\left( y_{t}^{\left( 2\right) }\right) ^{2}}} \end{aligned}$$

(20)

As before, we define for each group the measure of the angle between $\chi _{t}$ and $V_{0}^{\left( i\right) }$ as:

$$\begin{aligned} 1-2z_{t}^{\left( i\right) }=\chi _{t}.V_{0}^{\left( i\right) } \end{aligned}$$

Inserting these notations yields the dynamics for $\phi _{t}^{\left( i\right) }$ and $y_{t}^{\left( i\right) }$:

$$\begin{aligned} \phi _{t+1}^{\left( i\right) }= & {} \eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}}_{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) V_{0}^{\left( i\right) } \\&-\left( \frac{1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }+\eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}}_{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}^{\left( i\right) }\right) }\right) \frac{\chi _{t}+V_{0}^{\left( i\right) }}{2} \\&+\left( 1-\eta \right) \phi _{t}^{\left( i\right) } \\ y_{t+1}^{\left( i\right) }= & {} \eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}} _{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \frac{y_{t}^{\left( i\right) }}{\sqrt{\left( y_{t}^{\left( 1\right) }\right) ^{2}+\left( y_{t}^{\left( 2\right) }\right) ^{2}}} \\&-\left( \frac{1-\rho -\delta \left( \left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }+\eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}}_{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \right) }{1-z_{t}^{\left( i\right) }}\right) \\&\times \frac{ y_{t}^{\left( i\right) }\left( 1-z_{t}^{\left( i\right) }\right) +y_{t}^{\left( 3-i\right) }\left( \frac{1-2z_{t}^{\left( 3-i\right) }}{2}\right) }{\sqrt{\left( y_{t}^{\left( 1\right) }\right) ^{2}+\left( y_{t}^{\left( 2\right) }\right) ^{2}}} \\&+\left( 1-\eta \right) \frac{y_{t}^{\left( i\right) }\phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }+y_{t}^{\left( 3-i\right) }\phi _{t}^{\left( i\right) }.V_{0}^{\left( 3-i\right) }}{\sqrt{\left( y_{t}^{\left( 1\right) }\right) ^{2}+\left( y_{t}^{\left( 2\right) }\right) ^{2}}} \end{aligned}$$

However, we will not describe the dynamics in terms of $y_{t}^{\left( i\right) }$. We will rather focus on the dynamics of two other variables, and deduce the evolution of $y_{t}^{\left( i\right) }$ from these equations.

To do so, we will need the evolution of the personal value of the stock of agent i:

$$\begin{aligned} \phi _{t+1}^{\left( i\right) }.V_{0}^{\left( i\right) }= & {} \eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}}_{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \\&-\left( \frac{1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }+\eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}}_{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}^{\left( i\right) }\right) }\right) \\&\quad \times \frac{z_{t}^{\left( i\right) }+1}{2} +\left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) } \end{aligned}$$

as well as the evolution of the value attributed by agent $3-i$ to the stock of agent i:

$$\begin{aligned} \phi _{t+1}^{\left( i\right) }.V_{0}^{\left( 3-i\right) }= & {} -\left( \frac{ 1-\rho +\delta \rho \left( \left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }+\eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}} _{1}\left( \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) }\right) \right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}^{\left( i\right) }\right) }\right) \\&\times \frac{z_{t}^{\left( 3-i\right) }}{2}+\left( 1-\eta \right) \phi _{t}^{\left( i\right) }.V_{0}^{\left( 3-i\right) } \end{aligned}$$

We will denote these expressions by a letter x, that is:

$$\begin{aligned} x_{t}^{\left( i,i\right) }= & {} \phi _{t}^{\left( i\right) }.V_{0}^{\left( i\right) } \\ x_{t}^{\left( i,3-i\right) }= & {} \phi _{t}^{\left( i\right) }.V_{0}^{\left( 3-i\right) } \end{aligned}$$

We thus have the relations:

$$\begin{aligned} \phi _{t+1}^{\left( i\right) }.\chi _{t+1}=\frac{\left( \phi _{t}^{\left( i\right) }.\chi _{t}\right) \phi _{t+1}^{\left( i\right) }.V_{0}^{\left( i\right) }+\left( \phi _{t}^{\left( 3-i\right) }.\chi _{t}\right) \phi _{t+1}^{\left( i\right) }.V_{0}^{\left( 3-i\right) }}{\sqrt{\left( \phi _{t}^{\left( 1\right) }.\chi _{t}\right) ^{2}+\left( \phi _{t}^{\left( 2\right) }.\chi _{t}\right) ^{2}}} \end{aligned}$$

or,

$$\begin{aligned} y_{t+1}^{\left( i\right) }=\frac{\sum _{j}y_{t}^{\left( j\right) }x_{t+1}^{\left( i,j\right) }}{\sqrt{\sum _{j}y_{t}^{\left( j\right) }y_{t}^{\left( j\right) }}} \end{aligned}$$

(21)

which expresses $y_{t+1}^{\left( i\right) }$ as a function of the $x_{t+1}^{\left( i,j\right) }$. The dynamic system will then rather be expressed in terms of the variables $x_{t}^{\left( i,j\right) }$, where i and j run from 1 to 2. It is straightforward to get:

$$\begin{aligned} x_{t+1}^{\left( i,j\right) }= & {} \eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}} _{1}x_{t}^{\left( i,i\right) }\right) \delta _{i,j} \nonumber \\&-\left( \frac{1-\rho +\delta \rho \left( \left( 1-\eta \right) x_{t}^{\left( i,i\right) }+\eta y_{t}^{\left( i\right) }\left( 1+{\hat{K}} _{1}x_{t}^{\left( i,i\right) }\right) \right) }{\left( 1+\rho \right) \left( 1-z_{t}^{\left( i\right) }\right) }\right) \nonumber \\&\times \frac{z_{t}^{\left( j\right) }+\delta _{i,j}}{2}+\left( 1-\eta \right) x_{t}^{\left( i,j\right) } \end{aligned}$$

(22)

where $\delta _{i,j}$ is the Kronecker symbol, and $y_{t+1}^{\left( i\right) }$ is deduced from (21).

These equations have to be supplemented with the dynamics for the variables $z_{t+1}^{\left( i\right) }$. They are derived from the definition of (20):

$$\begin{aligned} 1-2z_{t+1}^{\left( i\right) }=\chi _{t+1}.V_{0}^{\left( i\right) }=\frac{ y_{t}^{\left( i\right) }}{\sqrt{\sum _{j}y_{t}^{\left( j\right) }y_{t}^{\left( j\right) }}} \end{aligned}$$

(23)

Having found the dynamic equations of the system, we are now interested in the equilibrium of the system. We only look for feasible equilibria, i.e. equilibria such that $y^{\left( i\right) }$, $x^{\left( i,j\right) }$, and $z^{\left( i\right) }$ have positive values. It corresponds to an equilibrium for which the equilibrium social value is located between $V_{0}^{\left( 1\right) }$ and $V_{0}^{\left( 2\right) }$. We will show that no equilibrium point exists (proposition 1), but only an equilibrium dynamics (proposition 2.a). Unfortunately, this equilibrium path is unstable (proposition 2.b and 2.c).

Proposition 1

For generic values of the parameters, there is no equilibrium for the system given by the Eqs. (21), (22) and (23).

Proof

The equations for a possible equilibrium are:

$$\begin{aligned} y^{\left( i\right) }= & {} \frac{\sum _{j}y^{\left( j\right) }x^{\left( i,j\right) } }{\sqrt{\sum _{j}y^{\left( i\right) }y^{\left( j\right) }}}\nonumber \\ x^{\left( i,j\right) }= & {} \eta y^{\left( i\right) }\left( 1+{\hat{K}} _{1}x^{\left( i,i\right) }\right) \delta _{i,j} \nonumber \\&-\left( \frac{1-\rho -\delta \left( \left( 1-\eta \right) x^{\left( i,i\right) }+\eta y^{\left( i\right) }\left( 1+{\hat{K}}_{1}x^{\left( i,i\right) }\right) \right) }{\left( 1+\rho \right) \left( 1-z^{\left( i\right) }\right) }\right) \nonumber \\&\times \frac{z^{\left( j\right) }+\delta _{i,j}}{2} +\left( 1-\eta \right) x^{\left( i,j\right) } \nonumber \\ 1-2z^{\left( i\right) }= & {} \frac{y^{\left( i\right) }}{\sqrt{\sum _{j}y^{\left( j\right) }y^{\left( j\right) }}} \end{aligned}$$

(24)

We focus on the equation for $i=1$. The proof is symmetric for $i=2$.

$$\begin{aligned} \eta x^{\left( 1,1\right) }= & {} \eta y^{\left( 1\right) }\left( 1+{\hat{K}} _{1}x^{\left( 1,1\right) }\right) -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x^{\left( 1,1\right) }\right. \right. \\&\left. \left. +\,\eta y^{\left( 1\right) }\left( 1+{\hat{K}} _{1}x^{\left( 1,1\right) }\right) \right) \right) \frac{z^{\left( 1\right) }+1}{2\left( 1+\rho \right) \left( 1-z^{\left( 1\right) }\right) } \\ \eta x^{\left( 1,2\right) }= & {} -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x^{\left( 1,1\right) }+\eta y^{\left( 1\right) }\left( 1+{\hat{K}} _{1}x^{\left( 1,1\right) }\right) \right) \right) \frac{z^{\left( 1\right) } }{2\left( 1+\rho \right) \left( 1-z^{\left( 1\right) }\right) } \end{aligned}$$

Which leads to express $x^{\left( 1,2\right) }$ as a function of $x^{\left( 1,1\right) }$, $z^{\left( 1\right) }$ and $y^{\left( 1\right) }$:

$$\begin{aligned} x^{\left( 1,2\right) }=\frac{z^{\left( 1\right) }x^{\left( 1,1\right) }\left( 1-{\hat{K}}_{1}y^{\left( 1\right) }\right) -y^{\left( 1\right) }z^{\left( 1\right) }}{z^{\left( 1\right) }+1} \end{aligned}$$

and $x^{\left( 1,1\right) }$, as a function of $z^{\left( 1\right) }$ and $y^{\left( 1\right) }$:

$$\begin{aligned} x^{\left( 1,1\right) }=\frac{2\left( \rho +1\right) \left( z^{\left( 1\right) }-1\right) \eta y^{\left( 1\right) }+\left( 1-\rho +\eta \rho y^{\left( 1\right) }\right) \left( z^{\left( 1\right) }+1\right) }{2\left( \rho +1\right) \left( z^{\left( 1\right) }-1\right) \eta \left( 1-y^{\left( 1\right) }{\hat{K}}_{1}\right) -\left( z^{\left( 1\right) }+1\right) \rho \left( y^{\left( 1\right) }\eta {\hat{K}}_{1}+1-\eta \right) } \end{aligned}$$

The condition $x^{\left( 1,2\right) }>0$ implies that $y^{\left( 1\right) }< \frac{1}{{\hat{K}}_{1}}$ and $x^{\left( 1,1\right) }>0$ leads to:

$$\begin{aligned} y^{\left( 1\right) }>\frac{\left( 1-\rho \right) \left( z+1\right) }{\eta \left( \rho -2z-3z\rho +2\right) } \end{aligned}$$

Inserting $x^{\left( 1,1\right) }$ in the expression for $x^{\left( 1,2\right) }$ allows to find another condition for $x^{\left( 1,2\right) }>0$ :

$$\begin{aligned} \left( \rho -\left( 1-\rho \right) {\hat{K}}_{1}\right) <0\text { and }y^{\left( 1\right) }>\frac{1-\rho }{\left( 1-\rho \right) {\hat{K}}_{1}-\rho } \end{aligned}$$

(25)

so that, combined with $y^{\left( 1\right) }<\frac{1}{{\hat{K}}_{1}}$, yields:

$$\begin{aligned} \frac{1}{{\hat{K}}_{1}}>y^{\left( 1\right) }>\frac{1-\rho }{\left( 1-\rho \right) {\hat{K}}_{1}-\rho } \end{aligned}$$

which is satisfied only if:

$$\begin{aligned} \left( \rho -\left( 1-\rho \right) {\hat{K}}_{1}\right) >0 \end{aligned}$$

a contradiction with the first inequality of (25). $\square$

Proposition 2

• (a) If $y_{0}^{\left( 1\right) }=y_{0}^{\left( 2\right) }$ and $x_{0}^{\left( i,3-i\right) }=x_{0}^{\left( 3-i,i\right) }$ there is a solution to the system with $1-2z_{t}^{\left( i\right) }=\frac{1}{\sqrt{2}}$, $x_{t}^{\left( i,3-i\right) }\rightarrow 0$ below a certain threshold (a lower bound for this threshold is given below), and $x_{t}^{\left( i,i\right) }\rightarrow \infty$ otherwise. This solution corresponds to a symmetric situation in which $\chi _{t}=\frac{V_{0}^{\left( 1\right) }+V_{0}^{\left( 2\right) }}{ \sqrt{2}}$, $y_{t}^{\left( 1\right) }=y_{t}^{\left( 2\right) }$.

• (b) This dynamics is unstable. If $1-2z_{0}^{\left( 1\right) }>\frac{1}{\sqrt{2}}$, then $1-2z_{t}^{\left( 1\right) }\rightarrow 1$ and $1-2z_{t}^{\left( 2\right) }\rightarrow 0$.

• (c) If $1-2z_{0}^{\left( 1\right) }< \frac{1}{\sqrt{2}}$, then $1-2z_{t}^{\left( 1\right) }\rightarrow 0$ and $1-2z_{t}^{\left( 2\right) }\rightarrow 1$.

Proof

• (a) Since the equations are symmetric between the 2 agents, the dynamics are identical if the initial conditions are equal, and then $1-2z_{t}^{\left( i\right) }=\frac{1}{\sqrt{2}}$. We can thus focus on the case $i=1$, and the result will be similar for $i=2$. For $1-2z_{t}^{\left( i\right) }=\frac{1}{\sqrt{2}}$, the system becomes:

  $$\begin{aligned} x_{t+1}^{\left( 1,1\right) }= & {} \eta \frac{x_{t}^{\left( 1,1\right) }+x_{t}^{\left( 1,2\right) }}{\sqrt{2}}\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \\&-\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }+\eta \frac{x_{t}^{\left( 1,1\right) }+x_{t}^{\left( 1,2\right) }}{\sqrt{2}}\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \right) \right) \\&\times \frac{3-2\sqrt{2} }{2\left( 1+\rho \right) }+\left( 1-\eta \right) x_{t}^{\left( 1,1\right) } \\ x_{t+1}^{\left( 1,2\right) }= & {} -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }+\eta \frac{x_{t}^{\left( 1,1\right) }+x^{\left( 1,2\right) }}{\sqrt{2}}\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \right) \right) \\&\times \frac{5-2\sqrt{2}}{34\left( 1+\rho \right) }+\left( 1-\eta \right) x_{t}^{\left( 1,2\right) } \end{aligned}$$

  From the second equation, one deduces that:

  $$\begin{aligned} x_{t+1}^{\left( 1,2\right) }-x_{t}^{\left( 1,2\right) }<0 \end{aligned}$$

  Since there is no equilibrium with $x_{t}^{\left( 1,2\right) }>0$, then $x_{t}^{\left( 1,2\right) }\rightarrow 0$. From the first equation $x_{t+1}^{\left( 1,1\right) }-x_{t}^{\left( 1,1\right) }>0$ only if:

  $$\begin{aligned}&\eta \frac{x_{t}^{\left( 1,1\right) }+x_{t}^{\left( 1,2\right) }}{\sqrt{2}}\left( 1- \frac{3-2\sqrt{2}}{2\left( 1+\rho \right) }\rho \right) \left( 1+{\hat{K}} _{1}x_{t}^{\left( 1,1\right) }\right) \\&\quad -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }\right) \right) \frac{3-2\sqrt{2}}{2\left( 1+\rho \right) }-\eta x_{t}^{\left( 1,1\right) }>0 \end{aligned}$$

  that is, if:

  $$\begin{aligned} x_{0}^{\left( 1,1\right) }>\frac{-a+\sqrt{a^{2}+b}}{c} \end{aligned}$$

  where

  $$\begin{aligned} a= & {} \left( \frac{1}{2}\sqrt{2}\eta \left( 1-\frac{3-2\sqrt{2}}{2\left( 1+\rho \right) }\rho \right) \left( 1+x^{\left( 1,2\right) }K_{1}\right) -\rho \frac{1-\eta }{2\rho +2}\left( 3-2\sqrt{2}\right) -\eta \right) \\ b= & {} -\,2\eta K_{1}\left( 1-\frac{3-2\sqrt{2}}{2\left( 1+\rho \right) }\rho \right) \left( x^{\left( 1,2\right) }\eta \left( 1-\frac{3-2\sqrt{2}}{ 2\left( 1+\rho \right) }\rho \right) \right. \\&\left. -\,\frac{1-\rho }{2\left( 1+\rho \right) } \left( 3-2\sqrt{2}\right) \right) \\ c= & {} \sqrt{2}\eta K_{1}\left( \frac{\rho }{2\rho +2}\left( 2\sqrt{2} -3\right) +1\right) \end{aligned}$$

  Thus, above a certain threshold depending on the initial conditions $x_{0}^{\left( 1,1\right) }$ and $x_{0}^{\left( 1,2\right) }$, the value of $x_{t}^{\left( 1,1\right) }$ will go to $\infty$, and $x_{t}^{\left( 1,1\right) }\rightarrow 0$ below this threshold. Set $x_{0}^{\left( 1,2\right) }=0$, then a lower bound for the threshold is:

  $$\begin{aligned} x_{0}^{\left( 1,1\right) }>\frac{-a^{\prime }+\sqrt{a^{\prime 2}+b^{\prime }} }{c} \end{aligned}$$

  with

  $$\begin{aligned} a^{\prime }= & {} \left( \frac{1}{2}\sqrt{2}\eta \left( 1-\frac{3-2\sqrt{2}}{ 2\left( 1+\rho \right) }\rho \right) -\rho \frac{1-\eta }{2\rho +2}\left( 3-2 \sqrt{2}\right) -\eta \right) \\ b^{\prime }= & {} 2\eta K_{1}\left( \left( 1-\frac{3-2\sqrt{2}}{2\left( 1+\rho \right) }\rho \right) \right) \left( \frac{1-\rho }{2\left( 1+\rho \right) } \left( 3-2\sqrt{2}\right) \right) \end{aligned}$$

• (b) Let us compute the variation of the dynamics with respect to a small change $\delta z^{\left( 1\right) }$ in $z_{t}^{\left( 1\right) }$ (the same reasoning is valid for $z_{t}^{\left( 2\right) }$). Starting from:

  $$\begin{aligned} x_{t+1}^{\left( 1,1\right) }-x_{t}^{\left( 1,1\right) }= & {} \eta y_{t}^{\left( 1\right) }\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }\right. \right. \\&\left. \left. +\,\eta y_{t}^{\left( 1\right) }\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \right) \right) \frac{z_{t}^{\left( 1\right) }+1}{2\left( 1+\rho \right) \left( 1-z_{t}^{\left( 1\right) }\right) } \\&-\,\eta x_{t}^{\left( 1,1\right) } \\ x_{t+1}^{\left( 1,2\right) }-x_{t}^{\left( 1,2\right) }= & {} -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }\right. \right. \\&\left. \left. +\,\eta y_{t}^{\left( 1\right) }\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \right) \right) \frac{ z_{t}^{\left( 1\right) }}{2\left( 1+\rho \right) \left( 1-z_{t}^{\left( 1\right) }\right) }-\eta x_{t}^{\left( 1,2\right) } \end{aligned}$$

  and expressing $y_{t}^{\left( 1\right) }$ as a function of $x_{t}^{\left( 1,1\right) }$ and $x_{t}^{\left( 1,2\right) }$, one has:

  $$\begin{aligned} x_{t+1}^{\left( 1,1\right) }-x_{t}^{\left( 1,1\right) }= & {} \eta \left( \left( 1-2z_{t}^{\left( 1\right) }\right) x_{t}^{\left( 1,1\right) }+\sqrt{1-\left( 1-2z_{t}^{\left( 1\right) }\right) ^{2}}x_{t}^{\left( 1,2\right) }\right) \\&\left( 1-\rho \frac{z_{t}^{\left( 1\right) }+1}{2\left( 1+\rho \right) \left( 1-z_{t}^{\left( 1\right) }\right) }\right) \left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \\&-\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }\right) \right) \frac{z_{t}^{\left( 1\right) }+1}{2\left( 1+\rho \right) \left( 1-z_{t}^{\left( 1\right) }\right) }-\eta x_{t}^{\left( 1,1\right) } \\ x_{t+1}^{\left( 1,2\right) }-x_{t}^{\left( 1,2\right) }= & {} -\left( 1-\rho +\rho \left( \left( 1-\eta \right) x_{t}^{\left( 1,1\right) }\right. \right. \\&\left. \left. +\eta y_{t}^{\left( 1\right) }\left( 1+{\hat{K}}_{1}x_{t}^{\left( 1,1\right) }\right) \right) \right) \frac{ z_{t}^{\left( 1\right) }}{2\left( 1+\rho \right) \left( 1-z_{t}^{\left( 1\right) }\right) }-\eta x_{t}^{\left( 1,2\right) } \end{aligned}$$

  Given that:

  $$\begin{aligned} \frac{\partial }{\partial z^{\left( 1\right) }}\left( 1-\rho \frac{z^{\left( 1\right) }+1}{2\left( 1+\rho \right) \left( 1-z^{\left( 1\right) }\right) } \right)< & {} 0 \\ -\frac{\partial }{\partial z^{\left( 1\right) }}\frac{z^{\left( 1\right) }+1 }{2\left( 1+\rho \right) \left( 1-z^{\left( 1\right) }\right) }< & {} 0 \end{aligned}$$

  one obtains, for $\delta z^{\left( 1\right) }<0$:

  $$\begin{aligned} \delta x_{t+1}^{\left( 1,1\right) }> & {} 0 \\ \delta x_{t+1}^{\left( 1,2\right) }> & {} 0 \end{aligned}$$

  Now, let $\cos u =\left( 1-2z_{t}^{\left( 1\right) }\right)$, so that $\delta u <0$ for $\delta z^{\left( 1\right) }<0$. We set $\cos u=\frac{1}{\sqrt{2}}$ to compute $\delta y_{t+1}^{\left( 1\right) }$:

  $$\begin{aligned} \delta y_{t+1}^{\left( 1\right) }= & {} \left( \cos u\right) \delta x_{t+1}^{\left( 1,1\right) }+\left( \sin u\right) \delta x_{t+1}^{\left( 1,2\right) }+\left( \left( \cos u\right) x_{t+1}^{\left( 1,2\right) }-\left( \sin u\right) x_{t+1}^{\left( 1,1\right) }\right) \delta u \\= & {} \frac{1}{\sqrt{2}}\left( \delta x_{t+1}^{\left( 1,1\right) }+\delta x_{t+1}^{\left( 1,2\right) }+\left( x_{t+1}^{\left( 1,2\right) }-x_{t+1}^{\left( 1,1\right) }\right) \delta u\right) \end{aligned}$$

  Note that $\left( x_{t+1}^{\left( 1,2\right) }-x_{t+1}^{\left( 1,1\right) }\right) <0$, so that:

  $$\begin{aligned} \left( x_{t+1}^{\left( 1,2\right) }-x_{t+1}^{\left( 1,1\right) }\right) \delta u>0 \end{aligned}$$

  Consequently, if $z_{t}^{\left( 1\right) }$ decreases, $x_{t+1}^{\left( 1,1\right) }$ and $x_{t+1}^{\left( 1,2\right) }$ increase and then $y_{t+1}^{\left( 1\right) }$ increases too. Symmetrically $y_{t+1}^{\left( 2\right) }$ decreases and from the definition of $\left( 1-2z_{t+1}^{\left( 1\right) }\right)$, we deduce that $\left( 1-2z_{t+1}^{\left( 1\right) }\right) -\left( 1-2z_{t}^{\left( 1\right) }\right) >0$ and then $z_{t+1}^{\left( 1\right) }-z_{t}^{\left( 1\right) }<0$. This reasoning applies for an arbitrary variation of $1-2z^{\left( 1\right) }$. As a consequence, starting with $1-2z_{0}^{\left( 1\right) }>\frac{1}{\sqrt{2}}$, or equivalently, departing from $1-2z_{0}^{\left( 1\right) }=\frac{1}{\sqrt{2}}$ with $\delta z_{0}^{\left( 1\right) }<0$ implies $z_{t}^{\left( 1\right) }\rightarrow 0$ and thus $1-2z_{t}^{\left( 1\right) }\rightarrow 1$. It also implies, from the definition of $1-2z_{t}^{\left( i\right) }$, that $1-2z_{t}^{\left( 2\right) }\rightarrow 0$.

• (c) This is the consequence of (b) by exchanging the role of (1) and (2).$\square$