Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria

Abstract

We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the cost function. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse gamma distribution as an equilibrium in the particular case of quadratic cost functions which has been previously considered in the literature.

Appendices

Appendix A: Proof of Lemma 3.5

(i) Introducing the change of variables (3.17) into (3.15) and using Green’s formula, we find (3.16). Green’s formula is applicable and the boundary terms disappear because of the assumptions of smoothness made on f and g.

(ii) We just let σ=φ in (3.16).

(iii) Taking σ=Constant in (3.16), the left-hand side vanishes. Therefore, if (3.18) is not satisfied, there cannot exist a solution. Supposing now that (3.18) is satisfied, we can restrict the set of test functions σ to ${\mathcal{H}}_{\varXi0}$ in the weak formulation (3.16). Indeed, from $\sigma \in{\mathcal{H}}_{\varXi0}$, we can construct an arbitrary test function in ${\mathcal{H}}_{\varXi}$ by simply adding a constant. But, because (3.18) is satisfied, the weak formulation (3.16) is still true for this test function. Now, because of the assumed Poincaré inequality (3.14), the left-hand side of (3.16) is a coercive bilinear form on ${\mathcal{H}}_{\varXi0}$ while, because of the assumption that $\psi\in{\mathcal{X}}_{\varXi}$, the right-hand side is a continuous linear form on ${\mathcal{H}}_{\varXi0}$. Therefore, Lax-Milgram’s theorem applies and there exists a unique solution $\varphi\in{\mathcal{H}}_{\varXi0}$ to problem (3.16). The most general solution is of the form φ+Constant because of point (ii). This ends the proof.

Appendix B: Proof of Lemma 3.13

Let v: $z \in{\mathbb{R}}_{+} \mapsto v(z) \in{\mathbb{R}}$ such that

$$\begin{aligned} & \int_0^\infty\bigl(|v(z)|^2 + | \partial_z v(z)|^2\bigr) \gamma_{\alpha,\beta }(z) dz < \infty, \end{aligned}$$

(B.1)

where γ _α,β (z) is the gamma distribution defined at (3.37). Then, formula (10) of [3] states that there exists a constant C _α,β >0 such that

$$\begin{aligned} & \int_0^\infty|v(z) - \bar{v}|^2 \gamma_{\alpha,\beta}(z) dz \leq C_{\alpha,\beta} \int_0^\infty| \partial_z v(z)|^2 \gamma_{\alpha ,\beta}(z) dz, \end{aligned}$$

(B.2)

where

$$\begin{aligned} & \bar{v} = \int_0^\infty v(z) \gamma_{\alpha,\beta}(z) dz. \end{aligned}$$

(B.3)

Then, we make the change of variables z=1/y in (B.1), (B.2), (B.3). We denote by u(y)=v(z) and use (3.38). We remark that ∂ _z v(z)=−y ² ∂ _y u(y). Therefore, we have, denoting by C _α,β generic constants only depending only on α and β:

$$\begin{aligned} \int_0^\infty|u(y)|^2 g_{\alpha,\beta}(y) \frac{dy}{y^2} =& \int_0^\infty|v(z)|^2 z^2 \gamma_{\alpha,\beta}(z) dz \\ =& C_{\alpha,\beta} \int_0^\infty|v(z)|^2 \gamma_{\alpha+2,\beta }(z) dz, \end{aligned}$$

and

$$\begin{aligned} \int_0^\infty|y^2 \partial_y u (y)|^2 g_{\alpha,\beta}(y) \frac{dy}{y^2} =& \int_0^\infty| \partial_z v (z)|^2 z^2 \gamma _{\alpha,\beta}(z) dz \\ =& C_{\alpha,\beta} \int_0^\infty| \partial_z v (z)|^2 \gamma _{\alpha+2,\beta}(z) dz, \end{aligned}$$

and finally,

$$\begin{aligned} \bar{u} = \int_0^\infty u (y) g_{\alpha,\beta}(y) \frac{dy}{y^2} =& \int_0^\infty v (z) z^2 \gamma_{\alpha,\beta}(z) dz \\ =& C_{\alpha,\beta} \int_0^\infty v (z) \gamma_{\alpha+2,\beta}(z) dz. \end{aligned}$$

Now, letting $(\alpha,\beta) = (\frac{\kappa+d}{d}, \frac{\kappa \varUpsilon}{d})$, we notice that v satisfies (B.1) (with α shifted to α+2) if and only if $u \in{\mathcal{H}}_{\varUpsilon}$. Furthermore, $\bar{v} = 0$ if and only if $u \in{\mathcal{H}}_{\varUpsilon0}$. Now, the Poincaré inequality (B.2) (with α shifted to α+2) leads to (3.14).