Strategic and stable pollution with finite set of economic agents and a finite set of consumption commodities: a Pareto comparison

Abstract

In this paper, we apply the framework of a public bad economy with a finite set of economic agents and a finite set of consumption commodities. The pollution of an agent is emitted into the air while consuming his consumption goods. A linear public bad economy model with a single private good had been introduced in Shitovitz and Spiegel (Econ Theory 22(1):17–31, 2003) and here we reconsider its extension to finite number of private goods and prove existence of a core ‘trading’ allocation (Perets et al. J Math Econ 48(3):163–169, 2012) that Pareto dominates the Nash allocation. This mathematical model embodies the restriction of consumption by all polluting agents, to decrease the amount of the public bad, affecting the whole economy, worldwide. Note specifically that the Lindahl allocation may not Pareto dominate the Nash allocation in some finite economy, in contrast to well-known asymptotic results.

Appendices

Appendix A: Strict ordinal normality of the utilities and the normality properties of the commodities

(A.1.) and (A.2.) under strict quasi-concave utilities are equivalent to $( {c_k } )_{k=1,\ldots ,K}$ being strictly normal commodities and the pollution G is a strict inferior commodity.

Proof

For $C^{2}$, strict quasi-concave utility function u( c, G ), on $( {0,W} ]^{K}\times [ {0,W} )$, we have that the marginal rate of substitution between G and $c_k$ is denoted by $\mathrm{MRS}_{G-c_k }\equiv -\frac{u_G}{u_{c_k }}( {c,G} )$ and depends only on $( {c_k ,G} )$.

1. Assume first that (A.1.) and (A.2.) hold, that is: $\frac{\partial \mathrm{MRS}_{G-c_k }}{\partial c_k }=-\frac{u_{c_k G} \cdot u_{c_k } -u_{c_k c_k } \cdot u_G }{\left( {u_{c_k } } \right) ^{2}}>0$ and $\frac{\partial \mathrm{MRS}_{G-c_k }}{\partial G}=-\frac{u_{GG} \cdot u_{c_k } -u_{c_k G} \cdot u_G }{( {u_{c_k } } )^{2}}>0$. Then, it follows that $u_{c_k G} \cdot u_{c_k } -u_{c_k c_k } \cdot u_G <0$ and $u_{GG} \cdot u_{c_k } -u_{c_k G} \cdot u_G <0$, for all $k=1,\ldots ,K$ on $( {0,W} ]^{K}\times [ {0,W} )$. Since $I>0$ and $(p_1 ,\ldots ,p_K )>>0$ it follows that the F.O.C. for the consumer maximization problem for $( {\hat{{c}}_i ,\hat{{G}}})\in R_{++}^{K+1}$ is $\frac{P_G }{P_{c_k } }=\frac{u_G ( {\hat{{c}}_1 ( I ),\ldots ,\hat{{c}}_K ( I ),\hat{{G}}( I )} )}{u_{c_k } ( {\hat{{c}}_1 ( I ),\ldots ,\hat{{c}}_K ( I ),\hat{{G}}( I )} )}$. Assuming the function is differentiable at $( {( {\hat{{c}}_i } )_{i\in N} ,\hat{{G}}} )\in ( {0,W} ]^{KN}\times [ {0,W} )$, the partial derivatives of both sides, with respect to I, implies that $u_{c_k } \left( {\sum \nolimits _{k=1}^K {u_{Gc_k } \cdot \frac{\partial \hat{{c}}_k }{\partial I}} +u_{GG} \cdot \frac{\partial \hat{{G}}}{\partial I}} \right) =u_G \left( {\sum \nolimits _{{k}'=1}^K {u_{c_{{k}'} c_k } \cdot \frac{\partial \hat{{c}}_k }{\partial I}} +u_{c_{{k}'} G} \cdot \frac{\partial \hat{{G}}}{\partial I}} \right)$.

That is equation $( \wedge )$

$\frac{\partial \hat{{c}}_{{k}'} }{\partial I}\cdot ( {u_{Gc_{{k}'} } \cdot u_{c_{{k}'} } -u_{c_{{k}'} c_{{k}'} } \cdot u_G } )+\sum _{k\ne {k}'} {( {\frac{\partial \hat{{c}}_k }{\partial I}( {u_{c_{{k}'} } \cdot u_{Gc_k } -u_G \cdot u_{c_k c_{{k}'} } } )} )} =\frac{\partial \hat{{G}}}{\partial I}\cdot ( {u_{c_{{k}'} G} \cdot u_G -u_{GG} \cdot u_{c_{{k}'} } } )$ Since $\frac{\partial \mathrm{MRS}_{G-c_{{k}'} }}{\partial c_k }=0,\forall k\ne {k}'$, we have $\frac{\partial \mathrm{MRS}_{G-c_{{k}'} } }{\partial c_k }=0$ which means $u_{Gc_k } \cdot u_{c_{{k}'} } -u_{c_k c_{{k}'} } \cdot u_G =0$. Combining the latter with $u_{c_{{k}'} G} \cdot u_{c_{{k}'} } -u_{c_{{k}'} c_{{k}'} } \cdot u_G <0$ and $u_{GG} \cdot u_{c_{{k}'} } -u_{c_{{k}'} G} \cdot u_G <0$, we have that $\frac{\partial \hat{{c}}_{{k}'} }{\partial I}>0$ and $\frac{\partial \hat{{G}}}{\partial I}<0$. Thus, for all $k=1,\ldots ,K \quad ( {\hat{{c}}_k } )_{k=1}^K$ are strictly normal goods and $\hat{{G}}$ is an inferior good. $\square$

2. Now assume that $\frac{\partial \hat{{G}}}{\partial p_G }<0$ and $\frac{\partial \hat{{c}}_k }{\partial p_k }>0$ for every $k=1,\ldots ,K$.

Since the utility function of each consumer is strictly quasi-concave, it follows that by the convexity of the preference relation induces by the utility $u(\hat{{c}}_{-k} ,c_k ,\hat{{G}})=\hat{{U}}$ is strictly quasi-concave in $( {c_k ,G} )$ which induces a strictly convex preference relation on $( {0,W} ]\times [ {0,W} )$. Since, $U(c_k ,G)\equiv u(\hat{{c}}_{-k} ,c_k ,G)$, as $u_k =\frac{\partial }{\partial c_k }U(c_k ,G)=\frac{\partial }{\partial c_k }u(\hat{{c}}_{-k} ,c_k ,G)$ it follows that $U_k = {\frac{\partial U( {c_1 ,\ldots ,c_K ,G} )}{\partial c_k }} |_{c_{-k} =\hat{{c}}_{-k} }$ and as we are only interested in ${\mathrm{MRS}_{G-c_k }^U } |_\wedge = {\mathrm{MRS}_{G-c_k }^u } |_\wedge = {\frac{dc_k (G)}{dG}} |_{\hat{{G}}}$, we will use the function $u(\hat{{c}}_{-k} ,c_k ,G)$ to finish the proof.

By (Mas-Colell et al. 1995 p. 938 Example M.D.2), we have that the determinant of the bordered Hessian satisfies:

$$\begin{aligned} \left| {{\begin{array}{llll} {U_{c_k c_k } }&{} {U_{c_k G} }&{} {U_{c_k } } \\ {U_{c_k G} }&{} {U_{GG} }&{} {U_G } \\ {U_{c_k } }&{} {U_G }&{} 0 \\ \end{array} }} \right| >0, \end{aligned}$$

the determinant equals:

$$\begin{aligned} 2U_{c_k } U_{c_k G} U_G -U_{c_k c_k } U_G U_G -U_{c_k } U_{GG} U_{c_k } >0. \end{aligned}$$

Note that all partial derivatives of U are actually the corresponding partial derivatives of u.

Which is $U_G ( {U_{c_k } U_{c_k G} -U_{c_k c_k } U_G } )+U_{c_k } ( {U_{c_k G} U_G -U_{c_k } U_{GG} } )>0$

By the derivation of the F.O.C. with respect to I, we get equation $(\wedge )$ which yields that $( {U_{c_k } U_{c_k G} -U_{c_k c_k } U_G } )$ and $( {U_{c_k G} U_G -U_{c_k } U_{GG} } )$ have opposite signs, combining with $U_G <0$ and $U_{c_k } >0$ and the last inequality, we may conclude that $( {U_{c_k G} U_G -U_{c_k } U_{GG} } )>0$ and $( {U_{c_k } U_{c_k G} -U_{c_k c_k } U_G } )<0$. Thus, the strict ordinal normality conditions for the utility u holds as $u_{c_k } =U_{c_k } ,u_{c_k G} =U_{c_k G}$ and $u_G =U_G$ where all other coordinates are $\hat{{c}}_{-k}$. Note that for each $( {c,G} )\in ( {0,W} ]^{K}\times [ {0,W} )$ by the strict quasi-concavity and differentiability, we have unique supporting hyperplane through ( c, G ) such that $( {c,G} )=( {\hat{{c}},\hat{{G}}} )$ is the interior solution for these prices. $\square$

Appendix B: Uniqueness of Nash equilibrium

Under assumptions (A.1.) and (A.2.) our pollutive economy admits a unique Nash equilibrium.

Proof

Assume by negation that there are two Nash equilibria $s^{*}$ and $s^{**}$, and we further assume $G^*\ge G^{**}$. Thus, if $\forall i\in N,\forall k\in \{ {1,\ldots ,K} \},c_k^{i*} \le c_k^{i**}$, then obviously by summation $s^{*}=s^{**}$. Else, $\exists i\in N,\exists k\in \{ {1,\ldots ,K} \},c_k^{i*} >c_k^{i**} >0$. Since, $\omega _{i,k} \ge c_k^{i*} >c_k^{i**}$, it follows that for $s^{**}$, from the Nash maximization problem for the interior solution $c_k^{i**}$, we have $u^{i}_{c_{{k}'}^i } (c_i ^{**},G^{**})+u^{i}_G (c_i ^{**},G^{**})=0$, implying $\mathrm{MRS}_{G-c_{{k}'}^i }^i ( {c_i^{**} ,G^{**}} )=1$. By (A.2), we have that $\mathrm{MRS}_{G-c_k^i }^i ( {c_i^*,G^{*}} )>\mathrm{MRS}_{G-c_k^i }^i ( {c_i^{**} ,G^{**}} )=1$. Therefore, $u^{i}_{c_k^i } (c_i ^{*},G^{*})+u^{i}_G (c_i ^{*},G^{*})<0$ implying there is $\varepsilon >0$ with $\omega ^{i}_k \ge c_{i,k}^*-\varepsilon >0$ such that $u^{i}(c_i ^{*}-\varepsilon e_k ,G^{*}-\varepsilon )>u^{i}(c_i ^{*},G^{*})$ contradicting that $s^{*}$ is a Nash equilibrium. $\square$