Pareto Efficiency, Inequality and Distribution Neutral Fiscal Policy—An Overview

Abstract

A structure of taxes and transfers that keep the income distribution unchanged even after positive or negative shocks to an economy, is referred as a Distribution Neutral Fiscal Policy. Marjit and Sarkar (Distribution neutral welfare ranking-extending pareto principle, 2017, [14]) referred this as a Strong Pareto Superior (SPS) allocation which improves the standard Pareto criterion by keeping the degree of inequality, not the absolute level of income, intact. In this paper we show the existence of a SPS allocation in a general equilibrium framework, and we provide a brief survey of distribution neutral fiscal policies existing in the literature. We also provide an empirical illustration with Indian Human Development Survey data.

Mathematical Appendix

Let $\{W^i\} :$ endowment vectors $i=1,2,\ldots ,n;\sum \limits _{i} W^i=W$.

Utility functions, real valued, assumed to be continuous, increasing, strictly quasiconcave : $\{\bar{U}^i\}$ The consumption possibility set is $\mathfrak {R}^n_+$ the non-negative orthant. The set of feasible allocation vectors $\mathfrak {F}=\{\{Y^i\}: Y^i\ge 0, \sum _i Y^i \le W\};~\{W^i\}\in \mathfrak {F}$; $\mathfrak {U}=\{\{U^i\} : \exists \{Y^i\}\in \mathfrak {F},U^i(Y^i)=\bar{U}^i\forall i\}$ is the set of feasible utilities. Notice $\{U^i(W^i)\}\in \mathfrak {U}$.

Statement 1 $\mathfrak {U}$ is a nonempty compact subset of $\mathfrak {R}^n_{+}$

Proof: The above follows given the continuity of $U^i$ over a compact set $\mathfrak {F}$.

Let $U^i(W^i)=\hat{U}^i\forall i$; there is a $P^*=(P^*_1,P^*_2,\ldots ,P^*_n)$ which is a competitive equilibrium i.e., $X^{i*}$ solves the problem $max U^i(.)$ subject to $P^*x\le P^*.W^i \forall i$ and $\sum _{i}X^{i*}=W$. Let $U^i(X^{i*}) = U^{i*}$.

Define $\mathfrak {U}^\mathfrak {P}=\{\{\bar{U}^i\} : \{\bar{U}^i\}\in \mathfrak {U},~\exists \{U^i\}\in \mathfrak {U} ~\text {such that}~,U^i>\bar{U}^i\forall i\}$: Pareto Frontier. The First Fundamental Theorem assures us that $\{U^{i*}\}\in \mathfrak {U}^\mathfrak {P}$.

Consider next $\bar{\theta }<\hat{\theta }$; from the property of the supremum, there is $\tilde{\theta }>\bar{\theta }$ such that $U(\tilde{\theta })\in \mathfrak {U}$ i.e., there is $\tilde{Y}^i$ such that $\{\tilde{Y}\}\in \mathfrak {F}$ and $U^i(\tilde{Y}^i)=\tilde{\theta }U^i(W^i) \forall i$. Now there must be a scalar $1>\alpha _i\ge 0$ such that $U^i(\alpha _i \tilde{Y}^i)=\bar{\theta }.U^i(W^i)$ since by shrinking the scalar $\alpha _i$ we can make the left hand side go to zero (U is increasing), whereas for $\alpha _i=1$, the left hand side is greater; also then we can claim $\{\alpha _i \tilde{Y}^i\}\in \mathfrak {F}$ since

$$\alpha _i \tilde{Y}^i\ge 0$$

and

$$\sum _{i}\alpha _i \tilde{Y}^i\le max_{i} \alpha _i \sum _{i}\tilde{Y}^i \le W$$

$\quad \square$

Statement 3 $U(\hat{\theta })\in \mathfrak {V}$

Thus one may conclude that $\{\hat{Y}^i\}$ is a Pareto optimal configuration with the property that $U^i(\hat{Y}^i)=\hat{\theta }U^i(W^i)\forall i$. And since $\hat{\theta }$ is uniquely determined, so is the configuration $\{\hat{Y}^i\}$.