Equity and efficiency in an overlapping generation model

Abstract

The paper addresses intergenerational and intragenerational equity in an overlapping generation economy. We aim at defining an egalitarian distribution of a constant stream of resources, relying on ordinal non-comparable information on individual preferences. We establish the impossibility of efficiently distributing resources while treating equally agents with same preferences that belong to possibly different generations. We thus propose an egalitarian criterion based on the equal-split guarantee: this requires all agents to find their assigned consumption bundle at least as desirable as the equal division of resources.

Appendix: Proofs

1.1 Theorem 4

Proof

Let $\varepsilon \in \left( 0,1\right]$. Consider a two-goods economy with one agent per generation (each agent is thus identified by its generation).

We shall first show as a lemma that for each $\bar{a}\in A^{es}$ with $\bar{a}\gg 0$, there exists an economy $E$ for which efficiency and no-domination can not be combined with making each agent at least as well-off than the $\varepsilon$ part of $\left( \bar{c},\bar{d}\right)$. Such economy requires a specific pattern of preferences for a finite number of periods. To proof the theorem, we then construct an economy with an infinite sequence of such preference patterns, one for each element of a dense grid on $A^{es}$. $\square$

Lemma 1

Let $\bar{a}\in A^{es}$ with $\bar{a}\gg 0$. On the domain $\mathcal{E}$, there exists no rule that satisfies efficiency, no-domination, and such that for each agent $\left( i,t\right) \in I\times \mathbb {N}_{>0}$, $a_{i}\left( t\right) \succsim _{i,t}\varepsilon \bar{a}_{i}\left( t\right) =\varepsilon \left( \bar{c},\bar{d}\right)$ and for each agent $i\in I$, $a_{i}\left( 0\right) \succsim _{i,0}\varepsilon \bar{a}_{i}\left( 0\right) =\varepsilon \bar{d}$.

Proof

There are two-goods and one agent per generation. Let $\bar{a}\in A^{es}$ be an equal-split allocation such that $\bar{a}\left( t\right) =\left( \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right) \gg 0$.

Step 1. Let the preferences of agents $t-1,t,t+1\in \mathbb {N}_{>0}$ be represented by the following linear functions:¹²

$$\begin{aligned} U_{t-1}\left( c_{t-1}^{1},c_{t-1}^{2},d_{t-1}^{1},d_{t-1}^{2}\right)&= \gamma \left( c_{t-1}^{1}+c_{t-1}^{2}\right) +\zeta d_{t-1}^{1}+d_{t-1}^{2}\\ U_{t}\left( c_{t}^{1},c_{t}^{2},d_{t}^{1},d_{t}^{2}\right)&= \delta c_{t}^{1}+c_{t}^{2}+d_{t}^{1}+\delta d_{t}^{2}\\ U_{t+1}\left( c_{t+1}^{1},c_{t+1}^{2},d_{t+1}^{1},d_{t+1}^{2}\right)&= c_{t+1}^{1}+\zeta c_{t+1}^{2}+\gamma \left( d_{t+1}^{1}+d_{t+1}^{2}\right) \end{aligned}$$

where $0<\gamma <\delta <\zeta <\frac{\varepsilon }{3}\min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] <1$. For each agent $i=t-1,t,t+1$ we denote by $U_{i}^{ES}$ the utility level achieved at the equal-split bundle. The $\varepsilon$ equal-split guarantee requires that $U_{i}\left( a_{i}\right) \ge \varepsilon U_{i}^{ES}$.

Part a) The distribution of goods available at time $t$ is represented in the Edgeworth box of Fig. 1a, where agent $t-1$’s origin is the bottom-left corner and agent $t$’s origin is the top-right corner. The equal-split bundle is denoted $ES$; its $\varepsilon$ component is $\varepsilon ES^{d}$ for consumption when old and $\varepsilon ES^{c}$ for consumption when young. By efficiency, the contract curve for the goods to distribute at period $t$ (among consumption when young of $t$ and consumption when old of $t-1$) is such that either $d_{t-1}^{1}=1$ or $d_{t-1}^{2}=0$ (this corner solution follows from the linearity of preferences with $\delta <\zeta$).

Fig. 1

Determining the allocation of agent $t$

When $d_{t-1}^{2}=0$, it is not possible to assign to agent $t-1$ a bundle that satisfies the $\varepsilon$ equal-split guarantee: the maximum utility when $d_{t-1}^{1}=c_{t-1}^{1}=c_{t-1}^{2}=1$ and $d_{t-1}^{2}=0$ is $U_{t-1}^{\max }=2\gamma +\zeta <\varepsilon \min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] \le \varepsilon \bar{d}^{2}<\varepsilon U_{t-1}^{ES}$. This is represented in the graph by $U_{t-1}^{\max }$; whereas, the indifference level satisfying the $\varepsilon$ equal-split guarantee, when consumption when young is maximum ($c_{t-1}^{1}=c_{t-1}^{2}=1$), is $\bar{U}_{t-1}^{d}$; since this is higher than the $U_{t-1}^{\max }$, the efficient allocations that guarantee to agent $t-1$ the equity constraint lie on the segment $\overline{F0_{t}^{c}}$: where $d_{t-1}^{1}=1$ and $d_{t-1}^{2}>0$.

Part b) The distribution of goods available at time $t+1$ is represented in the Edgeworth box of Fig. 1b, where agent $t$’s origin is the bottom-left corner and agent $t+1$’s origin is the top-right corner. The equal-split bundle is denoted $ES$; its $\varepsilon$ component is $\varepsilon ES^{d}$ for consumption when old and $\varepsilon ES^{c}$ for consumption when young. By efficiency, the contract curve for the goods to distribute at period $t+1$ (among consumption when young of $t+1$ and consumption when old of $t$) is such that either $c_{t+1}^{1}=0$ or $c_{t+1}^{2}=1$ (this corner solution follows from the linearity of preferences with $\delta <\zeta$).

When $c_{t+1}^{1}=0$, it is not possible to assign to agent $t+1$ a bundle that satisfies the $\varepsilon$ equal-split guarantee: the maximum utility when $c_{t+1}^{2}=d_{t+1}^{1}=d_{t+1}^{2}=1$ and $c_{t+1}^{1}=0$ is $U_{t+1}^{\max }=2\gamma +\zeta <\varepsilon \min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] \le \varepsilon \bar{c}^{2}<\varepsilon U_{t+1}^{ES}$. This is represented in the graph by $U_{t+1}^{\max }$; whereas, the indifference level satisfying the $\varepsilon$ equal-split guarantee, when consumption when old is maximum ($d_{t+1}^{1}=d_{t+1}^{2}=1$), is $\bar{U}_{t+1}^{c}$; since this is higher than the $U_{t+1}^{\max }$, the efficient allocations that guarantee to agent $t+1$ the equity constraint lie on the segment $\overline{0_{t}^{d}G}$: where $c_{t+1}^{1}>0$ and $c_{t+1}^{2}=1$.

Summing up, the lifetime consumption of agent $t$, $a\left( t\right)$, satisfies $c_{t}^{1}=0, c_{t}^{2}<1$, $d_{t}^{1}<1$, and $d_{t}^{2}=0$.

Step 2. Let the preferences of agents $\tau -1,\tau ,\tau +1,\tau +2\in \mathbb {N}_{>0}$ with $\tau =t+4$ be represented by the following utilities:

$$\begin{aligned} U_{\tau -1}\left( c_{\tau -1}^{1},c_{\tau -1}^{2},d_{\tau -1}^{1},d_{\tau -1}^{2}\right)&= c_{\tau -1}^{1}+c_{\tau -1}^{2}+d_{\tau -1}^{1}+\delta d_{\tau -1}^{2}\\ U_{\tau }\left( c_{\tau }^{1},c_{\tau }^{2},d_{\tau }^{1},d_{\tau }^{2}\right)&= c_{\tau }^{1}+\zeta c_{\tau }^{2}+\gamma \left( d_{\tau }^{1}+\delta d_{\tau }^{2}\right) \\ U_{\tau +1}\left( c_{\tau +1}^{1},c_{\tau +1}^{2},d_{\tau +1}^{1},d_{\tau +1}^{2}\right)&= \gamma \left( c_{\tau +1}^{1}+\zeta c_{\tau +1}^{2}\right) +\zeta d_{\tau +1}^{1}+d_{\tau +1}^{2}\\ U_{\tau +2}\left( c_{\tau +2}^{1},c_{\tau +2}^{2},d_{\tau +2}^{1},d_{\tau +2}^{2}\right)&= \delta c_{\tau +2}^{1}+c_{\tau +2}^{2}+d_{\tau +2}^{1}+d_{\tau +2}^{2} \end{aligned}$$

where $0<\gamma <\delta <\zeta <\frac{\varepsilon }{3}\min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] <1$. For each agent $i=\tau -1,\tau ,\tau +1,\tau +2$, $U_{i}^{ES}$ denotes the utility level at the equal-split bundle; then, the $\varepsilon$ equal-split condition requires that $U_{i}\left( a_{i}\right) \ge \varepsilon U_{i}^{ES}$.

Part a) The distribution of goods available at time $\tau$ is represented in the Edgeworth box of Fig. 2a, where agent $\tau -1$’s $\tau -1$’s origin is the bottom-left corner and agent $\tau$’s top-right corner. The equal-split bundle is denoted $ES$; its $\varepsilon$ component is $\varepsilon ES^{d}$ for consumption when old and $\varepsilon ES^{c}$ for consumption when young. By efficiency, the contract curve for the goods to distribute at period $\tau$ (among consumption when young of $\tau$ and consumption when old of $\tau -1$) is such that either $c_{\tau }^{1}=0$ or $c_{\tau }^{2}=1$ (this corner solution follows from the linearity of preferences with $\delta <\zeta$).

Fig. 2

Determining the allocations of agents $\tau$ and $\tau +1$

When $c_{\tau }^{1}=0$, it is not possible to assign to agent $\tau$ a bundle that satisfies the $\varepsilon$ equal-split guarantee: the maximum utility when $c_{\tau }^{2}=d_{\tau }^{1}=d_{\tau }^{2}=1$ and $c_{\tau }^{1}=0$ is $U_{\tau }^{\max }=\gamma \left( 1+\delta \right) +\zeta <\varepsilon \min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] \le \varepsilon \bar{c}^{1}<\varepsilon U_{\tau }^{ES}$. This is represented in the graph by $U_{\tau }^{\max }$; whereas, the indifference level satisfying the $\varepsilon$ equal-split guarantee, when consumption when old is maximum ($d_{\tau }^{1}=d_{\tau }^{2}=1$), is $\bar{U}_{\tau }^{c}$; since this is higher than the $U_{\tau }^{\max }$, the efficient allocations that guarantee to agent $\tau$ the equity constraint lie on the segment $\overline{0_{\tau -1}^{d}G'}$: where $c_{\tau }^{1}>0$ and $c_{\tau }^{2}=1$.

Part b) The distribution of goods available at time $\tau +2$ is represented in the Edgeworth box of Fig. 2b, where agent $\tau +1$’s origin is the bottom-left corner and agent $\tau +2$’s origin is the top-right corner. The equal-split bundle is denoted $ES$; its $\varepsilon$ component is $\varepsilon ES^{d}$ for consumption when old and $\varepsilon ES^{c}$ for consumption when young. By efficiency, the contract curve for the goods to distribute at period $\tau +2$ (among consumption when young of $\tau +2$ and consumption when old of $\tau +1$) is such that either $d_{\tau +1}^{1}=1$ or $d_{\tau +1}^{2}=0$ (this corner solution follows from the linearity of preferences with $\delta <\zeta$).

When $d_{\tau +1}^{2}=0$, it is not possible to assign to agent $\tau +1$ a bundle that satisfies the $\varepsilon$ equal-split guarantee: the maximum utility when $d_{\tau +1}^{1}=c_{\tau +1}^{1}=c_{\tau +1}^{2}=1$ and $d_{\tau +1}^{2}=0$ is $U_{\tau +1}^{\max }=\gamma \left( 1+\zeta \right) +\zeta <\varepsilon \min \left[ \bar{c}^{1},\bar{c}^{2},\bar{d}^{1},\bar{d}^{2}\right] \le \varepsilon \bar{d}^{2}<\varepsilon U_{\tau +1}^{ES}$. This is represented in the graph by $U_{\tau +1}^{\max }$; whereas, the indifference level satisfying the $\varepsilon$ equal-split guarantee when consumption when young is maximum ($c_{\tau +1}^{1}=c_{\tau +1}^{2}=1$) is $\bar{U}_{\tau +1}^{d}$; since this is higher than the $U_{\tau +1}^{\max }$, the efficient allocations that guarantee to agent $\tau +1$ the equity constraint lie on the segment $\overline{F'0_{\tau +2}^{c}}$: $d_{\tau +1}^{1}=1$ and $d_{\tau }^{2}>0$.

Step 3. The efficient distribution of resources available at $\tau +1$ requires that either:

(i) :

$c_{\tau +1}^{1}=0$ (and $d_{\tau }^{1}=1$); or

(ii) :

$c_{\tau +1}^{2}=1$ (and $d_{\tau }^{2}=0$).

From Step 1, $a\left( t\right)$ satisfies $c_{t}^{1}=0$, $c_{t}^{2}<1, d_{t}^{1}<1$, and $d_{t}^{2}=0$.

From Step 2, in case i), $a\left( \tau \right)$ is such that $c_{\tau }^{1}>0, c_{\tau }^{2}=1$, $d_{\tau }^{1}=1$ and $d_{\tau }^{2}\ge 0$: thus, $a\left( \tau \right)$ dominates $a\left( t\right)$.

In case ii), $a\left( \tau +1\right)$ is such that $c_{\tau +1}^{1}\ge 0, c_{\tau +1}^{2}=1, d_{\tau +1}^{1}=1$, and $d_{\tau +1}^{2}>0$: thus, $a\left( \tau +1\right)$ dominates $a\left( t\right)$. $\square$

In two dimensions, each allocation in $A^{es}$ can be mapped into a square with the proportion of each resource assigned to consumption when young on the two axes. We construct a dense grid over this square. First consider the center of the square, i.e. $\bar{a}\in A^{es}$ such that $\left( \bar{c}^{1},\bar{c}^{2}\right) =\left( \frac{1}{2},\frac{1}{2}\right)$. Second, by imaginary folding the square, once vertically and once horizontally, $4$ smaller (overlapping) squares are obtained; the center of each is such that $\left( \bar{c}^{1},\bar{c}^{2}\right) =\left( a,b\right)$ with $a,b\in \left\{ \frac{1}{4},\frac{3}{4}\right\}$. Folding these squares again, once vertically and once horizontally, $16$ smaller (overlapping) squares are obtained; the center of each is such that $\left( \bar{c}^{1},\bar{c}^{2}\right) =\left( a,b\right)$ with $a,b\in \left\{ \frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8}\right\}$. Repeating this folding operation, a cell-centered coarsening method is described.

Let $A^{es}\left( \nu \right)$ be the set of equal-split bundles that are a center of the squares obtained after $\nu$ folding iterations.¹³ For each $\nu \ge 1$, we can order the elements of $A^{es}\left( \nu \right)$ lexicographically (good $1$ first): for each $\bar{a},\tilde{a}\in A^{es}\left( \nu \right)$, $\bar{a}$ is ranked before $\tilde{a}$ if $\bar{c}^{1}<\tilde{c}^{1}$ or if $\bar{c}^{1}=\tilde{c}^{1}$ and $\bar{c}^{2}<\tilde{c}^{2}$. Let $lex\left( \bar{a};\nu \right)$ denote the rank of $\bar{a}$ in the set $A^{es}\left( \nu \right)$. Then each $\bar{a}$ in the grid is uniquely identified by a function $k$ that associates to it a number $n\in \mathbb {N}_{>0}$ such that $k\left( \bar{a}\right) =1$ for $\nu =0$ and $k\left( \bar{a}\right) =\frac{4^{\nu }-1}{4-1}+lex\left( \bar{a};\nu \right)$ for $\nu \ge 1$.

Let economy $E\in \mathcal{E}$ be such that for each $n\in \mathbb {N}_{>0}$ and the corresponding $\bar{a}=k^{-1}\left( n\right)$, preferences of agents $\theta -1,\theta ,\theta +1\in \mathbb {N}_{>0}$ with $\theta =2+10n$ are as the ones of $t-1,t,t+1$ in Step 1 of the above Lemma and preferences of agents $\theta +5,\theta +6,\theta +7,\theta +8\in \mathbb {N}_{>0}$ are as the ones of $\tau -1,\tau ,\tau +1,\tau +2$ in Step 2 of the Lemma (note that these preferences depend on the choice of $\bar{a}$ through the parameters $\gamma ,\delta ,\zeta$ that depend on it).

Assume by contradiction that there exists an equal-split allocation $\bar{a}\in A^{es}$ with $\bar{a}\gg 0$ that satisfies the axioms for economy $E$. Since the grid is dense and preferences are continuous, there exists an $n\in \mathbb {N}_{>0}$ such that for agents $\theta -1,\theta ,\theta +1,\theta +5,\theta +6,\theta +7,\theta +8\in \mathbb {N}_{>0}$ with $\theta =2+10n$ the clash of the Lemma holds.

1.2 Theorem 6

Proof

We construct an economy $E\in \mathcal{E}^{\kappa }$ for which no allocation satisfies efficiency and equal treatment of equals. Let $L=2$ and $\kappa \in \left( 0,1\right]$. Agents are of two kinds, $\alpha$ and $\beta$, with preferences represented by the following functions, with $\gamma >1$ and $\zeta \in \left( 0,1\right)$:

$$\begin{aligned} U_{\alpha }\left( c^{1},c^{2},d^{1},d^{2}\right) =\zeta \left( \gamma c^{1}+c^{2}\right) +\gamma d^{1}+d^{2}\\ U_{\beta }\left( c^{1},c^{2},d^{1},d^{2}\right) =\zeta \left( c^{1}+\gamma c^{2}\right) +d^{1}+\gamma d^{2} \end{aligned}$$

Let generations $t\in \left\{ 1,2,3,4,5\right\}$ consist of an $\alpha$ and a $\beta$ agent; let generations $t\in \left\{ 7,9,11,\ldots \right\}$ consist of two type $\alpha$ agents and generations $t\in \left\{ 8,10,12,\ldots \right\}$ consist of two type $\beta$ agents.

By equal treatment of equals, the $\alpha$ agents of generations $t\in \left\{ 7,9,11,\ldots \right\}$ should achieve the same utility level, say $\bar{U}_{\alpha }$. Similarly, the $\beta$ agents of generations $t\in \left\{ 8,10,12,\ldots \right\}$ should achieve the same utility level, say $\bar{U}_{\beta }$. The largest such utilities that can be feasibly assigned to these agents need to satisfy both the following constraints: $\bar{U}_{\alpha }\le \frac{1+\gamma }{2}+\zeta \left( \frac{1+\gamma }{2}-\bar{U}_{\beta }\right)$ and $\bar{U}_{\beta }\le \frac{1+\gamma }{2}+\zeta \left( \frac{1+\gamma }{2}-\bar{U}_{\alpha }\right)$. When generations $t\in \left\{ 1,2,3,4,5\right\}$ are efficiently assigned only the goods that cannot be assigned to other generations (generation $0$ and generation $6$), at least one agent ($\alpha$ or $\beta$) of these generations achieves a utility level larger than $U^{min}\equiv \gamma \frac{\zeta +\zeta ^{2}+\zeta ^{3}+\zeta ^{4}}{1+\zeta +\zeta ^{2}+\zeta ^{3}+\zeta ^{4}}$. Clearly, when $\zeta$ is close to $1$ and $\gamma$ is sufficiently large, $U^{\min }$ is larger than the utility level agents of generations $t\ge 7$ can equally achieve, i.e. $\gamma \frac{\zeta +\zeta ^{2}+\zeta ^{3}+\zeta ^{4}}{1+\zeta +\zeta ^{2}+\zeta ^{3}+\zeta ^{4}}>\frac{1+\gamma }{2}$. $\square$