Abstract In the setting of human capital accumulation we consider a problem of fair allocation of resources to be utilized for education. First we propose two axioms of envy-freeness, one presumes that nobody is responsible for his initial skill or his learning ability, the other presumes that everybody is perfectly responsible for his initial skill and his learning ability. Then we propose a series of axioms indexed by time period that allow intermediate levels of responsibility. Given that there are T periods, for each τ = 0, · · · , T we formulate an axiom of envy-freeness which presumes that the individuals are responsible for their skills and learning eﬀectiveness only after Period τ . We provide a family of rules such that for each τ the corresponding solution is eﬃcient and envy-free after Period τ .

1

Introduction

This paper is concerned with fair allocation of resources to be utilized for education, in particular for acquisition of skill. In contrast to material resources, skills or human capital in general are embodied to each individual, and cannot be transfered across individuals in the same way as the material ones, and normally they are not consumable by themselves. Skills are not unchanging fixed factors and can be acquired endogenously, though. This is nothing but one of the primary roles of education. It is true at the same time, however, ∗

Adam Smith Business School, University of Glasgow. email: [email protected]

1

that individuals have diﬀerent initial levels of skill, and that their learning outcomes depend on their learning abilities or talents which may be diﬀerent across them. Individuals are not “tabula rasa,” and we have to take such diﬀerences as given conditions. There is a long history of debates on whether the concept fairness should refer to equality of welfare/outcome or equality of opportunity, and also on what “equal” should mean in each case (see Roemer [6] and Phillips [5] for comprehensive discussions and references therein). Which view is appealing depends on the nature of environment and what factors each individual should be responsible for. In exchange economies without production, in which resources are falling from the heaven and nobody is responsible for that, the two views may be compatible with each other, since we can divide the total resources equally and let them exchange in a competitive market, which achieves the state in which everybody faces the same opportunity (budget set) and nobody envies anybody else. The two views conflict with each other, on the other hand, when there is production and people have diﬀerent levels of skill. There are two prominent definitions of fairness as absence of envy, one presumes nobody is responsible for his skill at all (see Pazner and Schmeidler [4]) and the other presumes everybody is responsible for his skill (see Varian [7]). Fleurbaey and Maniquet [2] propose a class of axioms indexed by reference points about which certain envy-freeness condition should be met, and provide characterization of allocation rules satisfying it and additional properties. Hayashi [3] provides characterization of classes of decentralizable allocations rules which allow a variety of degrees of responsibility. Yet they are in static settings and it has not been questioned where individuals’ skills are coming from and which part of the process of skill formation they are responsible for. Extension to a dynamic setting leads us to consider not only whether individuals are responsible for their skills at a fixed period but also how much they are responsible for their initial levels of skill and their learning abilities across periods. It leads us to consider a variety of degrees of responsibility about since which age people are taken to be responsible for their skills and eﬀectiveness of learning. This is indeed how we argue in real life until what age free and/or compulsory and/or uniform education should be provided in society. To our knowledge, Fleurbaey and Vallettay [1] is the only paper in the context of fair allocation which takes endogenous determination of human capital into account. They propose a leximin-type ranking over allocations and focus on how to save the most disad2

vantaged individual by taxation under incentive compatibility constraint, while we propose rules based on the idea of envy-freeness. In a model of human capital accumulation, we propose axioms which clarify what fairness should refer to depends on what should be the right degree of responsibility. This allows for quantifiable arguments on how people are responsible their skills and learning abilities. We include physical capital into the model as well, while we still try to keep the model as simple as possible, because we are concerned with the case that somebody has to work instead of learning because of poverty. Throughout the paper we assume that everybody is responsible for his preference. This might be unsatisfactory when we take a larger scope, where individuals’ preference formation depends on inheritance from their parents and/or on circumstances of early personality developments for which they cannot be responsible. Nevertheless we believe that the current exercise is an indispensable step in order to get observations which are suggestive about such general situations. First we propose two axioms of envy-freeness, one presumes that nobody is responsible for his initial skill or his learning ability, the other presumes that everybody is perfectly responsible for his initial skill and his learning ability throughout his life. These two form extreme points in the set of views about fairness, where equality of welfare/outcome corresponds to the first one and equality of opportunity corresponds to the second one. We show that the requirement of ex-ante eﬃciency is incompatible with the first one and compatible with the second one. Then we propose a series of axioms indexed by time period that allow intermediate levels of responsibility. Given that there are T periods, for each τ = 0, · · · , T , we formulate an axiom of envy-freeness which presumes that the individuals are responsible for their skills and learning eﬀectiveness only after Period τ . Note that when τ = 0 the condition reduces to envy-freeness under full responsibility and when τ = T it reduces to envy-freeness under full exemption. We provide a family of rules such that for each τ the corresponding solution is eﬃcient after Period τ and envy-free under responsibility after Period τ , while it is impossible to meet eﬃciency after earlier periods than Period τ . This shows that there is a trade-oﬀ between seeking eﬃciency to hold for longer periods and seeking to allow exemption for longer periods. 3

2

Set up

There are T periods, where individuals are born in Period 0, starts learning, working and consuming at Period 1, and live until Period T and there is no learning on this last period. Let I denote the set of individuals. There is one physical good which can be produced from labor or reproduced from the same good. For simplicity we assume that production of the physical good is linear: at each period t = 1, · · · , T , given each individual i’s human capital acquired in the previous period denoted by hi,t−1 , his labor hour ait , and the level of capital left in the previous period denoted by kt−1 , the physical good produced is ∑ hi,t−1 ait + rkt−1 , i∈I

where r is constant over time just for simplicity. The analysis does not change essentially (∑ ) as far as the production function takes the form Ft i∈I hi,t−1 ait , kt−1 , so without loss of generality we adopt the above form of production function. At Period t = 1, · · · , T −1 the physical good can be either consumed or used as resource for education, while at Period T it can only be consumed. At each period each individual is given 1 unit of time. At Period t = 1, · · · , T − 1 an individual’s time can be used either for labor or education or leisure, while at Period T it can be used either for labor or leisure only. For a given individual i, his life path denoted by xi = (hi0 , ci1 , ai1 , bi1 , ei1 , · · · , hi,t−1 , ciT , aiT ) is a sequence, where 1. hi0 denotes the given initial level of i’s human capital, 2. cit denotes i’s consumption of the physical good at Period t = 1, · · · , T , 3. ait denotes i’s labor hours at Period t = 1, · · · , T , 4. bit denotes i’s learning hours at Period t = 1, · · · , T − 1, 5. eit denotes the amount of the physical good used for i’s education at Period t = 1, · · · , T − 1, 6. hit denotes the level of i’s human capital at Period t = 1, · · · , T − 1, and the human capital accumulation follows hit = Git (bit , eit , hi,t−1 ) 4

where Git describes i’s learning ability at Period t. Each individual i ∈ I has preference over paths of consumptions, working hours and learning hours. Let ∆ = {(a, b) ∈ R2+ : a + b ≤ 1}. Let ≿i denote i’s preference over (R+ × ∆)T −1 × R+ × [0, 1], where (ci1 , ai1 , bi1 · · · , ciT , aiT ) ≿i (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ) is read as the individual i weakly prefers (ci1 , ai1 , bi1 · · · , ciT , aiT ) over (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ). Preferences are assumed to be convex, strongly increasing in consumptions and strongly decreasing in working hours and learning hours, and continuous. When an individual has no strict preference between working hours and learning hours the following condition holds. Indiﬀerence between Working and Learning: For all (ci1 , ai1 , bi1 · · · , ciT , aiT ) and (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ), if ait + bit = a′it + b′it for all t = 1, · · · , T − 1, aiT = a′iT and cit = c′it for all t = 1, · · · , T , then (ci1 , ai1 , bi1 · · · , ciT , aiT ) ∼i (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ) Given a profile of initial human capitals h0 = (hi0 )i∈I and the initial level of physical capital k0 , a social path x = (xi )i∈I is said to be feasible given k0 if there exist k1 , · · · , kT −1 ≥ 0 such that. ∑ i∈I

cit +

∑ i∈I

eit + kt ≤ rkt−1 + ∑

∑

hi,t−1 ait , t = 1, · · · , T − 1

i∈I

ciT ≤ rkT −1 +

∑

i∈I

hi,T −1 aiT

i∈I

hit = Git (bit , eit hi,t−1 ) i ∈ I, t = 1, · · · , T − 1

Here is the definition of ex-ante eﬃciency. Definition 1 A social path x is said to be ex-ante eﬃcient given k0 if there is no social path x′ which is feasible given k0 and satisfies (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ) ≿i (ci1 , ai1 , bi1 , · · · , ciT , aiT ) for all i ∈ I and the relation is strict for at least one i ∈ I. 5

We also consider a weaker eﬃciency concept that Pareto improvement is impossible after certain periods. Definition 2 A social path x is said to be conditionally eﬃcient after Period t given k0 if there is no social path x′ which is feasible given k0 and identical with x until Period t and satisfies (c′i1 , a′i1 , b′i1 , · · · , c′iT , a′iT ) ≿i (ci1 , ai1 , bi1 , · · · , ciT , aiT ) for all i ∈ I and the relation is strict for at least one i ∈ I. Note that Ex-ante eﬃciency is now a special case that an allocation is eﬃcient after Period 0.

3

Fairness under full exemption

First we consider a fairness concept under the presumption that individuals are taken to be fully exempted from responsibility for their skills and learning eﬀectiveness.1 Definition 3 Say that i envies j at x under full exemption if (cj1 , aj1 , bj1 , · · · , cjT , ajT ) ≻i (ci1 , ai1 , bi1 , · · · , ciT , aiT ) A social path x is envy-free under full exemption if nobody envies anybody there under full exemption. First we show that under full exemption eﬃciency and envy-freeness are incompatible even in a very small preference domain. Proposition 1 Ex-ante eﬃciency and Envy-freeness under full exemption are incompatible in the domain of linear (hence separable as well) preferences satisfying Indiﬀerence between Working and Learning. 1

It is a non-obvious question “which self” should be considered in order to to define a concept of fairness

in dynamic environments. While eﬃciency concepts are purely forward-looking notion, fairness concepts may not be, and they may take retrospective evaluation of life into account. In this paper we assume that the individuals are dynamically consistent, and there is no conflict between ex-ante evaluation of their life paths and retrospective ones to be made ex-post, which leads us to state conditions in terms of ex-ante preferences.

6

Proof. It is enough to provide a proof for the case of T = 2. We can show the proposition in several ways as there are several sources of impossibility. Let us isolate each of them below. Consider two individuals i and j. Their preferences are represented in the form ui (ci1 , ai1 , bi1 , ci2 , ai2 ) = ci1 + δi (1 − ai1 − bi1 ) + βi (ci2 + δi (1 − ai2 )) and uj (cj1 , aj1 , bj1 , cj2 , aj2 ) = cj1 + δj (1 − aj1 − bj1 ) + βj (cj2 + δj (1 − aj2 )) 1: Impossibility due to the diﬀerence of initial skills This is a straightforward translation of the known impossibility argument due to Pazner and Schmeidler [4]. Suppose that i’s learning ability is given by Gi1 (ei1 , bi1 , hi0 ) = hi0 and j’s learning ability is given by Gj1 (ej1 , bj1 , hj0 ) = hj0 Now suppose hi0 < δi δj < hj0 δi < δj βi = βj = β < 1/r Here Ex-ante eﬃciency implies ci1 = t1 , ai1 = 0, bi1 = 0, ei1 = 0, ci2 = t2 , ai2 = 0 and cj1 = hj0 + rk0 − t1 , aj1 = 1, bj1 = 0, ej1 = 0, cj2 = hj0 − t2 , ai2 = 1 for some t1 , t2 . Then, Envy-freeness under full exemption requires t1 + δi + β(t2 + δi ) ≥ hj0 + rk0 − t1 + β(hj0 − t2 ) 7

and t1 + δj + β(t2 + δj ) ≤ hj0 + rk0 − t1 + β(hj0 − t2 ) This implies 2(t1 + βt2 ) ≥ (1 + β)(hj0 − δi ) + rk0 and 2(t1 + βt2 ) ≤ (1 + β)(hj0 − δj ) + rk0 but it is impossible under δi < δj . 2: Impossibility due to the diﬀerence of eﬀectiveness of physical investment on education Suppose that i’s learning ability is given by Gi1 (ei1 , bi1 , hi0 ) = hi0 and j’s learning ability is given by Gj1 (ej1 , bj1 , hj0 ) = hj0 + γj ej1 Now suppose hi0 = hj0 = h0 h0 < δ i h0 < δj < h0 + γj rk0 βj γj r > 1 βi = βj = β δ i < δj Then Ex-ante eﬃciency implies ci1 = 0, ai1 = 0, bi1 = 0, ei1 = 0, ci2 = t, ai2 = 0 and cj1 = 0, aj1 = 0, bj1 = 0, ej1 = rk0 , cj2 = h0 + γj rk0 − t, ai2 = 1 for some t. Notice in particular that here the whole physical resources have to be invested on j’s human capital acquisition. 8

Then, Envy-freeness under full exemption requires δi + βi (t + δi ) ≥ βi (h0 + γj rk0 − t) and δj + βj (t + δj ) ≥ βj (h0 + γj rk0 − t) This implies (1 + β)t ≥ β(h0 + γj rk0 − δi ) and (1 + β)t ≤ β(h0 + γj rk0 − δj ) but it is impossible under δi < δj . 3: Impossibility due to the diﬀerence of eﬀectiveness of learning time Suppose that i’s learning ability is given by Gi1 (ei1 , bi1 , hi0 ) = hi0 and j’s learning ability is given by Gj1 (ej1 , bj1 , hj0 ) = hj0 + εj bj1 Now suppose hi0 = hj0 = h0 h0 < δ i h0 < δ j < h 0 + ε j βj ε j > δ j βi = βj = β < 1/r δ i < δj Then eﬃciency implies ci1 = t1 , ai1 = 0, bi1 = 0, ei1 = 0, ci2 = t2 , ai2 = 0 and cj1 = rk0 − t1 , aj1 = 0, bj1 = 1, ej1 = 0, cj2 = h0 + εj − t2 , ai2 = 1 9

for some t1 , t2 . Then, Envy-freeness under full exemption requires t1 + δi + β(t2 + δi ) ≥ rk0 − t1 + β(hj0 + εj − t2 ) and t1 + δj + β(t2 + δj ) ≤ rk0 − t1 + β(hj0 + εj − t2 ) This implies 2(t1 + βt2 ) ≥ rk0 + β(h0 + εj ) − (1 + β)δi and 2(t1 + βt2 ) ≤ rk0 + β(h0 + εj ) − (1 + β)δj but it is impossible under δi < δj . Here is a “market-like” solution which satisfies Envy-freeness under full exemption. Definition 4 A social path x e is said to be an equal education input/equal wage equilibrium if there exist (w e1 , · · · , w eT , eb1 , · · · ebT −1 , ee1 , · · · , eeT −1 ) and (e ki1 , · · · , e ki,T −1 )i∈I such that each individual’s life path with physical capital (e xi , e ki ) is optimal according to ≿i under the constraint cit + eet + ki,t = w et ait + rki,t−1 , t = 1, · · · , T − 1 ciT = w eT aiT + rki,T −1 hit = Git (e et , ebt , hi,t−1 ), t = 1, · · · , T − 1 ki,0 = k0 /n hi0 = given and ∑

e cit + ne et +

i∈I

∑

e kit = r

i∈I

∑ i∈I

∑

e ki,t−1 +

∑

i∈I

e ciT = r

∑

e hi,t−1e ait , t = 1, · · · , T − 1

i∈I

e ki,T −1 +

i∈I

∑

e hi,T −1e aiT

i∈I

Proposition 2 Equal education input/equal wage equilibrium satisfies Envy-freeness under full exemption. 10

Proof. It follows from i’s intertemporal budget constraints being consolidated into a single equation

T T T ∑ ∑ ∑ cit eet w et ait rk0 + = + , t−1 t−1 r r rt−1 n t=1 t=1 t=1

which is identical across all individuals.

4

Fairness under full responsibility

Here we introduce a fairness concept under the presumption that individuals are taken to be fully responsible for their skills and learning eﬀectiveness. Definition 5 Say that i envies j at x under full responsibility if there exists (hij0 , aij1 , bij1 , hij1 , · · · , hij,T −1 , aij,T ) such that aijt =

ajt hj,t−1 , t = 1, · · · , T hij,t−1

hij0 = hi0

( ) hijt = Git ejt , bijt , htj,t−1

aijt + bijt ≤ 1, t = 1, · · · , T − 1 aijT ≤ 1 and

(

) cj1 , aij1 , bij1 , · · · , cjT , aijT ≻i (ci1 , ai1 , bi1 , · · · , ciT , aiT )

A social path x is envy-free under full responsibility if nobody envies anybody there under full responsibility. The idea is that i envies j if i thinks I wish I had the same initial amount of physical resource as j has and I had been allowed to spend it and time on learning so as to acquire the skill that makes the same production as j’s.

11

Here envy is justified only for the reason that there is inequality of initial amounts of physical resource or inequality of constraints on how to utilize physical resource and time for education, not because of inequality of initial skills or learning abilities. Here is a market-based allocation rule which naturally corresponds to the above notion of fairness. Definition 6 A social path x e is said to be an equal education opportunity equilibrium if there exist (e ki1 , · · · , e ki,T −1 )i∈I such that each individual’s life path with physical capital (e xi , e ki ) is optimal according to ≿i under the constraint cit + eit + ki,t = hi,t−1 ait + rki,t−1 , t = 1, · · · , T − 1 ciT = hi,T −1 aiT + rki,T −1 hit = Git (eit , bit , hi,t−1 ) ki0 = k0 /n hi0 = given and ∑ i∈I

e cit +

∑

eeit +

i∈I

∑

e kit = r

i∈I

e ki,t−1 +

∑

e ciT = r

∑

e hi,t−1e ait , t = 1, · · · , T − 1

i∈I

i∈I

i∈I

∑

∑

e ki,T −1 +

i∈I

∑

e hi,T −1e aiT

i∈I

Proposition 3 Equal education opportunity equilibrium is ex-ante eﬃcient and envy-free under full responsibility. Proof. Let x be an allocation in equal education opportunity equilibrium. To see that x is eﬃcient, suppose it is Pareto-dominated by another feasible allocation x′ . Notice that the intertemporal budget constraints are consolidated into a single equation T T T ∑ ∑ ∑ cit eit hi,t−1 ait rk0 + = + , t−1 t−1 r r rt−1 n t=1 t=1 t=1

and also the feasibility conditions are consolidated into T T T ∑∑ ∑∑ ∑∑ eit hi,t−1 ait cit + = + rk0 t−1 t−1 t−1 r r r i∈I t=1 i∈I t=1 i∈I t=1

12

Since x′ is Pareto-dominating x we have T T T ∑ ∑ ∑ cit eit hi,t−1 ait rk0 + ≥ + , t−1 t−1 r r rt−1 n t=1 t=1 t=1

for all i and

T T T ∑ ∑ ∑ cit eit hi,t−1 ait rk0 + > + , t−1 t−1 r r rt−1 n t=1 t=1 t=1

for at least one i. Then, by adding up the individual constraints we obtain T T T ∑∑ ∑∑ ∑∑ cit eit hi,t−1 ait + > + rk0 t−1 t−1 r r rt−1 i∈I t=1 i∈I t=1 i∈I t=1

which is a violation of feasibility. To see that x is envy-free under full responsibility, suppose i envies j at x. Now it suﬃces to show that the corresponding life path (hi0 , cj1 , aij1 , bij1 , ej1 , hij1 , · · · , hij,T −1 , cj1 , aij1 ) is aﬀordable for i, which leads to a contradiction to i making optimal choice under the intetermporal budget constraint. By the assumption we have T T T ∑ ∑ ∑ cjt ejt hj,t−1 ajt rk0 + = + t−1 t−1 r r rt−1 n t=1 t=1 t=1

=

T ∑ hij,t−1 aijt t=1

5

rt−1

+

rk0 n

Fairness under responsibility for grown-ups

Here we consider a concept of fairness when individuals are taken to be responsible for their skills and learning eﬀectiveness only after passing certain age. Definition 7 Say that i envies j at x under responsibility after Period τ if there exists (aij,τ +1 , bij,τ +1 , hij,τ +1 , · · · , hij,T −1 , aijT ) 13

such that aijt =

ajt hj,t−1 , t = τ + 1, · · · , T hij,t−1

hijτ = hiτ

( ) hijt = Git ejt , bijt , htj,t−1 t = τ + 1, · · · , T

aijt + bijt ≤ 1, t = τ + 1, · · · , T − 1 aijT ≤ 1 and ( ) cj1 , aj1 , bj1 , · · · , cjτ , ajτ , bjτ , cj,τ +1 , aij,τ +1 , bij,τ +1 , · · · , cjT , aijT ≻i (ci1 , ai1 , bi1 , · · · , ciτ , aiτ , biτ , ci,τ +1 , ai,τ +1 , bi,τ +1 , · · · , ciT , aiT ) A social path x is envy-free under responsibility after Period τ if nobody envies anybody there under responsibility after Period τ . Note that when τ = 0 the condition reduces to envy-freeness under full responsibility and when τ = T it reduces to envy-freeness under full exemption. The following impossibility result follows from the similar reasoning for Proposition 1. Proposition 4 Envy-freeness under responsibility after Period τ and conditional eﬃciency after Period τ ′ are incompatible when τ ′ < τ . Here is a “market-like” solution concept which satisfies envy-freeness under responsibility as grown-ups. Definition 8 A social path x e is said to be an equal education input/wage under age τ equilibrium if there exist (w e1 , · · · , w eτ , eb1 , · · · ebτ , ee1 , · · · , eeτ ) and (e ki1 , · · · , e ki,T −1 )i∈I such that each individual’s life path with physical capital (e xi , e ki ) is optimal according to ≿i

14

under the constraint cit + eet + ki,t = w et ait + rki,t−1 , t = 1, · · · , τ cit + eit + ki,t = hi,t−1 ait + rki,t−1 , t = τ + 1, · · · , T − 1 ciT = hi,T −1 aiT + rki,T −1 if τ < T ciT = w eT aiT + rki,T −1 if τ = T hit = Git (e et , ebt , hi,t−1 ), t = 1, · · · , τ hit = Git (eit , bit , hi,t−1 ), t = τ + 1, · · · , T − 1 ki0 = k0 /n hi0 = given and ∑ ∑ i∈I

e cit + ne et +

i∈I

e cit +

∑ i∈I

∑

e kit = r

eeit +

e kit = r

i∈I

∑

∑

e ciT = r

∑

e hi,t−1e ait , t = 1, · · · , τ

i∈I

e ki,t−1 +

∑

e hi,t−1e ait , t = τ + 1, · · · , T − 1

i∈I

i∈I

i∈I

∑

e ki,t−1 +

i∈I

i∈I

∑

∑

e ki,T −1 +

i∈I

∑

e hi,T −1e aiT

i∈I

Proposition 5 Equal education input/equal wage under age τ equilibrium is conditionally eﬃcient after Period τ and envy-free under responsibility after Period τ . Proof. Let x be an allocation in equal education input/equal wage under age τ equilibrium. To see that x is conditionally eﬃcient after Period τ , suppose it is ex-post Paretodominated by another feasible allocation x′ which coincides with x up to Period τ . Notice that the intertemporal budget constraints are consolidated into a single equation T τ T τ T ∑ ∑ ∑ ∑ ∑ cit eet eit w et ait hi,t−1 ait rk0 + + = + + , t−1 t−1 t−1 t−1 t−1 r r r r r n t=1 t=1 t=τ +1 t=1 t=τ +1

and also the feasibility conditions are consolidated into τ T T T ∑ ∑ ∑ ∑∑ ∑∑ eet eit hi,t−1 ait cit +n + = + rk0 t−1 t−1 t−1 r r r rt−1 t=1 i∈I t=τ +1 i∈I t=1 i∈I t=1

Since x′ is Pareto-dominating x we have τ T τ T τ T ∑ ∑ ∑ ∑ ∑ ∑ h′i,t−1 a′it rk0 cit c′it eet e′it w et ait + + + ≥ + + , t−1 t−1 t−1 t−1 t−1 t−1 r r r r r r n t=1 t=τ +1 t=1 t=τ +1 t=1 t=τ +1

15

for all i and τ T τ T τ T ∑ ∑ ∑ ∑ ∑ ∑ h′i,t−1 a′it rk0 cit c′it eet e′it w et ait , + + + > + + rt−1 t=τ +1 rt−1 t=1 rt−1 t=τ +1 rt−1 rt−1 rt−1 n t=1 t=1 t=τ +1

for at least one i. Then, by adding up the individual constraints we obtain τ T T τ ∑∑ ∑ ∑ ∑ ∑ ∑ cit c′it eit eet + +n + t−1 t−1 t−1 r r r rt−1 t=1 i∈I t=1 i∈I t=τ +1 i∈I t=τ +1

>

τ ∑∑ w et ait i∈I t=1

rt−1

T ∑ ∑ h′i,t−1 ait + + rk0 rt−1 i∈I t=τ +1

Because x is an equilibrium we have

∑ i∈I

w et ait =

∑ i∈I

hi,t−1 ait for each t = 1, · · · , τ we

obtain τ T τ T ∑∑ ∑ ∑ ∑ ∑ ∑ cit c′it eet eit + +n + t−1 t−1 t−1 r r r rt−1 t=1 i∈I t=1 i∈I t=τ +1 i∈I t=τ +1

>

τ ∑∑ hi,t−1 ait i∈I t=1

rt−1

T ∑ ∑ h′i,t−1 ait + + rk0 , rt−1 i∈I t=τ +1

which is a violation of feasibility. To see that x is envy-free under responsibility after Period τ , suppose i envies j at x.

Now it suﬃces to show that the corresponding life path (hi0 , cj1 , aj1 , bj1 , ej1 , · · · , cj,τ +1 , aij,τ +1 , bij,τ +1 , ej is aﬀordable for i, which leads to a contradiction to i making optimal choice under the intertemporal budget constraint. By the assumption we have T τ T τ T ∑ ∑ ∑ ∑ ∑ hij,t−1 aijt rk0 cjt eet ejt w et ajt + + = + + . rt−1 t=1 rt−1 t=τ +1 rt−1 rt−1 rt−1 n t=1 t=1 t=τ +1

6

Conclusion

In the setting of human capital accumulation we have considered a problem of fair allocation of resources to be utilized for education. 16

First we proposed two axioms of envy-freeness, one presumes that nobody is responsible for his initial skill or his learning ability, the other presumes that everybody is perfectly responsible for his initial skill and his learning ability. We have shown that the first one is incompatible with the requirement of ex-ante eﬃciency while the second is compatible. Then we proposed a series of axioms indexed by time period that allow intermediate levels of responsibility. Given that there are T periods, for each τ = 0, · · · , T we formulate an axiom of envy-freeness which presumes that the individuals are responsible for their skills and learning eﬀectiveness only after Period τ . We have provided a family of rules such that for each τ the corresponding solution is eﬃcient after Period τ and envy-free under responsibility after Period τ , while it is impossible to require eﬃciency after earlier periods than Period τ . This shows that there is a trade-oﬀ between seeking eﬃciency to hold for longer periods and seeking to allow exemption for longer periods. A possible direction for future research is to find a more subtle way of sorting what aspects or which parts of life paths people are responsible for or exempted from responsibility for, not just sorting by age or time period. In particular, it will be questioned to what extent individuals are responsible for their preferences. Answering to it will require deeper understandings about how people develop their personalities in both individual and social dimensions.

References [1] Fleurbaey, Marc, and Giacomo Vallettay. “Fair optimal tax with endogenous productivities,” working paper 2012. [2] Fleurbaey, Marc, and Francois Maniquet. ”Fair allocation with unequal production skills: The no envy approach to compensation.” Mathematical Social Sciences 32.1 (1996): 71-93. [3] Hayashi, Takashi. “Decentralizability and responsibility under unequal skills” working paper, 2014. [4] Pazner, Elisha A., and David Schmeidler. ”A diﬃculty in the concept of fairness.” Review of Economic Studies (1974): 441-443. 17

[5] Phillips, Anne. ”Defending equality of outcome.” Journal of political philosophy 12.1 (2004): 1-19. [6] Roemer, John E. Equality of opportunity. Harvard University Press, 2009. [7] Varian, Hal R. ”Equity, envy, and eﬃciency.” Journal of Economic Theory 9.1 (1974): 63-91.

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