Comonotonic Allocation with Heterogeneous Agents and
Non-Linear-Discounting Utility
Tatsuya Tabata ∗July 21, 2023

Abstract
In this study, we employ non-linear discounting (NLD) utility in a multi-consumer
model and demonstrate that saving allocation in the stationary equilibrium is comonotonic to consumers attitudes toward future consumption. Time-additively separable(TAS) utility with a ﬁxed discount factor is widely employed in dynamic macroeconomics because of its tractability. However, this yields unrealistic results in a multiconsumer model. The number of savers in the economy is only one, while the others
live on a day-to-day basis. On the contrary, NLD utility allows consumers to vary
their discounting of future utility depending on their consumption plan. Due to thisproperty, a comonotonic saving allocation can occur.
Keywords: non-linear discounting utility; Comonotonic allocation

1

Introduction

Many macroeconomic models assume a representative consumer. The sum of
consumption is seen as the result of the ”representative agent.” Some critics
argue that the behavior of the representative consumer cannot be constructedfrom the results of each consumer’s utility maximization problem. Even if
the aggregability problem is solved, the multi-consumer model is signiﬁcantin discussing the equity of allocation or the eﬀect of redistributive policy.
Many scholars have discussed wealth allocation. Ramsey (1928) conjectedthat society will be divided into two classes, impatient laborers and patient capitalists. Becker (1980) and Mitra and Sorger (2013) formally proved
this. They employed time-additively separable (TAS) utility U (c0 , c1 . . .) =∑∞tt=0 β u(ct ) where c0 , c1 . . . is a consumption stream, and the constant β is∗

Faculty of Law, Tokoha University, Mizuochi 1-30, Aoi-ku , Shizuoka city, Shizuoka, 420-0831, Japan1

a discount factor. The discount factor of TAS utility remains constant andindependent of the consumption plan. A consumer with a higher discountfactor evaluates the utility from future consumption more than other consumers do. This means that they are more patient. Moreover, they maintaina high tendency to save even if their capital accumulation deepens. Hence,
they can oﬀer a lower interest rate, while the others are excluded from thecapital market.
non-linear discounting (NLD) utility is described with W (c, u) = v(c) +δ(u) where c is immediate utility, and u is future utility. This is a subclassof recursive utility suggested by Koopmans (1960). Letting δ(u) = βu, TAS
utility is included in the class of NLD utility, although we limit ”NLD” utility
to that of non-TAS to avoid confusion of diction. A consumer’s attitude
toward the future is the response of lifetime utility to the variance of future
utility. We deﬁne the time perspective as
W (c, u′ ) − W (c, u)δ (u) = lim.
u′ →uu′ − u′This discounts future utility as well as the discount factor of the TAS
utility. A consumer who is promised future welfare spares a larger portionof the additional income to immediate consumption. Time perspective istypically considered to decrease in future utility u. This property enables eachconsumer to adjust their consumption plan to satisfy the necessary conditionof utility maximization.
Some studies employing non-TAS utility, such as Lucas and Stokey (1984)
and Epstein and Hynes (1983), demonstrate that saving is owned by multiple agents. The former submits minimal knowledge on who saves and consumes more money or goods. The latter speciﬁes the utility function to the
Epstein-Hynes type, which describes the time perspective as dependent on
past consumption. In this study, we show that comonotonic saving allocation
can be viewed as the result of a consumer’s forward-looking behavior.
The rest of this paper is organized as follows. Section 2 introduces themodel and several assumptions. Section 3 examines the qualitative properties of stationary competitive equilibrium. A unique stationary competitive
equilibrium exists. The comonotonic relationship between saving and time
preferences always holds in the stationary competitive equilibrium. In Section 4, we show an example of stationary allocation by specifying the utility2

function to the Koopmans-Diamond-Williamson (KDW) utility, which is oneof the NLD utilities. The stationary consumption level of each consumer isdetermined by solving the Euler equation.
2

Model and Assumptions

We analyze a one-sector model and assume that time is discrete with index
t = 0, 1, 2 . . .. There are H consumers with indexes h = 1, 2 . . . H and only
one producer. This economy has capital and products markets, in which bothof are assumed to be competitive.
hh h∞Let ch = (ch0 , ch1 , . . .) ∈ R∞
+ and k = (k0 , k1 . . .) ∈ R+ be the hth consumer’s consumption and saving plans, respectively. We denote the consumption and saving paths from period t by t c = (ct , ct+1 . . .) and t k = (kt , kt+1 . . .),respectively. Let c̄ = (c̄, c̄ . . .) and k̄ = (k̄, k̄ . . .) be constant paths.
The producer lends capital Kt ∈ R++ to consumers with an interest ratert ∈ R++ at the beginning of period t. To analyze consumption and saving
distributions, we exclude zero input and output. For simplicity, the producer is assumed to have linear production function F (K) = AK (A > 1).
The interest rate is always r = A − 1; otherwise, the solution to the proﬁt
maximization problem does not exist. If the interest rate is r = A − 1, theproducer’s proﬁt is always zero, and the capital is never accumulated in theeconomy. The producer’s behavior is described by the solution to the static
proﬁt maximization problem (PMP)max AKt − (1 + rr )Kt

Kt ∈R++rt ∈ R++ : given
Consumers are given an initial endowment k0h in period 0. They lend saving∞[0, At k0 ]kth to the producer and obtain interest at rate rt . We deﬁne set Xt=1
as
∞
∀
t
Xt=1
[0, At k0 ] = {(c0 , c1 . . .) ∈ R∞
+ |( t = 1, 2 . . .) 0 ≤ ct ≤ A k0 }∞Any feasible consumption path starting from k0h is included in Xt=1
[0, At k0h ].
A lifetime utility function U h deﬁned over R∞+ is represented by3

U h (c) = v h (c0 ) + δ h [U h (1 c)]∞[0, At k0h ] → (R ∪ {+∞}).
where v h ; R+ → R and δ h : Xt=1
Functions v h , δ h , and U h are assumed to satisfy (U1)-(U7).

(U1) Strict Monotonicity
The functions v h and δ h are strictly increasing.
(U2) Existence and Negativeness of Second-Order Derivative
Functions v h and δ h are twice continuously diﬀerentiable over their domains. [It holds that] (v h )′′ (c) < 0 and (δ h )′′ (u) < 0 for any c ∈ R+ and∞u ∈ U h Xt=1
[0, At k0h ] .
Condition (δ h )′′ (u) < 0 implies that a consumer who is promised highfuture utility will increase immediate consumption rather than saving after
obtaining additional one unit of income.
(U3) Real Valued][ ∞[0, At k0h ] , δ h (u) takes a real value.
For any u ∈ U h Xt=1
(U4) Postponing [Damages Lifetime
Utility]∞t hhFor any u ∈ U Xt=1 [0, A k0 ] such that u > U h (0), it holds that u >v h (0) + δ h (u).
This implies that if a consumer postpones the beginning of their consumption plans and consumes nothing today, their lifetime utility decreases.
(U5) Compensation][ ∞ for PostponinghFor any u ∈ U Xt=1 [0, At k0h ] , there exists c′ ∈ R+ such that u ≤ v h (c′ ) +δ h (u).
The function v h is strictly increasing; hence, such c′ is unique. This implies that even if a consumer postpones the beginning of their consumption
plan, suﬃcient consumption today can compensate for the decrease in lifetime
utility caused by postponing.
(U6) Asymptotic Boundedness
It holds that∗ln{v h (c) + δ h (0)}is ﬁnite
c→+∞
ln c

¯
e = lim4W (c, u′ ) − W (c, u)
sup
<1
′−u
′
u
(c,u) u ∈(u,+∞)d¯ = sup¯∗(d∗ )Ae < 1∗¯ sup
where d = limu→+∞c

W (c, u′ ) − W (c, u)
supu′ − uu′ ∈(u,+∞)This condition and linear production function compose a suﬃcient condition of Lower Convergence suggested by Streufert (1990). Lower Convergencecan be described as follows:∞
For any k0 ∈ R+ and c ∈ Xt=0
[0, At k0 ], it holds thatlim U h (ct , 0) = U h (c).

t→∞This implies that an agent evaluates their distant future utilities muchless.
(U7) Range of Time Perspective][ ∞[0, At k0h ] such that
There exists some u′ ∈ U h Xt=1
(δ h )′ (u′ ) <1
AIt holds that1
< (δ h )′ [U h (0)] < 1
AThis implies that there exists an attainable future utility level u′ thatsatisﬁes the Euler equation (δ h )′ (u′ )A = 1.
Lemma 2.1. Given that (U 2) is satisﬁed, the utility function U h (c) is partially diﬀerentiable with respect to any ct , and the partial derivative is
Dct U (c) =t−1∏(δ ′ [U (s+1 c)]) · v ′ (ct )s=05Proof )This is derived by repeatedly applying the chain rule.
□
Consumers are assumed to perfectly foresee future interest rates. Thus,
the behavior of each consumer is described by the solution to the dynamic
utility maximization problem (UMP)max

ch ,kh ∈R∞+

U h (ch )hsub.to (∀ t = 0, 1 . . .) cht + kt+1
≤ (1 + rt )kth(∀ t = 0, 1, . . .) rt ∈ R++ : given
k0h ∈ R+ : given
The inner solution to the UMP is characterized by the Euler equation
t
∏

′

′

(δ [U (s+1 c)]) · v (ct+1 )(1 + rt ) =

s=0

t−1
∏

(δ ′ [U (s+1 c)]) · v ′ (ct )s=0

In particular, in the stationary state, the equation is reduced toδ ′ [U (c̄)] =3

1.
1 + r̄

Stationary Competitive Equilibriumh H
If the combination < (ch )H
h=1 , (k )h=1 , K, r > satisﬁes (1)–(5), then it is acompetitive equilibrium.

(1) Non Negativeness of Variables
In all periods t and for all consumers h, the hth consumer’s consumption
cht and saving kth are non-negative. The capital input Kt and interest rate rt
are positive for all t = 0, 1, . . ..
(2) Consumers’ Optimality
For all h = 1, 2 . . . H, (ch , kh ) is the solution for the h-th consumer’s UMP.
(3) Producer’s Optimality
6For all t = 0, 1 . . ., Kt is the solution for the PMP.
(4) Balance of the Capital Market∑hFor all t = 0, 1 . . ., it holds that H
h=1 kt = Kt .
(5) Balance of Products Market ∑∑H hhFor all t = 0, 1 . . ., it holds that H
h=1 kt+1 = F (Kt ).
h=1 ct +h H
If < (ch )H
h=1 , (k )h=1 , K, r > satisﬁes (6), in addition to (1)–(5), this iscalled a stationary competitive equilibrium.

(6) Constant Level of Consumption, Capital, and Input
The paths r, (ch , kh )H
h=1 , and K satisfy constancy.
”Stationarity” requires that all variables maintain the same level over time.
Due to the concavity of the production and utility functions, the inner
stationary competitive equilibrium is likely characterized by the following
conditions:(I) Interest Rate: r̄ = A − 11(II) Euler Equation: (∀ h = 1, 2, . . . H), (δ h )′ [U h (c̄)] =A(III) Budget Constraints
For all h = 1, 2, . . . H, it holds thatk̄ h = k0hc̄h + k̄ h = (1 + r̄)k̄ h∑h(IV) Balance of the Capital Market: H
h=1 k̄ = K̄Note that the balance of the products market is automatically achieved
due to Walras Law.
Lemma 3.1. Assume that the production function is F (K) = AK, and eachconsumer has a utility that satisﬁes (U 1)−(U 7). Then, < H, r, (c̄h , k̄h )Hh=1 , K̄ >satisfying (I)–(IV) is the unique inner stationary competitive equilibrium.
This theorem asserts the uniqueness of ”inner” and ”stationary” competitive equilibrium. Hence, corner or nonstationary equilibria may also exist.
Proof )First, we demonstrate the existence of the inner stationary competitive
equilibrium. Choose h ∈ {1, 2, . . . H} arbitrarily. We demonstrate that there
7exists a unique c̄h ∈ R++ satisfying (II). (U2) ensures the continuity of thederivative δ ′ . According to (U7) and the intermediate value theorem, there′′
exists uh ∈ R such that (δ h )′ (uh ) = 1/A . . . (a). From (U4), we have v h (0) <′′′′uh − δ h (uh ). From (U5), there exists c′ ∈ R+ such that uh − δ h (uh ) ≤ v h (c′ ).
′′Thus, v h (0) < uh − δ h (uh ) ≤ v h (c′ ). According to the intermediate value′′
theorem, there must be c̄h ∈ R++ such that v h (c̄h ) = uh − δ h (uh ) > v h (0).
Owing to the concavity of δ h , we obtain′′uh − U h (c̄h ) = {v h (c̄h ) + δ h (uh )} − {v h (c̄h ) + δ h [U h (c̄h )]}′≤ (δ h )′ [U h (c̄h )]{uh − U h (c̄h )}By repeating this inequality, we obtain′′uh − U h (c̄h ) ≤(δ h )′ [U h (c̄h )]{uh − U h (c̄h )}′≤{(δ h )′ [U h (c̄h )]}2 {uh − U h (c̄h )}..
.
′≤{(δ h )′ [U h (c̄h )]}t {uh − U h (c̄h )}for any t = 3, 4 . . . ,. Taking the limits of both sides with respect to t, thenit holds that′′uh − U h (c̄h ) ≤ lim {(δ h )′ [U h (c̄h )]}t {uh − U h (c̄h )} = 0t→∞′

Due to (a), (δ h )′ (uh ) is less than 1. We obtain′′′U h (c̄h ) − uh ≤ lim {(δ h )′ (uh )}t {U h (c̄h ) − uh } = 0
t→∞

′

′

by reversing uh and U h (c̄h ). Thus, we obtain uh = U h (c̄h ) and (II)(δ h )′ [U h (c̄)] =1
A

Set the interest rate to satisfy the equation (I). The producer’s optimality
automatically holds under condition (I). Therefore we can take each consumer’s savings to satisfy the equation (III). Capital input can also be set to8satisfy the balance of the capital market in the equation (IV).
The uniqueness of the stationary competitive equilibrium is derived directly from the condition (δ h )′′ < 0.
□
Theorem 3.1. Assume that there uniquely exists an inner stationary comh H
petitive equilibrium < (c̄h )Hh=1 , (k̄ )h=1 , K̄, r̄ >. If it holds that(δ 1 )′ [U 1 (c̃)] < (δ 2 )′ [U 2 (c̃)] < . . . < (δ H )′ [U H (c̃)] . . . (1)∀hfor any c̃ = (c̃, c̃, . . .) ∈ R∞++ satisfying ( h = 1, 2, . . . H) U (c̃) < +∞,then the allocation is comonotonic to the order in (1), that is,

0 < c̄1 < c̄2 < . . . < c̄H
0 < k̄ 1 < k̄ 2 < . . . < k̄ H .
Inequality (1) implies that if each consumer takes the same consumption
plan c̃, a consumer with a larger index exhibits a larger time perspective.
Proof )
Choose two consumers with indexes i, j(i < j) arbitrarily. In Lemma 3.1,
we show that c̄i and c̄j must satisfy(δ i )′ [U i (c̄i )] = (δ j )′ [U j (c̄j )] =1
. . . (a)A

If they take the same consumption path ci , (1) implies that(δ i )′ [U i (c̄i )] < (δ j )′ [U j (c̄i )] . . . (b)(a) and (b) yield(δ j )′ [U j (c̄j )] < (δ j )′ [U j (c̄i )]From δ ′′ < 0, it holds that U j (c̄i ) < U j (c̄j ). From (U1), c̄i < c̄j . The
equation (c) in Lemma 3.1 implies that it also holds that k̄ i < k̄ j .
□9The assertion of this theorem in a three-consumer economy is summarized
in Figure 1. The downward slope of each δ h curve originates from (δ h )′′ < 0.
Moreover, δ 3 is the highest level owing to the order in (1).
Figure 1: Consumer’s time perspective Du W (c, u) in a three-consumers economy(δ h )′ [U h (c̄)]

1/(1 + r̄)•••(δ 3 )′ [U 3 (c̄)](δ 2 )′ [U 2 (c̄)](δ 1 )′ [U 1 (c̄)]
c̄

c̄

4

1

c̄

2

c̄

3

An Example of Stationary Competitive Equilibrium

We assume that each consumer has KDW utility
U h (ch ) = (ch0 )a + lim dh log[1 + (ch1 )a + dh log{1 + (ch2 )a + . . . + dh log(1 + (cht )a )}]t→∞

where a ∈ (0, 1), dh ∈ (1/(A − log A), 1).
Parameter dh represents the intensity of a consumer’s time preference.
A consumer with a larger dh evaluates future utility more. A range of dh
is required to guarantee an inner solution. The second term of the righthand side is monotonically increasing in t; therefore, a limit exists if ∞ ispermitted for the limit value. The KDW utility has an aggregator function
W h (c, u) = ca +dh log(1+u) and a time perspective Du W h (c, u) = dh /(1+u).
Thus, it satisﬁes (U1)-(U7).
Assuming that d1 < d2 < . . . < dH . . . (b), The equation (1) in Theorem
3.1 holds.
The Euler equation at the stationary competitive equilibrium is10dh(1 + r̄) = 1
1 + U h (c̄h )The interest rate is r = A − 1; hence, we have U h (c̄h ) + 1 = dh A. We
substitute it into U h (c̄h ) = (c̄h )a + dh log{U h (c̄h ) + 1}, and then, we obtaindh A − 1 = (c̄h )a + dh log(dh A)⇒ c̄h = {dh A − 1 − dh log(dh A)}1/a . . . (c)dh ∈ (0, 1) implies that dh log(dh A) < log A, (d) yields
dh A − 1 − dh log(dh A) > dh A − 1 − log A ≥ 0
Hence, each consumer s stationary consumption level is strictly positive.
Next, we check for comonotonicity. By diﬀerentiating both sides of (c)with respect to dh , it follows that1
Ddh c̄h = {dh A − 1 − dh log(dh A)}1/a−1 {A − 1 − log(dh A)}a(d) and dh ∈ (0, 1) result in A−1−log(dh A) > 0. Therefore, the derivative
Ddh c̄h is positive. This implies that consumers with higher dh enjoy a larger
consumption that is
0 < c̄1 < c̄2 . . . < c̄H
From (III), we obtain k̄ h = c̄h /r̄. Thus, the order of saving follows that ofconsumption, that is,
0 < k̄ 1 < k̄ 2 < . . . < k̄ H

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