Implementation in Stationary Markov-Perfect
Equilibrium∗
Takashi Hayashi†Michele Lombardi‡

May 24, 2024

Abstract
This paper studies the implementation of society’s policy functions in stationary
Markov-perfect equilibrium. We identify a monotonicity condition, a natural
but nontrivial stationary recursive extension of Maskin monotonicity. We also
show that our condition is sufficient under a mild regularity condition when
there are three or more agents. We apply our result to a market solution forintertemporal trades and the recursive median voter solution for public capital
accumulation.

JEL classification: C79, D82∗

Part of this research was carried out when Takashi Hayashi was visiting the Department of
Economics and Statistics of the University of Napoli Federico II, whose financial support throughthe grant Prin 2022 HLPMKN is gratefully acknowledged. The usual disclaimer applies.
†
Adam Smith Business School, University of Glasgow. E-mail: takashi.hayashi@glasgow.ac.uk.
‡
University of Liverpool Management School. E-mail: michele.lombardi@liverpool.ac.uk. Department of Economics and Statistics and CSEF, University of Napoli Federico II.

1Contents
1 Introduction

3

2 The Environment

7

3 A Necessary Condition: Stationary Recursive Monotonicity

14

4 A Full Characterization for Economic Environments

18

5 Applications

21

5.1

Internalizing pecuniary externality of saving . . . . . . . . . . . . . .

5.2

Heterogeneous discounting and recursive median voter equilibrium inpublic capital accumulation . . . . . . . . . . . . . . . . . . . . . . .

6 Conclusions

21
25
26

2

1. Introduction
Stationary Markov-perfect equilibrium (SMPE) is a game theoretical solution conceptthat captures our day-to-day economic life well because we often cannot commit
to future actions. An SMPE is a refinement of the subgame perfect-equilibrium
to extensive form games where a state variable encodes the whole payoff-relevant
information and where strategies take the form of a time-independent function, called
policy function, which maps each present state variable into an action (Fudenberg andTirole (1991), Duffie et al. (1994), Maskin and Tirole (2001)).
While this refinement abstracts away complex dependence on histories, such as
those considered in the repeated game literature, it allows us to describe intertemporal trades by responding to state variables. Moreover, it is economically appealing
because it enables us to deal with capital accumulation and saving in a setting where
the set of social outcomes/allocations may endogenously change over time via theevolution of state variables (see, for instance, Levhari and Mirman (1980), Ericson
and Pakes (1995), Dockner and Sorger (1996) among many economic applications).
In the price-taking approach, this recursive method is common in the dynamic generalequilibrium theory (Prescott and Mehra (1980), Lucas Jr and Stokey (1984), Stokey
and Lucas Jr (1989), Ljungqvist and Sargent (2018)).
We study the implementation of social choice objectives in SMPEs. From a mechanism design viewpoint, in a world in which agents cannot commit to future outcomesor future actions, a “good mechanism” or a “stable mechanism” would be such thatpeople are fine to behave in an “automatic” way, period-by-period, without worrying
about writing lengthy contracts that directly bind their future lives in a complicatedmanner. Thus, at each period, the planner is constrained in his elicitation process
because he can elicit from agents only information about how society should advancein the next period and their current preferences.
We consider implementation problems in environments with complete information
among agents about their permanent types. As is shown by Hayashi and Lombardi3

(2019), even when agents have complete information about their permanent types,
lack of commitment to future actions creates severe problems because agents reveal
their information to the planner in a piece-by-piece manner, and this prevents theplanner from understanding the real intentions of such information revelation. For
instance, when an agent reports to the planner that she wants to save more, theplanner cannot distinguish whether this is because she is purely patient or because
she wants to manipulate equilibrium outcomes in the future by carrying over more
wealth. Also, when an agent reports she prefers a high tax rate today, where thesociety cannot commit to tax rates in the future, the planner cannot distinguishwhether this is because the agent purely prefers a high tax rate today or because she
prefers the outcome in the future that follows as a consequence of a high tax rate
today.
Hayashi and Lombardi (2019) show that the sequential or even recursive competitive equilibrium solution is not implementable (in subgame perfect-equilibria) because
the planner cannot distinguish between the real intentions for saving decisions, as explained above. This negative result is also due to their setup, which does not allow
internalizing pecuniary externalities due to saving decisions.1 A saving decision has
a pecuniary externality effect because it changes endowment/wealth allocations in
the next period while there is no consumption/production externality. This issue isparticularly relevant when we want to give a strategic foundation of the competitive
equilibrium in a dynamic environment where we cannot commit to future actions because we need to define prices and allocation of wealth/assets in future periods evenafter deviations.
We develop a setup that allows us to internalize such pecuniary externalities by
explicitly incorporating state variables for the next period as part of the object ofchoice for agents at each period. This lets us control saving decisions with pecuniary
externalities through taxation or subsidies. Thus, given the current state of society,1

The definition of recursive competitive equilibrium due to Prescott and Mehra (1980) rules out
this problem by assuming a continuum of traders.

4we collect information about people’s preferences over pairs, each specifying a current
social outcome and a state variable for the next period.
The basic idea of Maskin monotonicity (Maskin (1999)) is that the social choice isinvariant to any "unanimous lie." In static environments and in dynamic environments
without the stationarity condition, there is no restriction on how such a "unanimouslie" is. Indeed, Hayashi and Lombardi (2019) study an abstract dynamic environment
in which any backward induction argument works, and so there are no restrictions
on the relation between how people "unanimously lie" after some history and how
they "unanimously lie" after another history. In a stationary recursive environment,
however, even such "unanimous lies" must be told in a stationary Markov manner
while they do not need to be constant across histories.
We present a natural extension of Maskin monotonicity to our stationary recursive
environment, which we call Stationary Recursive Monotonicity (SRM). We show that
SRM is necessary for implementation in SMPEs. Moreover, under a natural regularity
condition, we show that SRM is sufficient for implementation.
We provide two applications. One is the recursive Lindahl solution, a marketlike solution that can fully internalize the pecuniary externalities of saving decisions
and achieve intertemporal trades. It is worth emphasizing that the collective valuefunction for a fully Pareto-efficient solution typically exhibits non-stationary timediscounting over sequences of consumption allocations when individuals’ discountrates differ. So, it generally runs into time-inconsistency problems (Zuber (2011),
Jackson and Yariv (2014),Jackson and Yariv (2015)). Nevertheless, we show that it is
implementable in stationary Markov-perfect equilibrium when the next period asset
allocation is “encoded” in the state variable, which restricts the society’s decisionin the next period. More precisely, we require that the society can commit to an
allocation of such assets to be carried over to the next period but not to a sequence ofconsumption allocations for all future periods; that is, when the society can commit
only period-by-period.
5The other is the recursive median voter solution in the public capital accumulation
problem due to Boylan and McKelvey (1995), Boylan et al. (1996), in which people
vote on how much of a reproducible public good to save in each period.
Related LiteratureThis paper contributes to the literature on dynamic implementation, e.g., Moore andRepullo (1988), Lee and Sabourian (2011), and Mezzetti and Renou (2017).
Moore and Repullo (1988) also study the implementation of social choice processes
in subgame perfect-equilibrium in a setting where agents’ preferences over outcomesare fixed. However, they assume that society commits to the chosen alternative/planwhen it reaches a decision. In our setup, a society selects a current outcome (e.g.,
consumption) and a society’s state for the next period in each period. As a result,society cannot commit to plans or future actions in our setting.
Lee and Sabourian (2011) study infinitely repeated implementation problems in
a complete information setting where agents’ preferences change in each period according to an independent and identical probability distribution, the time horizon is
infinite, and the discount factor is common across agents and is common knowledge
among the agents and the planner. Lee and Sabourian (2011) show the importanceof efficiency for repeated implementation and how any efficient social function canbe repeatedly implemented in Nash equilibrium when the discount factor is highenough. In a similar setting, Mezzetti and Renou (2017) connect the work of Lee
and Sabourian (2011) with Maskin’s static implementation environment and identify
a monotonicity-like condition, called dynamic monotonicity, that is necessary and, insome environments, also sufficient for repeated implementation in Nash equilibrium,
regardless of the value of the discount factor and whether the horizon is finite or not.
Although the implementation models of Lee and Sabourian (2011) and Mezzetti andRenou (2017) naturally fit in many applications, their i.i.d. assumption rules out intertemporal trades, which are common in dynamic contexts where agents’ preferences
6over sequences of outcomes are fixed. Moreover, since heterogeneity and information
about time discounting are critical in intertemporal trades, agents may disagree ontime discounting in our setup, and the planner needs to collect information aboutsuch disagreement and their per-period utilities. These modeling differences are crucial and motivated by the desire to give a strategic foundation to the dynamic generalequilibrium environment where intertemporal trades are present.
Our contribution significantly differs from the dynamic mechanism design literature
for the same reasons. Part of this literature studies the properties of profit-maximizing
mechanisms in settings where agents’ private information evolves stochastically overtime. Earlier contributions are Baron and Besanko (1984), Besanko (1985), and Riordan and Sappington (1987). More recent contributions include, for instance, Courty
and Li (2000), Battaglini (2005), Eso and Szentes (2007), and Kakade et al. (2013).
Another body of work investigates how to implement Pareto efficient allocations in a
dynamic setting where private information changes over time. Seminal works in thisarea are Bergemann and Valimaki (2010) and Atehy and Segal (2013). Pavan et al.
(2014) extend the above contributions to a general dynamic setting. Their approachto the design of optimal mechanisms can be thought of as the dynamic analog of theapproach proposed by Myerson (1981) for static settings.2 Finally, another body ofliterature studies the design of efficient and profit-maximizing mechanisms in settings
where agents’ information is fixed but where agents or objects arrive stochasticallyover time. The contributions of this third body are summarized in Gershkov andMoldovanu (2014).
2. The Environment
Preferences, Types, and Society State: We consider environments with a discrete and infinite time horizon that begins at time 1, a society I = {1, · · ·, n} thatconsists of a finite number n of infinitely-lived agents, a set of society types Θ describ2

For a recent introduction to the literature on dynamic mechanism design, see Bergemann and
Välimäki (2019).

7

ing agents’ preferences over sequences of outcomes in X ∞ , where X is the per-period
set of outcomes and a society state space A. We assume complete information amongplayers.3
Player i’s preferences over X ∞ at θ ∈ Θ are represented by an additively separable
utility function
Ui (x, θ) =∞X

βit−1 (θ) ui (xt , θ)

t=1

where βi (θ) is agent i’s discount factor at θ and x is a typical sequence in X ∞ .
Each society state a ∈ A specifies a set Y (a) ⊆ X × A, which consists of pairsS
specifying a current outcome and a next-period society state. Let Y = a∈A Y (a).
Example 1. In a borrowing-lending economy without production, a society state a ∈P
A specifies an allocation of net assets (credit/debt). Let A = a ∈ Rn+ : i∈I ai = 0
be the set of net asset allocations which add up to zero. Moreover, let us assume thateach agent i earns ei units of consumption good in each period, which is perishable.
For each society state a ∈ A, the set Y (a), defined by)(
Y (a) =x ∈ Rn+ :

X

xi =

X

ei

× A,

i∈I

i∈I

describes a set of pairs, each of which specifies an allocation of current consumptions
and an allocation of net assets for the next period.
Example 2. In an intertemporal production economy with physical capital, a society
state specifies an allocation of capital, so that A = RI+ . Let Q : R2+ → R bea production function that satisfies the standard regularity properties and exhibits
constant returns to scale. Then, for each society state a ∈ A, the set Y (a) defined by(
Y (a) =

(x, b) ∈ RI+ × RI+ :

!)
X

xi +

i∈I

X
i∈I

3

bi = Q

X
i∈I

ai ,

X

li

i∈I

Information is complete when agents’ preferences and outcomes are common knowledge among
all agents.

8

describes a set of pairs, each of which specifies a current consumption and an allocationof physical capital for the next period.
Example 3. There is one public good, which is reproducible over time. The social
choice problem is how much to save at each period. We assume people have the same
per-period utility function but disagree on time discounting.
Let X = R+ and A = R+ . Let Q : R+ → R be a production function which
satisfies the standard regularity properties. Since saving choice determines the current
consumption as well, the feasible set is
Y (a) = {(x, h) ∈ R2+ : x + b ≤ Q(a)}for each a ∈ A.
Assume there is u : X → R such that ui (z, θ) = u(z) holds for all θ ∈ Θ and alli ∈ I, where u satisfies the standard regularity properties. Hence, time discounting
is the only source of heterogeneity among the agents. Let us assume that Θ = (0, 1)I
and that βi (θ) = θi for each i ∈ I.
Stationary Policy Function: A stationary policy function is a function f : A → Y
such that f (a) ∈ Y (a) for all a ∈ A. Since Y (a) ⊆ X × A, a stationary policy
function f can be written as f = (z, α) where z : A → X and α : A → A are
functions such that f (a) = (z (a) , α (a)) for all a ∈ A. For each society state a ∈ A,
f (a) = (z (a) , α (a)) specifies the current outcome and the society state in the next
period. Let F (Y, A) denote the set of all stationary policy functions.
Fix any i ∈ I, any f = (z, α) ∈ F (A, X), any θ ∈ Θ and any a ∈ A. Let Wi (f, θ, a)be the continuation lifetime utility for agent i at θ when society follows f after thesocial state a, which is defined by the following recursive formula
Wi (f, θ, a) = ui (z (a) , θ) + βi (θ) Wi (f, θ, α (a)) .

9

Example 4. Let us consider the borrowing-lending economy without production
described in Example 1. Assume that ui (·, θ) : R+ → R is continuous, strongly
increasing and strictly concave.
First, let us note that the recursive competitive equilibrium solution cannot be awell-defined concept when there are pecuniary externalities. For, let us formulate
the recursive competitive equilibrium in our finite-agent setting. A stationary policy
function f would be seen as recursive competitive equilibrium if there is a function
q : A → R+ , which maps each state a into the price of asset q(a) being measured by
the consumption good in the given period, such that for each i ∈ I it holds(zi (a), αi (a)) ∈ arg max ui (xi , θ) + βi (θ)Wi (f, θ, (bi , α−i (a)))(xi ,bi )subject toxi + q(a)bi ≤ ai + ei ,where (xi , bi ) denotes individual i’s potential deviation to choose a pair of current
consumption and asset to carry over to the next period and α−i (a) denotes the otherindividuals’ asset holdings for the next period "in equilibrium."
But what is "(bi , α−i (a))"? If i deviates to bi , the asset/credit allocation (bi , α−i (a))
is not feasible, even in a hypothetical sense, and we cannot define the continuation
lifetime utility.
For that asset allocation to be feasible even after deviations, we must introduce a
Lindahl-type solution. A stationary policy function f is as recursive Lindahl equilibrium if there are functions qi : A → RI+ , for each i ∈ I, where qij (a) is the price ofindividual j’s additional saving that is measured by individual i’s consumption, suchP
that i∈I qij (a) = q(a) for all j ∈ I and it holds(zi (a), α(a)) ∈ arg max ui (xi , θ) + βi (θ)Wi (f, θ, b)(xi ,bi )10subject toxi + qi (a) · b ≤ ai + ei
for each i ∈ I.
Example 5. Let us consider the intertemporal production economy with the physicalcapital described in Example 2. Assume that ui (·, θ) : R+ → R is continuous, strongly
increasing and strictly concave. Assume that Q : R+ → R+ is continuous, strongly
increasing and concave.
In this setup, a stationary policy function f would is recursive Lindahl equilibrium
if there are functions r : A → R+ , w : A → R+ and qi : A → RI+ , for each i ∈ I, suchP
that i∈I qij (a) = q(a) for all j ∈ I and such that it holds(zi (a), α(a)) ∈ arg max ui (xi , θ) + βi (θ)Wi (f, θ, b)(xi ,bi )subject toxi + qi (a) · b ≤ r(a)ai + w(a)li
for each i ∈ I, and
!
X
i∈I

ai ,

X

li

∈ arg max Q(K, L) − r(a)K − w(a)L.

i∈I

Note that r(a) denotes the price of input and w(a) denotes the wage when the present
state is a, which are measured by the consumption good in the given period.
Example 6. Let us consider the public capital accumulation problem described in
Example 3. Given f = (z, α), one can define the value function by
Wi (f, θ, a) = u(z(a)) + βi (θ)Wi (f, θ, α(a)).

11

We can also define the preference relation Rif [θ] by(x, b)Rif [θ](x′ , b′ )⇔u(x) + βi (θ)Wi (f, θ, b) ≥ u(x′ ) + βi (θ)Wi (f, θ, b′ )In each period, given a society state a ∈ A, we run the majority rule over how
much to save between 0 and Q(a). f is a recursive Condorcet winner if f (a) is theCondorcet winner for the preference profile (Rif [θ])i∈I in such voting for each a ∈ A.
One can show that the recursive Condorcet winner must be equal to the optimal
policy function for the individual with the median discount factor; hence, it is unique.
It has been shown by Boylan and McKelvey (1995) and Boylan et al. (1996) in thefinite-period setting, and an analogous result holds in the stationary infinite-horizoncase.
The recursive Condorcet winner solution is "dictatorial" in the sense that it isalways optimal for the individual with median discount factor (Zuber (2011), Jackson
and Yariv (2014, 2015)), but it is a fair rule in the sense that it satisfies anonymity(Hayashi and Lombardi (2021)).
Social Choice Processes: A social choice process is a function ϕ : Θ → F (Y, A)
specifying a stationary policy function for each θ ∈ Θ. That is, for each θ ∈ Θ,
ϕ specifies a stationary policy function ϕ [θ] = (z [θ] , α [θ]) ∈ F (Y, A) such thatϕ [θ] (a) = (z [θ] (a) , α [θ] (a)) ∈ Y (a) for all a ∈ A.
Let ϕ[Θ] = {ϕ[θ] : θ ∈ Θ} be the set of stationary policy functions that areconsistent with some type under ϕ.
Mechanisms and Stationary Markov Strategies: Let B be any set. A mech
anism Γ is a tuple Γ = (Mi )i∈I , (g, h) , where for each i ∈ I, Mi : A → 2B is anonempty function such that Mi (a) ⊆ B is the set of messages available to agent i ata, and (g, h) : D → Y is an outcome function, where D = {(a, m) ∈ A × M : m ∈ M (a)},which assigns to each society state a and a feasible profile of messages at a, m ∈ M (a),12

a current outcome and a society state for the next period; that is, (g (a, m) , h (a, m)) ∈Y (a) for all (a, m) ∈ D.
Fix any mechanism Γ. A (stationary Markov) strategy for agent i is a function
σi : A → B such that σi (a) ∈ Mi (a) ⊆ B for all a ∈ A. A typical profile of strategies
is denoted by σ = (σi )i∈I . A strategy profile σ is often written as (σi , σ−i ).
Stationary Markow-Perfect Equilibrium (SMPE): Fix any mechanism Γ. Fix
any strategy profile σ, any θ ∈ Θ and any a ∈ A. Let WiΓ (σ, θ, a) be the continuation
lifetime utility for agent i at θ when society follows σ after the society state a, which
is defined by the following recursive formulaWiΓ (σ, θ, a) = ui (g (a, σ (a)) , θ) + βi (θ) WiΓ (σ, θ, h (a, σ (a))) ,which is uniquely determined by its recursive formulation.
A strategy profile σ is a Stationary Markov-Perfect Equilibrium (SMPE) of (Γ, θ)
if for all i ∈ I,
ui (g (a, σ (a)) , θ) + βi (θ) WiΓ (σ, θ, h (a, σ (a)))≥ ui (g (a, (mi , σ−i (a))) , θ) + βi (θ) WiΓ (σ, θ, h (a, (mi , σ−i (a))))for all a ∈ A and all mi ∈ Mi (a). The set of stationary Markov perfect equilibriumof (Γ, θ) is denoted by SMPE(Γ, θ)
Implementation in SMPEs: A mechanism Γ implements ϕ : Θ → F (A, X) in
SMPEs if for all θ ∈ Θ, SMPE(Γ, θ) ̸= ∅ and all σ ∈SMPE(Γ, θ),
ϕ [θ] (a) = (g (a, σ (a)) , h (a, σ (a)))
for all a ∈ A. When such a mechanism exists, ϕ is implementable in SMPEs.

13

3. A Necessary Condition: Stationary Recursive Monotonicity
We show that only stationary recursive monotonic social choice processes are implementable in SMPEs. To introduce this condition, we need additional notation.
Fix any f = (z, α) ∈ F (A, Y ) and any θ ∈ Θ. Agent i’s marginal preferencesinduced by the stationary policy function f at θ over Y , denoted by Rif [θ], is definedby(x, b) Rif [θ] (x′ , b′ )if and only if
ui (x, θ) + βi (θ) Wi (f, θ, b) ≥ ui (x′ , θ) + βi (θ) Wi (f, θ, b′ )for all (x, b) , (x′ , b′ ) ∈ X × A. That is, when society follows f at θ, agent i ranks
pairs specifying a current outcome and a society state in the next period according
to Rif [θ]. A preference profile of preference orderings induced by f at θ is a list ofagents’ preference orderings induced by the stationary policy function f at θ over Y
and it is denoted by Rf [θ].
Agent i’s lower contour set at (f, θ, x, b), denoted by Lfi ((x, b) , θ), is defined by
Lfi ((x, b) , θ) =

n
o
f
′ ′
′ ′
(x , b ) ∈ Y : (x, b) Ri [θ] (x , b ) .

To understand our necessary condition, let us call any map π : Θ × A → Θ as a(Markovian) society deception; that is, agents coordinates their reports to π (θ, a),though their preferences are described by θ and the society state is a. In the casethat agents’ reports are not coordinated, the planner detects a lie and can punish theagents.
Given a social choice process ϕ and a society deception π, they induce a social

14choice process, denoted by ϕ ◦ π, which is defined byϕ ◦ π (θ, a′ ) = ϕ [π (θ, a′ )] (a′ )for all (θ, a′ ) ∈ Θ × A. Since ϕ ◦ π(θ, ·) = (z ◦ π(θ, ·), α ◦ π(θ, ·)) ∈ F (Y, A) for eachθ, ϕ ◦ π can be equivalently defined byz ◦ π (θ, a′ ) = z [π (θ, a′ )] (a′ )α ◦ π (θ, a′ ) = α [π (θ, a′ )] (a′ )for all (θ, a′ ) ∈ Θ × A. When ϕ is the social choice process and the society follows
the deception π, the actual social choice process is ϕ ◦ π. This should not be confusedwith social choice process at π (θ, a), that is, ϕ [π (θ, a)], which appears below. Indeed,
ϕ [π (θ, a)] is the stationary policy function followed by the society when the society
always pretends that π (θ, a) describes their preferences; that is, π (θ, a) is fixed.
Definition 1. ϕ : Θ → F (A, Y ) satisfies Stationary Recursive Monotonicity (SRM)
provided that for all θ ∈ Θ and all π ∈ ΘΘ×A , if for all i ∈ I,
ϕ[π(θ,a)]Li

ϕ◦π(θ,·)(ϕ ◦ π (θ, a) , π (θ, a)) ∩ Y (a) ⊆ Li

(ϕ ◦ π (θ, a) , θ)for all a ∈ A, thenϕ [θ] = ϕ ◦ π(θ, ·).
Let us introduce SRM by considering the simple case where A = {a1 , a2 } and Θ ={θ, θ′ }. Let us suppose that θ is the actual society type and that agents coordinate
on a society deception π such that π(θ, a1 ) = θ and π(θ, a2 ) = θ′ . In other words,
agents truthful report the society type in state a1 , whereas they lie in state a2 . Sinceθ is the society type, the socially optimal stationary policy function is ϕ[θ].
Suppose that the social choice process ϕ assigns in state a1 the pair ϕ ◦ π (θ, a1 )when agents report π (θ, a1 ) and commit to continuation policy function ϕ [π (θ, a1 )]15and that it assigns in state a2 the pair ϕ ◦ π (θ, a2 ) when agents report π (θ, a2 ) andcommit to ϕ [π (θ, a2 )].
Moreover, suppose that:
ϕ[π(θ,a1 )]• the marginal preferences of each agent i change from Riϕ◦π(θ,·)

Ri[θ] in a monotonic way around ϕ◦π (θ, a1 ) (that is, whenever (ϕ ◦ π (θ, a1 ))ϕ[π(θ,a1 )]Ri

ϕ◦π(θ,·)[π (θ, a1 )] (x′ , a′ ), it holds that (ϕ ◦ π (θ, a1 )) Ri[θ] (x′ , a′ )); andϕ[π(θ,a2 )]• the marginal preferences of each agent i change from Riϕ◦π(θ,·)Ri[π (θ, a1 )] to[π (θ, a2 )] to[θ] in a monotonic way around ϕ ◦ π (θ, a2 ).

Then, SRM demands that the stationary policy function ϕ ◦ π(θ, ·) without commitment is the socially optimal stationary policy function at θ (that is, ϕ ◦ π(θ, ·) = ϕ[θ]).
Our first main result is that only social choice processes satisfying SRM are implementable in SMPEs.
Theorem 1. If ϕ : Θ → F (A, Y ) is implementable in SMPEs, then it satisfies SRM.
Proof. Suppose that Γ implements ϕ : Θ → F (A, Y ) in SMPEs. Fix any θ ∈ Θ andany π ∈ ΘΘ×A . Suppose that for all i ∈ I,
ϕ[π(θ,a)]Li

ϕ◦π(θ,·)(ϕ ◦ π (θ, a) , π (θ, a)) ∩ Y (a) ⊆ Li

(ϕ ◦ π (θ, a) , θ)(1)

for all a ∈ A. We show that ϕ [θ] = ϕ ◦ π(θ, ·).
For each a′ ∈ A, fix any σ [π (θ, a′ )] ∈SMPE(Γ, π (θ, a′ )). Let σ ◦ π (θ, ·) be definedbyσ ◦ π (θ, a) ≡ σ [π (θ, a)] (a)for all a ∈ A. Since Γ implements ϕ in SMPEs, it follows thatϕ [π (θ, a)] (a) = (g (a, σ [π (θ, a)] (a)) , h (a, σ [π (θ, a)] (a)))

16

(2)

for all a ∈ A, and soz [π (θ, a)] (a) = g (a, σ [π (θ, a)] (a))α [π (θ, a)] (a) = h (a, σ [π (θ, a)] (a))
for all a ∈ A. Since implementability implies that
Wi (ϕ [π (θ, a)] , π (θ, a) , α [π (θ, a)] (a)) = WiΓ (σ [π (θ, a)] , π (θ, a) , h (a, σ [π (θ, a)] (a)))
for all i ∈ I and all a ∈ A, it holds that
ϕ[π(θ,a)](g (a, (mi , σ−i [π (θ, a)] (a))) , h (a, (mi , σ−i [π (θ, a)] (a)))) ∈ Li

(ϕ ◦ π (θ, a) , π (θ, a))∩Y (a)(3)

for all i ∈ I, all a ∈ A and all mi ∈ Mi (a). For all i ∈ I and all a ∈ A, it follows
from (7) and (3) thatϕ◦π(θ,·)(g (a, (mi , σ−i [π (θ, a)] (a))) , h (a, (mi , σ−i [π (θ, a)] (a)))) ∈ Li

(ϕ ◦ π (θ, a) , θ)(4)

for all mi ∈ Mi (a). Since (8) holds for all i ∈ I, all a ∈ A and all mi ∈ Mi (a) andsince
Wi (ϕ ◦ π(θ, ·), θ, α [θ] (a)) = WiΓ (σ ◦ π(θ, ·), θ, h (a, σ [π (θ, a)] (a)))
for all i ∈ I and all a ∈ A, it follows that σ ◦ π(θ, ·) ∈ SM P E (Γ, θ). Since Γ
implements ϕ in SMPE, we have thatϕ [θ] (a) = (g (a, σ [π (θ, a)] (a)) , h (a, σ [π (θ, a)] (a)))(5)

for all a ∈ A. (2) and (5) imply that ϕ [θ] (a) = ϕ [π (θ, a)] (a) for all a ∈ A. Thus, ϕsatisfies SRM.

■
17

4. A Full Characterization for Economic Environments
To proceed with our construction of the mechanism, let Rϕ[θ] [θ] denote the profile ofagents’ preferences induced by the stationary policy function ϕ [θ] at θ over Y , that
is,


ϕ[θ]
Rϕ[θ] [θ] = Ri [θ]i∈I

for all θ ∈ Θ. Moreover, let
R (Θ) = Rϕ[θ] [θ] : θ ∈ Θ
be the domain of preferences induced by the social choice process ϕ over Θ.
Although SRM is a necessary condition for implementation in SMPEs, it is not
sufficient. Since our interest in this notion of implementation is driven mainly by its
economic applications, we tackle sufficiency in economic environments.
Definition 2 (Economic environment). The environment is economic if there is nostate θ ∈ Θ, stationary policy function f ∈ F (A, Y ) and society state a ∈ A such
that

(

)

i ∈ I : (x, b) ∈ max Y (a)Rif [θ]≥n−1

for some (x, b) ∈ Y (a).
Theorem 2. Assume n ≥ 3 and an economic environment. If ϕ : Θ → F (A, Y )satisfies SRM, then it is implementable in SMPEs.
The proof of Theorem 2 is based on the construction of a canonical mechanismthat is similar to the existing mechanisms in the literature of Nash implementation,
though it is modified appropriately to deal with the problem at hands.
Proof of Theorem 2. Assume n ≥ 3 and an economic environment. Suppose thatϕ : Θ → F (A, Y ) satisfies SRM.
18

For all i ∈ I and all a ∈ A, agent i’s message space is defined by
Mi (a) = R (Θ) × Y (a) × Z+ ,where Z+ is the set of non-negative integers, with (Ri , (xi , bi ) , k i ) as typical agent
i’s message. Let us define the outcome function (g, h) according the following threerules. For all a ∈ A and all m ∈ M (a),

RULE 1: If there exists Rϕ[θ̄] [θ̄], (x, b) ∈ R (Θ)×Y (a) such that mi = (Ri , (xi , bi ) , k i ) = Rϕ[θ̄] [θ̄], (x, b) , k i for all i ∈ I and (x, b) = ϕ θ̄ (a), then g (a, m) = x andh (a, m) = b.

RULE 2: For all i ∈ I, if there exists R [θ̄], (x, b) ∈ R (Θ) × Y (a) such that 
mj = Rϕ[θ̄] [θ̄], (x, b) , k j for all j ∈ I\ {i} and (x, b) = ϕ θ̄ (a) and (Ri , (xi , bi )) ̸=


ϕ[θ̄]
R [θ̄], (x, b) , then:ϕ[θ̄]

1. If (x, b) Ri

ϕ[θ̄][θ̄] (xi , bi ), then g (a, m) = xi and h (a, m) = bi .

2. Otherwise, g (a, m) = x and h (a, m) = b.
RULE 3: Otherwise, an integer game is played: identify the agent who plays thehighest integer (if there is a tie at the top, pick the agent with the lowest index
among them.) This agent is declared the winner of the game, and the alternative
implemented is the one she selects.
Suppose that θ ∈ Θ is the true state. Let us show that Γ = (Mi )i∈I , (g, h)implements ϕ in SMPEs. To show this, let us define the strategy profile σ by
σi (a) = Rϕ[θ] [θ], ϕ [θ] (a) , k i ∈ Mi (a)for all a ∈ A and all i ∈ I. It is clear that the above strategy profile is an SMPE for(Γ, θ); that is, σ ∈SMPE(Γ, θ).
19

To complete the proof, fix any σ ∈SMPE(Γ, θ). Given that we are in an economic
environment, it follows that there is no a ∈ A such that σ (a) falls either in Rule 2 orRule 3. Thus, for all a ∈ A, σ (a) falls into Rule 1. Then, there exists a deception
π ∈ ΘΘ×A such that for all a ∈ A and all i ∈ I,σi (a) = Rϕ[π(θ,a)] [π (θ, a)] , ϕ ◦ π (θ, a) , k i .
Rule 1 implies
ϕ [π (θ, a)] (a) = (g (a, σ (a)) , h (a, σ (a)))
for all a ∈ A, and soz [π (θ, a)] (a) = g (a, σ (a))α [π (θ, a)] (a) = h (a, σ (a))
for all a ∈ A.
Fix any society state a ∈ A. Moreover, fix any i ∈ I and any
ϕ[π(θ,a)]xi , bi ∈ Li
(ϕ ◦ π (θ, a) , π (θ, a)) ∩ Y (a) .
We show thatϕ◦π(θ,·)
xi , bi ∈ Li
(ϕ ◦ π (θ, a) , θ) .

(6)



Fix any mi = (Ri , (xi , bi ) , k i ) ∈ Mi (a) such that Ri , (xi , bi ) , k̄ i ≠ Rϕ[π(θ,a)] , ϕ ◦ π (θ, a) , k i .
Suppose that (Ri , (xi , bi )) ̸= Rϕ[π(θ,a)] , ϕ ◦ π (θ, a) . If
ϕ[π(θ,a)]ϕ [π (θ, a)] (a) Ri[π(θ, a)] xi , bi ,then Rule 2 implies that g (a, (mi , σ−i (a))) = xi and h (a, (mi , σ−i (a))) = bi . Oth-20

erwise, Rule 2 implies thatg (a, (mi , σ−i (a))) = z [π (θ, a)] (a)h (a, (mi , σ−i (a))) = α [π (θ, a)] (a) .
Suppose that (Ri , (xi , bi )) = Rϕ[π(θ,a)] , ϕ ◦ π (θ, a) . Then, Rule 1 implies thatg (a, (mi , σ−i (a))) = z [π (θ, a)] (a)h (a, (mi , σ−i (a))) = α [π (θ, a)] (a) .
Since σ ∈SMPE(Γ, θ), it follows thatϕ◦π(θ,·)ϕ [π (θ, a)] (a) Ri[θ] xi , bi ,and so (6) follows.
Since society state a, agent i, pair (xi , bi ) and i’s message mi were arbitrarily chosen,
it follows that
ϕ[π(θ,a)]Li

ϕ◦π(θ,·)(ϕ ◦ π (θ, a) , π (θ, a)) ∩ Y (a) ⊆ Li

(ϕ ◦ π (θ, a) , θ)for all i ∈ I and all a ∈ A. SRM implies that ϕ [θ] = ϕ ◦ π(θ, ·). Thus, Γ implements
ϕ in SMPEs.

■
5. Applications

In what follows, we provide two relevant economic applications.
5.1. Internalizing pecuniary externality of saving
This subsection shows that the recursive Lindahl solution, which allows internalizationof pecuniary externalities of saving decisions, is implementable in SMPE. We show this21

result under the simplifying assumption of no production. To this end, in what follows,
let us bear in mind the borrowing/lending economy with no production described in
Example 1 and Example 4 and the regularity conditions spelled out in those examples.
We have the following positive result.
Proposition 1. Assume n ≥ 3. The recursive Lindahl solution is implementable in
SMPEs provided that it is unique for all θ ∈ Θ.
Proof. Assume n ≥ 3. In light of Theorem 2, we need only to check that the Economic
Environment assumption is satisfied and that the recursive Lindahl solution satisfies
SRM.
Since consumptions are private, the Economic Environment assumption is automatically satisfied.
Finally, to show that the recursive Lindahl solution satisfies SRM, pick any a ∈ A
and any θ ∈ Θ and suppose that the premises of SRM are satisfied. Since ϕ[π(θ, a)](a)is the recursive Lindahl solution under type π(θ, a), it holds
ϕ[π(θ,a)]Bi (a) ⊆ Li

(ϕ ◦ π (θ, a) , π (θ, a)) ∩ Y (a)for all i ∈ I, where Bi (a) = {(x, b) ∈ Y (a) : xi + qi (a) · b = ai + ei } denotes thecorresponding budget set that is constrained by the feasible set.
Fix any i ∈ I. By the presumption of SRM, it holdsϕ◦π(θ,·)Bi (a) ⊆ Li

(ϕ ◦ π (θ, a) , θ) .

This shows that ϕ◦π(θ, a) is an optimal choice in Bi (a) for i with type θ, provided thatϕ ◦ π(θ, ·) is the continuation policy function. Since i and a were arbitrary, ϕ ◦ π(θ, ·)is a recursive Lindahl solution for θ. Under uniqueness of the Lindahl solution, we
obtain ϕ[θ] = ϕ ◦ π(θ, ·). Thus, the recursive Lindahl solution satisfies SRM.

■

How much does the solution internalize pecuniary externality? Here is an efficiency
22

property that is met by the recursive Lindahl solution in a direct logical manner, likein the first welfare theorem.
Definition 3. A policy function f is recursively efficient for type θ if for all a ∈ A
there is no (x, b) ∈ Y (a) such thatui (xi , θ) + βi (θ) Wi (f, θ, b) ≥ ui (z(a), θ) + βi (θ) Wi (f, θ, α(a))
for all i ∈ I andui (xi , θ) + βi (θ) Wi (f, θ, b) > ui (z(a), θ) + βi (θ) Wi (f, θ, α(a))
for some i ∈ I.
A social choice process ϕ is said to be recursively efficient if ϕ[θ] is recursively
efficient for every θ.
To see that the recursive Lindahl solution satisfies recursive efficiency, suppose that
there is a pair (x′ , b′ ) that improves over f (a) provided that the society follows f fromthe next period onwards. Then, it must be thatx′i + qi (a) · b′ ≥ ai + ei
for all i ∈ I andx′i + qi (a) · b′ > ai + ei
for some i ∈ I. By summing up the above inequalities, we obtain that!
Xqi (a)· b′ = q(a)i∈IXb′i > 0,i∈Iwhich is a contradiction.
Recursive efficiency is a weaker requirement than full efficiency of intertemporal23

consumption allocations, however.
Definition 4. A policy function f is Pareto-efficient for type θ if for all a ∈ A there
is no x ∈ X ∞ (a) such thatUi (xi , θ) ≥ Wi (f, θ, a)
for all i ∈ I andUi (xi , θ) > Wi (f, θ, a)for some i ∈ I, where X ∞ (a) denotes the set of sequences of consumption allocations
which are feasible given a.
A social choice process ϕ is said to be Pareto-efficient if ϕ[θ] is Pareto-efficient forevery θ.
In the next example, we show that recursive efficiency does not imply Pareto efficiency. Such efficiency loss can arise because of a lack of ability to commit to future
allocations, but the below is the most extreme one.
Example 7. Fix any per-period consumption allocation x∗ . Let f = (z, α) be apolicy function such that z(a) = x∗ for all a ∈ A and α is arbitrary. The continuation
lifetime utility of each i, that is, Wi (f, θ, b), is constant in b. Therefore, everybody
is indifferent in the allocation of assets for the next period and cares only about
current consumption. Under the assumption that there is just one physical good ateach period, (x∗ , α(a)) is recursively efficient, provided that f is the policy functionused from the next period onwards. Thus, f is recursively efficient but it is obviously
Pareto-inefficient.
We argue, however, that Pareto-efficient allocation can be obtained as a recursive Lindahl equilibrium. Hence, if recursive Lindahl equilibrium is unique it fullyinternalizes pecuniary externality of saving decisions.
To see this, assume the standard regularity conditions on preferences and earnings,

24and let x(b) denote the interior Pareto-efficient allocation such that
bi =

∞
X

pt (b)(xit (b) − ei )t=1

holds for each i, where p(b) = (p1 (b), p2 (b), p3 (b), · · · ) denotes the sequence of supporting prices with p1 (b) being normalized to 1.
Now consider the policy function f = (z, α) given byzi (a) = xi1 (a)bi − xi1 (b) + ei
αi (b) =p2 (b)
for each i. Then, the continuation lifetime utility Wi (f, θ, b) is concave in b, increasing
in bi , and decreasing in bj for all j ̸= i, following the argument analogous to Lucas Jr
and Stokey (1984).
Consider the preference induced over pairs of current consumption and asset allocation for the next period, which is represented in the formui (xi , θ) + βi (θ)Wi (f, θ, b).
Now, for each i and j, set qij (a) be equal to the marginal rate of substitution ofquantity bj for quantity xi at (zi (a), α(a)). Then (zi (a), α(a)) is individual i’s optimal
choice under the budget constraint
xi + qi (a) · b ≤ ai + ei .
5.2. Heterogeneous discounting and recursive median voter equilibrium in public
capital accumulation
Let us consider the public capital accumulation problem described in Example 3 andin Example 6 where the regularity conditions spelled out in those examples hold. We
25

show below that the recursive Condorcet winner is implementable in SMPEs when
agents always have distinct discount factors.
Proposition 2. Assume n ≥ 3. Assume that for each θ ∈ Θ, every agent i’s discountfactor differs from others. Then, the recursive Condorcet winner ϕ is implementable
in SMPEs.
Proof. Assume n ≥ 3. In light of Theorem 2, we need only to check that the Economic
Environment assumption is satisfied and that the recursive Condorcet winner satisfies
SRM.
The Economic Environment assumption is met when, for each θ ∈ Θ, every agent
i’s discount factor differs from those of others.
Finally, to show that the recursive Condorcet winner satisfies SRM, pick any a ∈ A
and any θ ∈ Θ. For each i, her upper contour set above the Condorcet winnerϕ ◦ π(θ, a) under type π(θ, a) when the continuation policy function is ϕ[π(θ, a)] isleft/right to ϕ◦π(θ, a) whenever her discount factor is smaller/larger than the median,which follows from Lemmata in the appendix.
Then, by the presumption of SRM, her upper contour set above ϕ ◦ π(θ, a) undertype θ when the continuation policy function is ϕ ◦ π(θ, a) is left/right to ϕ ◦ π(θ, a).
Hence, ϕ ◦ π(θ, a) remains the Condorcet winner for type θ when the continuation
policy function is ϕ ◦ π(θ, ·). Since the choice of a ∈ A was arbitrary, ϕ ◦ π(θ, ·) isa recursive Condorcet winner for θ. By the uniqueness of the recursive Condorcet
winner, we have ϕ[θ] = ϕ ◦ π(θ, ·).
Since the choice of θ ∈ Θ was arbitrary, we conclude that the recursive Condorcet
winner satisfies SRM.
■
6. Conclusions
A standard assumption in dynamic implementation theory is that the agents (andso the planner) can commit to future plans/actions once the social decision is made26

(see, for instance, Moore and Repullo (1988)). Although this assumption has led
to important insights, it is often made for technical convenience and is violated inmany situations. Another critical assumption is that agents’ preferences uncertainly
change over time in a transitory manner, and the planner’s objective is to repeatedly implement the same social objective upon realization of such uncertainty eachtime (see, for instance, Lee and Sabourian (2011) and Mezzetti and Renou (2017)).
Although this assumption naturally fits in many applications, it rules out intertemporal trades, which are common in dynamic contexts where agents’ preferences arepermanent. Moreover, most existing studies on dynamic mechanism design assume
that time discount is common and known, while heterogeneity and information abouttime discounting are critical in intertemporal trades. The classical dynamic generalequilibrium model involves intertemporal trades.
In this paper, we drop the above three assumptions and study the implementationof (single-valued) social choice processes in stationary Markov-perfect equilibrium(SMPE). This game theoretical solution concept captures our day-to-day economic
life well because we often cannot commit to future actions.
We develop a framework where agents’ preferences are permanent but cannot commit to future actions. Consequently, the planner can elicit agents’ preferences only
piece-by-piece, period-by-period. More precisely, in our framework, the current stateof the society is described by a state variable, and the objective of the planner is
to collect information about agents’ preferences over pairs, each specifying a current
social outcome and a state variable for the next period.
In this framework, we show that only stationary Markov monotonic social choiceprocesses are implementable in SMPEs. Moreover, we show that this condition is
also sufficient when a regularity condition is satisfied. This characterization result
allowed us to implement the recursive Lindhal solution and the recursive medianvoter solution in the public capital accumulation problem in SMPEs.

27

Appendices
Let W (f, a, β, u) denote the continuation lifetime utility of the agent with per-period
utility u, discount factor β, where the current capital amount is a, and the continuation policy function is f . The lemma below shows that the W satisfies the increasing
difference property in (a, β) for arbitrary increasing function f .
This property implies that the incremental gain for choosing a higher a (i.e., ãrather than a) is greater when β is higher. That is, W (f, ã, β, u) − W (f, a, β, u)is increasing in β. Equivalently, if β̃ > β, then W f, ã, β̃, u − W f, a, β̃, u isincreasing in a. In other words, saving and discount factors are complements of eachother. This result can be stated as follows.
Lemma 1. Assume that ui (·, θ) = u for every voter i = 1, ..., n and every θ ∈ Θ,
where u is a continuous, strictly increasing, and strictly concave utility function.
Assume that the policy function f is continuous and strictly increasing and assume
that it is such that difference Q(·) − f (·) is strictly increasing in the public goodholding. Let β̃ > β and ã > a. Then:W f, ã, β̃, u − W f, a, β̃, u > W (f, ã, β, u) − W (f, a, β, u)
and
h 


i
β̃ W f, ã, β̃, u − W f, a, β̃, u > β [W (f, ã, β, u) − W (f, a, β, u)] .
Proof. Let the premises hold. Then, inductively, f t (·) is continuous and strictlyincreasing. Moreover, since Q(f t−1 (·)) − f t (·) = Q(f t−1 (·)) − f (f t−1 (·)) and f t−1 (·)
is strictly increasing, it follows that Q(f t−1 (·)) − f t (·) is strictly increasing. Thus, by

28

our initial suppositions that β̃ > β and ã > a, we have thatW f, ã, β̃, u − W f, a, β̃, u
=
>

∞
X
t=1
∞
X

t−1

h

β t−1

h

β̃

u (Q (ft−1u (Q (f

t−1

t

t−1

t

t−1

(ã)) − f (ã)) − u (Q (f(ã)) − f (ã)) − u (Q (ft(a)) − f (a))t(a)) − f (a))

i

i

t=1

= W (f, ã, β, u) − W (f, a, β, u) ,and thath iβ̃ W f, ã, β̃, u − W f, a, β̃, u> β̃ [W (f, ã, β, u) − W (f, a, β, u)]> β [W (f, ã, β, u) − W (f, a, β, u)] .
■The strict increasing differences in (a, β) of voters’ continuation values allow us toshow the following result. Fix the saving peak and the discount factor of voter i.
Every other voter less patient than voter i dislikes savings more than voter i’s saving
peak. Every other voter more patient than voter i dislikes savings less than voter i’s
saving peak. This result can be stated as follows.
Lemma 2. Assume that ui (·, θ) = u for every voter i = 1, ..., n and every θ ∈ Θ,
where u is a continuous, strictly increasing, and strictly concave utility function.
Assume that the policy function f is continuous and strictly increasing and assume
that it is such that difference Q(·) − f (·) is strictly increasing in the public goodholding. Fix any βℓ ∈ {β1 , ..., βn } and any
a′ℓ ∈ arg ′ maxa ∈[0,Q(a)]



′

′

u(Q(a) − a ) + βℓ W (f, a , βℓ , u)

29



.

(7)

Then, for all a′ > a′ℓ and all β < βℓ , it holdsu(Q(a) − a′ℓ ) + βW (a′ℓ , ϕ, β, u) > u(Q(a) − a′ ) + βW (f, a′ , β, u) ,(8)

and for all a′ < a′ℓ and all β > βℓ , it holdsu(Q(a) − a′ℓ ) + βW (f, a′ℓ , β, u) > u(Q(a) − a′ ) + βW (f, a′ , β, u) .

(9)

Proof. Let the premises hold. Take any a′ > a′ℓ and any β < βℓ . Thus, by Lemma 1
and by the fact that a′ℓ is an element of (7), it follows thathi hiu(Q(a) − a′ ) + βW (f, a′ , β, u) − u(Q(a) − a′ℓ ) + βW (f, a′ℓ , β, u)
h
i
h
i
=
u(Q(a) − a′ ) − u(Q(a) − a′ℓ ) + β W (f, a′ , β, u) − W (f, a′ℓ , β, u)
i
i
h
h
′
′
′
′
+
β
<
W (f, a , βℓ , u) − W (f, aℓ , βℓ , u)u(Q(a) − a ) − u(Q(a) − aℓ )
ℓ
h
i h
i
′
′
′
=
−
u(Q(a) − a ) + βℓ W (f, a , βℓ , u)u(Q(a) − aℓ ) + βℓ W (f, aℓ , βℓ , u)
≦ 0,which establishes (8).
The case of a′ < a′ℓ and β > βℓ can be proved similarly.

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