Organizational Equilibrium with Capital Marco Bassetto, Zhen Huo, - - PowerPoint PPT Presentation

Organizational Equilibrium with Capital

Marco Bassetto, Zhen Huo, and José-Víctor Ríos-Rull Economic Growth and Fluctuations Barcelona GSE June 14, 2018

FRB of Chicago (Mpls), Yale University, University of Pennsylvania, UCL, CAERP 1

Introduction

Question

• Time inconsistency is a pervasive issue
• taxation, government debt, consumption-saving problem, . . .
• Benchmark: Markov equilibrium
• Can agents do better than Markov equilibrium?
• yes, with trigger strategies, self-punishment as a threat
• We show agents do better than Markov without “trigger strategies”
• continuation value independent of history

2

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal
• 2. Make their own proposal

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal
• 2. Make their own proposal
• 3. Wait for the next agent to make a proposal

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal
• 2. Make their own proposal
• 3. Wait for the next agent to make a proposal
• Organizational equilibrium

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal
• 2. Make their own proposal
• 3. Wait for the next agent to make a proposal
• Organizational equilibrium
• 1. Initial agent is willing to make a proposal, no delaying

3

This paper: Propose an Equilibrium Concept

• Agents not only decide their action: they can make proposals for the

future

• Future agents can
• 1. Follow the proposal
• 2. Make their own proposal
• 3. Wait for the next agent to make a proposal
• Organizational equilibrium
• 1. Initial agent is willing to make a proposal, no delaying
• 2. Future agents are willing to follow

3

Equilibrium Properties I

• No one can treat herself specially
• If a proposal favors the initial proposer, the next agent wants to copy the

idea and restart

• Cooperation across current and future decision makers has to be built

gradually

• Proposal starts with small deviation from Markov, otherwise temptation

to let future agents start the process

• Over time, greater departures from short-term best response, in

anticipation of more beneficial future outcomes

4

Equilibrium Properties II

• Compare with Markov equilibrium

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:
• 1. Same continuation value on or off equilibrium path

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:
• 1. Same continuation value on or off equilibrium path
• 2. No one wants to deviate and wait for a restart of the game

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:
• 1. Same continuation value on or off equilibrium path
• 2. No one wants to deviate and wait for a restart of the game
• Directly related to reconsideration-proof equilibrium

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:
• 1. Same continuation value on or off equilibrium path
• 2. No one wants to deviate and wait for a restart of the game
• Directly related to reconsideration-proof equilibrium
• Extend to a class of models with capital

5

Equilibrium Properties II

• Compare with Markov equilibrium
• Payoff only depends on (payoff relevant) state variables, like Markov

equilibrium

• Action can depend on history, different from Markov equilibrium
• Compare with trigger strategies
• History can be abolished, no self-punishment
• In terms of Subgame Perfect Refinements:
• 1. Same continuation value on or off equilibrium path
• 2. No one wants to deviate and wait for a restart of the game
• Directly related to reconsideration-proof equilibrium
• Extend to a class of models with capital
• Impose additional no-delaying condition (relevant for environments with

state variables)

5

Quantitative Findings

• Apply the equilibrium concept in
• Quasi-geometric discounting growth model
• Government taxation model
• Steady state
• Allocation is close to Ramsey outcome, much better than Markov

equilibrium

• Transition
• Allocation starts similar to Markov, converges to similar to Ramsey

6

Related Literature

• Markov equilibrium and GEE
• Currie and Levine (1993), Bassetto and Sargent (2005), Klein and Ríos-Rull (2003),

Klein, Quadrini and Ríos-Rull (2005), Krusell and Ríos-Rull (2008), Krusell, Kuruscu, and Smith (2010), Song, Storesletten and Zilibotti (2012)

• Sustainable plans
• Stokey (1988), Chari and Kehoe (1990), Abreu, Pearce and Stacchetti (1990), Phelan

and Stacchetti (2001)

• Quasi-geometric discounting growth model
• Strotz (1956), Phelps and Pollak (1968), Laibson (1997), Krusell and Smith (2003),

Chatterjee and Eyigungor (2015), Bernheim, Ray, and Yeltekin (2017), Cao and Werning (2017)

• Refinement of subgame perfect equilibrium
• Farrell and Maskin (1989), Kocherlakota (2008), Nozawa (2014), Ales and Sleet (2015)

7

Plan

• 1. An example: a growth model with quasi-geometric discounting
• 2. General definition and property
• 3. Application in government taxation problem

8

Growth model

The Environment

• Preferences: quasi-geometric discounting

Ψt = u(ct) + δ

∞

• τ=1

βτu (ct+τ)

9

The Environment

• Preferences: quasi-geometric discounting

Ψt = u(ct) + δ

∞

• τ=1

βτu (ct+τ)

• Period utility function u(c) = log c

9

The Environment

• Preferences: quasi-geometric discounting

Ψt = u(ct) + δ

∞

• τ=1

βτu (ct+τ)

• Period utility function u(c) = log c
• δ = 1 is the time-consistent case

9

The Environment

• Preferences: quasi-geometric discounting

Ψt = u(ct) + δ

∞

• τ=1

βτu (ct+τ)

• Period utility function u(c) = log c
• δ = 1 is the time-consistent case
• Technology

f (kt) = kα

t ,

kt+1 = f (kt) − ct.

9

Benchmark I: Markov Perfect Equilibrium

• Take future g(k) as given

max

k′

u[f (k) − k′] + δβΩ(k′; g)

• cont. value:

Ω(k; g) = u[f (k) − g(k)] + βΩ[g(k); g]

10

Benchmark I: Markov Perfect Equilibrium

• Take future g(k) as given

max

k′

u[f (k) − k′] + δβΩ(k′; g)

• cont. value:

Ω(k; g) = u[f (k) − g(k)] + βΩ[g(k); g]

• The Generalized Euler Equation (GEE)

uc = βu′

c [δf ′ k + (1 − δ) g ′ k] 10

Benchmark I: Markov Perfect Equilibrium

• Take future g(k) as given

max

k′

u[f (k) − k′] + δβΩ(k′; g)

• cont. value:

Ω(k; g) = u[f (k) − g(k)] + βΩ[g(k); g]

• The Generalized Euler Equation (GEE)

uc = βu′

c [δf ′ k + (1 − δ) g ′ k]

• The equilibrium features a constant saving rate

k′ = δαβ 1 − αβ + δαβ kα = sM kα

10

Benchmark II: Ramsey Allocation with Commitment

• Choose all future allocations at period 0

max

k1

u[f (k0) − k1] + δβΩ(k1)

• cont. value:

Ω(k) = max

k′

u[f (k) − k′] + βΩ(k′)

11

Benchmark II: Ramsey Allocation with Commitment

• Choose all future allocations at period 0

max

k1

u[f (k0) − k1] + δβΩ(k1)

• cont. value:

Ω(k) = max

k′

u[f (k) − k′] + βΩ(k′)

• The sequence of saving rates is given by

st =        sM =

αδβ 1−αβ+δαβ , t = 0

sR = αβ, t > 0

11

Benchmark II: Ramsey Allocation with Commitment

• Choose all future allocations at period 0

max

k1

u[f (k0) − k1] + δβΩ(k1)

• cont. value:

Ω(k) = max

k′

u[f (k) − k′] + βΩ(k′)

• The sequence of saving rates is given by

st =        sM =

αδβ 1−αβ+δαβ , t = 0

sR = αβ, t > 0

• Steady state capital in Markov equilibrium is lower than Ramsey’s

sM < sR

11

Elements of Org Equil: Proposal and Value Function

• A proposal is a sequence of saving rates {s0, s1, s2, . . .}

12

Elements of Org Equil: Proposal and Value Function

• A proposal is a sequence of saving rates {s0, s1, s2, . . .}
• Everyone in the future can implement the same proposal

12

Elements of Org Equil: Proposal and Value Function

• A proposal is a sequence of saving rates {s0, s1, s2, . . .}
• Everyone in the future can implement the same proposal
• Given an initial capital k0, the proposal induces a sequence of capitals

k1 = s0kα k2 = s1kα

1 = kα2 0 s1sα

. . . kt = kαt

0 Πt−1 j=0 sαt−j−1 j 12

Proposal and Value Function

• Given a proposal (sequence of saving rates) {s0, s1, s2, . . .} and an

initial capital k0, we get the sequence of capitals kt = kαt

0 Πt−1 j=0 sαt−j−1 j 13

Proposal and Value Function

• Given a proposal (sequence of saving rates) {s0, s1, s2, . . .} and an

initial capital k0, we get the sequence of capitals kt = kαt

0 Πt−1 j=0 sαt−j−1 j

• The lifetime utility for the agent who makes the proposal is

U(k0, s0, s1, . . .) = log[(1 − s0)kα

0 ] + δ ∞

• j=1

βj log

• (1 − sj)kα

j

• =α(1 − αβ + δαβ)

1 − αβ log k0 + log(1 − s0) + δ

∞

• j=1

βj log

• (1 − sj)Πj−1

τ=0sαj−τ τ

• ≡φ log k0 + V (s0, s1, . . .)

13

Proposal and Value Function

• The lifetime utility for agent at period t is

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff

14

Proposal and Value Function

• The lifetime utility for agent at period t is

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff
• There is a Separability property between capital and saving rates

14

Proposal and Value Function

• The lifetime utility for agent at period t is

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff
• There is a Separability property between capital and saving rates
• True for the initial proposer and all subsequent followers

14

Proposal and Value Function

• The lifetime utility for agent at period t is

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff
• There is a Separability property between capital and saving rates
• True for the initial proposer and all subsequent followers
• This property is crucial to our equilibrium concept

14

Proposal and Value Function

• The lifetime utility for agent at period t is

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff
• There is a Separability property between capital and saving rates
• True for the initial proposer and all subsequent followers
• This property is crucial to our equilibrium concept
• What type of proposals can be implemented?

14

Can the Ramsey Outcome be Implemented?

• If the initial agent with k0 proposes {sM, sR, sR, . . .}, which implies

k1 = sMkα

15

Can the Ramsey Outcome be Implemented?

• If the initial agent with k0 proposes {sM, sR, sR, . . .}, which implies

k1 = sMkα

• By following the proposal, the next agent’s payoff is

U

• k1, sR, sR, sR, . . .
• = φ log k1 + V
• sR, sR, sR, . . .
• 15

Can the Ramsey Outcome be Implemented?

• If the initial agent with k0 proposes {sM, sR, sR, . . .}, which implies

k1 = sMkα

• By following the proposal, the next agent’s payoff is

U

• k1, sR, sR, sR, . . .
• = φ log k1 + V
• sR, sR, sR, . . .
• By copying the proposal, the next agent’s payoff is

U

• k1, sM, sR, sR, . . .
• = φ log k1 + V
• sM, sR, sR, . . .
• > φ log k1 + V
• sR, sR, sR, . . .
• 15

Can the Ramsey Outcome be Implemented?

• If the initial agent with k0 proposes {sM, sR, sR, . . .}, which implies

k1 = sMkα

• By following the proposal, the next agent’s payoff is

U

• k1, sR, sR, sR, . . .
• = φ log k1 + V
• sR, sR, sR, . . .
• By copying the proposal, the next agent’s payoff is

U

• k1, sM, sR, sR, . . .
• = φ log k1 + V
• sM, sR, sR, . . .
• > φ log k1 + V
• sR, sR, sR, . . .
• Copying is better than following:

Ramsey outcome cannot be implemented

15

Can a Constant Saving Rate be Implemented?

• Suppose the initial agent proposes {s, s, s . . .}

16

Can a Constant Saving Rate be Implemented?

• Suppose the initial agent proposes {s, s, s . . .}
• By following the proposal, the payoff for agent in period t is

U (kt, s, s, . . .) =φ log kt + V (s, s, . . .) where V (s, s, . . .) ≡ H(s) =

• 1 +

βδ 1 − β

• log(1−s)+

δαβ (1 − αβ)(1 − β) log(s)

16

Can a Constant Saving Rate be Implemented?

• Suppose the initial agent proposes {s, s, s . . .}
• By following the proposal, the payoff for agent in period t is

U (kt, s, s, . . .) =φ log kt + V (s, s, . . .) where V (s, s, . . .) ≡ H(s) =

• 1 +

βδ 1 − β

• log(1−s)+

δαβ (1 − αβ)(1 − β) log(s)

• To be followed, the constant saving rate has to be

s∗ = argmax H(s)

16

Optimal Constant Saving Rate

sM < s∗ < sR

17

Can {s∗, s∗, . . .} be Implemented?

• If the initial agent proposes {s∗, s∗, . . .}, no one has incentive to copy

18

Can {s∗, s∗, . . .} be Implemented?

• If the initial agent proposes {s∗, s∗, . . .}, no one has incentive to copy
• But, she prefers to choose sM, and wait the next to propose

{s∗, s∗, . . .} U

• k0, sM, s∗, s∗, . . .
• = φ log k0 + V
• sM, s∗, s∗, . . .
• > φ log k0 + V (s∗, s∗, s∗, . . .)

18

Can {s∗, s∗, . . .} be Implemented?

• If the initial agent proposes {s∗, s∗, . . .}, no one has incentive to copy
• But, she prefers to choose sM, and wait the next to propose

{s∗, s∗, . . .} U

• k0, sM, s∗, s∗, . . .
• = φ log k0 + V
• sM, s∗, s∗, . . .
• > φ log k0 + V (s∗, s∗, s∗, . . .)
• Constant s∗ proposal cannot be implemented, no one wants to

propose it

18

Can {s∗, s∗, . . .} be Implemented?

• If the initial agent proposes {s∗, s∗, . . .}, no one has incentive to copy
• But, she prefers to choose sM, and wait the next to propose

{s∗, s∗, . . .} U

• k0, sM, s∗, s∗, . . .
• = φ log k0 + V
• sM, s∗, s∗, . . .
• > φ log k0 + V (s∗, s∗, s∗, . . .)
• Constant s∗ proposal cannot be implemented, no one wants to

propose it

• But, something else can be implemented, which converges to s∗

18

Can {s∗, s∗, . . .} be Implemented?

• If the initial agent proposes {s∗, s∗, . . .}, no one has incentive to copy
• But, she prefers to choose sM, and wait the next to propose

{s∗, s∗, . . .} U

• k0, sM, s∗, s∗, . . .
• = φ log k0 + V
• sM, s∗, s∗, . . .
• > φ log k0 + V (s∗, s∗, s∗, . . .)
• Constant s∗ proposal cannot be implemented, no one wants to

propose it

• But, something else can be implemented, which converges to s∗
• For this, we proceed to define the organizational equilibrium

18

Organizational Equilibrium

Definition A sequence of saving rates {sτ}∞

τ=0 is organizationally admissible if

• 1. V (st, st+1, st+2, . . .) is (weakly) increasing in t
• 2. The first agent has no incentive to delay the proposal.

V (s0, s1, s2, . . .) ≥ max

s

V (s, s0, s1, s2, . . .) Within organizationally admissible sequences, the sequence that attains the maximum of V (s0, s1, s2, . . .) is an organizational equilibrium.

19

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion
• Same continuation value on and off equilibrium path

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion
• Same continuation value on and off equilibrium path
• No one can be better off by deviating and counting on others to restart

the game

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion
• Same continuation value on and off equilibrium path
• No one can be better off by deviating and counting on others to restart

the game

• In equilibrium:

U(kt, st, st+1, . . .) = φ log kt + V (st, st+1, . . .) = φ log kt + V ∗

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion
• Same continuation value on and off equilibrium path
• No one can be better off by deviating and counting on others to restart

the game

• In equilibrium:

U(kt, st, st+1, . . .) = φ log kt + V (st, st+1, . . .) = φ log kt + V ∗

• Total payoff only depends on capital, not a trigger with self-punishment

20

Remarks on Organizational Equilibrium

• Organizational Equilibrium is the outcome of some Subgame Perfect

Equilibrium (SPEq)

• SPEq example: if someone deviates, next agent restarts from s0
• SPEq refinement criterion
• Same continuation value on and off equilibrium path
• No one can be better off by deviating and counting on others to restart

the game

• In equilibrium:

U(kt, st, st+1, . . .) = φ log kt + V (st, st+1, . . .) = φ log kt + V ∗

• Total payoff only depends on capital, not a trigger with self-punishment
• Agents’ action depend on past actions, not a Markov equilibrium

20

Construct the Organizational Equilibrium

• Look for a sequence of saving rates {s0, s1, . . .}

21

Construct the Organizational Equilibrium

• Look for a sequence of saving rates {s0, s1, . . .}
• Every generation obtains the same V

V (st, st+1, . . .) = V (st+1, st+2, . . .) = V which induces the following difference equation β(1 − δ) log(1 − st+1) = δαβ 1 − αβ log st + log(1 − st) − (1 − β)V

21

Construct the Organizational Equilibrium

• Look for a sequence of saving rates {s0, s1, . . .}
• Every generation obtains the same V

V (st, st+1, . . .) = V (st+1, st+2, . . .) = V which induces the following difference equation β(1 − δ) log(1 − st+1) = δαβ 1 − αβ log st + log(1 − st) − (1 − β)V

• We call this difference equation “the proposal function”

st+1 = q(st; V )

21

Construct the Organizational Equilibrium

• Look for a sequence of saving rates {s0, s1, . . .}
• Every generation obtains the same V

V (st, st+1, . . .) = V (st+1, st+2, . . .) = V which induces the following difference equation β(1 − δ) log(1 − st+1) = δαβ 1 − αβ log st + log(1 − st) − (1 − β)V

• We call this difference equation “the proposal function”

st+1 = q(st; V )

• The maximal V and an initial s0 are needed to determine {sτ}∞

τ=0 21

Determine V ∗

• As V increases, the proposal function q(s; V ) moves upwards
• The highest V = V ∗ is achieved when q(s; V ) is tangent to the 45 degree

line (at s∗)

22

Determine the Initial Saving Rate s0

• The first agent should have no incentive to delay the proposal

max

s

V (s, s0, s1, s2, . . .) = V (sM, s0, s1, s2, . . .)

• s0 has to be such that

V ∗ = V (s0, s1, s2, . . .) ≥ V (sM, s0, s1, s2, . . .) − → s0 ≤ q∗ sM

• We select s0 = q∗

sM , which yields the highest welfare during transition

23

Org Equil in the Quasi-Geometric Discounting Growth Model

The organizational equilibrium {sτ}∞

τ=0 is given recursively by the proposal

function q∗ st = q∗(st−1) = 1 − exp

• −(1 − β)V ∗ +

δαβ 1−αβ log st−1 + log(1 − st−1)

β(1 − δ)

• where the initial saving rate s0, the steady state s∗, and V ∗ are given by

s0 = q∗ sM s∗ = δαβ (1 − β + δβ)(1 − αβ) + δαβ V ∗ = 1 − β + δβ 1 − β log(1 − s∗) + αδβ (1 − β)(1 − αβ) log s∗

24

Transition Dynamics

• The equilibrium starts from s0, and monotonically converges to s∗.

25

Remarks

• 1. To solve proposal function, no agent can treat herself specially,

Vt = Vt+1 Thank you for the idea, I will do it myself

• 2. To determine the initial saving rate, the agent starts from low saving

rate Goodwill has to be built gradually

• 3. We will show how the outcome compared with the Markov and

Ramsey We do much better than Markov equilibrium

26

Comparison: Steady State

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

• Organizational equilibrium is much better than the Markov equilibrium

27

Comparison: Allocation in Transition

• Organizational equilibrium: starts low, converges to being close to

Ramsey

28

Comparison: Payoff in Transition

U(kt, st, st+1, . . .)

• total payoff

= φ log kt + V (st, st+1, . . .)

• action payoff

29

Organizational Equilibrium for Weakly Separable Economies

General Definition

• An infinite sequence of decision makers is called to act

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A
• State evolves kt+1 = F(kt, at)

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A
• State evolves kt+1 = F(kt, at)
• Preferences: U(kt, at, at+1, at+2, . . .)

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A
• State evolves kt+1 = F(kt, at)
• Preferences: U(kt, at, at+1, at+2, . . .)
• 1. At any point in time t, the set A is independent of the state kt

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A
• State evolves kt+1 = F(kt, at)
• Preferences: U(kt, at, at+1, at+2, . . .)
• 1. At any point in time t, the set A is independent of the state kt
• 2. U is weakly separable in k and in {as}∞

s=0

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). and such that v is strictly increasing in its second argument.

30

General Definition

• An infinite sequence of decision makers is called to act
• State k ∈ K
• Action a ∈ A
• State evolves kt+1 = F(kt, at)
• Preferences: U(kt, at, at+1, at+2, . . .)
• 1. At any point in time t, the set A is independent of the state kt
• 2. U is weakly separable in k and in {as}∞

s=0

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). and such that v is strictly increasing in its second argument.

• 3. V is weakly separable in a0 and {as}∞

s=1

V (a0, a1, a2, ...) ≡ V (a0, V (a1, a2, . . .)), with V strictly increasing in its second argument.

30

On the Choice of Actions

• Weak separability and state independence of A depend on the

specification of the action set

• Example: hyperbolic discounting. If the choice is c, feasible actions

depend on k

• So, sometimes a problem may look nonseparable, but may become

separable by rescaling actions appropriately

31

Organizational Equilibrium

Definition A sequence of actions {at}∞

t=0 is organizationally admissible if

• 1. V (at, at+1, at+2, . . .) is (weakly) increasing in t
• 2. The first agent has no incentive to delay the proposal.

V (a0, a1, a2, . . .) ≥ max

a∈A V (a, a0, a1, a2, . . .)

Within organizationally admissible sequences, the sequence that attains the maximum of V (a0, a1, a2, . . .) is an organizational equilibrium.

32

Organizational Equilibrium vs. Subgame-Perfect Equilibrium

• 1. Org Equil is the equilibrium path of a sub-game perfect equilibrium
• 2. It can be implemented through various strategies. Examples:
• Restart from the beginning when someone deviates
• Use difference equation to make each player indifferent between

deviating and following the equilibrium strategy (over a range)

33

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

• Rationale for renegotiation/reconsideration-proofness: reject threats

that are Pareto-dominated ex post

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

• Rationale for renegotiation/reconsideration-proofness: reject threats

that are Pareto-dominated ex post

• Similar spirit for no-delay condition:

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

• Rationale for renegotiation/reconsideration-proofness: reject threats

that are Pareto-dominated ex post

• Similar spirit for no-delay condition:
• If agents coordinate on Pareto-dominant equilibrium (s∗) right away...

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

• Rationale for renegotiation/reconsideration-proofness: reject threats

that are Pareto-dominated ex post

• Similar spirit for no-delay condition:
• If agents coordinate on Pareto-dominant equilibrium (s∗) right away...
• ... then they should do the same next period (independent of past

history)...

34

Organization Equil vs. Reconsideration-Proof Equil

• Reconsideration-proof equilibria =

⇒ Value for all current and future players independent of past history

• Org Equil: same property only for action payoff:

U(k, a0, a1, a2, . . .) ≡ v(k, V (a0, a1, a2, . . .)). Future players affected by different state

• Without state variables, Org Equil is the outcome of a

Reconsideration-Proof Equil

• Rationale for renegotiation/reconsideration-proofness: reject threats

that are Pareto-dominated ex post

• Similar spirit for no-delay condition:
• If agents coordinate on Pareto-dominant equilibrium (s∗) right away...
• ... then they should do the same next period (independent of past

history)...

• =

⇒ no discipline for first player

34

Existence

• Under separability and other weak conditions, Organizational

Equilibrium exists

• (to be completed) Organizational Equilibrium admits a recursive

structure with a fixed point at+1 = q∗(at)

35

A Class of Separable Economies

• Most economies do not satisfy separability condition
• Our strategy: approximate the original economy by separable ones
• First order approximation satisfies the separable property

Ψt = u(kt, at) + δ

∞

• τ=1

βτu (kt+τ, at+τ) subject to u(kt, at) = Γ10 + Γ11h(kt) + Γ12m(at) h(kt+1) = Γ20 + Γ21h(kt) + Γ22g(at)

• h(k), m(a), g(a) can be any monotonic functions

36

Example

• Original economy

Ψt = log(ct) + δ

∞

• τ=1

βτ log(ct+τ) s.t. ct + it = kα

t

kt+1 = (1 − d)kt + it

• The approximated economy

Ψt = log(ct) + δ

∞

• τ=1

βτ log(ct+τ) s.t. ct + it = kα

t

kt+1 = k k1−d

t

id

t 37

Government Policy

A Simple Version: Govnt Expd with k Inc Taxation

• Preferences:

∞

t=0 βt[γc log ct + γg log gt] 38

A Simple Version: Govnt Expd with k Inc Taxation

• Preferences:

∞

t=0 βt[γc log ct + γg log gt]

• Technology:

f (kt) = kα

t ,

kt+1 = f (kt) − ct − gt.

38

A Simple Version: Govnt Expd with k Inc Taxation

• Preferences:

∞

t=0 βt[γc log ct + γg log gt]

• Technology:

f (kt) = kα

t ,

kt+1 = f (kt) − ct − gt.

• Consumers’ budget constraint:

ct + kt+1 = (1 − τt)rtkt + πt

38

A Simple Version: Govnt Expd with k Inc Taxation

• Preferences:

∞

t=0 βt[γc log ct + γg log gt]

• Technology:

f (kt) = kα

t ,

kt+1 = f (kt) − ct − gt.

• Consumers’ budget constraint:

ct + kt+1 = (1 − τt)rtkt + πt

• Prices:

rt = fk(kt), πt = f (kt) − rkkt

38

A Simple Version: Govnt Expd with k Inc Taxation

• Preferences:

∞

t=0 βt[γc log ct + γg log gt]

• Technology:

f (kt) = kα

t ,

kt+1 = f (kt) − ct − gt.

• Consumers’ budget constraint:

ct + kt+1 = (1 − τt)rtkt + πt

• Prices:

rt = fk(kt), πt = f (kt) − rkkt

• Government budget constraint:

gt = τtrtkt

38

Difference from Previous Setup

• In the quasi-geometric discounting, only one player per period
• Here, gov’t + private sector
• Need to deal with the competitive equilibrium component

39

Payoffs

• Given an arbitrary {τt}∞

t=0, Euler equation has to hold in equilibrium

u′(ct) = β(1 − τt+1)f ′(kt+1)u′(ct+1)

40

Payoffs

• Given an arbitrary {τt}∞

t=0, Euler equation has to hold in equilibrium

u′(ct) = β(1 − τt+1)f ′(kt+1)u′(ct+1)

• It induces a sequence of saving rates (and allocations) such that

st 1 − st − ατt = αβ(1 − τt+1) 1 − st+1 − ατt+1

40

Payoffs

• Given an arbitrary {τt}∞

t=0, Euler equation has to hold in equilibrium

u′(ct) = β(1 − τt+1)f ′(kt+1)u′(ct+1)

• It induces a sequence of saving rates (and allocations) such that

st 1 − st − ατt = αβ(1 − τt+1) 1 − st+1 − ατt+1

• Total payoff with initial capital k

U(k, τ0, τ1, τ2, . . .) = α(1 + γ) 1 − αβ log k + V (τ0, τ1, τ2, . . .)

40

Payoffs

• Given an arbitrary {τt}∞

t=0, Euler equation has to hold in equilibrium

u′(ct) = β(1 − τt+1)f ′(kt+1)u′(ct+1)

• It induces a sequence of saving rates (and allocations) such that

st 1 − st − ατt = αβ(1 − τt+1) 1 − st+1 − ατt+1

• Total payoff with initial capital k

U(k, τ0, τ1, τ2, . . .) = α(1 + γ) 1 − αβ log k + V (τ0, τ1, τ2, . . .)

• Action payoff

V ({τt}∞

t=0) = ∞

• t=0

βt

• log(1−ατt −st)+γ log ατt + αβ(1 + γ)

1 − αβ log st

• 40

Organizational Equilibrium in Government Taxation Problem

Definition A sequence of tax rates {τt}∞

t=0 is organizationally admissible if

• V (τt, τt+1, τt+2, . . .) is (weakly) increasing in t

41

Organizational Equilibrium in Government Taxation Problem

Definition A sequence of tax rates {τt}∞

t=0 is organizationally admissible if

• V (τt, τt+1, τt+2, . . .) is (weakly) increasing in t
• The implementability constraint is satisfied

41

Organizational Equilibrium in Government Taxation Problem

Definition A sequence of tax rates {τt}∞

t=0 is organizationally admissible if

• V (τt, τt+1, τt+2, . . .) is (weakly) increasing in t
• The implementability constraint is satisfied
• Government has no incentive to delay the proposal.

V (τ0, τ1, τ2, . . .) ≥ max

τ

V (τ, τ0, τ1, τ2, . . .) Within organizationally admissible sequences, any sequence that attains the maximum of V (τ0, τ1, τ2, . . .) is an organizational equilibrium.

41

Proposal Function in Organizational Equilibrium

• The equilibrium starts from τ0, and monotonically converges to τ ∗.

42

43

Comparison: Payoff in Transition

U(kt, τt, τt+1, . . .)

• total payoff

= α(1 + γ) 1 − αβ log kt + V (τt, τt+1, . . .)

• action payoff

44

Quantitative Version: depreciation and leisure

• Preferences

∞

• t=0

βt[γc log ct + γg log gt + γℓ log(1 − ℓt)]

45

Quantitative Version: depreciation and leisure

• Preferences

∞

• t=0

βt[γc log ct + γg log gt + γℓ log(1 − ℓt)]

• Consumers’ budget constraint

ct + kt+1 = kt + (1 − τ ℓ

t − τt)wtℓt + (1 − τ k t − τt)(rt − δ)kt 45

Quantitative Version: depreciation and leisure

• Preferences

∞

• t=0

βt[γc log ct + γg log gt + γℓ log(1 − ℓt)]

• Consumers’ budget constraint

ct + kt+1 = kt + (1 − τ ℓ

t − τt)wtℓt + (1 − τ k t − τt)(rt − δ)kt

• Technology:

f (kt) = kα

t ℓ1−α t

, kt+1 = (1 − δ)kt + it

45

Quantitative Version: depreciation and leisure

• Preferences

∞

• t=0

βt[γc log ct + γg log gt + γℓ log(1 − ℓt)]

• Consumers’ budget constraint

ct + kt+1 = kt + (1 − τ ℓ

t − τt)wtℓt + (1 − τ k t − τt)(rt − δ)kt

• Technology:

f (kt) = kα

t ℓ1−α t

, kt+1 = (1 − δ)kt + it

• Government budget constrain

gt = τ k

t (rt − δ)kt + τ ℓ t wtℓt + τt(wtℓt + (rt − δ)kt) 45

Labor Income Tax

Aggregate statistics Labor income tax Pareto Ramsey Markov Organization y 1.000 0.790 0.794 0.792 k/y 2.959 2.959 2.959 2.959 c/y 0.583 0.583 0.600 0.591 g/y 0.180 0.180 0.164 0.172 c/g 3.240 3.240 3.662 3.435 ℓ 0.320 0.253 0.254 0.253 τ 0.281 0.256 0.269

Parameters: α = 0.36, β = 0.96, δ = 0.08, γg = 0.09, γc = 0.27, γℓ = 0.64

46

Capital Income Tax

Aggregate statistics Capital income tax Pareto Ramsey Markov Organization y 1.000 0.685 0.570 0.660 k/y 2.959 1.972 1.360 1.824 c/y 0.583 0.722 0.697 0.716 g/y 0.180 0.120 0.195 0.138 c/g 3.240 6.018 3.580 5.188 ℓ 0.320 0.275 0.282 0.277 τ 0.594 0.774 0.645

Parameters: α = 0.36, β = 0.96, δ = 0.08, γg = 0.09, γc = 0.27, γℓ = 0.64

47

Total Income Tax

Aggregate statistics Total income tax Pareto Ramsey Markov Organization y 1.000 0.764 0.769 0.767 k/y 2.959 2.676 2.698 2.687 c/y 0.583 0.601 0.612 0.606 g/y 0.180 0.185 0.173 0.179 c/g 3.240 3.240 3.542 3.379 ℓ 0.320 0.259 0.259 0.259 τ 0.236 0.220 0.228

Parameter: α = 0.36, β = 0.96, δ = 0.08, γg = 0.09, γc = 0.27, γℓ = 0.64

48

Conclusion

• Propose organizational equilibrium for economy with state variables

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself
• Goodwill has to be built gradually

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself
• Goodwill has to be built gradually
• Outcome is close to Ramsey allocation, much better than Markov

equilibrium

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself
• Goodwill has to be built gradually
• Outcome is close to Ramsey allocation, much better than Markov

equilibrium

• Future agenda:

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself
• Goodwill has to be built gradually
• Outcome is close to Ramsey allocation, much better than Markov

equilibrium

• Future agenda:
• Idea can be used to generalize renegotiation-proofness in games with

multiple players

49

Conclusion

• Propose organizational equilibrium for economy with state variables
• Three properties
• No one is special: thank you for the idea, I will do it myself
• Goodwill has to be built gradually
• Outcome is close to Ramsey allocation, much better than Markov

equilibrium

• Future agenda:
• Idea can be used to generalize renegotiation-proofness in games with

multiple players

• Further analysis of approximation options

49