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Introduction Growth model Organizational Equilibrium Government Policy
Organizational Equilibrium with Capital
Marco Bassetto, Zhen Huo, and Jose-Vıctor Rıos-Rull
FRB of Chicago, Yale University, University of Pennsylvania, UCL, CAERP
Banque de France
April 15, 2019
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 1/52
Introduction Growth model Organizational Equilibrium Government Policy
Question
Time inconsistency is a pervasive issue taxation, government debt, consumption-saving problem, monetary policy, . . .
Two Benchmarks: Markov equilibrium Sequential equilibrium/sustainable plan
Markov equilibrium: Interesting comparative statics Outcome determined by fundamentals ... but can be largely improved upon
Sequential equilibrium: Can often attain very good outcomes (folk theorem) Can also attain very bad outcomes (folk theorem again) Relies on self-punishment as a threat Weak predictions (big set of equilibria)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 2/52
Introduction Growth model Organizational Equilibrium Government Policy
Question
Time inconsistency is a pervasive issue taxation, government debt, consumption-saving problem, monetary policy, . . .
Two Benchmarks: Markov equilibrium Sequential equilibrium/sustainable plan
Markov equilibrium: Interesting comparative statics Outcome determined by fundamentals ... but can be largely improved upon
Sequential equilibrium: Can often attain very good outcomes (folk theorem) Can also attain very bad outcomes (folk theorem again) Relies on self-punishment as a threat Weak predictions (big set of equilibria)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 2/52
Introduction Growth model Organizational Equilibrium Government Policy
Question
Time inconsistency is a pervasive issue taxation, government debt, consumption-saving problem, monetary policy, . . .
Two Benchmarks: Markov equilibrium Sequential equilibrium/sustainable plan
Markov equilibrium: Interesting comparative statics Outcome determined by fundamentals ... but can be largely improved upon
Sequential equilibrium: Can often attain very good outcomes (folk theorem) Can also attain very bad outcomes (folk theorem again) Relies on self-punishment as a threat Weak predictions (big set of equilibria)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 2/52
Introduction Growth model Organizational Equilibrium Government Policy
Our View
Good institutions and social norms do not evolve overnight
Collaboration across cohorts of decision makers builds slowly
It probably also erodes slowly
Look for equilibrium concept that captures this, and addresses shortcomingsof Markov & Best Sequential Eq.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 3/52
Introduction Growth model Organizational Equilibrium Government Policy
Organizational Equilibrium
Reconsideration-Proof Equilibrium (Kocherlakota, 1996); OrganizationalEquilibrium (Prescott-Rıos-Rull, 2005)
Based on renegotiation-proofness (Farrell and Maskin, 1989)
Punisher does not suffer from punishing past misdeeds
Retains meaningful comparative statics
Improves on Markov Equilibrium
... but does not deal with state variables (only repeated games, no dynamicgames)
This is where we come in
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 4/52
Introduction Growth model Organizational Equilibrium Government Policy
Organizational Equilibrium
Reconsideration-Proof Equilibrium (Kocherlakota, 1996); OrganizationalEquilibrium (Prescott-Rıos-Rull, 2005)
Based on renegotiation-proofness (Farrell and Maskin, 1989)
Punisher does not suffer from punishing past misdeeds
Retains meaningful comparative statics
Improves on Markov Equilibrium
... but does not deal with state variables (only repeated games, no dynamicgames)
This is where we come in
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 4/52
Introduction Growth model Organizational Equilibrium Government Policy
Equilibrium Properties
Compare with Markov equilibrium
payoff only depends on state variables, like Markov equilibrium
action can depend on history, different from Markov equilibrium
Compare with sequential equilibrium
no self-punishment
Refinement I: same continuation value on or off equilibrium path
Refinement II: no one wants to deviate and wait for a restart of the game
New issues with state variables
how to induce stationary environment
Player preferences no longer purely forward-looking (new role for no-delayingcondition)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 5/52
Introduction Growth model Organizational Equilibrium Government Policy
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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 6/52
Introduction Growth model Organizational Equilibrium Government Policy
Related Literature
Markov equilibrium and GEE
Currie and Levine (1993), Bassetto and Sargent (2005), Klein and Rıos-Rull (2003), Klein, Quadrini and
Rıos-Rull (2005), Krusell and Rıos-Rull (2008), Krusell, Kuruscu, and Smith (2010), Song, Storesletten
and Zilibotti (2012)
Sustainable plan
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 (1996), Prescott and Rıos-Rull (2005), Nozawa (2014), Ales and
Sleet (2015)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 7/52
Introduction Growth model Organizational Equilibrium Government Policy
Plan
1 An example: a growth model with quasi-geometric discounting
2 General definition and property
3 Application in government taxation problem
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 8/52
Introduction Growth model Organizational Equilibrium Government Policy
Part I: A Growth Model
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 9/52
Introduction Growth model Organizational Equilibrium Government Policy
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) = kαt , kt+1 = f(kt)− ct.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 10/52
Introduction Growth model Organizational Equilibrium Government Policy
Benchmark I: Markov Perfect Equilibrium
Take future g(k) as given
maxk′
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 + δαβkα = sM kα
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 11/52
Introduction Growth model Organizational Equilibrium Government Policy
Benchmark II: Ramsey Allocation with Commitment
Choose all future allocations at period 0
maxk1
u[f(k0)− k1] + δβΩ(k1)
cont. value: Ω(k) = maxk′
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
sM < sR
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 12/52
Introduction Growth model Organizational Equilibrium Government Policy
Elements of Organization Equilibrium: Proposal
A proposal is a sequence of saving rates s0, s1, s2, . . .
Given an initial capital k0, the proposal induces a sequence of capital
k1 = s0kα0
k2 = s1kα1 = kα
2
0 s1sα0
...
kt = kαt
0 Πt−1j=0s
αt−j−1
j
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 13/52
Introduction Growth model Organizational Equilibrium Government Policy
Proposal and Value Function
A proposal is a sequence of saving rates s0, s1, s2, . . .
The lifetime utility for the agent who makes the proposal is
U(k0, s0, s1, . . .)
= log[(1− s0)kα0 ] + δ∞∑j=1
βj log[(1− sj)kαj
]=α(1− αβ + δαβ)
1− αβlog k0 + log(1− s0) + δ
∞∑j=1
βj log[(1− sj)Πj−1
τ=0sαj−ττ
]≡φ log k0 + V (s0, s1, . . .)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 14/52
Introduction Growth model Organizational Equilibrium Government Policy
Separability
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?
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 15/52
Introduction Growth model Organizational Equilibrium Government Policy
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, . . .) ≥ maxsV (s, s0, s1, s2, . . .)
Within organizationally admissible sequences, a sequence that attains themaximum of V (s0, s1, s2, . . .) is an organizational equilibrium.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 16/52
Introduction Growth model Organizational Equilibrium Government Policy
Remarks on Organizational Equilibrium
OE is outcome of some SPE
SPE example: if someone deviates, next agent restarts from s0
SPE refinement criterion
same continuation value on and off equilibrium path (for V component)
no one 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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 17/52
Introduction Growth model Organizational Equilibrium Government Policy
Can the Ramsey Outcome be Implemented?
Imagine the initial agent with k0 proposes sM , sR, sR, . . ., which implies
k1 = sMkα0
By following the proposal, the next agent’s payoff is
U(k1, s
R, sR, sR, . . .)
= φ log k1 + V(sR, sR, sR, . . .
)By copying the proposal, the next agent’s payoff is
U(k1, s
M , 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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 18/52
Introduction Growth model Organizational Equilibrium Government Policy
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∗ = argmaxH(s)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 19/52
Introduction Growth model Organizational Equilibrium Government Policy
Optimal Constant Saving Rate
sM < s∗ < sR
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 20/52
Introduction Growth model Organizational Equilibrium Government Policy
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, s
M , s∗, s∗, . . .)
= φ log k0 + V(sM , s∗, s∗, . . .
)> φ log k0 + V (s∗, s∗, s∗, . . .)
Constant s∗ proposal cannot be implemented, incentive to delay
But, something else can be implemented, which converges to s∗
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 21/52
Introduction Growth model Organizational Equilibrium Government Policy
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 as the proposal function
st+1 = q(st;V )
The maximal V and an initial s0 are needed to determine sτ∞τ=0
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 22/52
Introduction Growth model Organizational Equilibrium Government Policy
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 (ats∗)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 23/52
Introduction Growth model Organizational Equilibrium Government Policy
Determine the Initial Saving Rate s0
The first agent should have no incentive to delay the proposal
maxsV (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 for period t+ 1
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 24/52
Introduction Growth model Organizational Equilibrium Government Policy
Organizational Equilibrium in Quasi-Geometric Discounting Growth Model
Proposition
The organizational equilibrium sτ∞τ=0 is given recursively by the proposal function q∗
st = q∗(st−1) = 1− exp
−(1− β)V ∗ + δαβ1−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− βlog(1− s∗) +
αδβ
(1− β)(1− αβ)log s∗
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 25/52
Introduction Growth model Organizational Equilibrium Government Policy
Transition Dynamics
The equilibrium starts from s0, and monotonically converges to s∗.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 26/52
Introduction Growth model Organizational Equilibrium Government Policy
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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 27/52
Introduction Growth model Organizational Equilibrium Government Policy
Comparison: Steady State
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
Organizational equilibrium is much better than the Markov equilibrium
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 28/52
Introduction Growth model Organizational Equilibrium Government Policy
Comparison: Allocation in Transition
Organizational equilibrium: starts low, converges to being close to Ramsey
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 29/52
Introduction Growth model Organizational Equilibrium Government Policy
Comparison: Payoff in Transition
U(kt, st, st+1, . . .)︸ ︷︷ ︸total payoff
= φ log kt + V (st, st+1, . . .)︸ ︷︷ ︸action payoff
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 30/52
Introduction Growth model Organizational Equilibrium Government Policy
Part II: Organizational Equilibrium for
Weakly Separable Economies
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 31/52
Introduction Growth model Organizational Equilibrium Government Policy
General DefinitionAn 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, . . .)
Assumption
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.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 32/52
Introduction Growth model Organizational Equilibrium Government Policy
On the Choice of Actions
Weak separability and state independence of A depend on the specificationof the action set
Example: hyperbolic discounting. If the choice is c, feasible actions dependon k
So, sometimes a problem may look nonseparable, but may become separableby rescaling actions appropriately
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 33/52
Introduction Growth model Organizational Equilibrium Government Policy
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, . . .) ≥ maxa∈A
V (a, a0, a1, a2, . . .)
Within organizationally admissible sequences, the sequence that attains themaximum of V (a0, a1, a2, . . .) is an organizational equilibrium.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 34/52
Introduction Growth model Organizational Equilibrium Government Policy
Organizational Equilibrium (OE) vs. Subgame-Perfect Equilibrium
1 OE 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 andfollowing the equilibrium strategy (over a range)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 35/52
Introduction Growth model Organizational Equilibrium Government Policy
OE vs. Reconsideration-Proof Equilibrium
Reconsideration-proof equilibria =⇒ Value for all current and future playersindependent of past history
OE: 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, OE is the outcome of a reconsideration-proofequilibrium
Rationale for renegotiation/reconsideration-proofness: reject threats that arePareto-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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 36/52
Introduction Growth model Organizational Equilibrium Government Policy
OE vs. Reconsideration-Proof Equilibrium
Reconsideration-proof equilibria =⇒ Value for all current and future playersindependent of past history
OE: 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, OE is the outcome of a reconsideration-proofequilibrium
Rationale for renegotiation/reconsideration-proofness: reject threats that arePareto-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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 36/52
Introduction Growth model Organizational Equilibrium Government Policy
Existence and Properties
Under separability and other weak conditions, OE exists
Assume that continuation utility is recursive:
V (a1, a2, ...) = W (a1, V (a2, a3, ....))
Then: OE admits a recursive structure
at+1 = q∗(at)
Equilibrium converges to a steady state
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 37/52
Introduction Growth model Organizational Equilibrium Government Policy
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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 38/52
Introduction Growth model Organizational Equilibrium Government Policy
Example
Original economy
Ψt = log(ct) + δ
∞∑τ=1
βτ log(ct+τ )
s.t. ct + it = kαt
kt+1 = (1− d)kt + it
The approximated economy
Ψt = log(ct) + δ
∞∑τ=1
βτ log(ct+τ )
s.t. ct + it = kαt
kt+1 = k k1−dt idt
Let ct = (1− s)kαt , it = skαt , the economy is separable between k and s
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 39/52
Introduction Growth model Organizational Equilibrium Government Policy
Part III: Government Taxation Problem
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 40/52
Introduction Growth model Organizational Equilibrium Government Policy
A Simple Version
Preference:∑∞t=0 β
t[γc log ct + γg log gt]
Technology: f(kt) = kαt , 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
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 41/52
Introduction Growth model Organizational Equilibrium Government Policy
Difference from Previous Setup
In the quasi-geometric discounting, only one player per period
Here, gov’t + private sector
Need to short-circuit competitive equilibrium component
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 42/52
Introduction Growth model Organizational Equilibrium Government Policy
Payoff
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)
Induce a sequence of saving rates such that
st1− st − ατt
=αβ(1− τt+1)
1− st+1 − ατt+1
From saving rate, get allocation
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 logατt +αβ(1 + γ)
1− αβ log st
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 43/52
Introduction Growth model Organizational Equilibrium Government Policy
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, . . .) ≥ maxτ
V (τ, τ0, τ1, τ2, . . .)
Within organizationally admissible sequences, any sequence that attains themaximum of V (τ0, τ1, τ2, . . .) is an organizational equilibrium.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 44/52
Introduction Growth model Organizational Equilibrium Government Policy
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, . . .) ≥ maxτ
V (τ, τ0, τ1, τ2, . . .)
Within organizationally admissible sequences, any sequence that attains themaximum of V (τ0, τ1, τ2, . . .) is an organizational equilibrium.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 44/52
Introduction Growth model Organizational Equilibrium Government Policy
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, . . .) ≥ maxτ
V (τ, τ0, τ1, τ2, . . .)
Within organizationally admissible sequences, any sequence that attains themaximum of V (τ0, τ1, τ2, . . .) is an organizational equilibrium.
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 44/52
Introduction Growth model Organizational Equilibrium Government Policy
Proposal Function in Organizational Equilibrium
The equilibrium starts from τ0, and monotonically converges to τ∗.
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Introduction Growth model Organizational Equilibrium Government Policy
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 46/52
Introduction Growth model Organizational Equilibrium Government Policy
Comparison: Payoff in Transition
U(kt, τt, τt+1, . . .)︸ ︷︷ ︸total payoff
=α(1 + γ)
1− αβ log kt + V (τt, τt+1, . . .)︸ ︷︷ ︸action payoff
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 47/52
Introduction Growth model Organizational Equilibrium Government Policy
A Quantitative Version
Preference∞∑t=0
βt[γc log ct + γg log gt + γ` log(1− `t)]
Consumers’ budget constraint
ct + kt+1 = kt + (1− τ `t − τt)wt`t + (1− τkt − τt)(rt − δ)kt
Technologyf(kt) = kαt `
1−αt , kt+1 = (1− δ)kt + it
Government budget constraint
gt = τkt (rt − δ)kt + τ `twt`t + τt(wt`t + (rt − δ)kt)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 48/52
Introduction Growth model Organizational Equilibrium Government Policy
A Quantitative Version
Preference∞∑t=0
βt[γc log ct + γg log gt + γ` log(1− `t)]
Consumers’ budget constraint
ct + kt+1 = kt + (1− τ `t − τt)wt`t + (1− τkt − τt)(rt − δ)kt
Technologyf(kt) = kαt `
1−αt , kt+1 = (1− δ)kt + it
Government budget constraint
gt = τkt (rt − δ)kt + τ `twt`t + τt(wt`t + (rt − δ)kt)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 48/52
Introduction Growth model Organizational Equilibrium Government Policy
A Quantitative Version
Preference∞∑t=0
βt[γc log ct + γg log gt + γ` log(1− `t)]
Consumers’ budget constraint
ct + kt+1 = kt + (1− τ `t − τt)wt`t + (1− τkt − τt)(rt − δ)kt
Technologyf(kt) = kαt `
1−αt , kt+1 = (1− δ)kt + it
Government budget constraint
gt = τkt (rt − δ)kt + τ `twt`t + τt(wt`t + (rt − δ)kt)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 48/52
Introduction Growth model Organizational Equilibrium Government Policy
A Quantitative Version
Preference∞∑t=0
βt[γc log ct + γg log gt + γ` log(1− `t)]
Consumers’ budget constraint
ct + kt+1 = kt + (1− τ `t − τt)wt`t + (1− τkt − τt)(rt − δ)kt
Technologyf(kt) = kαt `
1−αt , kt+1 = (1− δ)kt + it
Government budget constraint
gt = τkt (rt − δ)kt + τ `twt`t + τt(wt`t + (rt − δ)kt)
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 48/52
Introduction Growth model Organizational Equilibrium Government Policy
Labor Income Tax
Aggregatestatistics
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
` 0.320 0.253 0.254 0.253
τ 0.281 0.256 0.269
Parameter: α = 0.36, β = 0.96, δ = 0.08, γg = 0.09, γc = 0.27, γ` = 0.64
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 49/52
Introduction Growth model Organizational Equilibrium Government Policy
Capital Income Tax
Aggregatestatistics
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
` 0.320 0.275 0.282 0.277
τ 0.594 0.774 0.645
Parameter: α = 0.36, β = 0.96, δ = 0.08, γg = 0.09, γc = 0.27, γ` = 0.64
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 50/52
Introduction Growth model Organizational Equilibrium Government Policy
Total Income Tax
Aggregatestatistics
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
` 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, γ` = 0.64
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 51/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52
Introduction Growth model Organizational Equilibrium Government Policy
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 multipleplayers Further analysis of approximation options
Bassetto, Huo, Rıos-Rull Organizational Equilibrium 52/52