Organizational Equilibrium with Capital

Transcription

1 Organizational Equilibrium with Capital Marco Bassetto, Zhen Huo, and José-Víctor Ríos-Rull FRB of Chicago, Yale University, University of Pennsylvania, UCL, CAERP Fiscal Policy Conference Mar 20, 2018 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 1/49

2 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 2/49

3 This paper: Propose an Equilibrium Concept Agents not only decide their action, but may also make proposals for future Future agents can follow the proposal make their own proposal wait for the next agent to make a proposal Organizational equilibrium initial agent is willing to make a proposal, no delaying future agents are willing to follow Bassetto, Huo, Ríos-Rull Organizational Equilibrium 3/49

4 Equilibrium Properties I No one can treat herself specially if a proposal favors the initial, the next wants to copy the idea and restart Goodwill has to be built gradually proposal starts with low sacrifice, otherwise wait for the next to propose more sacrifice is made overtime, anticipate the next to make more sacrifice Bassetto, Huo, Ríos-Rull Organizational Equilibrium 4/49

5 Equilibrium Properties II Compare with Markov equilibrium payoff only depends on state variables, like Markov equilibrium action can depend on history, different from Markov equilibrium Compare with trigger strategy history can be abolished, no self-punishment SPE refinement I: same continuation value on or off equilibrium path SPE refinement II: 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 5/49

6 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, Ríos-Rull Organizational Equilibrium 6/49

7 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 plan Stokey (1988), Chari and Kehoe (1990), Abreu, Pearce and Stacchetti (1990), Benhabib and Rustichini (1997), 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), Ales and Sleet (2015) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 7/49

8 Plan 1 An example: a growth model with quasi-geometric discounting 2 General definition and property 3 Application in government taxation problem Bassetto, Huo, Ríos-Rull Organizational Equilibrium 8/49

9 Part I: A Growth Model Bassetto, Huo, Ríos-Rull Organizational Equilibrium 9/49

10 The Environment Preferences: quasi-geometric discounting Ψ t = u(c t) + δ β τ u (c t+τ ) τ=1 period utility function u(c) = log c δ = 1 is the time-consistent case Technology f(k t) = k α t, k t+1 = f(k t) c t. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 10/49

11 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) u c = βu [ c δf k + (1 δ) g k] The equilibrium features a constant saving rate k = δαβ 1 αβ + δαβ kα = s M k α Bassetto, Huo, Ríos-Rull Organizational Equilibrium 11/49

12 Benchmark II: Ramsey Allocation with Commitment Choose all future allocations at period 0 max k 1 u[f(k 0) k 1] + δβω(k 1) cont. value: Ω(k) = max k u[f(k) k ] + βω(k ) The sequence of saving rates is given by s M αδβ =, t = 0 1 αβ+δαβ s t = s R = αβ, t > 0 Steady state capital in Markov equilibrium is lower than Ramsey s M < s R Bassetto, Huo, Ríos-Rull Organizational Equilibrium 12/49

13 Elements of Organization Equilibrium: Proposal and Value Function A proposal is a sequence of saving rates {s 0, s 1, s 2,...} Everyone in the future can implement the same proposal Given an initial capital k 0, the proposal induces a sequence of capital k 1 = s 0k α 0 k 2 = s 1k1 α = k0 α2 s 1s α 0. k t = k0 αt Π t 1 j=0sj αt j 1 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 13/49

14 Proposal and Value Function A proposal is a sequence of saving rates {s 0, s 1, s 2,...} Given an initial capital k 0, the proposal induces a sequence of capital k t = k0 αt Π t 1 j=0sj αt j 1 The lifetime utility for the agent who makes the proposal is U(k 0, s 0, s 1,...) = log[(1 s 0 )k α 0 ] + δ α(1 αβ + δαβ) = 1 αβ φ log k 0 + V (s 0, s 1,...) β j log [ (1 s j )kj α ] j=1 log k 0 + log(1 s 0 ) + δ j=1 [ ] β j log (1 s j )Π j 1 τ=0 sαj τ τ Bassetto, Huo, Ríos-Rull Organizational Equilibrium 14/49

15 Proposal and Value Function The lifetime utility for agent at period t is U(k t, s t, s t+1,...) }{{} total payoff = φ log k t + V (s t, s t+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, Ríos-Rull Organizational Equilibrium 15/49

16 Can the Ramsey Outcome be Implemented? Imagine the initial agent with k 0 proposes {s M, s R, s R,...}, which implies k 1 = s M k α 0 By following the proposal, the next agent s payoff is ( ) ( ) U k 1, s R, s R, s R,... = φ log k 1 + V s R, s R, s R,... By copying the proposal, the next agent s payoff is ( ) ( ) U k 1, s M, s R, s R,... = φ log k 1 + V s M, s R, s R,... ( ) > φ log k 1 + V s R, s R, s R,... Copying is better than following, Ramsey outcome cannot be implemented Bassetto, Huo, Ríos-Rull Organizational Equilibrium 16/49

17 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 where V (s, s,...) H(s) = U (k t, s, s,...) =φ log k t + V (s, s,...) ( 1 + βδ ) δαβ log(1 s) + 1 β (1 αβ)(1 β) log(s) To be followed, the constant saving rate has to be s = argmax H(s) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 17/49

18 Optimal Constant Saving Rate s M < s < s R Bassetto, Huo, Ríos-Rull Organizational Equilibrium 18/49

19 Can {s, s,...} be Implemented? If the initial agent proposes {s, s,...}, no one has incentive to copy But, she prefers to choose s M, and wait the next to propose {s, s,...} ( ) ( ) U k 0, s M, s, s,... = φ log k 0 + V s M, s, s,... > φ log k 0 + 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 19/49

20 Organizational Equilibrium Definition A sequence of saving rates {s τ } τ=0 is organizationally admissible if 1 V (s t, s t+1, s t+2,...) is (weakly) increasing in t 2 The first agent has no incentive to delay the proposal. V (s 0, s 1, s 2,...) max V (s, s s 0, s 1, s 2,...) Within organizationally admissible sequences, the sequence that attains the maximum of V (s 0, s 1, s 2,...) is an organizational equilibrium. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 20/49

21 Remarks on Organizational Equilibrium OE is outcome of some SPE SPE example: if someone deviates, next agent restarts from s 0 SPE refinement criterion same continuation value on and off equilibrium path no one better off by deviating and counting on others to restart the game In equilibrium, U(k t, s t, s t+1,...) = φ log k t + V (s t, s t+1,...) = φ log k t + 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, Ríos-Rull Organizational Equilibrium 21/49

22 Construct the Organizational Equilibrium Look for a sequence of saving rates {s 0, s 1,...} Every generation obtains the same V V (s t, s t+1,...) = V (s t+1, s t+2,...) = V which induces the following difference equation β(1 δ) log(1 s t+1) = δαβ log st + log(1 st) (1 β)v 1 αβ We call this difference equation as the proposal function s t+1 = q(s t; V ) The maximal V and an initial s 0 are needed to determine {s τ } τ=0 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 22/49

23 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 with 45 degree line at s Bassetto, Huo, Ríos-Rull Organizational Equilibrium 23/49

24 Determine the Initial Saving Rate s 0 The first agent should have no incentive to delay the proposal max V (s, s s 0, s 1, s 2,...) = V (s M, s 0, s 1, s 2,...) s 0 has to be such that V = V (s 0, s 1, s 2,...) V (s M, s 0, s 1, s 2,...) s 0 q ( s M ) We select s 0 = q ( s M ), which yields the highest welfare during transition Bassetto, Huo, Ríos-Rull Organizational Equilibrium 24/49

25 Organizational Equilibrium in Quasi-Geometric Discounting Growth Model Proposition The organizational equilibrium {s τ } τ=0 is given recursively by the proposal function q (1 β)v + δαβ s t = q 1 αβ (s t 1 ) = 1 exp log s t 1 + log(1 s t 1 ) β(1 δ) where the initial saving rate s 0, the steady state s, and V are given by s 0 = q ( s M ) s = δαβ (1 β + δβ)(1 αβ) + δαβ V = 1 β + δβ 1 β log(1 s ) + αδβ log s (1 β)(1 αβ) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 25/49

26 Transition Dynamics The equilibrium starts from s 0, and monotonically converges to s. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 26/49

27 Remarks 1 To solve proposal function, no agent can treat herself specially, V t = V t+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, Ríos-Rull Organizational Equilibrium 27/49

28 Comparison: Steady State Organizational equilibrium is much better than the Markov equilibrium Bassetto, Huo, Ríos-Rull Organizational Equilibrium 28/49

29 Comparison: Allocation in Transition Organizational equilibrium: starts low, converges to being close to Ramsey Bassetto, Huo, Ríos-Rull Organizational Equilibrium 29/49

30 Comparison: Payoff in Transition U(k t, s t, s t+1,...) }{{} total payoff = φ log k t + V (s t, s t+1,...) }{{} action payoff Bassetto, Huo, Ríos-Rull Organizational Equilibrium 30/49

31 Part II: Organizational Equilibrium for Weakly Separable Economy Bassetto, Huo, Ríos-Rull Organizational Equilibrium 31/49

32 General Definition An infinite sequence of decision makers is called to act state k K action a A state evolves k t+1 = F (k t, a t) preferences: U(k t, a t, a t+1, a t+2,...) Assumption 1 At any point in time t, the set A is independent of the state k t 2 U is weakly separable in k and in {a s} s=0 U(k, a 0, a 1, a 2,...) v(k, V (a 0, a 1, a 2,...)). and such that v is strictly increasing in its second argument. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 32/49

33 Organizational Equilibrium Definition A sequence of actions {a t} t=0 is organizationally admissible if 1 V (a t, a t+1, a t+2,...) is (weakly) increasing in t 2 The first agent has no incentive to delay the proposal. V (a 0, a 1, a 2,...) max V (a, a0, a1, a2,...) a A Within organizationally admissible sequences, the sequence that attains the maximum of V (a 0, a 1, a 2,...) is an organizational equilibrium. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 33/49

34 Property of Organizational equilibrium (OE) 1 OE is the equilibrium path of a sub-game perfect equilibrium restart from the beginning when someone deviates 2 OE is a reconsideration-proof equilibrium on or off equilibrium leads to the same continuation value reconsideration-proof equilibrium may not be an OE, needs no-delay condition 3 Under separability and other weak conditions OE always exists OE admits a recursive structure with a fixed point a t+1 = q (a t) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 34/49

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 subject to Ψ t = u(k t, a t) + δ β τ u (k t+τ, a t+τ ) τ=1 u(k t, a t) = Γ 10 + Γ 11h(k t) + Γ 12m(a t) h(k t+1) = Γ 20 + Γ 21h(k t) + Γ 22g(a t) h(k), m(a), g(a) can be any monotonic functions Bassetto, Huo, Ríos-Rull Organizational Equilibrium 35/49

36 Example Original economy Ψ t = log(c t) + δ β τ log(c t+τ ) τ=1 s.t. c t + i t = k α t k t+1 = (1 d)k t + i t The approximated economy Ψ t = log(c t) + δ β τ log(c t+τ ) τ=1 s.t. c t + i t = k α t k t+1 = k k 1 d t i d t Let c t = (1 s)kt α, it = skα t, the economy is separable between k and s Bassetto, Huo, Ríos-Rull Organizational Equilibrium 36/49

37 Comparison Parameter: β = 0.96, d = 0.1, α = 0.36 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 37/49

38 Part III: Government Taxation Problem Bassetto, Huo, Ríos-Rull Organizational Equilibrium 38/49

39 A Simple Version Preference: t=0 βt [γ c log c t + γ g log g t] Technology: f(k t) = k α t, k t+1 = f(k t) c t g t. Consumers budget constraint: c t + k t+1 = (1 τ t)r tk t + π t Prices: r t = f k (k t), π t = f(k t) r k k t Government budget constraint: g t = τ tr tk t Bassetto, Huo, Ríos-Rull Organizational Equilibrium 39/49

40 Payoff Given an arbitrary {τ t} t=0, Euler equation has to hold in equilibrium u (c t) = β(1 τ t+1)f (k t+1)u (c t+1) Induce a sequence of saving rates satisfies (Implementability Constraint) s t αβ(1 τt+1) = 1 s t ατ t 1 s t+1 ατ t+1 Total payoff with initial capital k U(k, τ 0, τ 1, τ 2,...) = α(1 + γ) 1 αβ log k + V (τ0, τ1, τ2,...) Action payoff V ({τ t} t=0) = t=0 { } β t αβ(1 + γ) log(1 ατ t s t) + γ log ατ t + log st 1 αβ Bassetto, Huo, Ríos-Rull Organizational Equilibrium 40/49

41 Organizational Equilibrium in Government Taxation Problem Definition A sequence of tax rates {τ t} t=0 is organizationally admissible if V (τ t, τ t+1, τ t+2,...) is (weakly) increasing in t The implementability constraint is satisfied Government has no incentive to delay the proposal. V (τ 0, τ 1, τ 2,...) max V (τ, τ τ 0, τ 1, τ 2,...) Within organizationally admissible sequences, any sequence that attains the maximum of Φ(τ 0, τ 1, τ 2,...) is an organizational equilibrium. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 41/49

42 Proposal Function in Organizational Equilibrium The equilibrium starts from τ 0, and monotonically converges to τ. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 42/49

43 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 43/49

44 Comparison: Payoff in Transition α(1 + γ) U(k t, τ t, τ t+1,...) = }{{} 1 αβ total payoff log kt + V (τt, τt+1,...) }{{} action payoff Bassetto, Huo, Ríos-Rull Organizational Equilibrium 44/49

45 A Quantitative Version Preference β t [γ c log c t + γ g log g t + γ l log(1 l t)] t=0 Consumers budget constraint c t + k t+1 = k t + (1 τ l t τ t)w tl t + (1 τ k t τ t)(r t δ)k t Technology f(k t) = k α t l 1 α t, k t+1 = (1 δ)k t + i t Government budget constrain g t = τ k t (r t δ)k t + τ l t w tl t + τ t(w tl t + (r t δ)k t) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 45/49

46 Labor Income Tax Aggregate statistics Labor income tax Pareto Ramsey Markov Organization y k/y c/y g/y c/g l τ Parameter: α = 0.36, β = 0.96, δ = 0.08, γ g = 0.09, γ c = 0.27, γ l = 0.64 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 46/49

47 Capital Income Tax Aggregate statistics Capital income tax Pareto Ramsey Markov Organization y k/y c/y g/y c/g l τ Parameter: α = 0.36, β = 0.96, δ = 0.08, γ g = 0.09, γ c = 0.27, γ l = 0.64 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 47/49

48 Total Income Tax Aggregate statistics Total income tax Pareto Ramsey Markov Organization y k/y c/y g/y c/g l τ Parameter: α = 0.36, β = 0.96, δ = 0.08, γ g = 0.09, γ c = 0.27, γ l = 0.64 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 48/49

49 Conclusion Propose organizational equilibrium for economy with state variables The equilibrium concept applies to government policy problem 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 49/49