Organizational Equilibrium with Capital

Transcription

1 Organizational Equilibrium with Capital Marco Bassetto, Zhen Huo, and José-Víctor Ríos-Rull FRB of Chicago, Yale University, Penn, UCL, CAERP Bristol Macroeconomic Workshop, May 31, / 57

2 Question Time inconsistency is a pervasive issue Benchmark outcome: Markov perfect equilibrium Can agents do better than Markov equilibrium? Yes, with trigger strategies We show agents do better than Markov equilibrium without trigger strategies 2 / 57

3 This paper Propose an equilibrium concept for economies with state variables where Initial agents can make proposals for the future Future agents can follow the proposal, or 1. copy the proposal 2. wait for the next agent to make a proposal 3. make their own proposal Equilibrium: a proposal that all future agents are willing to follow Organizational Equilibrium (OE) (Prescott and Rios-Rull, 2000) is a special case 3 / 57

4 OLG Model: U(c y, c o ) = log(c y ) + log(c o ), ω y = 3, ω o = 1 Consumption when old: c o U(c y,c o ) = U(3,1) U(c y,c o ) = U(2,2) U(c y,c o ) = U(3,2) c y +c o = ω y +ω o 2 3 Consumption when young: c y 4 / 57

5 This paper Propose organizational equilibrium for economies with state variables Apply organizational equilibrium to quasi-geometric discounting model large welfare improvement compared with Markov equilibria Extend the organizational equilibrium to government policy problems 5 / 57

6 Equilibrium Properties I No one can treat herself specially if a proposal favors the initial, the next wants to copy the idea and restart Thank you for the idea, I will do it myself Goodwill has to be built gradually deviation from Markov is like making a sacrifice proposal starts with low sacrifice, otherwise wait for the next to propose more sacrifice is made overtime, anticipate the next to make more sacrifice The long-run outcome is close to the case with commitment 6 / 57

7 Equilibrium Properties II Comparison with Markov Equilibrium difference: action also depends on history in organizational equilibrium similarity: utility level only depends on pay-off relevant state variables History can be abolished, no trigger strategy in equilibrium 7 / 57

8 Related Literature Markov equilibrium and GEE Currie and Levine (1993), Bassetto and Sargent (2005), Klein, Krusell and Rios-Rull (2008), 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) 8 / 57

9 Brendon and Ellison (2016) Analyze optimal policy in the Ramsey tradition, but they restrict the planner to choose policies that satisfy a recursive Pareto criterion: this criterion disallows sequences that benefit policymakers in the early periods but are dominated for all policymakers from a given time onward. Like them, we also reject policies that allow early decision makers to dictate future paths that lead to early benefits but are undesirable ex post into the. Rather than developing an optimality criterion, we propose a solution concept aimed at positive analysis, where implicit cooperation across policymakers at different times builds over time. Because of this different motivation, our "no-restarting condition" is imposed on a period-by-period basis. Our papers also differ in the treatment of physical state variables, where we rely on weak separability to define "state-free" notions of government actions. 9 / 57

10 Plan 1. An example: a growth model with quasi-geometric discounting 2. General organizational equilibrium for separable economies 3. Extension to a government taxation problem 10 / 57

11 1 Growth model with Quasi-geometric Discounting 11 / 57

12 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. 12 / 57

13 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(f (k) g(k)) = βu c (f [g(k)] g[g(k)] δf k [g(k)] + (1 δ) g k [g(k)] u c = βu c [ δf k + (1 δ) g k ] The equilibrium features a constant saving rate k = δαβ 1 αβ + δαβ k α = s M k α 13 / 57

14 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 (k ) in Markov equilibrium is lower than Ramsey k M = (s M) 1 α 1 < k R = (s R) 1 α 1 14 / 57

15 Elements of Organization Equilibrium: Proposal and Value Function Proposals are Rescalable Actions, this is actions that have a domain that is independent of the state (k) and describe the whole set of possible actions. Here the natural set of Rescalable Actions is savings rates. Hence 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 1 = s 0k α 0 k 2 = s 1k α 1. = k α2 0 s 1s α 0 k t = k0 αt Π t 1 j=0 sj αt j 1 15 / 57

16 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=0 sj 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 )k0 α ] + δ β j log[(1 s j )kj α ] α(1 αβ + δαβ) = 1 αβ φ log k 0 + V (s 0, s 1,...) 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 ] Note the separability of lifetime utility into a term with the state and a term with the proposal. 16 / 57

17 Proposal and Value Function Given an initial capital k 0, the proposal induces a sequence of capital k t = k0 αt Π t 1 j=0 sj αt j 1 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 Crucially, there is a Separability property between capital and the rescaled actions for the lifetime utility of the proposer and all subsequent followers. 17 / 57

18 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 18 / 57

19 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 (k t, s, s,...) =φ log k t + V (s, s,...) = φ log k t + H(s) where H(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) 19 / 57

20 Optimal Constant Saving Rate s M < s R 20 / 57 < s

21 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 1 + V s M, s, s,... > φ log k 1 + 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 21 / 57

22 Organizational Equil in Quasi-Geometric Discounting Model Definition A sequence of saving rates {s τ } τ=0 is organizationally admissible if V (s t, s t+1, s t+2,...) is (weakly) increasing in t 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, any sequence that attains the maximum of V (s 0, s 1, s 2,...) is an organizational equilibrium. This definition guarantees that 1. no incentive to copy previous agent s proposal 2. no incentive to wait for the next agent to propose 3. no incentive to start a new proposal 22 / 57

23 Is the Rescaled Action Unique? Consider other rescaled action r defined as s = ψ(r, k) U(k 0, r 0, r 1,...) = φ log k 0 + V (ψ(r 0, k 0), ψ(r 1, k 1),...) Is the utility separable between state k and action r? Proposition The class of rescaled actions that admits separability between capital and actions are monotonic transformation of saving rates, i.e., s = ψ(r) Furthermore, the allocation in an equilibrium with action r is identical to that in an equilibrium with action s. 23 / 57

24 Remarks on Organizational Equilibrium Separability between capital and saving rates allows a global maximum Payoff only depends on capital, not a trigger strategy U(k t, s t, s t+1,...) = φ log k t + V (s t, s t+1,...) = φ log k t + V Agents need to know previous agents action, not a Markov equilibrium 24 / 57

25 Construct the Organizational Equilibrium An organizational equilibrium is a sequence of saving rates {s 0, s 1,...} To prevent copying the idea, 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 25 / 57

26 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 26 / 57

27 Determine the Initial Saving Rate s 0 The first agent should have no incentive to delay the proposal V (s 0, s 1, s 2,...) 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 27 / 57

28 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) 28 / 57

29 Transition Dynamics The equilibrium starts from s 0, and monotonically converges to s. 29 / 57

30 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 30 / 57

31 Comparison: Steady State Recall that the payoff with constant saving rate U(k, s, s,...) = φ log k + H(s) where H(s) = 1 β + βδ 1 β log(1 s) + δαβ (1 αβ)(1 β) log(s) When δ = 1 R(s) H(s) = 1 1 β log(1 s) + αβ (1 αβ)(1 β) log(s) 31 / 57

32 Comparison: Steady State s maximizes H(s): always with additional discounting s R maximizes R(s): no additional discounting in the long run 32 / 57

33 Comparison: Steady State Organizational equilibrium is much better than the Markov equilibrium 33 / 57

34 Comparison: Allocation in Transition Organizational equilibrium: starts low, converges to being close to Ramsey 34 / 57

35 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 35 / 57

36 2 Organizational Equilibrium for Weakly Separable Economies 36 / 57

37 General Setup An infinite sequence of decision makers is called to act State variable k K, action x X (k) Preferences: Ψ(k t, x t, x t+1, x t+2,...) We are looking for a particular class of economies that are weakly separable can be separated into a term that depends on the state and a term that depends in the sequence of rescaled actions. 37 / 57

38 General Setup: Our class of Economies Assumption There exists a set A and a function γ : A K A with the following properties 1. The action can be recalled: X (k) = {a : a A such that x = γ(a, k)} 2. Define preferences over rescaled actions as follows: U(k, a t, a t+1, a t+2,...) := Ψ(k, x t, x t+1, x t+2,...), where for t 0, x t is computed recursively as x t(k) = γ(a t, k t(k)), k t+1 (k) = F (k t(k), x t). and U is weakly separable in k and in {a τ } τ=t, i.e., there exist v and V U(k, a t, a t+1, a t+2,...) = v(k, V (a t, a t+1, a t+2,...)). 38 / 57

39 Organizational Equilibrium Definition A sequence of actions {a τ } τ=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, any sequence that attains the maximum of V (a 0, a 1, a 2,...) is an organizational equilibrium. 39 / 57

40 Some Examples of Separable Economies Consumption-saving problem Ψ t = c1 σ t 1 σ + δ τ=1 β τ ct+τ 1 σ 1 σ, s.t. c + a = Ra + y. Growth model with elastic labor supply and slow adjustment of capital Ψ t = log c t + (1 lt)1 γ 1 γ + δ τ=1 β τ ( log c t+τ + ) (1 lt+τ )1 γ 1 γ subject to c t + i t = k α t l 1 α t k t+1 = k 1 µ t i µ t OLG model with arbitrary endowment process T Ψ t = ω τ log(c t+τ ) τ=0 40 / 57

41 But most Economies are Non-separable In general, economies with physical state variables are not separable In the sense that rescaled actions are not uniquely defined. Approximate a non-separable economy by a separable economy Choose the separable economy and associated rescaled action to minimize approximation error along the equilibrium 41 / 57

42 A Particularly Useful Class of Separable Economies The following type of economies are separable economies 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) Where h(k), m(a), g(a) are any monotonic functions, say logs. 42 / 57

43 A Strategy to Approximate A Non-separable Economies Set monotonic functions h, g, m (e.g., log functions) and rescaled actions a (e.g., saving rates), then the original economy is Ψ t = u(k t, a t) + δ β τ u (k t+τ, r t+τ ) τ=1 where k t+1 = F (k t, r t) All other rescaled actions given by r = φ(a, k), cab be (first order) approximated by r a + φ k (k k) Its steady state is characterized by φ k (1 β(1 δ))u 2 (1 βf k ) + δβu 1 F 2 + (1 β)(1 δ)βu 2 F 2 φ 1 k = 0 1. For any φ k choose Γ in the separable economy to match the steady-state level and first derivative 2. Given Γ and φ k, compute the separable economy with action r 3. Compute approximation error: t=0 ωt u(kt, at) û(kt, φ 1 (r t, k t)) 4. Choose φ k to minimize approximation error 43 / 57

44 Example Original economy where Ψ t = log(c t) + δ β τ log(c t+τ ) τ=1 c t + k t+1 = kt α + (1 d)k t The separable economy Ψ t = û(k t, a t) + δ β τ û(k t, a t) τ=1 where û(k t, a t) = Γ 10 + Γ 11 log(k t) + Γ 12 log(1 a t) log(k t+1 ) = Γ 20 + Γ 21 log(k t) + Γ 22 log(a t) Notice that in principle we could have also approximated by functions other than logs 44 / 57

45 Approximated Equilibrium Capital in origianl economy Capital in approximated economy Error in percentage: log(c t) û(k t,a t) log(c t) Time t Time t Capital starts from 10% lower than its steady state value 45 / 57

46 3 Government Policy 46 / 57

47 Part III: Government Policy Problem 47 / 57

48 An Example of a Government Policy Problem A strategic government v.s. a continuum of competitive agents Government finances public goods via capital income taxes with a period by period balanced budget constraint, but cannot commit to a sequence of taxes at time 0 Apply organizational equilibrium to this environment Notice that Tax Rates are its Natural Rescaling Actions. Significant but graduate welfare improvement over Markov equilibrium 48 / 57

49 Environment Preferences: t=0 βt [log c t + γ 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 49 / 57

50 Payoff Given an arbitrary {τ t} t=0, the Euler equation has to hold in equilibrium u (c t) = β(1 τ t+1)f (k t+1)u (c t+1) Which induces 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 α(1 + γ) log(1 ατ t s t) + γ log ατ t + log st 1 αβ 50 / 57

51 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. 51 / 57

52 Comparison: Steady State Total payoff in steady state U(k, τ, τ,...) = α(1 + γ) 1 αβ log k + V (τ, τ,...) = α(1 + γ) 1 αβ log k + H(τ) Action payoff in steady state H(τ) = log(1 ατ s) + γ log ατ + with s = αβ(1 τ) α(1 + γ) 1 αβ log s Tax rates in steady state τ : organizational equilibrium τ R : Ramsey problem τ M : Markov equilibrium 52 / 57

53 Comparison: Steady State Organizational equilibrium maximizes the steady state action payoff 53 / 57

54 Proposal Function in Organizational Equilibrium The equilibrium starts from τ 0, and monotonically converges to τ. 54 / 57

55 55 / 57

56 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 56 / 57

57 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 The equilibrium concept can also be applied to government policy problem 57 / 57