Organizational Equilibrium with Capital

Transcription

1 Organizational Equilibrium with Capital Marco Bassetto, Zhen Huo, and José-Víctor Ríos-Rull FRB of Chicago, New York University, University of Pennsylvania, UCL, CAERP HKUST Summer Workshop in Macroeconomics June 22,2016 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 1/47

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 strategy Bassetto, Huo, Ríos-Rull Organizational Equilibrium 2/47

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 the same 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 3/47

4 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 problem Bassetto, Huo, Ríos-Rull Organizational Equilibrium 4/47

5 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 well defined notion of thank you for the idea : proposals independent of capital Goodwill has to be built gradually deviation from Markov is like making a sacrifice proposal starts with low sacrifice, otherwise waiting for the next to propose more sacrifice is made overtime, knowing the next will make more sacrifice The long-run outcome is close to the case with commitment Bassetto, Huo, Ríos-Rull Organizational Equilibrium 5/47

6 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 6/47

7 Related Literature Markov equilibrium and GEE Currie and Levine (1993), Bassetto and Sargent (2005), Klein, Krusell and Rios-Rull (2008) 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) Refinement of subgame perfect equilibrium Farrell and Maskin (1989), Kocherlakota (2008), Ales and Sleet (2015) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 7/47

8 Plan 1 An example: a growth model with quasi-geometric discounting. 2 General Organizational equilibrium for separable economies 3 Approximating any economy with a separable economy 4 A government taxation problem Bassetto, Huo, Ríos-Rull Organizational Equilibrium 8/47

9 Part I: A Growth Model Bassetto, Huo, Ríos-Rull Organizational Equilibrium 9/47

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 10/47

11 Benchmark I: Markov Perfect Equilibrium Consider the differentiable Markov equilibrium with GEE (Krusell and Smith, 2003) Recursive formulation: V (k; g) = max k u[f(k) k ] + δβω(k ; g) Continuation value: Ω(k; g) = u[f(k) g(k)] + βω[g(k); g] The Generalized Euler Equation (GEE) )[ ] u (f(k) g(k)) = βu (f[g(k)] g[g(k)] δf [g(k)] + (1 δ) g [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 11/47

12 Benchmark II: Ramsey Allocation with Commitment First period: max k1 u[f(k 0) k 1] + δβω(k 1) Continuation 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 ( k M = αδβ 1 αβ + δαβ ) 1 1 α < k R = (αβ) 1 1 α Bassetto, Huo, Ríos-Rull Organizational Equilibrium 12/47

13 Optimal Constant Saving Rate with Quasi-Geometric Utility Value function with constant saving rate s V (k; s) = u[(1 s)f(k)] + δβ Ω[sf(k); s] Continuation value Ω(k; s) = u[(1 s)f(k)] + β Ω(k; s) By guess-and-verify, the value function is V (k; s) = α(1 αβ + δαβ) 1 αβ ( log k+ 1 + βδ ) log(1 s)+ 1 β δαβ (1 αβ)(1 β) log(s) Which constant saving rate is the optimal one? Bassetto, Huo, Ríos-Rull Organizational Equilibrium 13/47

14 How does V (k; s) look like? V(k;s) s M = s M s s R αδβ 1 αβ + δαβ < δαβ s = (1 β + δβ)(1 αβ) + δαβ < sr = αβ Bassetto, Huo, Ríos-Rull Organizational Equilibrium 14/47

15 Towards the Organizational Equilibrium: Can s be achieved? Imagine the proposal is to choose s all the time 1 the next agent has no incentive to copy the proposal (it is a constant) 2 all agents have incentive to choose s M, and wait the next to propose 3 no other proposals yield higher utility for all Constant s proposal cannot be implemented, violating no waiting condition But, something else can be implemented, which converges to s For this, we need to proceed to define the organizational equilibrium Bassetto, Huo, Ríos-Rull Organizational Equilibrium 15/47

16 Proposals 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 1 = s 0 k α 0 k 2 = s 1 k α 1 = kα2 0 s 1s α 0. k t = k0 αt Π t 1 j=0 sαt j 1 j 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 j=1 log k 0 + log(1 s 0 ) + δ α(1 αβ + δαβ) = log k 0 + Φ(s 0, s 1,...) 1 αβ j=1 β j log[(1 s j )Π j 1 τ=0 sαj τ τ ] Bassetto, Huo, Ríos-Rull Organizational Equilibrium 16/47

17 Proposals 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 agent at period t is α(1 αβ + δαβ) U(k t, s t, s t+1,...) = log k t + Φ(s t, s t+1,...) 1 αβ }{{}}{{} due to action due to capital Crucially, capital k t and saving rates {s t, s t+1, s t+2,...} are separable Bassetto, Huo, Ríos-Rull Organizational Equilibrium 17/47

18 Organizational Equilibrium in Quasi-Geometric Discounting Growth Model Definition A sequence of saving rates {s τ } τ=0 is organizationally admissible if Φ(s t, s t+1, s t+2,...) is (weakly) increasing in t The first agent has no incentive to delay the proposal. Φ(s 0, s 1, s 2,...) max Φ(s, s s 0, s 1, s 2,...) Within organizationally admissible sequences, any sequence that attains the maximum of Φ(s 0, s 1, s 2,...) is an organizational equilibrium. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 18/47

19 Organizational Equilibrium in Quasi-Geometric Discounting Growth Model Definition A sequence of saving rates {s τ } τ=0 is organizationally admissible if Φ(s t, s t+1, s t+2,...) is (weakly) increasing in t The first agent has no incentive to delay the proposal. Φ(s 0, s 1, s 2,...) max Φ(s, s s 0, s 1, s 2,...) Within organizationally admissible sequences, any sequence that attains the maximum of Φ(s 0, s 1, s 2,...) is an organizational equilibrium. The admissible sequence of saving rates and the maximum of Φ make sure 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 18/47

20 Remarks on Organizational Equilibrium Separability between capital and saving rates allows a global maximum Payoff only depends on capital, not a trigger strategy α(1 αβ + δαβ) U(k t, s t, s t+1,...) = log k t + Φ(s t, s t+1,...) 1 αβ α(1 αβ + δαβ) = log k t + Φ 1 αβ Agents need to know previous agents action, not a Markov equilibrium Bassetto, Huo, Ríos-Rull Organizational Equilibrium 19/47

21 Construct the Organizational Equilibrium An organizational equilibrium is a sequence of saving rates {s 0, s 1,...} To prevent copying the idea, every generation obtain the same Φ Φ(s t, s t+1,...) = Φ(s t+1, s t+2,...) = Φ which induces the following difference equation β(1 δ) log(1 s t+1) = δαβ log st + log(1 st) (1 β)φ 1 αβ We call this difference equation as the proposal function s t+1 = q(s t) The maximal Φ and an initial s 0 are needed to determine {s τ } τ=0 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 20/47

22 Determine Φ q (s) s As Φ increases, the proposal function q(s) move upwards The highest Φ = Φ is achieved when q(s) is tangent with 45 degree line at s Bassetto, Huo, Ríos-Rull Organizational Equilibrium 21/47

23 Determine the Initial Saving Rate s 0 The first agent should have no incentive to delay the proposal Φ(s 0, s 1, s 2,...) max Φ(s, s s 0, s 1, s 2,...) Given the next agent will propose the organizational equilibrium max Φ(s, s s 0, s 1, s 2,...) = Φ(s M, s 0, s 1, s 2,...) s 0 has to be such that Φ = Φ(s 0, s 1, s 2,...) Φ(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 the transition Bassetto, Huo, Ríos-Rull Organizational Equilibrium 22/47

24 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)φ + 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 Φ are given by s 0 = q ( s M ) s = δαβ (1 β + δβ)(1 αβ) + δαβ Φ = 1 β + δβ 1 β log(1 s ) + αδβ log s (1 β)(1 αβ) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 23/47

25 Transition Dynamics q (s) 45 degree line s M s 0 s s R The equilibrium starts from s 0, and monotonically converges to s. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 24/47

26 Remarks 1 To solve proposal function, no agent can treat herself specially, Φ t = Φ 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 25/47

27 Comparison: Steady State The organizational equilibrium is very like the Ramsey allocation 1 Steady state capital k Organizational Equilibrium Ramsey Allocation Markov Equilibrium δ Parameter values: α = 0.4, β = 0.96 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 26/47

28 Comparison: Transition Recall that the lifetime utility is given by U(k t, s t, s t+1,...) = α(1 αβ + δαβ) 1 αβ log k t + Φ(s t, s t+1,...) For the initial agent, starting with the same capital Ramsey allocation: highest Φ(s 0, s 1, s 2,...) Markov equilibrium: lowest Φ(s 0, s 1, s 2,...) Organizational equilibrium: close to but smaller than Ramsey allocation For subsequent agents, Ramsey allocation: lower Φ(s t,...), highest capital Markov equilibrium: lowest Φ(s t,...) and capital Organizational equilibrium: highest Φ(s t,...), much higher capital than Markov Bassetto, Huo, Ríos-Rull Organizational Equilibrium 27/47

29 Capital kt Organizational Equilibrium Ramsey Allocation Markov Equilibrium Time t Parameter values: α = 0.4, β = 0.96, and δ = 0.9 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 28/47

30 25.35 Total utility Ut Organizational Equilibrium Ramsey Allocation Markov Equilibrium Time t Total utility U(k t, s t, s t+1,...) = α(1 αβ + δαβ) 1 αβ log k t + Φ(s t, s t+1,...) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 29/47

31 Organizational Equilibrium Ramsey Allocation Utility due to action Φt Time t Utility due to action U(k t, s t, s t+1,...) = α(1 αβ + δαβ) 1 αβ log k t + Φ(s t, s t+1,...) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 30/47

32 Next In this example, saving rate permits the lifetime utility to be separable between actions and state variables any monotonic transformation of saving rate yields the same allocation We define organizational equilibrium for the class of economy that permits the separable property between action and states We then show how to approximate a non-separable economy with a separable economy locally Bassetto, Huo, Ríos-Rull Organizational Equilibrium 31/47

33 Part II: Organizational Equilibrium for Weakly Separable Economy Bassetto, Huo, Ríos-Rull Organizational Equilibrium 32/47

34 General Setup An infinite sequence of decision makers is called to act state variable k K, action x X(k) preference: Ψ(k t, x t, x t+1, x t+2,...) 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 Φ U(k, a t, a t+1, a t+2,...) = v(k, Φ(a t, a t+1, a t+2,...)). Bassetto, Huo, Ríos-Rull Organizational Equilibrium 33/47

35 Organizational Equilibrium Definition A sequence of actions {a τ } τ=0 is organizationally admissible if 1 Φ(a t, a t+1, a t+2,...) is (weakly) increasing in t 2 The first agent has no incentive to delay the proposal. Φ(a 0, a 1, a 2,...) max Φ(a, a0, a1, a2,...) a A Within organizationally admissible sequences, any sequence that attains the maximum of Φ(a 0, a 1, a 2,...) is an organizational equilibrium. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 34/47

36 Recursive Structure and Proposal Function Assumption There exist functions w 0 : A R R, w : A R R and W : A R, s.t. Φ(a 0, a 1, a 2,...) = w 0 (a 0, W (a 1, a 2,...)) W (a t, a t+1, a t+2,...) = w (a t, W (a t+1, a t+2,...)), t 1 for all sequences {a 0 } t=0. Proposition Assume w and w 0 are smooth functions, then an organizational equilibrium {a t} t=0 satisfies the following recursive equation a t+1 = q(a t) Moreover, if A is bounded, then there exists a fixed point a, s.t. a = q(a ) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 35/47

37 Part III: Approximated Organizational Equilibrium for Non-separable Economy Bassetto, Huo, Ríos-Rull Organizational Equilibrium 36/47

38 First Order Approximation Original economy: Ψ t = u(k t, x t) + δ τ=1 βτ u (k t+τ, x t+τ ) where k t+1 = F (k t, x t) Use the following separable economy to approximate the original economy Ψ t = û(k t, x t) + δ β τ û (k t+τ, x t+τ ) τ=1 where û(k, x) = h(k) + m(x) h(k ) = αh(k) + g(x) Around any (x, y), this approximation matches the level and first derivative Bassetto, Huo, Ríos-Rull Organizational Equilibrium 37/47

39 Example Original economy Preferences: U t = log(c t) + δ τ=1 βτ u (c t+τ ) Technology: k t+1 = k α t + (1 d)kt ct Approximate the resource constraint around k by k α + (1 d)k = ξ(k)k θ(k) The new separable economy is Preferences: U t = log(c t) + δ τ=1 βτ u (c t+τ ) Technology: k t+1 = ξ(k)k θ(k) c t Additional requirement: k is the steady state in organizational equilibrium Bassetto, Huo, Ríos-Rull Organizational Equilibrium 38/47

40 Approximated Production Function k α + (1 d)k ξ(k)k θ(k) 2 Production Capital k The approximation works well around the steady state Bassetto, Huo, Ríos-Rull Organizational Equilibrium 39/47

41 Part IV: Government Policy Problem Bassetto, Huo, Ríos-Rull Organizational Equilibrium 40/47

42 Government Policy Problem A strategic government v.s. a continuum of competitive agents Government cannot commit to a policy rule at time 0 Apply organizational equilibrium to this environment Significant welfare improvement over Markov equilibrium, but gradually Bassetto, Huo, Ríos-Rull Organizational Equilibrium 41/47

43 Environment Preference: 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 42/47

44 Government Problem Government proposal: a sequence of capital tax rates {τ t} t=0 Given {τ t} t=0, allocation in competitive equilibrium is unique k α t = c t + k t+1 + g t g t = ατ tk α t u (c t) = β(1 τ t+1 )αk α 1 t+1 u (c t+1 ) Government preference can be written as U(k, {τ t} t=0) = β t [log c t + γ log g t] = t=0 α(1 + γ) 1 αβ log k + Φ({τt} t=0) Bassetto, Huo, Ríos-Rull Organizational Equilibrium 43/47

45 Organizational Equilibrium in Government Taxation Problem Definition A sequence of tax rates {τ t} t=0 is organizationally admissible if Φ(τ t, τ t+1, τ t+2,...) is (weakly) increasing in t Government has no incentive to delay the proposal. Φ(τ 0, τ 1, τ 2,...) max Φ(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 44/47

46 Optimal Constant Tax Rate 1.2 Welfare: Φ(τ) τ τ M Taxation in Markov equilibrium is too high Constant τ cannot be implemented directly Bassetto, Huo, Ríos-Rull Organizational Equilibrium 45/47

47 Transition Dynamics in Organizational Equilibrium q(τ) 45 degree line τ M τ τ 0 The equilibrium starts from τ 0, and monotonically converges to τ. Bassetto, Huo, Ríos-Rull Organizational Equilibrium 46/47

48 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 Bassetto, Huo, Ríos-Rull Organizational Equilibrium 47/47