Deviant Behavior in Monetary Economics

Transcription

1 Deviant Behavior in Monetary Economics Lawrence Christiano and Yuta Takahashi July 26, 2018

2 Multiple Equilibria Standard NK Model Standard, New Keynesian (NK) Monetary Model: Taylor rule satisfying Taylor principle. Passive fiscal policy; various frictions (investment, adjustment costs). Literature focuses on unique equilibrium local to positive inflation steady state. Referred to as desired equilibrium here. In practice, that equilibrium is pretty good in a welfare sense. But, we have reasons to think that there are other equilibria in NK model: BSGU(01,JET) showed there are two steady states. In simple monetary models there are also other equilibria: Hyperinflation, deflation, cycling, and chaos.

3 Achieving Uniqueness by Escape Clause Intuitive motivation: In a hyperinflation equilibrium, money growth high. Just declare we refuse to allow high money growth. In deflation, money growth slow. Just declare we refuse to allow slow (negative) money growth. Taylor rule with an escape clause (Christiano-Rostagno 2001, BSGU). While inside an inflation monitoring range, follow Taylor rule. Escape clause triggered if monitoring range violated, Switch to some form of money growth rule. Basic escape clause idea originates with Obstfeld and Rogoff (83,JPE). Practical examples of escape clauses: Exigent circumstances clause 13.3 in Federal Reserve Act. Authority to declare martial law. Deposit insurance.

4 Push Back Against Dramatic Conclusions in Two Papers Cochrane, Journal of Political Economy, Argues that proofs of equilibrium uniqueness with the escape clause correct, but not economically interesting (involves blow up the world threat). He s right, but his model assumes an endowment economy. In a production economy like New Keynesian model, his blow-up-the-world criticism does not apply, while the uniqueness result remains. Atkeson-Chari-Kehoe, Quarterly Journal of Economics, ACK suggest shrinking the monitoring range to a singleton and letting the escape clause do all the work to uniquely implement the desired equilibrium. Taylor principle rendered essentially irrelevant under their proposal. Our results suggest that ACK proposal lacks robustness: small trembles could lead the economy to select poor equilibrium outcomes. Conclusion Taylor principle with escape clause and wide monitoring range useful.

5 Model Representative household and Dixit-Stiglitz production. Government levies taxes, provides monetary transfers: and balances budget in each period. ( µ t 1) M t 1, µ t = M t / M t 1, Monetary policy: { µ t } selected so that, in equilibrium, { ( ) } φ R t = max 1, R πt π, π t+1 P t+1, P R π /β, t where π = µ 1 and R are desired inflation and interest rate.

6 Representative Household HH problem: max {c t,l t,m t,b t} t=0 s.t. [ ] β t ct 1 γ 1 γ l t 1+ψ, γ > 1, ψ > ψ t=0 m t + b t = W t l t + m t 1 P t 1 c t 1 + R t 1 b t 1 + T t P t c t m t HH first order conditions: W t P t = c γ t l ψ t c γ t R = βc γ t t+1, π t+1 (R t 1) [m t P t c t ] = 0 plus transversality condition.

7 Firms Competitive, final good firm production and profits: [ 1 Y t = 0 ] ε Y ε 1 ε 1 ε i,t di, ε > 1 i th intermediate good firm production: Y t,i = l t,i. Demand curve: Optimizing price: p i,t = ( ) ε [ 1 pi,t Y i,t = Y t P t = P t 0 ] 1 p 1 ε 1 ε i,t di ε subsidy, τ, neutralizes monopoly power ε 1 (1 τ) W {}}{ t = W t

8 Market Clearing and Households in Equilibrium Market Clearing: 1 0 l i,t = l t, c t = Y t = l t, b t = 0, m t = M t Firms: p i,t = p j,t = W t for all i, j, so P t = W t. Labor market clearing: Bond market clearing: 1 = W t P t = c γ t l ψ t = c γ+ψ t c t = 1. Intertemporal Euler Equation ( Fisher equation ): 1 = β R t π t+1.

9 (Monopolistically) Competitive Equilibrium Let: a t = ( l t, {l i,t } c t, π t, R t, W t, µ t, M t, m t, b t ). Definition 1 A competitive equilibrium under the Taylor rule is a sequence, (a t ) t=0, that satisfies, for t 0, (i) intermediate good firm optimality, (ii) final good firm optimality, (iii) household optimization, conditional on m 1 P 1 c 1 + R 1 b 1, P 1 ; (iv) government policy and (v) market clearing.

10 Scaling Scaled, logged Fisher equation β R t = π t+1 : ( ) Rt R t = π t+1, ( ) where R t ln, π R t+1 ln Monetary policy: Pre-transformed: R t = max { ( ) } φ 1, R πt π Transformed : R t = max { R l, φπ t }, ( ) where R l ( πt+1 Combining ( ) and ( ), yields equilibrium difference equation: π ). ( ) 1 = ln R. Scaled money growth: µ t = log ( µ t π ) π t+1 = max { R l, φπ t }.

11 Properties of Taylor Rule Equilibrium

12 Multiplicity and Local Uniqueness of Desired Equilibrium π t+1 π t+1 = max { R l, φπ t } 45 ( ) β ln π R l φ π l π u π t R l Multiple equilibria, {π t }, each indexed by π 0 Desired equilibrium is unique equilibrium that never violates monitoring range, [π l, π u ].

13 Taylor rule with escape clause Follow Taylor rule with Taylor principle, φ > 1, during normal times when inflation remains inside the monitoring range, π t [π l, π u ]. Escape clause: if for some t, π t / [π l, π u ]. then, switch forever to constant money growth, µt+s = µ 1, for s 1. Result: under Taylor rule with escape clause desired equilibrium allocations only allocations that satisfy the equilibrium conditions.

14 Three steps in uniqueness proof 1. Establish existence and uniqueness of continuation equilibrium, after escape clause activated. 2. Show that activation of escape clause inconsistent with equilibrium conditions. 3. Uniqueness follows from uniqueness of equilibria in which bounds, [π l, π u ], never touched.

15 Escape clause never activated in equilibrium Lemma 2 Consider the case in which monetary policy is the Taylor rule with escape clause, φ > 1, Rl φ < π l 0 π u <, and µ [π l, π u ]. An equilibrium has the following property: π t [π l, π u ] for t 0. Proof: Suppose not, so that π T / [π l, π u ], for some T. Consider the case, π T > π u and π j [π l, π u ], j < T. 1. Taylor rule: R T = φπ T > π u. 2. Escape clause and property of continuation equilibrium: π T +1 = µ π u. 3. So, R T π T +1 > 0, violating Fisher equation, contradiction. Next, consider the case π T < π l. A similar argument shows that R T π T +1 < 0, violating the Fisher equation at T. QED.

16 Uniqueness of Equilibrium Under Escape Clause Strategy Proposition 3 Suppose that φ > 1, Rl φ < π l 0 π u <, and µ [π l, π u ]. Suppose monetary policy is governed by the Taylor rule with escape clause. The only equilibrium is the desired equilibrium. π t+1 = max { R l, φπ t } π t+1 45 ( ) β ln π R l φ π l π u π t R l

17 Cochrane s critique of uniqueness argument Cochrane (JPE11) has no technical problem with the proof. His problem has to do with the economics of how non-desired equilibria are prevented from occurring. Asserts that uniqueness result not due to escape strategy per se, but really reflects: Government commitment to do something infeasible in case undesired allocations occur, i.e., make R T π T He calls doing something that is infeasible, blowing up the world. He concludes that the uniqueness result economically uninteresting. Rejects the escape clause as a resolution to multiplicity in standard NK model.

18 Blowing up the World, Really? Cochrane raises interesting question: How, exactly, did the escape clause rule out the other equilibria? Problem: standard concept of equilibrium is silent on this question: It simply says that something is an equilibrium and something else is not. Need new concept of equilibrium. Follow Bassetto (2005) and Atkeson-Chari-Kehoe (2010). Introduce an exit ramp, off of the equilibrium path Ask why people in the economy will choose not to take the exit ramp.

19 Exit Ramp Off Equilibrium

20 New Approach to Equilibrium Want a concept of equilibrium that can be used to answer question: What is it about the escape clause that makes people look down the exit ramp and say I will not go down there? Specific question: why don t people get off the path and jump to hyperinflation? When they contemplate the possibility of hyperinflation, they realize the government will raise the real interest rate sharply and create a recession with low costs. But, in that case, they realize that no one would raise prices. So, no inflation.

21 Back to Model: a timing assumption Morning : intermediate good producers simultaneously and without observing what others do, post their price, p i,t, i [0, 1]. i th producer s p i,t decision based on a belief about what others are doing, and what will happen in afternoon. Afternoon : pi,t, i [0, 1] taken as given. Production occurs, labor markets clear, monetary policy implements Taylor rule with escape clause. [ ] ε di. Pt = 0 p1 ε i,t

22 Intermediate Good Firm Beliefs Morning of period t. i th producer sets price, p i,t = W e t superscript e indicates belief about value of W t in afternoon absence of i subscript indicates assumption, consensus. Expand: p i,t = P e t ( ) W e t = P e Pt e t reflects labor supply and l t = c t {}}{ (c e t ) γ+ψ where Pt e is the i th firm s belief about p j,t for all j i (assumption, symmetry ). c e t is the i th firm s belief about consumption in the continuation equilibrium that is implied by P e t : c e t = c e t (P e t ).

23 Consistent Beliefs From previous slide, firm i s price decision: p i,t = P e t (c e t (P e t )) γ+ψ. Firm i believes everyone else thinks the same way that it does, so P ee t = p i,t, and: P ee t = P e t (c e t (P e t )) γ+ψ. Assumption: can t hold two different beliefs about the same thing at the same time P e t is a consistent of belief (also, Nash equilibrium ) if: P e t = P e t (c e t (P e t )) γ+ψ.

24 Consistent Beliefs Divide aggregate best response function by µ P t 1 : P ee t µ P t 1 = Pe t µ P t 1 (c e t ) γ+ψ Taking logs: π ee t = π e t + (γ + ψ) ln (c e t ) F (π e t ). People choose their beliefs, π e t, to be consistent: π e t = F (π e t ). To find consistent beliefs, firms have to try out a lot of them. In each case, must compute continuation equilirium, conditional on each belief. From this, can discover the economics of why escape clause rules out hyperinflation.

25 Competitive Equilibrium Recall: a t = ( l t, {l i,t } c t, π t, R t, W t, µ t, M t, m t, b t ). Definition 4 A competitive equilibrium with a Taylor rule an escape clause is a sequence, (a t ) t=0, that satisfies, for t 0, (i) intermediate good firm optimality, (ii) final good firm optimality, (iii) household optimization, conditional on m 1 P 1 c 1 + R 1 b 1, P 1 ; (iv) government policy and (v) market clearing. In a competitive equilibrium, beliefs are consistent. Another word for consistent : rational expectations.

26 Strategy Equilibrium Again: a t = ( l t, {l i,t } c t, π t, R t, W t, µ t, M t, m t, b t ). Definition 5 A strategy equilibrium is a sequence, a t, for t 0, with two properties: (i) it is a competitive equilibrium with the Taylor rule and escape clause and (ii) for each date t and each πt e, there is a well defined continuation equilibrium conditional on h t 1 = (a 0,..., a t 1, πt e ). Part (ii) requires that off equilibrium continuation equilibria exist. Now, we can ask in a coherent way, why does the escape strategy kill non-desired equilibria?

27 Exit Ramp Off Equilibrium

28 Why Isn t High Inflation an Equilibrium? If π e t > π u, easy to show that actual inflation would be: π t = π e t + (γ + ψ) Intuition behind the above expression: ln c e t {[ }} ]{ φ 1 γ πe t, γ > 1. If individual firms believe other firms will set high prices (i.e., π e t high), then firms believe R t will be high because of Taylor principle. firms believe π t+1 will be low because of low t + 1 money growth. Believe that high real rate will be associated with low ct. Low ct means low demand for labor low W t/p t. Low real wage, means nominal wage rises by less than Pt: W t = (W t/p t) P t. So, individual firms don t raise prices much if they believe others will. High inflation not consistent belief. Cannot occur in equilibrium.

29 Bottom Line High inflation not an equilibrium. Suppose everyone believed everyone else would raise prices a lot. Then, they know the government would create a Volcker-style recession: inflation would in fact rise by less or even fall. Certainly feasible: they actually did it in the 1980s. No blowing up the world involved. So, where did Cochrane go wrong? He made his argument using an endowment economy. So, the Fisher equation holds on and off equilibrium, high real rate infeasible. He s right when he says that, from the perspective of his underlying model, the escape clause works by a threat to blow up the economy. He s right to conclude that his model is not interesting. But, he s wrong when he suggests that this has anything to do with New Keynesian models. The essence of those models is that a high real rate causes a recession.

30 Conclusion Our analysis suggests that φ > 1 and a wide inflation monitoring range is a good monetary policy. The escape clause idea may deserve further attention as a device for selecting the unique, locally bounded equilibrium in standard New Keynesian models.