Uncertainty Traps. Pablo Fajgelbaum 1 Edouard Schaal 2 Mathieu Taschereau-Dumouchel 3. March 5, University of Pennsylvania

Transcription

1 Uncertainty Traps Pablo Fajgelbaum 1 Edouard Schaal 2 Mathieu Taschereau-Dumouchel 3 1 UCLA 2 New York University 3 Wharton School University of Pennsylvania March 5, /59

2 Motivation Large uncertainty events as measured by the VIX volatility index: Question: how do large uncertainty events affect the business cycle? 2/59

3 Uncertainty-driven Business Cycles Bloom (ECMA 2009) studies impact of stock market volatility on output Stock market volatility correlates with other measures of uncertainty such as cross-sectional dispersion of firm growth rates, stock returns, GDP forecasts, etc. VAR impact of stock market volatility shock on industrial production 3/59

4 Uncertainty-driven Business Cycles Proposed interpretations: Wait-and-see effects: in times of high uncertainty, people prefer to delay large, costly investments: option value of waiting investment drops, consumption of durables declines, etc; Precautionary motives: save more, consumption drops; Risk premium effects: volatility may implies a larger probability of default, rise in cost of finance: investment drops. 4/59

5 Uncertainty-driven Business Cycles Bloom (2009) proposes a story based on wait-and-see effects with: Time-varying volatility in aggregate and idiosyncratic productivity Partial irreversibilities in investment (fixed adjustment costs) Individual firm s production: where y it = A t z it F (k it,l it ) loga t = loga t 1 +σ t ε A t, εa t N (0,1) logz it = logz it 1 +σ t ε z it, ε z it N (0,1) and σ t {σ H,σ L } follows a two-state Markov chain 5/59

6 Option Value of Waiting Optimal timing of investment model à la Dixit-Pindyck (1994) Two periods: t and t +1 Decision at t +1 value of investingv i (z t+1 ) value of not investing = 0 z t+1 f (fixed cost) 6/59

7 Option Value of Waiting Decision at t value of investingv i (z t ) z t value of waitingv w (z t ) = βe[max(0,v i (z t+1 )] wait invest 7/59

8 Option Value of Waiting Decision at t high uncertainty low uncertainty z t wait invest 8/59

9 Option Value of Waiting In other words: (S,s) bands widen with uncertainty High uncertainty leads firms to wait more: Embedded in the value of waiting is the option to cut downside risk This option value is larger when uncertainty is high Some form of irreversibility (putty-clay, sunk costs, etc.) is essential to this effect 9/59

10 Uncertainty-driven Business Cycles Impulse response to 1-sd. volatility shock in Bloom (2009): 10/59

11 Uncertainty-driven Business Cycles Large recent literature on uncertainty-driven cycles Empirical: Measurement: Ludvigson et al. (2013) Identification and causality: Bachmann, Elstner and Sim (2013) Quantitative and theoretical: Wait-and-see effects: Bloom (2009), Bloom, Floetotto et al. (2012), Bachmann and Bayer (2009), Basu and Bundick (2011), Schaal (2012) Uncertainty and financial frictions: Arellano, Bai and Kehoe (2012), Gilchrist, Sim and Zakrajsek (2010) Policy uncertainty: Fernandez-Villaverde et al. (2013), Baker, Bloom and Davis (2012) Source of volatility/uncertainty: Bachmann and Moscarini (2011), Boedo and D Erasmo (2014), Orlik and Veldkamp (2014) 11/59

12 This Paper: Uncertainty Traps This paper proposes a theory of endogenous uncertainty to explain the persistence of recessions Uncertainty varies countercyclically because of social learning Social learning: agents learn from the actions of others Similar to Van Nieuwerburgh and Veldkamp (2005, 2006): more information is released in good vs. bad times Endogenous uncertainty provides an additional propagation mechanism Interaction of social learning and wait-and-see effects in model of irreversible investment 12/59

13 Mechanism 13/59

14 Mechanism 14/59

15 Mechanism 15/59

16 Mechanism 16/59

17 Mechanism Uncertainty traps: Self-reinforcing episodes of high uncertainty and low economic activity 17/59

18 Roadmap Today: focus on simple stylized model Isolate how key forces interact to create uncertainty traps Establish conditions for their existence and policy implications The paper then extends the model to more standard RBC environment Compare an economy with and without endogenous uncertainty The mechanism generates substantial persistence for reasonable parameters 18/59

19 Theoretical Model Infinite horizon model in discrete time N atomistic firms indexed by n { 1,..., N } producing a homogeneous good Each firm n has a unique investment opportunity and must decide to either do the project today or wait for the next period The project produces output x n = θ +ε x n Aggregate productivity (the fundamental) θ follows a random walk and ε θ iidn ( 0,γ 1 θ θ = θ +ε θ ), ε x n iidn ( ) 0,γ 1. x 19/59

20 Investment and Adjustment Costs Firms have CARA preferences over wealth u(x) = 1 a ( 1 e ax ) Firms face a random fixed investment cost f cdf F, iid, with variance σ f Firms that invest are immediately replaced by firms with new investment opportunities N {1,,N} is the endogenous number of firms that invest. 20/59

21 Information Firms do not observe θ directly, but receive noisy signals: 1 Public signal that captures the information released by media, agencies, etc. Y = θ +ε y, with ε y N ( 0,γy 1 ) 2 Output of all investing firms Note: Each individual signal ( x n = θ +ε x n, with ε x n iidn 0,γ 1 x can be summarized by the aggregate signal: X 1 x n = θ + 1 ε x n N (0,(Nγ x) 1) N N n I No bounded rationality: firms use all available information efficiently n I No asymmetric information 21/59 )

22 Timing Each firm start the period with common beliefs 1 Firms draw investment cost f and decide to invest or not 2 Production takes place, public signals X and Y are observed 3 Agents update their beliefs and θ is realized 22/59

23 Law of Motion for Information Before observing signals, firms share the same beliefs about θ θ I N ( µ,γ 1) After observing signals X and Y, the posterior about θ is θ I,X,Y N ( µ post,γpost 1 ) with µ post = γµ+γ yy +Nγ x X γ +γ y +Nγ x γ post = γ +γ y +Nγ x Next period s beliefs about θ = θ +ε θ is µ = µ post ( 1 γ = + 1 ) 1 Γ(N,γ) γ post γ θ 23/59

24 Firm Problem Firms choose whether to invest or not V(µ,γ,f) = max VW (µ,γ),v I (µ,γ) f }{{}}{{} wait invest Decision is characterized by a threshold f c (µ,γ) such that firm invests f f c (µ,γ) 24/59

25 Firm Problem Value of waiting Value of investing V W (µ,γ) = βe µ,γ [V (µ,γ,f )] V I (µ,γ) = E µ,γ [u(x) µ,γ] 25/59

26 Aggregate Consistency The aggregate number of investing firms N is N = n 1I(f n f c (µ,γ)) Firms have the same ex-ante probability to invest p(µ,γ) = F (f c (µ,γ)) The number of investing firms follows a binomial distribution N (µ,γ) Bin [ N,p(µ,γ) ] 26/59

27 Recursive Equilibrium Definition An equilibrium consists of the threshold f c (µ,γ), value functions V(µ,γ,f), V W (µ,γ) and V I (µ,γ), and a number of investing firms N (µ,γ,{f n }) such that 1 The value functions and policy functions solve the Bellman equation; 2 The number of investing firms N satisfies the consistency condition; 3 Beliefs (µ,γ) follow their laws of motion. 27/59

28 Characterizing the Evolution of Beliefs: Mean Mean beliefs µ follow µ = γµ+γ yy +Nγ x X γ +γ y +Nγ x Lemma For a given N, mean beliefs µ follow a random walk with time-varying volatility s, µ µ,γ = µ+s(n,γ)ε, with s N s > 0 and γ < 0 and ε N (0,1). 28/59

29 Characterizing the Evolution of Beliefs: Precision Precision of beliefs γ follow ( γ 1 = Γ(N,γ) = + 1 γ +γ y +Nγ x γ θ ) 1 Lemma 1) Belief precision γ increase with N and γ, 2) For a given N, Γ(N,γ) admits a unique stable fixed point in γ. 29/59

30 Characterizing the Evolution of Beliefs Precision of beliefs γ follow γ = Γ(N,γ) γ N γ 30/59

31 Characterizing the Evolution of Beliefs Precision of beliefs γ follow γ = Γ(N,γ) γ N 2 > N 1 N γ 31/59

32 Equilibrium Characterization Proposition Under some weak conditions and for γ x small, 1) The equilibrium exists and is unique; 2) The investment decision of firms is characterized by the cutoff f c (µ,γ) such that: firm with cost f invests f f c (µ,γ) 3) f c is a strictly increasing function of µ and γ. 32/59

33 Aggregate Investment Pattern N E[N] 0 µ γ 33/59

34 Uncertainty Traps We now examine the existence of uncertainty traps Long-lasting episodes of high uncertainty and low economic activity We now take the limit as N, N N = F (f c (µ,γ)) Details Given the equilibrium pattern of investment, we can study the dynamics of uncertainty 34/59

35 Equilibrium Dynamics of Belief Precision Precision of beliefs γ follow γ = Γ(N,γ) γ N.6 N.4 N.2 N N = γ 35/59

36 Equilibrium Dynamics of Belief Precision Precision of beliefs γ follow γ = Γ(N(µ,γ),γ) γ N.6 N.4 N.2 N N = 0 N/N = F(f c ) γ 36/59

37 Equilibrium Dynamics of Belief Precision Precision of beliefs γ follow γ = Γ(N(µ,γ),γ) γ γ l γ h γ 37/59

38 Equilibrium Dynamics of Belief Precision Precision of beliefs γ follow γ = Γ(N(µ,γ),γ) lowµ γ γ l γ h γ 38/59

39 Equilibrium Dynamics of Belief Precision Precision of beliefs γ follow γ = Γ(N(µ,γ),γ) γ high µ low µ 0.2 γ l γ h γ 39/59

40 Equilibrium Dynamics of Belief Precision To summarize: There is a range of µ [µ l,µ h ] for which the dynamics of belief precision admits two stable stationary points which we call regimes: a low activity/high uncertainty regime, γl a high activity/low uncertainty regime, γh For µ high, only the high activity regime exists, For µ low, only the low activity regime exists. 40/59

41 Existence of Uncertainty Traps Definition Given mean beliefs µ, there is an uncertainty trap if there are at least two locally stable fixed points in the dynamics of beliefs precision γ = Γ(N(µ,γ),γ). Does not mean that there are multiple equilibria The equilibrium is unique, The past history of shocks determines which regime prevails 41/59

42 Existence of Uncertainty traps Proposition For γ x and σ f low enough, there exists a non-empty interval [µ l,µ h ] such that, for all µ 0 (µ l,µ h ), the economy features an uncertainty trap with at least two stable steady states γ l (µ 0 ) < γ h (µ 0 ). Equilibrium γ l (γ h ) is characterized by high (low) uncertainty and low (high) investment. The dispersion of fixed costs σ f must be low enough to guarantee a strong enough feedback from information on investment 42/59

43 Uncertainty Traps: Falling in the Trap We now examine the effect of a negative shock to µ May arise from negative shock to θ or signals X or Y Hit the economy at t = 5 and last for 5 periods We consider small, medium and large shocks Under what conditions can the economy fall into an uncertainty trap? 43/59

44 Uncertainty Traps: Falling in the Trap Impact of a small negative shock to µ γ γ 44/59

45 Uncertainty Traps: Falling in the Trap µt γt N(µt,γt) N t 45/59

46 Uncertainty Traps: Falling in the Trap Impact of a medium-sized negative shock to µ γ γ 46/59

47 Uncertainty Traps: Falling in the Trap Impact of a medium-sized negative shock to µ γ γ 47/59

48 Uncertainty Traps: Falling in the Trap Impact of a medium-sized negative shock to µ γ γ 48/59

49 Uncertainty Traps: Falling in the Trap µt γt N(µt,γt) N t 49/59

50 Uncertainty Traps: Falling in the Trap Impact of a large negative shock to µ γ γ 50/59

51 Uncertainty Traps: Falling in the Trap Impact of a large negative shock to µ γ γ 51/59

52 Uncertainty Traps: Falling in the Trap Impact of a large negative shock to µ γ γ 52/59

53 Uncertainty Traps: Falling in the Trap µt γt N(µt,γt) N t 53/59

54 Uncertainty Traps: Escaping the Trap We now start after a full switch of the economy towards the low regime How can the economy escape the trap? 54/59

55 Uncertainty Traps: Escaping the Trap µt γt N(µt,γt) N t 55/59

56 Uncertainty Traps Economy with endogenous regime switches that displays strong non-linearities: for small fluctuations, uncertainty does not matter much, only large or prolonged declines in productivity (or signals) lead to self-reinforcing uncertainty events: uncertainty traps In such events, the economy may remain in a depressed state even after mean beliefs about the fundamental recover (µ) Jobless recoveries, high persistence in aggregate variables The economy can remain in such a trap until a large positive shock hits the economy 56/59

57 Remarks Uncertainty is a by-product of recessions, not the cause of recessions, yet still matters The model highlights one potential difficulty about recent attempts to use forecast data to proxy for uncertainty Agents can be uncertain about the fundamental and yet quite certain about other variables: output, investment, etc. Survey of Professional Forecasters: large forecast errors at the end of 2008, which disappeared very quickly in /59

58 Welfare Implications The economy is inefficient because of an informational externality Firms do not internalize the effect of their investments on public information Proposition The following results hold: 1) The competitive equilibrium is inefficient. The socially efficient allocation can be implemented with positive investment subsidies τ (µ, γ); 2) In turn, uncertainty traps may still exist in the efficient allocation. 58/59

59 Extensions The paper extends this model along the following dimensions: Neoclassical production functions with capital and labor Mean-reverting process for θ Long-lived firms that accumulate capital over time Firms receive investment opportunities stochastically Numerical exercise: Endogenous uncertainty substantially increase the persistence of recessions versus constant uncertainty, The additional persistence is large for a wide range of values for γx, it is however key that γ y is not too high for uncertainty to matter Key challenge: how to identify/measure the informational parameters in the data for full quantitative evaluation 59/59