Uncertainty Traps. Edouard Schaal NYU. July 8, 2013 [ PRELIMINARY AND INCOMPLETE ] Abstract

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1 Uncertainty Traps Pablo Fajgelbaum UCLA Edouard Schaal NYU July 8, 03 Mathieu Taschereau-Dumouchel Wharton PRELIMINARY AND INCOMPLETE ] Abstract We develop a quantitative theory o endogenous uncertainty and business cycles. In the model, higher uncertainty about undamentals discourages investment but agents can learn rom the actions o others. Thereore, in times o low activity inormation lows slowly and uncertainty stays high, urther discouraging investment. This creates room or uncertainty traps sel-reinorcing episodes o high uncertainty and low activity. We characterize conditions that give rise to these events. Negative shocks to average productivity or belies may have permanent eects on the level o activity through the persistence o uncertainty. We also characterize optimal policy interventions. The socially eicient allocation can be implemented with aggregate-belies dependent subsidies, but under certain conditions it necessarily eatures uncertainty traps. We embed these orces into a standard quantitative model o the business cycle to evaluate the impact o uncertainty traps. JEL Classiication: E3, D83

2 Introduction One o the central eatures o macroeconomic activity is its high persistence. The NBER deinition o business cycles implies that it takes or the U.S. economy close to 40 months on average to recover rom a through until the next peak. Business cycles are also asymmetric: it takes about 7 months or the economy to move rom peak to through, so that recoveries last on average more than twice as long as downalls. These eatures have been visible during in the crisis. The unemployment rate increased rom 4.4% in May 007 to 0% in October 009 and has barely decreased to a level 7.9% in early 03. What explains these prolonged declines in economic activity? In this paper, we develop a quantitative theory o endogenous uncertainty and business cycles to explain these phenomena. The theory captures two key orces. First, higher uncertainty about undamentals discourages investment. Second, economic agents can learn rom the actions o others. The interaction between these two orces creates room or uncertainty traps sel reinorcing episodes o high uncertainty and low activity. In times o low activity inormation lows slowly and uncertainty stays high, urther discouraging investment. This explains why low activity may persist under good undamentals. We irst develop a baseline theory that includes only the essential eatures o the mechanism, and then we extend the model in various dimensions or a quantitative evaluation. In the model, irms choose to undertake an irreversible investment whose return depends on an imperectly observed undamental. Belies about that undamental are common to all irms, but can be regularly updated using various signals. Formally, we deine uncertainty as the variance o the prior about the undamental. Inormation, in turn, diuses through a simple social learning channel: the higher the number o irms that invests, the larger the number o signals received by irms and the stronger the reduction in their uncertainty. This environment naturally produces an interaction between belies and economic activity. Firms are more likely to invest i they hold more optimistic or less uncertain belies about the undamental. Thereore, low uncertainty is associated with a high investment rate. At the same time, the law o motion or belies depends on the investment rate through social learning. When ew irms invest, uncertainty rises and the irms optimism, captured by the mean o the belies distribution, is less likely to luctuate. Using this setup we demonstrate the existence o uncertainty traps. Formally, we deine an uncertainty trap as the coexistence o multiple stationary points in the joint dynamics o uncertainty andeconomicactivity oragivenmeanothedistributionobelies. Wereertotheseixedpointsas regimes. Due to the complementarity between investment and inormation diusion, in high-activity regimes there is low uncertainty and in low-activity regimes there is high uncertainty. Despite this multiplicity, the recursive equilibrium is uniquely pinned down by the stochastic evolution o the mean level o belies. But, because o it, the unique equilibrium is prone to nonlinear dynamics and asymmetries. For example, the long-run response to a temporary negative shock becomes considerably more protracted when its magnitude is above some threshold. The economy quickly recovers ater a small temporary shock, but it may permanently shit into a low activity regime

3 ater a large shock o the same duration. In turn, a positive temporary shock o suicient magnitude can put the economy back on track. As in other theories o social learning, there are ineiciently low levels o investment because agents do not internalize the eect o their actions on common inormation. This ineiciency naturally creates room or welare-enhancing policy interventions. To ind these policies, we study the problem o a constrained planner that is subject to the same inormational constraints as private agents. We ind that the socially constrained-eicient allocation can be implemented with aggregate-belies dependent subsidies. For example, it could be desirable to subsidize investment in times o high uncertainty and low activity. However, under certain conditions, the optimal policy does not eliminate the uncertainty traps. Thereore, while policy interventions may be desirable, they do not necessarily eradicate the nonlinearities generated by the complementarity between uncertainty and economic activity. Ater characterizing the model, we evaluate the quantitative importance o the uncertainty traps. For that, we extend the baseline model to bring it closer to general real business cycle models. Among other eatures, we generalize the capital accumulation process by adding an intensive margin. We also introduce a risk-averse representative household with endogenous labor supply. To estimate the importance o uncertainty traps, we compare the outcomes rom our extended model with a restricted setup in which uncertainty is not allowed to adjust endogenously. In preliminary numerical exercises, we ind that uncertainty traps make economic downturns more persistent and pronounced relative to a ramework with ixed uncertainty. The emphasis on the wait-and-see eect o uncertainty on investment is shared with a recent literature that studies how changes in the volatility o productivity shocks aects the economy, such as Arellano et al. 0, Bachmann and Bayer 009, Bloom 009, Bloom et al. 0, and Schaal 0. Two eatures set us apart rom that literature. First, these papers ocus on uncertainty induced by time-varying volatility in productivity. In contrast, our learning approach enables us to dissociate subjective uncertainty rom volatility in undamentals. While in our setup volatility generates uncertainty, there can also be periods o high uncertainty with constant volatility. Second, in our analysis the movements in uncertainty are endogenous. That literature ocuses, in contrast, on exogenous volatility shocks to productivity. These two distinguishing eatures create the additional propagation o shocks that we explore in the paper. The notion o uncertainty in this paper seems justiied in the ace o systematic reerences by businessmen and commentators to high levels o uncertainty in the atermath the recession despite the decline in several measures o volatility. Indeed, our theory allows or uncertainty to persist in a context with low volatility. The advantage o allowing or endogenous uncertainty movements as opposed to exogenous volatility shocks is that endogenous uncertainty is better able to deliver persistent macroeconomic series. Because high volatility events are short-lived, models that ocus on that type o shock are hard to reconcile with the persistence o recessions. In contrast, Measures o aggregate and idiosyncratic volatility such as the VIX volatility index have substantially declined since 009, as shown in Schaal 0. Another interesting source o uncertainty suggested in the literature, rom which we abstract in this paper, is policy uncertainty. See Baker et al. 0; Fernández-Villaverde et al. 0. 3

4 subjective uncertainty traps can deliver persistence in a low-volatility context. Endogenous movements in uncertainty can be modeled in dierent ways. We make this notion operative using a simple concept o social learning. Intuitively, we envision irms holding bits and pieces o inormation about a shared undamental; when a irm invests or hires, its actions reveal inormation about the state o the economy to other agents. Hence, our analysis relates to papers on ads and herding in the tradition o Banerjee 99, Bikhchandani et al. 99, and Chamley and Gale 994. A number o studies, such as Cunningham 004, Kaustia and Knüper 009, Khang 0, and Patnam 0, empirically document the relevance o social learning in various contexts such as investment in the semiconductor industry, stock market entry decisions, housing purchases, and R&D expenditures. Social learning about technology has also been demonstrated to be important in other contexts such as economic development, as shown by Foster and Rosenzweig 995 and Hausmann and Rodrik 003. Our analysis also relates to a theoretical macroeconomic literature that studies environments with learning rom market outcomes such as Rob 99, Caplin and Leahy 993, Zeira 994, Veldkamp005, Ordonez009, and Amador and Weill 00, as well as to papers on endogenous volatility over the business cycle, such as Bachmann and Moscarini 0 and DErasmo and Boedo 0. Specially related is the analysis in Van Nieuwerburgh and Veldkamp 006. In their model, agents hold belies about a undamental and the signal-to-noise ratio varies procyclically; this delays recoveries because agents discount new inormation more heavily in recessions. However, in that paper, uncertainty about the undamental provides a weak eedback and the economy quickly learns its way out o a recession. The key eature that distinguishes our analysis is the presence o irreversible investments. The option value created by irreversibilities oers a strong additional motive or agents to deer investment in uncertain times. The interaction between social learning and irreversible investment leads, in our setup, to nearly permanent eects o uncertainty on economic activity. The paper is structured as ollows. Section presents the baseline model and the deinition o the recursive equilibrium. Section 3 characterizes the partial-equilibrium investment decision o an individual irm, the uniqueness o the equilibrium, the existence o uncertainty traps, and the welare implications. Section 4 eatures the preliminary quantitative analysis using an extended model. Section 5 concludes. Proos are relegated to the appendix. Suggestive Evidence The central channel in the theory is the eedback between uncertainty and investment. We argue that the inactivity o irms during recessions slows down the diusion o inormation, creating uncertainty and discouraging urther investment. Thereore, the model predicts that recessions are times where both uncertainty and irm inactivity are high, and that these eatures may persist even i productivity has recovered. In this section we provide irst-pass evidence consistent with these eatures o the model. 4

5 . Uncertainty over the Business Cycle The literature that studies the impact o uncertainty shocks establishes that the variance in idiosyncratic shocks to productivity increases during bad times. Bloom et al. 0 demonstrate that the dispersion o plant- and industry- level shocks to productivity is counter-cyclical and peaks in recessions. Other commonly used measures o irm-level volatility, such as the VIX index o volatility in stock market returns or the dispersion in irm level sales, reproduce the same pattern. Our theory is more speciically concerned with subjective uncertainty. Direct measures o subjective uncertainty are also available and exhibit similar counter-cyclical patterns. Dierent surveys ask respondents to assess the main reasons why they preer to postpone economic decisions. According to the National Federation o Independent Business 0, 40% o answers rank economic uncertainty as the most critical problem that they aced in 0. A more systematic evidence comes rom the Michigan Survey o Consumers, which shows a peak during recessions in the percent o consumers who state that uncertain uture is the main reason to postpone purchases o durable goods in the United States see igure. A similar pattern is observed in the UK, where irm s perceived uncertainty increases during recessions according to the CBI Industrial Trends Survey. Leduc and Liu 0 argue that these subjective measures o uncertainty are countercyclical. Other measures o subjective uncertainty with strong counter-cyclical patterns are the variance o ex-post orecast errors about economic conditions and the dispersion o belies eatured in Bachmann et al. 03. In bad times, agents hold more heterogeneous belies about uture economic conditions. Our model allows or short-lived dispersion o belies among economic agents within each period, and predicts that the within-period variance o belies is larger when uncertainty is high and economic activity is low. Share o Consumers Responding Uncertain Future Uncertainty Year Figure : Subjective Uncertainty over Time Source: Michigan Survey o Consumers 5

6 . Share o Zeros in Investment over the Business Cycle A second piece o evidence consistent with the basic mechanism that we present concerns the incidence o irm inactivity during recessions. While aggregate investment naturally is countercyclical, we emphasize a microeconomic channel based on the inactivity o irms. Because irms ace indivisible investment choices, their incentives to invest are low when uncertainty is high. In turn, we posit that lack o activity slows down diusion o inormation. Thereore, we expect to see a higher raction o irms that do not invest when the level o activity is low. The literature on lumpy adjustment studies the distribution o investment rates over the business cycle, but it ocuses more on investment spikes rather than zeros. In this literature, Cooper and Haltiwanger 006 report that 8% o plant-year observations in the United States between 97 and 998 have investment rates below % in absolute value. For us, the important question is how this raction varies over the business cycle. Gourio and Kashyap 007 report the share o investment zeros or the US and Chile between 975 and 000, arguing that in both countries the share o exact or near-zeros is strongly countercyclical. They report a correlation between aggregate investment and the share o investment rates close to zero o in the US and in Chile. Since aggregate investment is strongly procyclical, this implies that the share o zeros or near-zeros in irm-level investment is countercyclical. We complement this evidence with data on the prevalence o exact and near-zeros in investment or a longer time series that includes the current recession. For that, we use quarterly data rom Compustat between 975 and 0. We ollow Eiseldt and Rampini 006 in using the variable Property, Plant and Equipment as proxy or physical capital at the irm level. Figure shows the share o irms with zero or near-zero investment rates. To compute these igures, we irst restrict the Compustat dataset to irms with non-missing investment rates at the quarterly level and to quarters with at least 500 such irms. Then, we calculate the share o zeros or near-zeros in each quarter. The igure shows the average shares across all quarters in each year, distinguishing between all irms and irms in manuacturing only. Share o Zeros Share o Exact Zeros over the Business Cycle Year Share o Near Zeros Share o Near Zeros over the Business Cycle Year All Firms Manuacturing Only All Firms, Investment<=% All Firms, Investment<=% Manuacturing Only, Investment<=% Manuacturing Only, Investment<=% Figure : Share o irms with Zero or Near-Zero Investment Source: Compustat On average, % o irms display zero investment rates at the quarterly level. In turn, the 6

7 share o inactive irms is countercyclical. For the years in which the series overlaps with the Gourio and Kashyap 007 data, both series display similar properties. Interestingly, or the recent recession investment inactivity spikes and remains relatively high ater economic activity has recovered. In 0, 4.9% o irms display zero change in capital. Similar patterns are observed or the share o irms near zero investment in absolute value. For example, the average share o irms with investment rate below % is 33%, and with investment rate below % is 53%, and both measures peak in bad times. Figure 3 shows the positive correlation between uncertainty rom the Michigan Consumer Survey and the share o irms with zero investment rom Compustat. The years corresponding to the crisis and its atermath appear on the upper right area o the graph. Zero Investment Compustat Correlation: Uncertainty Michigan Survey Data Lowess Fit Figure 3: Uncertainty and Share o Firms with Zero Investment 3 Baseline Model We present a stylized model that eatures only the necessary ingredients to generate uncertainty traps. The intuitions rom this simple model as well as the laws o motion governing the dynamics o uncertainty will carry through to the ull model that we use or quantitative analysis. 3. Population and Technology There is a large, ixed number o irms N indexed by n {,..., N }. Each irm has an investment opportunity that produces output x n. Output x n is the sum o two components: a persistent aggregate component θ which denotes the economy s undamental as well as an idiosyncratic Gourio and Kashyap 007 use establishment level data rom the Census Bureau s Annual Survey o Manuacturers. The correlation between the share o irms with investment rate below % in their data and in Compustat is

8 transitory component ε x n, x n = θ+ε x n. Time is discrete. The aggregate term ollows a random walk, so that the next-period s undamental is θ = θ +ε θ. The innovations ε θ, ε x n are independent and normally distributed, ε θ N 0,γ θ and ε x n N 0,γx. We let uxbe the low payo to the irm when it invests. Firms have constant absolute riskaversion 3, ux = e ax, a where a is the coeicient o absolute risk aversion. To produce, irms need to incur a ixed cost. This cost is an i.i.d. draw in every period rom the continuous cumulative distribution F with mean and variance σ. Thereore, irms solve an optimal stopping time problem. In each period, given the ixed cost and the belies about the returns to the investment that we speciy below, irms can either wait or invest. When a irm invests, it pays the ixed cost, produces x n, and exits the economy. We assume that investing irms are immediately replaced with new irms that hold an investment opportunity. In this way, the mass o irms remains constant over time Timing and Inormation At the beginning o a period irms do not observe the undamental θ but hold belies about its distribution. We call uncertainty the variance o the prior belies about θ. Because the undamental and the signals are normally distributed and inormation is public, all irms start the period with a common, normally distributed prior about θ, θ I N µ,γ, where I is the inormation set at the beginning o the period. The mean o the belies distribution µ captures the optimism o agents about the state o the economy, while γ is the precision o inormation. A lower γ means that irms have higher uncertainty. In each period, the aggregate state space o the economy reduces to the common belies µ,γ. 3 The assumption o risk aversion is not necessary or the results. We include it or technical reasons in the general-equilibrium uniqueness proos. In the simulation o the baseline model, we show that the key properties o the model carry through with risk neutrality. In the ull quantitative model, risk aversion arises rom a standard stochastic discount actor derived rom risk averse households. 4 The assumption that irms exit when they invest is or tractability o the baseline model and it can be relaxed. 8

9 Firms may learn about the undamental θ in various ways as time unolds. First, irms learn by producing. We let N {,..., N } be the endogenous number o irms that invests in a given period and I be the set o such irms. When irm n invests, output x n is observed by every irm but the undamental θ cannot be distinguished rom the idiosyncratic term ε x n. Thereore, production is a noisy signal o the undamental. Because o the Gaussianity assumption, the inormation about the undamental conveyed by each irm s output is summarized by the public signal where X x n = θ +ε X N N, n I ε X N ε x n N N n I 0,Nγ x. Average output, X, summarizes all the inormation provided by the distribution o irm-level output. It is also important that the precision o this signal, Nγ x, increases with the number o investing irms, N. The higher is N, the more precise is the inormation collected by each individual irm. In addition, irms learn about the undamental rom a public signal Y observed at the end o each period, Y = θ +ε y, ε y N 0,γy. 3 This captures the inormation released by statistical agencies or the media. The timing o events is summarized in Figure 4. t N irms decide to invest based on common belies and investment costs Production takes place; Public signalsx and y are observed Figure 4: Timing o events Belies are updated t Firm Problem The value o a irm with an investment opportunity that starts the period with belies µ,γ and a draw o the ixed investment cost is V µ,γ, = max { V W µ,γ,v I µ,γ }, 4 where V W µ,γ is the value o the irm i it waits until the next period and V I µ,γ is the value o the irm ater incurring the investment cost. We assume that the number o irms N is large enough that irms behave competitively. Speciically, they do not internalize the impact o their decisions on aggregate inormation. The irm s 9

10 problem yields an investment rule χµ,γ, {0,}, such that 5 i invests V I µ,γ V W µ,γ χµ,γ, = 0 otherwise. 5 When a irm waits, it starts the next period with new belies µ,γ about the undamental and a new ixed cost draw. Thereore, the value o waiting is V W µ,γ = βe V µ,γ, ]. 6 In turn, when a irm invests it receives output x and exits. Thereore, the value o investing, net o the ixed cost, is V I µ,γ = Eux µ,γ] = E e a x ] µ,γ = a a 3.4 Law o Motion or Common Belies µ,γ e aµ+a γ + γx. 7 Firms start the period with belies µ, γ. By period s end, they have observed the public signals X and Y deined in and 3. Ater that, standard rules or Bayesian updating imply that the common posterior belie about θ is normally distributed with mean and precision o inormation equal to µ post = γ µ+γ y Y +Nγ x X γ +γ y +Nγ x, γ post = γ +γ y +Nγ x. These standard updating rules have a straightorward interpretation: the mean o the posterior belie is a precision-weighted average o the past belie µ and the new signals, Y and X, whereas its precision is the sum o the precision o the prior belie, γ, and the precision o the the new signals. {µ post,γ post } deine the belies by period s end. Beore the next period starts, the undamental θ receives an innovation as deined in. Thereore, at the beginning o the new period the prior about the undamental θ is normally distributed with mean and inverse o uncertainty equal to µ = µ post = γ µ+γ y Y +Nγ x X, 8 γ +γ y +Nγ x γ ΓN,γ = + = +. 9 γ post γ θ γ +γ y +Nγ x γ θ Conditions 8 and 9 are the laws o motions or belies. The irst moment, µ, depends on 5 We assume that irms choose to invest in the case o indierence. This assumption is innocuous as these events happen with probability 0. 0

11 the aggregate public signals X and Y. The number o investing irms, N, determines the quality o the public signal X. In turn, the precision o inormation γ solely evolves based on N. The higher is N, the more precise the public signal X is and the higher the precision o the prior belies on θ. For uture reerence, and because it is a key object in our analysis, we let ΓN,γ in 9 be the law o motion o the precision o inormation. 3.5 Law o Motion or the Number o Investing Firms N We have so ar introduced the irm s problem and the law o motion or the aggregate state given a number o investing irms N. O course, the process or N must be consistent with the individual choices o irms. The number o irms that invests satisies N µ,γ = N χµ,γ, n = N n= n= I V I µ,γ n V W µ,γ. 0 Because the investment rule χµ,γ, n is a random unction o the ixed cost, the number o investing irms is stochastic and depends on the realization o the shocks { n } n N. Thereore, at the beginning o a period and beore these shocks are realized, N is perceived as ollowing a binomial distribution, N µ,γ Bin N,pµ,γ. The probability o investment or each irm, pµ,γ, must equal the probability o receiving a ixed cost such that investment is proitable. From 5, this implies a ixed cost below the threshold V I µ,γ V W µ,γ, so that pµ,γ = F V I µ,γ V W µ,γ. As the total number o irms grows large, the raction o irms that invests in every period becomes a deterministic unction o aggregate belies, N µ,γ N a.s pµ,γ. This dependence o the level o activity on aggregate belies justiies our initial statement that {µ,γ} are the sole aggregate states o the economy. From the perspective o an individual irm, when the total number o irms is large, N is a deterministic unction o aggregate belies. In turn, the mean o aggregate belies µ evolves stochastically according to 8, but the law o motion or the precision o inormation γ = ΓN µ,γ,γ, deined in 9, is a deterministic unction o {µ,γ}. 3.6 Recursive Equilibrium We are ready to deine a recursive equilibrium Deinition. An equilibrium consists o the policy unction χµ,γ,, value unctions V µ,γ,,

12 V W µ,γ, V I µ,γ, laws o motions or aggregate belies {µ,γ }, and a number o investing irms N µ,γ, such that. the value unction V µ,γ, solves 4, with V W µ,γ and V I µ,γ deined according to 6 and 7, generating a policy unction χµ,γ, in 5;. the aggregate belies {µ, γ} evolve according to 8 and 9; and 3. the number N µ,γ o irms that invest is given by 0. 4 Equilibrium Characterization and Uncertainty Traps We start by characterizing the partial-equilibrium investment decision o a irm given the laws o motions or belies. Because o the irreversibility o the investment, irms are less likely to invest when uncertainty is high. Ater establishing the existence and uniqueness o the recursive equilibrium, we characterize its key properties. Speciically, we examine the interaction between the option value o investment and social learning. This interaction creates episodes o sel-sustaining uncertainty and low activity, which we call uncertainty traps. We provide suicient conditions on the parameters that guarantee the existence o uncertainty traps, we discuss the type o aggregate dynamics that they imply, and we characterize their policy implications. 4. Investment Rule Given the Evolution o Belies The investment rule χµ,γ, crucially depends on how belies evolve. We establish two simple lemmas about the aggregate belies process. Evolution o Mean Belies Using 8 we can characterize the stochastic process or the mean o the prior distribution o belies about the undamental. Lemma. Mean belies µ ollow a random walk with time-varying volatility s, where sn,γ = γ γ+γ y+nγ x µ = µ+sn,γε, and ε N 0,. Mean belies capture the optimism o agents about the undamental, and they evolve stochastically due to the the arrival o new inormation. The volatility o mean belies is time-varying because the amount o inormation that irms collect over time is endogenous. The volatility sn,γ depends negatively on the current precision o belies γ. In times o low uncertainty, when the precision o belies is high, Bayesian updaters place less weight on new inormation, making mean belies less sensitive to the cycle. The volatility o optimism also depends positively on the number o active irms, N. When N is large, inormation lows aster, making belies more likely to jump. Through this eect, the volatility o mean belies is lower in recessions.

13 Evolution o Uncertainty The precision o belies γ is specially important as it embodies the dynamics o uncertainty. The precision o belies is random only as a result o the initeness o the number o irms, which vanishes as their total number grows large. Conditioning on the realization o N, the dynamics o precision γ is deterministic and allows or a simple analytical characterization. Lemma. The precision o next-period belies γ increases with N and γ. For a given N, the law o motion or the precision o belies γ = ΓN,γ admits a unique stable ixed point in γ. γ N = N N =.6 N N =.4 N N =. N γ N = 0 γ Γ Nγ,γ γ Figure 5: Example o dynamics or belie precision γ Figure 5 depicts ΓN,γ or dierent values o N that range rom N = 0 to N = N. As stated in Lemma, an increase in the level o activity raises the next-period precision o inormation γ or any precision o inormation in the current period. In the example o the igure, it is evident that the support o the ergodic distribution o γ must be bounded between γ and γ, i.e., the levels or the inverse o uncertainty corresponding to N = 0 and N = N. In equilibrium, N is endogenous and varies with γ. Assuming momentarily that N γ is an increasing step unction, the igure illustrates how the eedback rom uncertainty to investment opens up the possibility o multiple stationary points in the dynamics o belies precision or its inverse, uncertainty. For the chosen path o N γ, the unction γ = ΓN γ,γ depicted by the solid line in the igure has three stable ixed points. Below, we ormally establish that this type o multiplicity is a generic eature o the recursive equilibrium. Optimal Timing o Investment How does the individual investment decision depend on belies? Intuitively, more optimistic priors, in the orm o higher mean belies µ, should raise 3

14 aggregate investment due to the higher opportunity cost o delays. In turn, uncertainty may reduce investment or two reasons. First, higher uncertainty reduces the expected payo o the investment due to risk aversion. Second, because the investment is costly and irreversible there is an option value o waiting. This creates an extra reason to wait when uncertainty is high in order to gather additional inormation and avoid downside risks. More ormally, this delay occurs because, as highlighted in Lemma, mean belies are more volatile when uncertainty is high. Since the dierence between the value o waiting deined in 6 and the value o investing deined in 7 is a convex unction o mean belies, the higher volatility in mean belies caused by higher uncertainty makes waiting more attractive than investing. c c µ,γ F c µ,γ df p 0 F c µ,γ µ or γ Figure 6: Investment probability as a unction o belies The ollowing proposition ormally establishes that this intuition is valid and provides a characterization o the optimal investment behavior. Proposition. Under the regularity condition in assumption stated in the appendix, given a random number o investing irms N µ,γ Bin N,pµ,γ or some pµ,γ and or γ x suiciently low, there exists a unique cuto or the ixed costs c µ,γ R {, } such that irms invest i and only i c µ,γ. The cuto c µ,γ is strictly increasing with µ and γ. This partial-equilibrium result characterizes the investment rule given the random number o investing irms N µ,γ and the laws o motion or µ and γ. Firms invest i and only i the idiosyncratic ixed cost alls below the threshold c µ,γ. Figure 6 depicts how the probability o investment is aected by the belies µ and γ. The upper-let panel shows the threshold as unction o the mean level o belies or the precision o inormation. For each level o belies, the bottom 4

15 panel shows the probability o investing, which corresponds to the shaded area below the ixed-cost distribution in the right panel. Crucially or what ollows, the probability o investment decreases with uncertainty. 4. Equilibrium Uniqueness WehaveestablishedinLemmasandhowbeliesdependonthenumberoinvestingirms, and in Proposition how irms investment decision is aected by belies. From the latter, a randomly chosen irm invests with probability F c µ,γ given an arbitrary unction pµ,γ. To ulill the recursive equilibrium deinition, we need that pµ,γ = F c µ,γ. The next proposition states that a unique equilibrium satisies this condition. Proposition. Under the regularity conditions in assumptions and stated in the appendix, and or γ x small enough, a recursive equilibrium exists and is unique. The expected number o investing irms is increasing in mean belies µ and decreasing in uncertainty i.e., increasing in the precision o belies γ. Figure 7 depicts the expected number o investing irms as a unction o belies µ,γ. The partial-equilibrium results rom Proposition carry through to the general equilibrium: investment is more likely as irms are more optimistic about the undamental µ high or less uncertain γ high. In turn, as we illustrated in the example o igure 5, when the number o irms that invests increases with γ there may be multiple ixed points in the joint dynamics o activity and uncertainty. In what ollows, we demonstrate that this multiplicity is a generic eature o the equilibrium and that it leads to persistent dynamics. 4.3 Uncertainty Traps We describe here the core mechanism o the paper. We assume at this point that the total number o irms N is large enough, so that N µ,γ N pµ,γ = F c µ,γ. With this assumption we can treat N as a deterministic unction o belies, ignoring luctuations due to the initeness in the number o irms. 6 We ormally deine an uncertainty trap as the existence o multiple stationary points in the dynamics o belie precision γ given a level o mean belies µ. Deinition. Given mean belies µ 0, there is an uncertainty trap i there are at least two locally stable ixed points in the dynamics o belies precision γ = ΓN µ 0,γ,γ. 6 O course, we must be careul in taking this limit to ensure that agents remain uncertain. See the appendix or a ormal statement. 5

16 N EN] 0 µ γ Figure 7: Example o aggregate investment pattern Uncertainty traps, deined as multiple ixed points in the dynamics o uncertainty given a level o mean belies, do not imply that there are multiple equilibria in the model. In act, in Proposition we have already established that the recursive equilibrium is unique. While multiple values o γ may satisy the requirement that γ = ΓN µ,γ,γ given a value o µ, the recursive equilibrium is unique because mean belies µ evolve stochastically, as characterized in Lemma. We deined to each ixed point in the dynamics o belies as a regime. Do uncertainty traps necessarily arise? We ormally establish the existence o a range o values or µ or which the economy necessarily eatures uncertainty traps. An important condition on the parameters which guarantees the existence o uncertainty traps is that the dispersion in the distribution o ixed costs, σ, is not too large. This ensures a strong enough eedback rom inormation to investment. 7 Proposition 3. Under the conditions o Proposition and or σ small enough, there exists a non-empty interval µ l,µ h ] such that, or all µ 0 µ l,µ h, the economy eatures an uncertainty trap with at least two regimes γ l µ 0 < γ h µ 0. Regime γ l γ h is characterized by high low uncertainty and low high investment. Figure 8 oers an example or the law o motion o γ given dierent values o µ. For the range o values o µ in µ l,µ h ], the dynamics o belie precision admits two locally stable regimes. This is the range highlighted in Proposition 3. For values o µ above µ h, the dynamics o belies only 7 Intuitively, as the distribution o ixed costs becomes less dispersed, the number o investing irms N µ,γ becomes steeper with changes in belies. See the appendix or details. 6

17 0.8 N = 0 N = N N/N = F c γ µ > µ h µ = µ h µ = µ l µ < µ l 0. γ l γ h γ Figure 8: Example o dynamics or precision γ admits the high-activity regime, while or values below µ l, it only admits the low-activity regime. Hence, or a non-negligible range o mean belies, the economy may luctuate between extremes o no activity or ull activity. O course, the mean o the belies distribution evolve in the background ollowing a random walk. Thereore, the economy may remain in one o the two regimes or a while, but eventually escaping i µ drits suiciently ar away. Proposition 3 establishes that the situation depicted in this igure is a generic eature o the equilibrium. Uncertainty traps give rise to non-linear aggregate dynamics, business cycle asymmetries and shocks that may have near permanent eect on the economy. Figure 9 illustrates these eects though various simulations based on the example rom Figure 8. The top panel presents three dierent series o shocks to the mean belies µ. The three series start rom the high activity/low uncertainty regime. The economy is hit at t=5 by a negative shock to mean belies, either due to a particularly bad realization o the public signals or the undamental. The economy returns to normal rom t=0 onwards. Across the three series, what varies is the magnitude o the initial drop. The middle and bottom panels show the response o belie precision γ and the number o investing irms N. The total number o irms that invest prior to the shock equals N. The solid gray line represents a small temporary shock. Ater the shock hits, irms still ind it proitable to invest, the number o investing irms remains equal to N, and the precision o belies is unaected. When the economy is hit by a temporary shock o slightly larger magnitude, some irms stop investing, leading to a gradual increase in uncertainty. As uncertainty rises, investment goes down even urther and the economy starts to drit towards the low regime. However, when mean belies 7

18 0.6 µt γt Nµt,γt N t Figure 9: Persistent Eects o Temporary Shocks recover, the precision o inormation and the number o active irms quickly return to the highactivity regime. In contrast, when the economy is hit by an even larger temporary shock, such as the dotted line, the number o irms delaying investment is large enough to produce a sel-sustaining increase in uncertainty. The economy quickly shits to the low-activity regime and remains trapped there even ater mean belies recover to the starting position. How does the economy escape an uncertainty trap? Figure 0 depicts the evolution o the economy aterit ishitby thelargeshock romfigure9. Asbeore, thelarge negative shockhitting the economy rom periods 5 to 0 pushes the economy into the low activity-high uncertainty regime. Eventually, the economy receives positive signals that lead to a temporary increase in mean belies between periods 0 and 5, maybe because o positive realizations o the undamental. When the temporary increase in average belies is not suiciently strong, the recovery is short-lived. However, when the temporary increase is suiciently large, the economy reverts back to the high-activity regime. Once again, temporary shocks o suicient magnitude to the undamental may lead to permanent eects on the economy. A number o additional lessons can be drawn rom these simulations. First, uncertainty only matters in cases where irms do not have an overwhelming preerence or either investing or waiting. Second, in this ramework uncertainty is a by-product o recessions. This result echoes some o the empirical indings rom Bachmann et al. 03. Third, as in models with learning in the spirit o Van Nieuwerburgh and Veldkamp 006, this theory provides an explanation or asymmetries in business cycles. Finally, the simulations also highlight that agents can be uncertain about the undamentals without necessarily being uncertain about endogenous variables. For example, in 8

19 µt γt Nµt,γt N t Figure 0: Escaping an Uncertainty Trap the low activity regime, irms can predict the level o investment because they know that they are uncertain and trapped in the bad equilibrium. This highlights a potential diiculty with identiying uncertainty in the data. As implied by the model, accuracy o orecasts may possibly be a bad proxy or uncertainty about undamentals. 4.4 Policy Implications The existence inormation rictions raises the question o eiciency. In the decentralized equilibrium irms invest less oten than they should rom a social perspective because they do not internalize that their actions release inormation to the rest o the economy. Proposition 4 shows that the decentralized economy is constrained ineicient. But a simple policy instrument such as an investment subsidy that only depends on current belies µ, γ is suicient to make irms internalize their impact on the rest o the economy and implements the eicient allocation. Proposition 4. The decentralized competitive equilibrium is constrained ineicient and the symmetric, socially eicient allocation can be implemented with positive investment subsidies τ µ, γ. In turn, when γ x and σ are small, the eicient allocation is still subject to uncertainty traps. Proposition 4 implies that irms are more likely to invest in the eicient allocation. It does not mean, though, that uncertainty traps cannot arise in the constrained-eicient allocation. Even though such situations seem to occur due to a lack o coordination between irms, there are cases when the planner cannot do any better than the decentralized economy and ends up trapped in a similar ashion. I the planner does not have any additional inormation than agents in the economy, it is still optimal to wait when uncertainty is too high. Hence, there still exists a strong complementarity between inormation and the level o activity in the constrained eicient-allocation, although uncertainty traps might be less likely to arise in that case. 9

20 5 Quantitative Analysis Preliminary] To quantiy the impact o uncertainty traps we enrich the model in several dimensions. We introduce a Cobb-Douglas production unction with labor and capital as inputs. We also introduce an intensive margin or investment, so that irms can now precisely choose their capital stock, and a representative household that supplies labor and owns the irms. These additions allow us to evaluate the eect o uncertainty traps on employment and wages. 5. Technology Now, each irm operates a Cobb-Douglas production unction to produce the unique consumption good. A irm n employing l units o labor and using k units o capital produces output where with distributions q n = e y k α nl α n, y = θ+ε y θ = ρ θ θ+ε θ ε θ N 0, ρ θ γ θ ε y N 0,γy. As beore, the stochastic process θ is the undamental. Now, it ollows an AR process instead o a random walk as in our baseline model. Next-period capital or surviving irms evolves according to k n = δ +i n k n, where δ denotes the depreciation rate. When a irm invests, it must pay a ixed cost n k n in units o the inal good and a variable cost ci n k n. As in the baseline model, 0 is an i.i.d. random variable with density g. The unction c is strictly convex and continuously dierentiable. By making the ixed and variable investment costs proportional to the current capital stock, we can solve the investment problem independently rom the capital stock and preserve tractability. A irm exits with probability ω. In that case, a raction ξ o its capital is destroyed while the remainder is assigned to a new entering irm. A irm can only invest i it has an investment opportunity. Firms without an opportunity randomly receive one with probability q. A irm that does not exert its investment opportunity carries it into the ollowing period, but each irm can only hold one investment opportunity at a time. 0

21 5. Timing and Inormation The timing and the inormation diusion process closely ollow rom the baseline model. At the beginning o a period, irms hold belies about the state o the undamental, θ I N µ,γ. As beore, µ and γ denote the mean and the precision o these belies. A irm that invests obtains a signal x n = θ+ǫ x n about theundamental. Asinthebaselinemodel, wedenoteby N theendogenous number o irms that decide to invest in a given period. We assume that, at the end o the period, a irm observes the decision o all the other irms in the economy. This is equivalent to observing the public signal X = x n = θ +ε X N N n I where I is the set o irms that invest, and ε X N 0,Nγ N x. Importantly, the precision o this aggregate public signal is increasing in the number o irms that decide to invest in the current period. Firms also learn about the undamental by observing the aggregate productivity y. The timing o events is as ollow:. All irms share the same prior distribution over the undamental θ.. Firms that do not hold an investment opportunity receive one with probability q. 3. Firms that hold an investment opportunity observe their ixed cost n and decide whether or not to invest. I they pay the cost, they invest i n. 4. The irms that invested receive a signal: Each irm n that invests observes a private signal x n = θ +ε x n. Ater observing x n, each irm chooses labor l n. In equilibrium, this 5. The common shock y is revealed and observed by everyone. Actions are observed. All irms produce and markets clear. 6. Each irm survives with probability ω. I a irm dies, it is replaced by a new irm with capital equal to a raction ξ o the exiting irm. 7. Agents update their belies or next period. The key dierence with the baseline model is that, now, the labor decision o irms is revealing o their private inormation. As beore, all irms decide whether or not to invest based on their common inormation and ixed cost. Those who invest, also decide how much capital to add. Once the investment has taken place, irms observe a signal and, based on that signal, hire a number o workers. Because the number o workers hired by a irm is monotonic in the signals, and because

22 labor is observed, the inormation diused through the economy again increases linearly with the number o irms that invest. 5.3 Representative Household A representative household consumes C units o the inal good and supplies L units o labor to irms in a competitive environment. The household holds wealth in irm shares and maximizes lietime expected utility given its budget constraint. Optimality leads to standard labor-consumption choices satisying and as well as, wω,x,y = U LCΩ,X,y,LΩ,X,y U C CΩ,X,y,LΩ,X,y pω,x,y = U C CΩ,X,y,LΩ,X,y, where we denote the state o the economy at the beginning o each period by Ω. With preerences given by U = C σ σ and the price satisies hl, the wage is wω,x,y = h C σ pω,x,y = C σ. 5.4 Firm Problem We letk be the average stock o capital in the economy and Q be the raction o irms with an investment opportunity. Then, the aggregate state at the beginning o the period as Ω = {µ,γ,k,q}. The value unction o a irm with an opportunity to invest is where the value o waiting is { V Ω,k = E max V w Ω,k,maxE x V i Ω,x,i,,k ]}], i V w Ω,k = max l E y,x pω,x,y e y k α l α lwω,x,y +ωβv Ω, δk Ω ] and the value o investing is

23 V i Ω,x,i,,k = max l E y,x pω,x,y e y k α l α lwω,x,y cik k +ωβ qv Ω, δ +ik + qv 0 Ω, δ +ik ] Ω,x ]. Similarly, the value o a irm without an investment opportunity is V 0 Ω,k = max l E y,x pω,x,y e y k α l α lwω,x,y + ωβ qv Ω, δk + qv 0 Ω, δk ] Ω ]. To simpliy these expressions, it is useul to introduce the ollowing notation and w = E y,x wω,x,y Ω], ep = E y,x e y pω,x,y Ω], pw = E y,x pω,x,ywω,x,y Ω] w x = E y,x wω,x,y Ω,x], ēp x = E y,x e y pω,x,y Ω,x], pw x = E y,x pω,x,ywω,x,y Ω,x]. The labor demand decision o irms implies that all value unctions are linear in k. We can thereore write the values per unit o capital as { v Ω = E max v 0 Ω = α ep α v w Ω = α ep α }], v w Ω,maxE x v i Ω,x,i ] p i α pw α α α + δωβey,x v Ω Ω ], pw α α α px ci pw x + δ +iωβe y,x qv Ω + qv 0 Ω Ω,x ]. v i Ω,x,i = α ep x α Finally, the policy unction or investment is α α + δωβey,x qv Ω + qv 0 Ω Ω ], ωβ iω,x = c E y,x qv Ω + qv 0 Ω Ω ], p x 3

24 and a irm decides to invest i and only i E x v i Ω,x,i ] v w Ω > p. 5.5 Aggregation To aggregate the model analytically we take the limit as the total number o irms goes to ininity and the precision o each individual signal goes to 0 in a way that keeps the precision o inormation unaltered. 8 In the limit, each irm disregards its private inormation at the hiring stage. Thereore we ignore the value unction s dependence on x and proceed with the aggregation. Under these assumptions, the laws o motion o inormation perceived by agents replicate those in the baseline model. The only dierences is that, now, the next-period prior is adjusted by the act that the undamental ollows an AR process instead o a random walk, µ = ρ γµ+γ yy +Nγ x X γ +γ y +Nγ x γ ρ = + ρ σθ. γ +γ y +Nγ x The raction o irms that invest, Ñ Ω, satisies ÑΩ = QG maxe x v i Ω,i ] v w Ω / p. i The average capital by irm K evolves according to K Ω = Kω δ +NΩiΩ+ ωξ. The law o motion o the raction o irms with an investment opportunity is: The average output is given by Q Ω = ωnq +Q N+ Qq+ ωq YΩ,X,y = Ke y α ep n pw n Finally, the consumption o the representative household is ˆ CΩ,X,y = YΩ,X,y K g ciω+ α α. v w Ω < maxe x v i Ω,i ] p i d. 8 More precisely, we assume that the precision o each private signal is γ x = N γ x, and then take the limit as N but. See the appendix or details 4

25 5.6 Simulation Preliminary The model can be solved numerically by iterating on the aggregate variables w, p and N, and the laws o motion K and Q. All these objects are unctions o the state space Ω. The algorithm consists o an outer loop that iterates on the vector w,p,ñ,k,q and an inner loop that iterates on the value unction and the policy decision. The linear structure o the model allows us to solve the ull general equilibrium exactly without using approximations in the spirit o Krussell-Smith. To evaluate the quantitative inluence o uncertainty traps we perorm a preliminary simulation. The goal o the simulation is to illustrate that uncertainty traps can create additional persistence in the macroeconomic aggregates. 9 The parameters values or the simulation are in Table. Parameter Value Preerence or leisure h = Discount actor β = 0.95 Capital intensity α = /3 Depreciation rate δ = % Persistence o undamental ρ θ = 0.98 Precision o ergodic distribution o undamental γ θ = 0 Precision o productivity shocks γ y = 0 Precision o public signal i all irms enter γ x = 00 Distribution o ixed cost pd ζ or 0, max ] Parameter o distribution o ixed cost ζ = 00 Upper bound o the support o ixed cost max = 0.0 Probability o receiving an investment opportunity q = 0% Fraction o capital that new irms keep ξ = 0% Table : Parameter values or the numerical simulations Once the value unctions and the prices have been computed, we can examine the entry behavior o irms or dierent belies µ,γ about the undamental. Figure shows the number o active irms as unction o belies. As in the baseline model, the number o active irms increases both with mean belies and with the precision o inormation. As we demonstrated in the simpler setup, this opens up the possibility o uncertainty traps. Theeect o the uncertainty trap more clear whenthe shocks to the economy are large. To illustrate this, consider the evolution o the economy when it is hit by a small shock to the undamental, equal to /0 o its standard deviation. The impulse response unctions to this shock are shown in the solid lines o Figure. The dashed line represents the evolution o this economy when the precision o belies is kept ixed at its initial level. Thereore, the dierence between the two lines corresponds to the the additional impact o endogenous uncertainty. Because a small raction o irms stops investing, the impact on uncertainty is not suiciently strong and the economy behaves similarly with ixed and with endogenous uncertainty. 9 In current work, we are developing a proper calibration exercise. So ar, this is a simulation or illustrative purposes. 5