Frictional unemployment with stochastic bubbles

European Economic Review 122 (2020) 103352

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European Economic Review journal homepage: www.elsevier.com/locate/euroecorev

Frictional unemployment with stochastic bubblesR Guillaume Vuillemey a,b, Etienne Wasmer b,c,∗ a b c

HEC Paris, France CEPR, United Kingdom New York University in Abu Dhabi, United Arab Emirates

a r t i c l e

i n f o

Article history: Received 30 April 2019 Accepted 1 December 2019 Available online 13 December 2019 Keywords: Bubbles Labor frictions Unemployment volatility

a b s t r a c t We show that the volatility puzzle in labor economics (Shimer, 2005) stems from the inability of technology shocks to generate sufficient volatility of firm value. We introduce non-fundamental shocks to firm value, akin to bubbles, into an otherwise standard searchand-matching model. When calibrated to stock market data, stochastic bubbles significantly improve the ability of the matching model to quantitatively explain the volatility of the US labor market. An extension with multiple sectors improves the persistence of simulated labor market variables. © 2019 Published by Elsevier B.V.

1. Introduction Economic history shows a number of events of seemingly irrational dynamics of asset prices, such as bubbles, crashes and panics (Kindleberger, 1978). In standard macroeconomic models, bubbles – that is, deviation of asset prices from their fundamental value – are ruled out by assuming that individual rationality is common knowledge. Indeed, all agents foresee that bubbles cannot survive forever, due to future resource constraints or transversality conditions. As a result, models that aim to explain the high volatility of financial variables, and the associated real effects, have to rely on additional assumptions: liquidity constraints, high intertemporal elasticities of substitution, habit formation, or other shocks. Recently, influential papers have renewed the interest in bubbles as determinants of business cycles (Carvalho et al., 2011; Martin and Ventura, 2012; 2018; Fahri and Tirole, 2012) and of the sectoral allocation of workers (Cahuc and Challe, 2012). Labor economics has studied unemployment volatility, albeit with very limited references to asset price bubbles. Shimer (2005) convincingly showed that the standard search-and-matching model of the labor market (Pissarides, 1985; 20 0 0) fails to generate enough volatility of labor market tightness with only technological shocks. To motivate our analysis, we first show that, while failing to deliver sufficient volatility, the standard model generates the right ratio between R We are grateful to the editor (Peter Rupert), one anonymous associate editor and two anonymous referees for excellent feedback. We also thank participants at the NBER 2018 SI, 2017 Bubbles in Macroeconomics at CREI, the 2016 SaM conference (Amsterdam), OFCE-SKEMA in Nice, IIES Stockholm, CREI at Pompeu Fabra, U. Autonoma Barcelona, IUE Florence, UC Irvine Seminar, Stanford University, UC Santa Barbara, NYU AD and University of Southern California and HEC lunch seminar, in particular Vladimir Asriyan, Gadi Barlevy, Edouard Challe, Tom Cooley, Juan Dolado, Jordi Gali, Pablo Guerron-Quintanan, Wouter den Haan, Bob Hall, Christian Haefke, Tomohiro Hirano, Ryo Jinnai, Ramon Marimon, Finn Kydland, Albert Marcet, Alberto Martin, Guido Menzio, Sebastian Merkel, Alessandra Pelloni, Torsten Persson, Monika Piazzesi, Christopher Pissarides, Jean-Marc Robin, Guillaume Rocheteau, Peter Rupert, Robert Shimer, Eric Swanson, Jean Tirole, Jaume Ventura, Pengfei Wang, Philippe Weil and John Wooders. The paper was partly written during a stay at Stanford University (Etienne Wasmer) and at Harvard University (Guillaume Vuillemey), the hospitality of which is gratefully acknowledged. Financial support fromANR-11-LABX-0091 and ANR-11-IDEX- 0 0 05-02 is gratefully acknowledged. ∗ Corresponding author. E-mail addresses: [email protected] (G. Vuillemey), [email protected] (E. Wasmer).

https://doi.org/10.1016/j.euroecorev.2019.103352 0014-2921/© 2019 Published by Elsevier B.V.

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the volatility of labor market variables and the volatility of the firm value. Therefore, the main issue to be resolved is the low volatility of firm value in the matching model. Hall (2017) recognized the link between financial variables and unemployment and explains the fluctuations of both sets of variables with exogenous changes in stochastic discount factors. We propose that bubbles affecting firms’ value (akin to stock market bubbles) can help resolve a significant part of the Shimer (2005) puzzle. Our first contribution is to introduce bubbles in an otherwise standard search-and-matching model of the labor market. We solve the model in closed form and obtain explicit formulas for all quantities of interest. Bubbles arise stochastically, are associated with explosive dynamics of firms’ value, and burst stochastically. During bubbles, firm value grows exponentially, more jobs are created even though their real productivity is unchanged, and labor market tightness increases. Wages for new matches also increase, reflecting the change in market tightness. When the bubble bursts, firm value drops immediately, but unemployment increases at a slower pace reflecting the rate of job destruction. In the state where the bubble is latent, firm value remains higher than the fundamental value, because it incorporates expectations of capital gains due to future bubbles. Blanchard and Fischer (1989, Chapter 5) argued that ruling out bubbles requires a non-realistic degree of rationality and foresight. The simplest assumption that we make is that some (new-born) agents are not aware that the bubble needs to burst with probability one due to future resource constraints. They only form probabilities on the parameters of the bubble (stochastic appearance, burst and speed of propagation) that they observe from history. Older agents may be aware of the impossibility for bubbles to survive in the long term but face stochastic liquidity needs and randomly sell their assets to young agents. Young agents have no way to invalidate their priors about the structure of the economy. An alternative interpretation leading to the same equations is that all agents are fully rational, but full rationality that could lead to elimination of bubbles is not common knowledge. It is enough that agents believe that other agents do not foresee the low-probability event in which the resource constrained may bind. Then, we calibrate the model to assess its quantitative performance. Our calibration of bubbles is disciplined in two respects. First, we fix the Poisson parameters at which bubbles appear and disappear using historical data on investor sentiment by Baker and Wurgler (2006). Second, conditional on a bubble arising, there are an infinite number of solutions associated with divergent expectations, each of them characterized by a constant of integration. Recent empirical work by Giglio et al. (2016) enables us to restrict the set of admissible values for this constant, even though we also explore alternative values. Regarding labor market variables, we follow the calibration by Shimer (2005). Overall, the degrees of freedom in the calibration are thus very limited. Based on this calibration, our second contribution is to show that bubbles significantly improve the quantitative ability of the search-and-matching model to explain US data. First, the calibration of bubbles enables us to match the empirical volatility of firm value, even though it is not a moment that we explicitly target. The log standard deviation of firm value equals 0.153 in our model, as compared to 0.151 in the data (and only 0.017 in the model without bubbles). Second, the model considerably improves the ability of the matching model to replicate labor market moments – again, these moments are not targeted by our calibration. As an example, the standard model explains less than 10% of the log standard deviation of market tightness (0.035, as compared to 0.371 in the data). Instead, our model explains more than 60% of the variation (log standard deviation equal to 0.232). The empirical performance of the matching model also increases for other relevant moments. That said, our paper remains exploratory in several respects, and we highlight dimensions along which further improvements are needed. Finally, we propose an extension of the model with two sectors, with frictions affecting the reallocation of workers across sectors. This model helps understanding the sectoral mis-allocation of workers that follows from bubbles, as documented in empirical work by Charles et al. (2018), and following the theory by Cahuc and Challe (2012). This extension also increases the persistence of labor market variables, bringing them closer to data. The paper is organized as follows. Section 2 reviews the relevant literature, while Section 3 emphasizes the importance of financial volatility as a determinant of employment volatility. Section 4 derives the search-and-matching model with bubbles. We study its quantitative properties in Section 5, and a two-sector extension in Section 6. 2. Related literature The literature linking financial bubbles to labor market outcomes is small.1 Kocherlakota (2011) blends the OLG model of Samuelson (1958) with a matching model that does not satisfy the usual equilibrium job creation condition. In his model, the burst of bubbles does not affect unemployment when monetary policy is sufficiently accommodative – a result which differs from ours, since bubbly states in our model are necessarily characterized by allocative distortions. In Miao et al. (2016), credit constraints give rise to multiple equilibria, so that shifts in beliefs can create unemployment. Cahuc and Challe (2012) study an OLG economy in which bubbles induce skilled workers to move away from the productive sector to the financial sector. Guerron-Quintana et al. (2018) model a set of recurrent bubbles, similar to ours, and study crowding out effects on investments and growth. In contrast with these models, we build from the standard searchand-matching model (Pissarides, 1985; Mortensen and Pissarides, 1994). We use it to characterize fundamental and bubbly 1 A large literature studies the possibility of bubbles in fully rational models (Samuelson, 1958; Tirole, 1985; Santos and Woodford, 1997). Such bubbles reduce dynamically inefficient investment. In Martin and Ventura (2012), bubbles driven by shocks to investor sentiment can raise investment.

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solutions, and to explore their quantitative properties. As such, our paper contributes to the large body of research that proposes potential resolutions of the “unemployment volatility puzzle” highlighted by Shimer (2005).2 A set of papers relate pricing in financial markets to labor market outcomes, in the absence of bubbles. Most importantly, Hall (2017) shows that changes in stochastic discount factors explain a significant part of unemployment fluctuations. In particular, when the discount rate rises, the present value of expected profits from filled vacancies falls, so that job creation falls. In his work, a Markovian matrix with five states corresponding to different stochastic discount factors is able to replicate the empirical volatility of employment. While changes in discount rates can explain aggregate unemployment fluctuations, they cannot rationalize sectoral shifts in the market valuation of firms, as we document below. Furthermore, the ultimate source of variation in discount rates remains unexplained, and one of the five aggregate states features a negative month-to-month interest rate, a property that is akin to bubbles, a justification for exploring non-fundamental shocks. Related, Kilic and Wachter (2018) show that time-varying disaster risk can explain the joint behavior of stock market valuations and unemployment. Recently, Branch and Silva (2019) replicate the comovement between unemployment and asset prices in a search-and-matching model with liquidity shocks and limited commitment. Relative to these papers, we focus on non-fundamental fluctuations in asset prices. Finally, a related literature studies the impact of non-fundamental shocks to stock prices on investment in physical assets, rather than on hiring decisions. Gilchrist et al. (2005) suggest that firms can undo the effect of bubbles by issuing shares, so that the impact on investment is limited. More recently, a set of papers, reviewed by Bond et al. (2012), show that shocks to stock prices affect real investment. This is because firm managers learn from stock prices, so that a higher stock price leads managers to reassess the value of the capital expenditures, and to invest more. Interestingly, the empirical evidence shows that managers invest more in response to both fundamental and non-fundamental shocks to asset prices (Chen et al., 2007; Dessaint et al., 2019). 3. Motivating facts While the search-and-matching model (Pissarides, 20 0 0) is appealing theoretically, its ability to match the data has been challenged by Shimer (2005). In the standard model with exogenous wages, where the only source of fluctuations are productivity shocks, the model generates a volatility of labor market variables that is about 10 times lower than in US data. We replicate and confirm this finding below (Section 5.3). The failure of the standard model to generate enough volatility can be understood by studying its main equation, which characterizes the value of a job J 0 ,

J0 =

x−w γ = , r+s q (θ )

(1)

where θ is the equilibrium value of labor market tightness for a given productivity x and wage w, q(θ ) is the job filling rate for a given matching function, γ is a recruiting cost, s is the job destruction rate, and r is the discount rate. Eq. (1), later derived explicitly, has an intuitive interpretation: in continuous time, the present value of a job equals a flow of future profit (x − w) discounted by the interest rate plus the job destruction rate (r + s). With a constant wage, a standard CobbDouglas matching function, and considering log deviations around the steady state, the elasticity of labor market tightness with respect to productivity is

d. ln θ = d. ln x

1

x

η (θ ) x − w

,

(2)

where η (θ ) = −θ q (θ )/q(θ ) ∈ (0, 1 ) the elasticity of the hiring rate. In (2), we clearly see the puzzle highlighted by Shimer (2005): the volatility of labor market tightness is low relative to the volatility of productivity. Using the same logic, we highlight a relation that has remained unnoticed so far. Regardless of the wage-setting rule, the following relation between the volatility of labor market tightness and the volatility of job value holds,

d. ln θ = d. ln J 0

1

η (θ )

.

(3)

Eq. (3) leads to a reassessment of the Shimer (2005) puzzle. Indeed, the elasticity of the matching function η is often estimated to be around 0.5. Therefore, the model predicts a volatility of labor market tightness that is twice as large as that of firm value.3 Using stock market data (S&P Composite Index) to compute the volatility of firm value, we show below that this relation holds almost exactly in the data: the empirical volatility of labor market tightness is 0.371, while that of firm value equals 0.151 (Panel A of Table 2). Therefore, to explain the Shimer (2005) puzzle, the main issue to be resolved is the low volatility of firm value. Our proposed solution to the employment volatility puzzle is the existence of non-fundamental shocks to the value of firms, akin to bubbles. Economic history is full of asset pricing events which are hard to explain using technology shocks 2 Other solutions to the puzzle include adding wage rigidity (Hall, 2005), financial frictions (Wasmer and Weil, 2004) or alternative calibrations (Hagedorn and Manovskii, 2008), among others. 3 In the single-worker firms of the standard matching model, J is both job value and firm value.

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Panel A: Stock market valuations and unemployment

20

0

10

-1 1950

1970

1980

1990

2000

20

0

10

-1 1950

0 1960

1

2010

Unemployment rate (%)

1

P/E ratio (log st. dev.)

P r ice-ea r n in g r a t io

Unemployment rate (%)

S&P Composite (log st. dev.)

S&P C om p osit e

0 1960

1970

1980

1990

2000

2010

Panel B: Stock market valuations and job finding rate

0.8

1

0.6

0

0.4

-1 1950

0.2 1960

1970

1980

1990

2000

2

0.8

1

0.6

0

0.4

-1 1950

Job finding rate (%)

2

P/E ratio (log st. dev.)

P r ice-ea r n in g r a t io

Job finding rate (%)

S&P Composite (log st. dev.)

S&P C om p osit e

0.2 1960

1970

1980

1990

2000

Fig. 1. Aggregate stock market valuation and labor markets. Panel A plots the US unemployment rate against both the stock market valuation (left panel) and the price-earning ratio (right panel) over the period from 1951Q1 to 2014Q4. Panel B plots the US job finding rate against both the stock market valuation (left panel) and the price-earning ratio (right panel) over the period from 1951Q1 to 2007Q4. Stock market data is for the S&P Composite Stock Price Index. For the stock market level and P/E ratio, we show cyclical variation obtained by HP-filtering the log of these variables, with smoothing parameter equal to 10 0,0 0 0. Unemployment is in percentage of the labor force. NBER recession periods are shaded. Details on the construction of the data series are provided in Appendix A.

only, and are thus often referred to as “bubbles”.4 Recent examples include the Black Monday crash of 1987, the Dot-com bubble of 1999–20 0 0, and the stock market crash of 20 08–20 09. Furthermore, Fig. 1 shows evidence linking stock market valuations to unemployment over the 1951–2014 period (Panel A). Panel B shows that these changes in unemployment are closely tied to changes in job finding rates, consistent with the evidence by Shimer (2012).

4. Bubbles in a search-and-matching model We present our baseline search-and-matching model with bubbles.

4

Shiller (1981) shows that the volatility of stock prices cannot be explained by the volatility of dividends.

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4.1. Setup There is a measure one of risk-neutral agents discounting future cash flows at rate r > 0. At a given point in time, v vacant positions exist, n workers are employed and u = 1 − n are unemployed. Unmatched entrepreneurs post vacancies, hire one worker and produce, until they retire and sell the firm at market value.5 The tightness of the labor market is θ = v/u. We assume the existence of a matching function m(v, u) with constant returns to scale. Let q be the job filling rate and f the job finding rate, which satisfy respectively

q (θ ) =

m ( v, u )

v

f (θ ) =

and

m ( v, u ) = θ q(θ ). u

Existing jobs are destroyed at an exogenous Poisson rate s > 0, so that the law of motion of unemployment is

∂u = s(1 − u ) − θ q(θ )u. ∂t

(4)

When opening and keeping a vacancy unfilled, a firm incurs posting costs γ > 0 per unit of time. A successful match yields a constant per-period output flow x, which we interpret as labor productivity. Furthermore, x is assumed to exceed the value of unemployment benefits z > 0, so that there are bilateral gains from matching. The firm pays an employed worker a per-period wage w, and earns a profit flow x − w. We assume that agents form time-varying expectations about the market value of firms. There are two aggregate states, corresponding to different expectations about the dynamics of firm value. Bubbly states are characterized by expectations of a positive growth rate of firm value, in the absence of expected growth in profits. New unconstrained investors expect capital gains and do not consider that a resource constraint will bind in finite time. In contrast, latent states are characterized by stationary expectations of all agents. Transitions between states are modeled as exogenous shifts to the beliefs of agents, and can be interpreted as changes in investor sentiment (Baker and Wurgler, 2006). The instantaneous probability that a bubbly state arises from the latent state is ϒ ≥ 0, while bubbly states end stochastically, with instantaneous probability  ≥ 0.6 Bubbles affect all firms simultaneously, that is, we assume that shares in all firms are publicly traded (or, equivalently, that no friction prevents secondary market trading for private firms).7 Furthermore, agents cannot short-sell. Whenever ϒ = 0 or  → +∞, the model boils down to the fundamental model of Pissarides (20 0 0). Indeed, in these cases, either bubbles never appear or they disappear at infinite speed. 4.2. Firm value and equilibrium Variables in the latent and bubbly state are denoted with superscripts L and B. In the bubbly state, asset values depend on the time elapsed since divergent expectations have started. The age of a bubble is denoted τ , and JB (t, τ ) is the date-t value of a firm τ periods after the start of the bubble. We denote an investor by i and subjective expectations Ei . The value of filled and unfilled vacancies are denoted generically J and Jv . An investor values a job in state L as

JiL (t ) = (x − w )dt + +

  1 × ϒ dtEi JiB (t + dt, 0 ) + sdtEi JvL,i (t + dt ) 1 + rdt

  1 × (1 − sdt − ϒ dt )Ei JiL (t + dt ) , 1 + rdt

(5)

and in state B when the bubble has age τ as

JiB (t, τ ) = (x − w )dt + +

  1 × dtEi JiL (t + dt ) + sdtEi JvB,i (t + dt, τ + dt ) 1 + rdt

  1 × (1 − sdt − dt )Ei JiB (t + dt, τ + dt ) . 1 + rdt

(6)

Similarly, the asset value of a vacancy in state L is

JvL,i (t ) = − γ dt + +

 1  ϒ dtEi JvB,i (t + dt, 0 ) + q(θt )dtEi JiL (t + dt ) 1 + rdt

 1  [1 − q(θt )dt − ϒ dt ]Ei JvL,i (t + dt ) , 1 + rdt

(7)

5 Retirement of entrepreneurs occurs at a Poisson rate ν . Upon this shock, entrepreneurs exit the market and sell the firm, whether the vacancy is filled or not. There is no disruption for labor or for vacancies, so that the liquidity/retirement shock ν has no implication for the equations of the model. It simply implies that a mass of firms is traded each period in financial markets. New-born investors acquire these firms at their market value. 6 A mnemonic to remember notation is as follows: ϒ resembles the opening of a bottle of Champagne and its bubbles, while  resembles the superposition of letters E and X, as in “explosion”. 7 We relax this assumption in Section 6. Based on US data, employment in listed Compustat firms represents 48% of civilian employment (series CE16OV in FRED). Non-listed firms can be affected by bubbles through both a higher likelihood of IPO (so-called “IPO waves”) and through higher expectations of the resale value in a secondary market.

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and in a bubble of age τ ,

JvB,i (t, τ ) = − γ dt + +

  1 × dtEi JiL (t + dt ) + q(θt )dtEi JiB (t + dt, τ + dt ) 1 + rdt

  1 × [1 − q(θt )dt − dt ]Ei JvB,i (t + dt, τ + dt ) . 1 + rdt

(8)

Definition of equilibrium. Given the two aggregate states S = {L, B}, an equilibrium with constant wages and free-entry for vacancies is a list



     θt,S τ , JS (t, τ ), JvS (t, τ ), E JS (t + dt, τ + dt ) , E JvS (t + dt, τ + dt ) , wS (t, τ ), u ,

satisfying, for all i, t and τ , four entry conditions,

Ei JvL,i (t + dt ) = Ei JvL,i (t ) = 0,

(9)

Ei JvB,i (τ + dt, t + dt ) = (1 + Rt dt )Ei JvL,i (t ) = 0,

(10)

where Rt denotes the growth rate of asset values, two coordination equations,

Ei JiL (t + dt ) = Ei JiL (t ) = J L (t ),

(11)

Ei JiB (τ + dt, t + dt ) = (1 + Rt dt )Ei JiB (τ , t ) = J B (τ , t ),

(12)

two asset value equations for jobs 5 and (6), two asset value equations for vacancies 7 and (8), one wage rule for each aggregate state (derived in Section 4.4), and the steady state equation on unemployment flows (4).8 In rational models with infinite horizons, two conditions prevent bubbly solutions from existing. The first one is the transversality condition, often thought of as an individual rationality condition by analogy with a finite horizon model: at the terminal time, risk-averse agents want zero-cash holding and liquidate their assets. In infinite horizon, this does not hold and the conditions under which the transversality condition is an individual rationality condition typically imply curvature in utility and decreasing returns in production.9 We have linear value functions and this argument does not apply here. The second condition is a feasability condition. In standard rational models, agents know that the economy cannot sustain a bubble with a growth rate above the growth rate of the economy (Santos and Woodford, 1997). Indeed, after some date T, all resources will be exhausted. If T is common knowledge, agents will liquidate their assets before T. To allow bubbles to survive, we make one assumption. At each date, a fraction of entrepreneurs needs to sell their firms, which requires them to find new-born entrepreneurs to buy at the current price. We assume that new-born entrepreneurs know that the bubble may burst stochastically, but that they do not know that a feasibility constraint will bind in finite time. This sustains the equilibrium. Even if older investors, holding the firm, know that there is such a T, they also know that they will find an investor as long as the economy is at date t < T. The fact that new-born investors do not know about the feasibility constraint captures the idea that they have limited experience and a short history of realizations of bubbles to form expectations. 4.3. Solutions We now let dt tend to zero to get continuous-time solutions with a well-defined growth rate for bubbly assets. Asset values for filled and unfilled vacancies are









rJ L = x − w + s JvL − J L + ϒ J B (0 ) − J L +







∂ JL , ∂t

(13)



rJ B (τ ) = x − w + s JvB (τ ) − J B (τ ) +  J L − J B (τ ) +

∂ JB , ∂t

 L  L L    ∂ JL θt J − Jv + ϒ JvB (0 ) − JvL + v , ∂t      ∂ J B (τ ) rJvB (τ ) = −γ + q θtB J B (τ ) − JvB (τ ) +  JvL − JvB (τ ) + v . ∂t rJvL = −γ + q

(14) (15) (16)

We maintain the assumption of a free-entry for vacancies, so that JvL = 0 and JvB (τ ) = 0 for any τ . It follows that ∂ JvL /∂ t = 0 and ∂ JvB (τ )/∂ t = 0. Transitions from bubbly to latent stages affect the sentiment of all agents and therefore the values JB or JL of all firms. 8 A path with positive Rt is possible. Given our assumptions, rational investors do not bet against the market to eliminate bubbly pricing by backward induction. 9 See Dorfman (1969) and Ekeland and Scheinkman (1986).

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4.3.1. Firm value in the latent state We keep wages unspecified here.10 A steady-state equilibrium in the latent state implies that ∂ J L /∂ t = 0. The equilibrium job creation condition, obtained from (13) and (15), is

γ x − w + YJ B ( 0 )   = JL = , r+s+Y q θL

(17)

where θ L is the steady-state labor market tightness in the latent state. Eq. (17) states that the expected cost of a vacancy in the latent state equals the present value of future profits, plus the present value of capital gains due to future bubbles, which may start with probability ϒ . Future flows are discounted to account for the fact that both job destruction and the appearance of bubbly episodes are stochastic events. The latent state is characterized by a value of filled vacancies which is strictly higher than in the fundamental state (for which we denote variables with a superscript 0). Indeed, θ L > θ 0 whenever ϒ > 0. This is because the value of a vacancy incorporates expectations of capital gains due to future bubbles. Hence, the latent state differs from the fundamental state of the economy. In this sense, a bubble is always present, but in a “latent” form. 4.3.2. Firm value in the bubbly state We make no assumption on wage here, except that it is fixed between two aggregate states, whether the firm was created in the latent state (in which case the wage can possibly be renegotiated when a bubbly episode starts, see Appendix B.1 for calculations) or in the bubbly state (in which case the wage for new hires can depend on the duration of the bubbly state but remains constant after job creation). Hence, the asset value of a filled job in the bubbly state (14) rewrites as a differential equation,

(r + s + )JB (τ ) = x − w + JL +

∂ J B (τ ) . ∂t

(18)

Solving for (18), the job creation condition in the bubbly state satisfies

J B (τ ) =

x − w + J L + KeRτ , r+s+

(19)

where we denote by

R=r+s+

(20)

the growth rate of the bubble. This growth rate of firm value is the sum of the risk-free-rate and the destruction rate of the bubbly asset, consistent with Blanchard and Watson (1982). The first term in (19) is the present discounted value of future profits, plus that of future capital losses arising from the stochastic end of the bubbly episode. The second term is the value of the bubble τ periods after it started. K is a constant of integration. It is indeterminate but can be interpreted as capturing the magnitude of the initial jump in firm value. 4.3.3. Jumps in firm value across states The appearance of bubbly episodes is characterized by a jump in firm value, followed by an exponential increase until the bubble bursts. To characterize this dynamics, let

(τ ) = JB (τ ) − JL ,

(21)

denote the capital gain from a bubble of duration τ , relative to the latent state. Using the definition of the fundamental firm value J0 , one has

JL = J0 +

ϒ r+s

(0 ).

(22)

Relative to the fundamental value, the value of a filled vacancy in the latent state is distorted upward, since the possibility that bubbles start, with probability ϒ , is priced. Moreover, the value of a filled position in the bubbly state is

J B (τ ) = J 0 +

R  KeRτ −

(τ ), r+s r+s

(23)

which equals the fundamental value J0 , plus the capitalized value of the bubble after τ periods, net of the present value of future capital losses due to the burst of the bubble. The values of the capital gains from the bubble, (τ ), and that of the initial jump, (0), can be derived in closed-form. Replacing (22) and (23) into (21) yields

(τ ) = 10

R  ϒ KeRτ −

(τ ) −

(0 ). r+s r+s r+s

Below, we derive solutions for several wage determination rules.

(24)

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After some calculations (see Appendix B.1), the value of (0) follows,

(0 ) = K

R . R+ϒ

(25)

Therefore, with constant wages, the jump in firm value at the inception of the bubble is proportional to the constant of integration K divided by the sum of all relevant discount factors. Furthermore, for any wage profile,

 

(τ ) − (0 ) = K eRτ − 1 ,

(26)

that is, firm value grows exponentially after the emergence of a bubble. Together, (25) and (26) are central equations describing the dynamics of bubbles in firm value. A question is whether K can be pinned down under the assumption of continuity in firm value. Eq. (26) shows that it is not possible. Indeed, solving for J B (0 ) = J L yields K = 0. Imposing continuity of the firm value function rules out bubbles. Therefore, when K = 0, our model boils down to the fundamental search-and-matching economy by Pissarides (1985, 20 0 0). Instead, the appearance of a bubble needs to be associated with a jump in firm value that K characterizes fully. Thereafter, we treat K as a parameter. Our restriction to positive bubbles implies K > 0.11 4.3.4. Labor market tightness across states We now characterize the dynamics of labor market tightness. In the bubbly state, by (16), the value of a filled job satisfies

J B (τ ) =

q

γ  . θτB

(27)

Since the same job creation condition prevails in the latent state (Eq. (17)), the inception of a bubble is therefore associated with a jump in labor market tightness. In the latent state, the value of labor market tightness is constant, even immediately after a bubble bursts from a higher value. Indeed, tightness immediately converges to the final steady-state (due to free entry), while vacancies and unemployment slowly converge. Using (22) and replacing J0 by its value yields

γ γ ϒ R  =  + K, L 0 r + s R + ϒ q θ q θ

(28)

where θ 0 is the fundamental value of labor market tightness. Tightness in the latent state is higher than in the fundamental state, by a term increasing in K. Intuitively, firms post more vacancies in the latent state than in the fundamental state, in order to already operate and benefit from capital gains when a bubble emerges. After the bubble starts, one can characterize market tightness for τ = 0, using (25), (25) and (28),

 γ γ ϒ R  B  =   + 1+ K. r+s R+ϒ q θτ =0 q θ0

After this jump, as seen in Appendix B.1, it is easy to show the exponential increase of labor market tightness for any duration τ of the ongoing bubble satisfies

 γ γ ϒ Rτ   =   +K e + . (R + ϒ )(r + s ) q θτB q θ0

Bubbles distort the fundamental allocation by increasing the tightness of the labor market above the fundamental and over time. As workers and firms are matched at a higher rate, this induces unemployment to decrease. 4.4. Wages We show that bubbles exist when wages are the endogenous outcome of a Nash bargaining process. We can now introduce superscripts L and B for wages. To begin with, the value functions of employed workers (WL and WB ) and of unemployed workers (WuL and WuB ) are









rW L = wL + s WuL − W L + ϒ W B − W L +







∂W L , ∂t 

rWuL = z + θ q(θ ) W L − WuL + ϒ WuB − WuL +

∂ WuL , ∂t

(29)

(30)

11 Another question pertains to the interpretation of K, which is partly behavioral. It can be interpreted as presbyopia (difficulty to see small objects), rather than myopia about the future, in the following sense. When agents see changes in asset prices below some detection threshold, they do not infer that sustainable growth in asset prices will follow. In contrast, when they see a positive change in prices above the detection threshold (e.g., 2% in our calibration), they coordinate on the positive value of K. In that sense, the value of K can be interpreted as the smallest possible fluctuation in asset prices above which some agents coordinate on the bubbly path.

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352









rW B = wB + s WuB − W B +  W L − W B +







∂W B , ∂t 

rWuB = z + θ q(θ ) W B − WuB +  WuL − WuB +

∂ WuB . ∂t

9

(31) (32)

In contrast with asset equations for firm value, bubbles can be ruled out in asset equations for workers. Indeed, bubbles in firm value exist because a firm’s stock can be bought and sold in a perfectly liquid market. There is no such liquid market for labor, so that all time derivatives in (29) to (32) are equal to zero. When bargaining occurs, the surplus of a match is assumed to be shared according to the Nash rule. We denote by α > 0 the bargaining power of workers. The wage wL is time-independent and solves



 

wL = argmax W L (w ) − WuL α J L (w ) − JvL

1−α

,

yielding a surplus-sharing rule

  (1 − α ) W L − WuL = α JL .

In the latent state, replacing value functions and solving for wL yields





wL = (1 − α )z + α x + θ L γ ,

(33)

as proved in Appendix B.2. In the bubbly state, labor market tightness increases over time, which is reflected in the solution to the bargaining problem at the time of job creation. Let τ  ≥ τ denote the time at which the different quantities are evaluated, where τ is the time elapsed since the start of a bubbly state. The solution for wages maintains the asset value of a job as a linear combination of a constant (in τ ) and of a term growing as exponential of Rτ  . Let JB (0, τ  ) denote the value of a job created at the time when the bubbly state started and evaluated after τ  periods, and wB (0, τ  ) the wage of a job re-negotiated at the time the bubbly state started and considered after τ  periods. We have





wB (0, 0 ) = (1 − α )z + α x + θτB=0 γ . wB (0,

(34)

τ  ) is allowed to evolve over time, firm value would no longer be a combination of a

We prove in Appendix B.4 that if constant and of an exponential term. It would instead be the solution of a generalized Riccati equation, with finite limits at infinity despite interesting endogenous dynamics; see also Vuillemey and Wasmer (2019). These solutions however do not qualify as a bubble. Therefore, we restrict attention to solutions we call “semi-flex wages”, where wages are bargained at the time of job creation, and remain constant until the aggregate state (latent or bubbly) changes. When this occurs, wages are renegotiated based on the current value of labor market tightness.12 Consistent with this specification, Section 6 shows evidence that wages are relatively insensitive to bubbly episodes, as opposed to employment. Based on these assumptions, simpler notation can be used, wB (0 ) = wB (0, 0 ). The entry wage for a firm created during a bubbly episode with duration τ is





wB (τ ) = (1 − α )z + α x + θτB γ .

(35)

It follows that the value of a firm created when a bubbly state emerges is

J B ( 0, τ  ) =

x − w B ( 0 ) + J L  + KeRτ . r+s+

For a job created after the start of a bubbly state, when τ > 0, the value of a new firm has a similar form. However, since the current value of θτB is higher, the wage negotiated at the time of job creation reflects these better conditions (for workers). Therefore, the value of a new job created τ periods after the bubbly state started and considered in τ  > τ , is

J B (τ , τ  ) =

x − w B ( τ ) + J L  + KeRτ . R

(36)

The equilibrium job creation condition is obtained by substituting the wage rule in the job creation condition. In the latent state, using the jump function (0) and rearranging yields

γ ϒ (1 − α )(x − z ) − αθ L γ  = +

(0 ). L r+s r+s q θ

(37)

As compared with the entry equation in the fundamental model, expectations of future bubbles and associated capital gains

(0) lead to a higher number of vacancies being posted, thus to higher equilibrium tightness θ L > θ 0 . In the bubbly state, at the time of job creation τ , substituting the wage in the job creation condition yields, after some calculations (in Appendix B.3),

γ  (x − z )(1 − α ) − αθτB γ  = + KeRτ + [K − (0 )]. r+s r+s q θτB 12

Appendix B.5 discusses the conditions under which workers may want to quit in the semi-flex case.

(38)

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G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352

v θτB+ dτ C

θτB=0

B

θL

A D u Fig. 2. Dynamics of the economy with stochastic bubbles. This figure illustrates dynamic labor market adjustments in the (u, v) space during a bubble. The blue line is the Beveridge curve. The red lines are job creation conditions. When a bubble starts, the economy jumps from A to B, and evolves to C until the bubble explodes (state D) and returns slowly to the latent steady state A. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.5. Short-run dynamics We illustrate the workings of the model during bubbles. In Fig. 2, we plot the Beveridge curve together with the equilibrium job creation condition.13 The ergodic state in the latent state is given by the intersection of the Beveridge curve and of the job creation condition in that state (point A). The appearance of a bubble leaves the Beveridge curve unchanged but rotates the value of equilibrium market tightness counterclockwise, since θτB=0 > θ L . The adjustment of market tightness is instantaneous. Indeed, vacancies immediately jump to their new optimal value, pinned down by the free-entry condition. Unemployment adjusts more slowly, at the speed of convergence given by transition rates s and f(θ ). At τ = 0, u remains unchanged, so the economy jumps to point B. Then, if the bubble were not to grow, the (v, u) locus would slowly converge to the intersection between the new job creation condition and the Beveridge curve. Instead, since the bubble expands, there are two joint and competing dynamics characterizing the (v, u) locus. First, firm value γ /q(θτB ) grows at an exponential rate R. The increase in market tightness induces firms to post more and more vacancies. Second, there is also partial convergence to the Beveridge curve. The economy evolves along the dotted line that leads to point C. Convergence is never achieved when θτB increases along with the bubble. When the bubble bursts (point C), vacancies jump down, and unemployment then converges to its equilibrium latent value, characterized by a market tightness θ L . The convergence of the (v, u) locus to the Beveridge curve is achieved only in the latent state. These dynamics are obviously absent in a model without bubbles.14 5. Calibration and quantitative properties We calibrate the model and assess its quantitative performance. 5.1. Extension to productivity shocks To calibrate the model, we first assume that aggregate productivity x varies stochastically, according to a Poisson process with intensity λ. The distribution of productivity is characterized by a c.d.f. G(x) with support [xmin , xmax ]. The value of a filled vacancy in the latent and bubbly states is respectively

(r + s + ϒ + λ )JL (x ) = x − wL + λ



xmax xmin

J L ( x ) dG ( x ) + ϒ J B ( τ = 0, x ),

(39)

The Beveridge curve is obtained by setting ∂ u/∂ t = 0 in (4). In an extension of the model with permanent productivity shocks (as in Section 5.1), the model dynamics in the (v, u) space follows the counter-cyclical loops described by Pissarides (1985). 13 14

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352

and

(R + λ )JB (τ , x ) = x − wB + JL (x ) + λ



xmax xmin

J B ( τ , x ) dG ( x ) +

∂ JB . ∂t

11

(40)

We postulate the existence of solution of the form a(x + b) + Kegτ where firm value grows exponentially at a constant rate g between two productivity changes. It is easy to verify that g = r + s +  = R, as before, and that a = (r + s + )−1 . Similarly, looking for linear solutions for J L (x ) = a (x + b ), one easily verifies that a = (r + s + ϒ ). 5.2. Calibration To calibrate the model, we follow Shimer (2005) to a large extent. One period of the model is assumed to represent one quarter. At this frequency, the quarterly discount rate is r = 0.012. We set the separation rate s to 0.1, implying an average job duration of 2.5 years. We set the value of leisure z to 0.4, which can therefore be interpreted as unemployment benefits. The cost of posting a vacancy is set to γ = 0.213. We also set workers’ bargaining power to α = 0.72.15 Furthermore, we model labor productivity as an AR(1) in logs,

log (xt ) = ρ log (xt−1 ) + t ,

(41)

with ρ ∈ [0, 1] and t ∼ N (0, σ ). We transform (41)

into a discrete-time

 Markov chain using Tauchen (1986)’s method, letting log (xt ) take 35 points of support in − 5σ / 1 − ρ 2 , +5σ / 1 − ρ 2 . Parameters ρ and σ are calibrated so as to match the empirical standard deviation and autocorrelation of labor productivity (see Appendix A for details on data used). We obtain ρ = 0.92 and σ = 0.011. The only departure from Shimer (2005) is with respect to the matching function. Instead of using a Cobb-Douglas specification, we follow den Haan et al. (20 0 0), by using m(u, v ) = vu/(vζ + uζ )1/ζ . In a discrete-time calibration, this matching function has the appealing feature of being bounded between 0 and 1. During bubbles, since labor market tightness may rise well above its fundamental value, this feature is particularly relevant. Following den Haan et al. (20 0 0), we calibrate ζ = 1.27, which allows replicating empirical matching probabilities.16 To calibrate ϒ and , information on the frequency of occurrence of bubbles, as well as on their average duration, is needed. We follow the historical account of stock market bubbles by Baker and Wurgler (2006), whose definition is close to ours, since they explain the aggregate behavior of the stock market by shifts in investor sentiment. They document four main bubbles over the 1961–2002 period: i) 1961 to 1962, ii) 1967 to 1968, iii) late 1970s to 1983, iv) 1999–20 0 0. This account suggests that there is on average one bubbly episode every 10 years, and that a bubbly episode lasts on average 2 years. We take this approximation and consider that, within an average period of 10 years, the economy spends 2 years in the bubbly state and 8 years in the latent state. Consequently, we calibrate, at a quarterly frequency, ϒ = 1/32 and  = 1/8. To address potential limitations, we also consider alternative calibrations below.17 Finally, while K can in principle take any value, we fix it based on recent work by Giglio et al. (2016). Focusing on the housing market, they design tests of bubbles by comparing the value of freeholds with that of leaseholds (which have maturities of up to 700 years). The difference in value between these two types of contracts can be interpreted as K in our model. While the difference in the value between freeholds and leaseholds is generally not significant, we use for K a value of 2% that falls within their confidence interval, near their median estimate. The interpretation of K in this case, which determines the size (0) of the bubble at its inception, is as a behavioral threshold for agents: they do not update expectations about the aggregate state when changes in firm value are below a threshold. All calibrated parameter values are summarized in Table 1. 5.3. Baseline results We compare moments simulated from the model with both the data and with the fundamental model (i.e., with K = 0, to rule out bubbles). Moments computed from US data over the 1951–2014 period are displayed in Panel A of Table 2.18 Then, to simulate each model, we let the economy start at its non-stochastic steady state, simulate 300 periods of data, and drop the first 50 periods. The resulting sample contains 250 periods of simulated data (62.5 years), and therefore closely resembles our US data series, which last for 252 quarters (63 years). 10 0 0 such samples are simulated. Standard deviations for the moments of interest are computed across these simulated samples. All moments are computed using log deviations from an HP trend with smoothing parameter 10 0,0 0 0. As an illustration, Fig. 3 plots 50 years of simulated data both for the fundamental model and for the model with bubbles. 15 The value of α in Shimer (2005) comes from the Hosios rule with a Cobb-Douglas matching function. When the matching function is not Cobb-Douglas, as we assume, the elasticity of the matching technology is not constant anymore. We thus study an alternative value of α = 0.5 in Panel B of Table 3, and find that the results are close to unchanged. 16 With this matching function, the job filling rate is q(θ ) = (1 + θ ζ )−1/ζ . 17 Potential shortcomings include: (i) the limited length of the available stock market series, (ii) the difficulty to precisely define bubbly episodes, as well as their duration, and (iii) the inability to detect potentially short-lived bubbles. 18 For consistency with the literature, we follow Shimer (2005) by converting all variables at a quarterly frequency and HP-filtering them with a smoothing parameter equal to 10 0,0 0 0. Standard deviations are from HP trends.

12

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352

0.5

1.08 1.06

0.45 1.04 1.02

0.4

1 0.35

0.98 0.96

0.3 0.94 0.92

0.25 0

20

40

60

80

100

120

140

160

180

200

1.6

20

40

60

80

100

120

140

160

180

200

20

40

60

80

100

120

140

160

180

200

0.88

1.5 0.87

1.4 1.3

0.86

1.2 0.85

1.1 1

0.84

0.9 0.8

0.83

0.7 0.82

0.6 20

40

60

80

100

120

140

160

180

200

Fig. 3. Simulated time series data. This figure plots one simulation of the model over a period of 50 years (200 quarters). In each chart, the blue line corresponds to the fundamental model and the red line to the model with bubbles. Both are simulated using the baseline calibration of Table 1. The same path of productivity shocks is used to simulate both models. We report the simulated time series data of productivity (x), firm value (J), labor market tightness (θ ) and employment (n). Table 1 Calibrated parameter values. Parameter

Definition

Value

Source

s r z q (θ )

Separation rate Discount rate Value of leisure Matching function Workers’ bargaining power Cost of posting a vacancy

0.1 0.012 0.4 (1 + θ 1.27 )−1/1.27 0.72 0.213

Shimer (2005) Shimer (2005) Shimer (2005) den Haan et al. (2000) Shimer (2005) Shimer (2005)

ϒ 

Probability that a bubbly state starts Probability that a bubbly state ends Initial value of bubble on firm value

1/32 1/8 2% · J0

Baker and Wurgler (2006) Baker and Wurgler (2006) US data

Productivity Standard deviation of productivity Autocorrelation of productivity

Stochastic 0.011 0.92

See (41) US data US data

α γ

K x

σ ρ

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352

13

Table 2 Data vs. simulated moments. This table contains moments both computed from the data (Panel A) and simulated from two models: the fundamental model with no bubbles (Panel B) and the model with bubbles (Panel C). Data is at a quarterly frequency, over the 1951–2015 period. In both models, the wage is computed based on the labor market tightness at the time of job creation and updated with productivity shocks. Moments for all variables are computed using log deviations from an HP trend with smoothing parameter 10 0,0 0 0. Calibrated parameter values are summarized in Table 1. Bootstrapped standard errors are in parentheses. Standard errors are computed using 10 0 0 model solutions for 250 quarters. Persistence refers to the autocorrelation of the first differences for each variable. Panel A: Data moments — 1951–2014 Labor market moments

St. dev. Autocorr. Persistence u v

Correlation matrix

θ f x J

Financial moments

u

v

θ

f

x

J

J/(x − w )

0.200 0.948 0.588

0.187 0.943 0.634

0.371 0.947 0.670

0.136 0.935 0.120

0.020 0.897 0.102

0.151 0.923 0.333

0.161 0.931 0.350

1 – – – – –

−0.875 1 – – – –

−0.964 0.970 1 – – –

−0.950 0.826 0.913 1 – –

−0.330 0.359 0.356 0.327 1 –

−0.485 0.431 0.468 0.535 0.178 1

−0.493 0.448 0.483 0.505 0.335 0.938

Panel B: Fundamental model (wages depend on current productivity and entry θ ) St. dev. Autocorr. Persistence

u v

Correlation matrix

θ f x J

0.011 (0.001) 0.921 (0.010) 0.271 (0.034)

0.018 (0.001) 0.607 (0.036) −0.280 (0.030)

0.026 (0.002) 0.837 (0.021) −0.083 (0.035)

0.014 (0.001) 0.837 (0.021) −0.083 (0.035)

0.020 (0.001) 0.848 (0.020) −0.046 (0.035)

0.011 (0.001) 0.837 (0.021) −0.083 (0.035)

0.027 (0.003) 0.959 (0.009) 0.147 (0.044)

1 – – – – –

−0.557 1 – – – –

−0.819 0.933 1 – – –

−0.819 0.933 0.999 1 – –

−0.825 0.920 0.993 0.993 1 –

−0.819 0.932 0.999 0.999 0.994 1

0.306 −0.057 −0.172 −0.172 −0.177 −0.172

Panel C: Model with bubbles (wages depend on current productivity and entry θ ) St. dev. Autocorr. Persistence

u v

Correlation matrix

θ f x J

0.069 (0.014) 0.912 (0.012) 0.286 (0.015)

0.179 (0.045) 0.666 (0.040) −0.212 (0.017)

0.232 (0.059) 0.811 (0.017) −0.050 (0.010)

0.082 (0.016) 0.841 (0.021) −0.025 (0.016)

0.020 (0.001) 0.848 (0.020) −0.047 (0.035)

0.153 (0.042) 0.793 (0.017) −0.059 (0.010)

0.356 (0.101) 0.929 (0.027) 0.352 (0.149)

1 – – – – –

−0.690 1 – – – –

−0.828 0.977 1 – – –

−0.822 0.940 0.969 1 – –

−0.118 0.083 0.099 0.154 1 –

−0.811 0.973 0.992 0.927 0.068 1

0.287 0.013 0.098 0.017 0.012 0.011

Simulated moments are also collected in Table 2. We focus first on financial moments. Interestingly, the log standard deviation of firm value obtained from the model is extremely close to its data equivalent (0.153 versus 0.151), and significantly higher than in the fundamental model without bubbles (log standard deviation of 0.017). This result is remarkable since our calibration did not explicitly try to match the volatility of firm value. Then, turning to the volatility of labor market variables, the model provides significant quantitative improvement relative to the fundamental matching model. The volatility of labor market tightness θ is 9 times higher than in the fundamental model with no bubbles and a similar wage rule (0.232 instead of 0.026). The volatility of the job finding rate is 6 times higher than in the fundamental model (0.082 vs. 0.014). Both are reasonably close to the data (0.371 and 0.136, respectively), even though they are slightly lower. The volatility of vacancies (equal to 0.179) is also close to that observed in the data (0.187), and much higher than the one generated by the fundamental model (0.018). Finally, unemployment in our model is six times more volatile than in the fundamental model (0.069 vs. 0.011). Next, we report correlation matrices between all main variables (Table 2). It is noticeable that the introduction of bubbles breaks the near-perfect correlation between productivity shocks and labor market tightness that characterizes the fun-

14

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352 Table 3 Simulated moments - Alternative specifications. This table contains simulation results based on an alternative wage rule (Panel A) and on alternative parameterizations (Panel B). In Panel A, the wage is computed based on the productivity and the labor market tightness at the time of job creation, and is constant afterwards. Panel B studies wider sets for parameters governing bubbles, changing on parameter at the time, keeping all others at their baseline value. Moments for all variables are computed using log deviations from an HP trend with smoothing parameter 10 0,0 0 0. Calibrated parameter values are summarized in Table 1. Bootstrapped standard errors are in parentheses. Standard errors are computed using 10 0 0 model solutions for 250 quarters. Persistence refers to the autocorrelation of the first differences for each variable. Panel A: Alternative wage rule Labor market moments u St. dev. Autocorr. Persistence

St. dev. Autocorr. Persistence

v

θ

Financial moments f

x

J

J/(x − w )

Fundamental model (wages fixed with entry productivity and entry θ ) 0.055 0.086 0.124 0.071 0.020 0.054 (0.005) (0.005) (0.010) (0.007) (0.001) (0.004) 0.923 0.607 0.836 0.834 0.849 0.838 (0.011) (0.037) (0.022) (0.023) (0.020) (0.021) 0.286 −0.275 −0.084 −0.085 −0.046 −0.083 (0.037) (0.032) (0.036) (0.039) (0.035) (0.035)

0.066 (0.007) 0.912 (0.018) −0.054 (0.040)

Model with bubbles (wages fixed with entry productivity and entry θ ) 0.087 0.199 0.262 0.107 0.020 0.150 (0.012) (0.041) (0.051) (0.014) (0.001) (0.040) 0.917 0.657 0.817 0.837 0.849 0.799 (0.010) (0.036) (0.016) (0.019) (0.019) (0.015) 0.290 −0.225 −0.056 −0.049 −0.047 −0.061 (0.024) (0.022) (0.019) (0.029) (0.035) (0.015)

0.385 (0.094) 0.934 (0.024) 0.378 (0.159)

Panel B: Alternative parameterizations Volatility

ϒ

Baseline



α

K

f x

0.069 0.179 0.232 0.082 0.020

1/24 0.076 0.199 0.257 0.090 0.020

1/40 0.063 0.162 0.210 0.075 0.020

1/4 0.055 0.169 0.207 0.067 0.020

1/12 0.068 0.152 0.204 0.082 0.020

0.5% 0.049 0.096 0.132 0.059 0.020

1% 0.052 0.119 0.157 0.062 0.020

2% 0.069 0.179 0.232 0.082 0.020

3% 0.079 0.223 0.285 0.094 0.020

4% 0.085 0.278 0.312 0.099 0.020

5% 0.096 0.345 0.358 0.111 0.020

0.5 0.067 0.172 0.228 0.080 0.020

J J/(x − w )

0.153 0.356

0.170 0.398

0.138 0.323

0.145 0.263

0.124 0.428

0.078 0.279

0.096 0.298

0.153 0.356

0.195 0.399

0.243 0.451

0.299 0.532

0.149 0.351

u v

θ

damental model – a feature that was most criticized. In the fundamental model, this correlation is 0.993, while it falls to 0.099 with bubbles (as compared to 0.356 in the data). Bubbles create autonomous dynamics of labor market tightness due to expectations of growth in firm value.19 Productivity is no longer the only variable that moves incentives to post vacancies.20 Importantly, the additional volatility generated by bubbles reflects changes in the quantity of labor demanded by firms, due to changes in expectations, and not hidden wage stickiness. Indeed, given our choice of the wage rule, the model with bubbles embeds additional flexibility of wages than the fundamental model. This is because wages are renegotiated when bubbles start or end. This feature of the model biases us against finding additional volatility. Indeed, it is well-documented, at least since (Shimer, 2005), that lower wage flexibility is needed to generate higher volatility. Furthermore, the fact that the effect of bubbles on the labor market operates primarily through hirings is consistent with the aggregate evidence in Fig. 1: there is a strong empirical correlation between bubbles and hiring decisions. It is also consistent with recent empirical evidence. For example, Charles et al. (2018) shows that, during the US housing boom in the 20 0 0s, college attendance declined, due to improved labor market opportunities, particularly in the real estate sector. It is also consistent with the evidence that we bring, in Section 6, on the technology boom of 20 0 0.21 Finally, we stress that the model fails to deliver a high enough persistence (i.e., autocorrelation of growth rates) of the labor market variables. This is also a shortcoming of the fundamental model. Again, this is a consequence of the fact that 19 Note that the model does not make specific predictions regarding the correlation between θ and J. Indeed, this correlation is close to 1 both in the fundamental model (0.999) and in the model with bubbles (0.992). This is a general feature of the search-and-matching model, where the only incentive to post vacancies arises from a high value of J, whether due to good productivity shocks or to bubbles. This induces a mechanical correlation between J and θ . 20 In Appendix C.1, we study the social planner problem, and show that a low correlation between labor market variables and productivity highlights that deviations from the constrained efficient allocation are potentially large. Indeed, it means that a sizable part of the observable labor market dynamics does not contribute to increasing the fundamental values of utilities and profits. 21 Empirically, the effect of the boom does not need to come only from hiring decisions by existing firms. The creation of new firms in booming industries is also consistent with the model.

G. Vuillemey and E. Wasmer / European Economic Review 122 (2020) 103352

15

vacancies immediately jump to their new equilibrium value after a shock. Unemployment, which is a state variable, is the only variable with positive and significant persistence, yet lower than in the data (0.286, as compared to 0.588 in the data and to 0.271 in the fundamental model). In our model, expansion during a bubbly episode generates additional persistence. However, the burst of the bubble is associated with an extremely fast drop in vacancies and market tightness. Together, these two forces are not sufficient to generate higher persistence. 5.4. Alternative parameterizations Next, we briefly explore how alternative specifications affect the simulated moments. As a first exercise, we adopt an alternative wage rule. The wage depends on the level of productivity at the time of job creation and remains constant afterwards. Therefore, wages are less flexible than in our baseline case. As is well-known, this generates higher volatility in the fundamental model. In Panel A of Table 3, we show that this alternative wage rule indeed augments the volatility of labor market tightness by a significant factor. It now reaches 0.124 in the fundamental model, i.e., a third of the value observed in the data. In the model with bubbles, the same wage rule generates a volatility of labor market tightness which is twice as high, equal to 0.262. It is therefore even closer to the data. From this exercise, we again see that the model with bubbles does not rely on wage rigidity to generate higher volatility. Instead, the autonomous dynamics of job creation through expectations is sufficient to create large volatility. Second, using our benchmark wage rule, we explore in Panel B the sensitivity of simulated moments to the three parameters capturing the occurrence and magnitude of bubbles, ϒ ,  and K. We vary one parameter at the time, while keeping all other parameters at their baseline value. The main takeaway is that simulated volatilities are not overly sensitive to the choice of the parameters governing bubbles. In particular, the introduction of bubbles provides significant quantitative improvement to the fundamental model even if bubbles have a low probability of appearing, a low average duration, or are associated with small initial jumps in firm value. Therefore, the ability of a model with bubbles to generate higher volatility is not due to the specifics of our baseline calibration. For example, reducing ϒ from 1/32 to 1/40 (i.e., a bubble starts after 10 years in the latent state on average) generates standard deviations of θ and f of 0.210 and 0.063 (as compared to 0.026 and 0.014 in the fundamental model, and 0.371 and 0.136 in the data), respectively. Similarly, even with an average duration of one year ( = 1/4), bubbles generate substantial volatility of market tightness and of the job finding rate. With respect to K, we see that even with a very low initial jump in firm value (K = 0.5%), the model still leads to significant improvement relative to the fundamental model: the volatility of θ is 0.132, as opposed to 0.232 with the baseline calibration and 0.026 in the fundamental model. Not surprisingly, increasing the frequency of bubbles (higher ϒ ) or the initial jump in value (higher K) makes it possible to generate higher volatility levels, close to the data. However, this comes at the cost of a volatility of the financial variables (firm value and the P/E ratio) which is larger than in the data. 6. Two-sector model We now study an extension of the baseline model to two sectors. 6.1. Empirical evidence on sectoral bubbles We consider cases where only one sector faces bubbles, and reallocation of workers across sector is subject to frictions. There are two empirical interpretations of this model. First, it can be that only listed firms are directly affected by bubbles, while non-listed firms are only affected indirectly through reallocation. In this case, the bubbly sector can be interpreted as the set of listed firms. Second, our preferred interpretation is that bubbles may affect only one sector of the economy, rather than the aggregate stock market. To motivate this second interpretation, we use graphical evidence from the technology boom of 1999–20 0 0. First, Panel A of Fig. 4 shows that the market valuation of technology stocks (Nasdaq) relative to other stocks (Dow Jones Industrial Average) increased considerably in the months leading up to the March 20 0 0 crash. Then, Panel B shows that the total number of workers employed in the technology sector (normalized to 100 in January 1995) increased at a faster pace than the number of workers in other sectors in the years leading to 20 0 0 crash. After 20 0 0, it decreased back to a level similar to that of the pre-bubble period. The same holds true after detrending the series (right figure). Finally, Panel C shows that financial fluctuations affect labor markets primarily through hiring decisions rather than through wages. Indeed, after detrending, we see no significant difference between wages in the technology sector relative to other industries.22 6.2. Model with sectoral bubbles The main equations of the model on the firm’s side are unchanged. Indexing by 0 the fundamental value of firms in the sector without bubble, and by L and B the bubbly episodes in the sector affected by bubbles, the solution, at a given wage, are identical to those of the one-sector model. The value of firms in the fundamental sector is not necessarily stationary, as 22

The wage data are described in Appendix A. They include some, but not all stock options, and therefore partly underestimate wage volatility.

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Fig. 4. Stock market valuations, employment and wages during the technology bubble. Panel A plots the Dow Jones Industrial Average and the Nasdaq Composite Index over the 1995–2005 period (“technology bubble”). The Nasdaq is concentrated on technology firms while the DJIA is a general index. We normalize both series to 100 in January 1995. Panel B plots employment in the technology sector and in all other sectors over the same period. The left figure plots the total number of employed workers, normalized to 100 in January 1995. The right figure plots the same variables in log deviations from trends. Panel C plots hourly earnings in the technology sector and in all other sectors over the same period. The left figure plots the level of hourly earnings, normalized to 100 in January 1995. The right figure plots the same variables in log deviations from trends. Trends are obtained using an HP filter with smoothing parameter equal to 10 0,0 0 0. Details on the construction of the data series are provided in Appendix A.

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Table 4 Simulated moments – Sectoral model. This table contains our simulation results for the model with sectoral bubbles. The wage is computed based on the labor market tightness at the time of job creation and updated with productivity shocks. Moments for all variables are computed using log deviations from an HP trend with smoothing parameter 10 0,0 0 0. Calibrated parameter values are summarized in Table 1. Bootstrapped standard errors are in parentheses. Standard errors are computed using 10 0 0 model solutions for 250 quarters. Persistence refers to the autocorrelation of the first differences for each variable. Labor market moments u

θ

v

Financial moments f

x

Wages depend on current productivity and entry θ 0.049 0.153 0.189 0.075 0.020 (0.013) (0.042) (0.055) (0.015) (0.001) 0.916 0.703 0.803 0.821 0.848 (0.015) (0.016) (0.012) (0.018) (0.020) 0.353 −0.054 0.087 0.111 −0.046 (0.011) (0.014) (0.015) (0.013) (0.035)

St. dev. Autocorr. Persistence

J

J/(x − w )

0.133 (0.042) 0.760 (0.015) 0.054 (0.010)

0.297 (0.087) 0.910 (0.024) 0.222 (0.0135)

Persistence of θ with alternative parameterizations

Persistence of θ

=0

 = 1/160

 = 1/32

 = +∞

−0.052 (0.018)

0.087 (0.015)

0.198 (0.021)

−0.040 (0.016)

it is affected by a growing bubble in the other sector. We assume that workers reallocate across sectors and denote by  the speed of reallocation between the two sectors. This parameter may depend on the difference in unemployment values across sectors. Employed workers cannot quit, to simplify the analysis. One has









rWuB = z + θ q(θ ) W B − WuB +  WuL − WuB + (Wu0 − WuB )1Wu0 >WuL + rWu0 = z + θ 0 q

∂ WuB , ∂t

 0  0  ∂ Wu0 , θ W − Wu0 + (WuL − Wu0 )1WuL >Wu0 + ∂t

where 1X>Y is a dummy taking value 1 if the inequality X > Y is satisfied and 0 otherwise. Total labor force across sectors is normalized to one. Employed and unemployed workers are denoted respectively n and u in the bubbly sector, and n0 and u0 in the fundamental sector, with u + u0 + n + n0 = 1. There are two regimes for unemployment flows, one in which unemployed workers in the fundamental sector move to the bubbly sector, and one in which the opposite happens. Labor flows are represented by

∂u = sn − f (θ )u − 1Wu0 >Wu  u + 1Wu0 Wu  u − 1Wu0 Wu0 and from u to u0 when Wu0 > Wu . In the absence of mobility, when  = 0, the two sectors are solved separately and the equations of Section 4 hold. The total unemployment rate is a weighted average of the unemployment rate of each sector. Weights are given by the population inherited from the past in each sector. In the polar case in which  = +∞, mobility is perfect across sectors. In this case, the equilibrium condition is WuS = Wu0 , S = L, B. We report in Table 4 the simulation results for  = 0 and  = +∞ (no reallocation and frictionless reallocation, respectively), as well as intermediate cases  = 1/32 and  = 1/160, when workers who get trained in one sector have difficulties moving to another sector. With either no reallocation or perfect reallocation, the persistence of θ remains close to zero. Persistence increases for intermediate values of θ , yielding an inverted U-shaped relationship. Intuitively, persistence arises from slow reallocation across sectors. When reallocation is subject to heavy frictions, there is persistence during downturns. However, labor flows are also limited in the bubbly state, so the magnitude of the distortions to be corrected is limited. The opposite is true when reallocation is frictionless: distortions are large during the boom, but they are corrected quickly during busts, generating little persistence. We conclude that, while the persistence of the labor market tightness θ can be raised in the model with sectoral bubbles, up to about 0.198, the value remains far from its empirical counterpart (0.670). For other moments, the sectoral models brings only limited improvement. 7. Conclusion We show that non-fundamental shocks to asset values can be an important determinant of unemployment volatility. When calibrating financial market bubbles with historical data, the quantitative ability of the search-and-matching model to reproduce the empirical volatility of labor market variables is greatly improved. That said, our work also opens new

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questions. First, while the model generates higher volatility, it fails to generate higher persistence. One potential avenue to increase persistence would be to further extend the model with sectoral bubbles. Second, another avenue to better match some of the moments of interest would be to bring our model of bubbles closer to recent works linking labor market volatility either to fluctuations in stochastic discount factors or to financial market imperfections (Hall, 2017; Branch and Silva, 2019). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.euroecorev.2019. 103352. References Baker, M., Wurgler, J., 2006. Investor sentiment and the cross-section of stock returns. J. Financ. 61, 1645–1680. Blanchard, O.J., Fischer, S., 1989. Lectures on Macroeconomics. MIT Press. Blanchard, O. J., Watson, M. W., 1982. Bubbles, rational expectations and financial markets. NBER Working Paper 945. Bond, P., Edmans, A., Goldstein, I., 2012. The real effects of financial markets. Annu. Rev. Financ. Econ. 4, 339–360. Branch, W., Silva, M., 2019. Unemployment and the stock market when households lack commitment. Working paper. Cahuc, P., Challe, E., 2012. Produce or speculate? asset bubbles, occupational choices and efficiency. Int. Econ. Rev. 53, 1005–1035. Carvalho, V., Martin, A., Ventura, J., 2011. Bubbly business cycles. Working paper. Charles, K.K., Hurst, E., Notowidigdo, M.J., 2018. Housing booms, labor market opportunities, and college attendance. Am. Econ. Rev. 108, 2947–2994. Chen, Q., Goldstein, I., Jiang, W., 2007. Price informativeness and investment sensitivity to stock price. Rev. Financ. Stud. 20, 619–650. den Haan, W.J., Ramey, G., Watson, J., 20 0 0. Job destruction and propagation of shocks. Am. Econ. Rev. 90, 482–498. Dessaint, O., Foucault, T., Fresard, L., Matray, A., 2019. Noisy stock prices and corporate investment. Rev. Financ. Stud. 32, 2625–2672. Dorfman, R., 1969. An economic interpretation of optimal control theory. Am. Econ. Rev. 59, 817–831. Ekeland, I., Scheinkman, J.A., 1986. Transversality conditions for some infinite horizon discrete time optimization problems. Math. Oper. Res. 11, 216–229. Fahri, E., Tirole, J., 2012. Bubbly liquidity. Rev. Econ. Stud. 79, 678–706. Giglio, S., Maggiori, M., Stroebel, J., 2016. No-bubble condition: model-free tests in housing markets. Econometrica 84, 1047–1091. Gilchrist, S., Himmelberg, C., G., H., 2005. Do stock price bubbles influence corporate investment? J. Monet. Econ. 52, 805–827. Guerron-Quintana, P.A., Hirano, T., Jinnai, R., et al., 2018. Recurrent Bubbles, Economic Fluctuations, and Growth. Technical Report. Bank of Japan. Hagedorn, M., Manovskii, I., 2008. The cyclical behavior of equilibrium unemployment and vacancies revisited. Am. Econ. Rev. 98, 1692–1706. Hall, R.E., 2005. Employment fluctuations with equilibrium wage stickiness. Am. Econ. Rev. 95, 50–65. Hall, R.E., 2017. High discounts and high unemployment. Am. Econ. Rev. 107, 305–330. Kilic, M., Wachter, J.A., 2018. Risk, unemployment, and the stock market: a rare-event-based explanation of labor market volatility. Review of Financial Studies 31, 4762–4814. Kindleberger, C., 1978. Manias, Panics, and Crashes. Basic. Kocherlakota, N., 2011. Bubbles and unemployment. Working paper. Martin, A., Ventura, J., 2012. Economic growth with bubbles. Am. Econ. Rev. 102, 3033–3058. Martin, A., Ventura, J., 2018. The macroeconomics of rational bubbles. a user’s guide. Annu. Rev. Econ. 10, 505–539. Miao, J., Wang, P., Xu, L., 2016. Stock market bubbles and unemployment. Econo. Theory 61, 273–307. Mortensen, D.T., Pissarides, C.A., 1994. Job creation and job destruction in the theory of unemployment. Rev. Econ. Stud. 61, 397–415. Pissarides, C.A., 1985. Short-run equilibrium dynamics of unemployment, vacancies and real wages. Am. Econ. Rev. 75, 676–690. Pissarides, C.A., 20 0 0. Equilibrium Unemployment Theory. MIT Press. Samuelson, P., 1958. An exact consumption-loan model of interest with or without the social contrivance of money. J. Political Econ. 66, 467–482. Santos, M.S., Woodford, M., 1997. Rational asset pricing bubbles. Econometrica 65, 19–57. Shiller, R., 1981. Do stock prices move too much to be justified by subsequent changes in dividends? Am. Econ. Rev. 71, 421–436. Shimer, R., 2005. The cyclical behavior of equilibrium unemployment and vacancies. Am. Econ. Rev. 95, 25–49. Shimer, R., 2012. Reassessing the ins and outs of unemployment. Rev. Econ. Dyn. 15, 127–148. Tauchen, G., 1986. Finite state Markov-chain approximations to univariate and vector autoregressions. Econ. Lett. 20, 177–181. Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica 53, 1499–1528. Vuillemey, G., Wasmer, E., 2019. Exact dynamic solutions in search-and-matching models. Working Paper. Wasmer, E., Weil, P., 2004. The macroeconomics of labor and credit market imperfections. Am. Econ. Rev. 94, 944–963.