Endogenous Disasters
Nicolas Petrosky-Nadeau
Lu Zhang
Lars-Alexander Kuehn
September 2015∗
Abstract
The textbook Diamond-Mortensen-Pissarides model of equilibrium unemployment,
once solved accurately with a globally nonlinear algorithm, gives rise endogenously
to rare disasters. Intuitively, in the presence of large negative shocks, inelastic wages
remain relatively high, reducing profits. The marginal costs of hiring run into downward rigidity, which is an inherent attribute of the matching process, and fail to decline.
The falling profits interact with the rigid hiring costs to stifle job creation flows, depressing the economy into disasters. The disaster dynamics in the baseline model are
robust to extensions to home production, capital accumulation, and recursive utility.
JEL Classification: E21, E24, E40, G12
Keywords: Search and matching, rare disasters, dynamic stochastic general equilibrium, unemployment, home production, capital, recursive utility, the equity premium
Petrosky-Nadeau is with Federal Reserve Bank of San Francisco, Zhang is with The Ohio State University
and NBER, and Kuehn is with Carnegie Mellon University. For helpful comments, we thank our discussants Michele Boldrin, Robert Dittmar, Nicolae Garleanu, Francois Gourio, Howard Kung, Lars Lochstoer,
Rodolfo Prieto, Matias Tapia, and Stan Zin, as well as Hang Bai, Ravi Bansal, Frederico Belo, Jonathan Berk,
Nicholas Bloom, Andrew Chen, Jack Favilukis, Michael Gallmeyer, Bob Hall, Urban Jermann, Aubhik Khan,
Xiaoji Lin, Laura Xiaolei Liu, Stavros Panageas, Vincenzo Quadrini, Tomasz Strzalecki, René Stulz, Julia
Thomas, José Ursúa, Neng Wang, Michael Weisbach, Ingrid Werner, Chen Xue, Amir Yaron, and seminar
participants at Boston University, Columbia Business School, Shanghai University of Finance and Economics,
the Federal Reserve Bank of New York, the Federal Reserve Board, the 19th Mitsui Finance Symposium on
“Financial Market Implications of the Macroeconomy” at University of Michigan, the 2010 CEPR European
Summer Symposium on Financial Markets, the 2010 Human Capital and Finance Conference at Vanderbilt
University, the 2010 Society of Economic Dynamics Annual Meetings, the 2011 American Finance Association Annual Meetings, the 2012 Canadian Economics Association Annual Meetings at University of Calgary,
the 2012 National Bureau of Economic Research (NBER) Summer Institute Asset Pricing Meeting, the 2012
Financial Intermediation Research Society Conference, the 2012 Society for Financial Studies Finance Cavalcade, the 2012 University of British Columbia Phillips, Hager, and North Centre for Financial Research
Summer Finance Conference, the 26th Annual Meeting of the Canadian Macroeconomics Study Group: Recent Advances in Macroeconomics, the 2nd Tepper/LAEF Laboratory for Aggregate Economics and Finance
Advances in Macro-Finance Conference at Carnegie Mellon University, the “New Developments in Macroeconomics” Conference at University College London, The Ohio State University, Universidad Catolica de Chile
2nd International Finance Conference, University of Montreal, University of Texas at Austin, and University
of Wisconsin. This paper supersedes our work previously circulated as “An equilibrium asset pricing model
with labor market search” (NBER Working Paper 17742) and “Endogenous disasters and asset prices.”
∗
1
Introduction
The textbook Diamond-Mortensen-Pissarides search model of equilibrium unemployment,
once solved accurately with a globally nonlinear projection algorithm, gives rise endogenously to rare disasters per Rietz (1988) and Barro (2006).
We calibrate the baseline model to the Barro-Ursúa (2008) historical cross-country panel
of output and consumption, extended through 2013. The model implies an output growth
volatility of 5.31% per annum, close to 5.63% in the cross-country panel, as well as an output
growth kurtosis of 12.8, close to 11.87 in the data. In addition, the mean unemployment rate
in the model is 6.28%, close to 6.98% in the 1929–2013 U.S. data, and the unemployment
volatility in the model is 23.41%, close to 21.76% in the U.S. sample.
Most important, applying the Barro-Ursúa (2008) peak-to-trough measurement on simulated data, we find that the output disasters in the baseline model have the same average
size, about 22%, as in the data. The disaster probability is 5.04% in the model, which is
somewhat lower than 7.83% in the data (adjusted for trend growth). For consumption disasters, the disaster probability is 2.87% in the model, which is lower than 8.57% in the data,
but the average size is 25.58% in the model, which is comparable with 23.16% in the data.
Through comparative statics, we find that two key ingredients, including small profits
and matching frictions, combine to give rise to endogenous disasters.
First, we use a
relatively high flow value of unemployment activities, implying realistically small profits.
More important, a high flow value of unemployment also makes wages inelastic. In bad
times, output falls, but wages do not fall as much, causing profits to drop disproportionately.
Second, matching frictions induce downward rigidity in the marginal costs of hiring. If
one side of the labor market becomes more abundant than the other side, it will be increasingly difficult for the abundant side to meet and trade with the other side (which becomes
increasingly scarce). In particular, expansions are periods in which many vacancies compete
for a small pool of unemployed workers. The entry of an additional vacancy can cause a pronounced drop in the probability of a given vacancy being filled. This effect raises the marginal
costs of hiring, slowing down job creation flows, and making expansions more gradual.
1
Conversely, recessions are periods in which many unemployed workers compete for a
small pool of vacancies. Filling a vacancy occurs quickly, and the marginal costs of hiring
are lower. However, in recessions, the congestion in the labor market affects unemployed
workers, rather than vacancies. The entry of a new vacancy has little impact on the probability of a given vacancy being filled. Consequently, although the marginal costs of hiring rise
rapidly in expansions, the marginal costs decline only slowly in recessions. This downward
rigidity is further reinforced by fixed matching costs per Pissarides (2009). By putting a
constant component into the marginal costs of hiring, the fixed costs restrict the marginal
costs from declining in recessions, further hampering job creation flows.
To see how the key ingredients combine to endogenize disasters, consider a large negative
deviation of productivity from its mean. The profits, which are small to begin with, become
even smaller as productivity falls. Wages are also inelastic, remaining relatively high, and
reducing the small profits still further. To make a bad situation worse, the marginal costs
of hiring run into downward rigidity, an inherent attribute of the matching process, which
is further buttressed by fixed matching costs. As the marginal costs of hiring fail to decline
to counteract the impact of shrinking profits, the incentives of hiring are suppressed, and
job creation flows stifled. All the while, jobs are destroyed at a steady rate. Consequently,
aggregate employment falls off a cliff, giving rise to endogenous disasters.
The disaster dynamics are robust to several model extensions. We first microfound the
high flow value of unemployment via home production. The extended model implies an output disaster probability of 9.95%, which is even higher than that in the data, as well as a consumption disaster probability of 7.52%, which is still lower than that in the data. However,
the average size is slightly smaller, 18.58% and 18.06% for output and consumption, respectively. We also find that the disaster dynamics are stronger when market and home goods
are less substitutable, and when market goods are weighted less in the household’s utility.
In the second extension, we incorporate capital into the baseline model. The disaster
dynamics are similar to those in the home production model. The disaster probabilities are
largely aligned with those in the data, but the average size is slightly smaller. We also show
that capital adjustment costs amplify consumption dynamics, dampen investment dynamics,
2
while leaving employment and output dynamics largely unaffected. Finally, we incorporate
recursive utility into the baseline model. The disaster dynamics are quantitatively similar
to those in the baseline model. A high risk aversion allows this extended model to match
the equity premium in the data. However, the high risk aversion has little impact on the
disaster moments and the employment dynamics, a finding that echoes Tallarini (2000).
Our work is built on the macro labor literature.1 Our projection algorithm allows us to
quantify the disaster dynamics. In contrast, these dynamics have been largely ignored so far
in this literature, in which models are routinely solved with loglinearization. We also contribute to the disasters literature (Rietz 1988; Barro 2006) by providing an endogenous mechanism for disasters. Almost all prior studies specify disasters exogenously as large negative
shocks to aggregate productivity. In contrast, log productivity in our work follows a standard
first-order autoregressive process with homoscedastic shocks. Finally, our work lends support
to prior studies on the important role of sticky wages in business cycle fluctuations (Christiano, Eichenbaum, and Evans 2005; Gertler and Trigari 2009) as well as in the Great Depression (Bernanke and Carey 1996; Bordo, Erceg, and Evans 2000; Cole and Ohanion 2004).
The rest of the article unfolds as follows. Section 2 shows disaster dynamics in the baseline
search model. Section 3 shows that the disaster dynamics are robust to extensions to home
production, capital accumulation, and recursive utility. Section 4 concludes. We furnish data
sources, proofs, computation details, and supplementary results in the Online Appendix.
2
The Baseline Model
We present a textbook Diamond-Mortensen-Pissarides (DMP) model of equilibrium unemployment in Section 2.1, and discuss its globally nonlinear solution algorithm in Section 2.2.
We calibrate the model in Section 2.3, and quantify its disaster dynamics in Section 2.4.
1
Diamond (1982), Mortensen (1982), and Pissarides (1985) lay the theoretical foundations for the search
model. Merz (1995) and Andolfatto (1996) embed search frictions into the real business cycle framework.
Shimer (2005) argues that the unemployment volatility in the search model is too low relative to that in the
data. Hall (2005) uses sticky wages, Hagedorn and Manovskii (2008) use tiny profits, and Pissarides (2009)
uses fixed matching costs to address the unemployment volatility puzzle.
3
2.1
Environment
The model is populated by a representative household and a representative firm that uses
labor as the single productive input. The household has log utility, log(Ct ), meaning that
its stochastic discount factor is given by Mt+1 = β(Ct /Ct+1 ), in which Ct is consumption,
and β is the time discount factor. Following Merz (1995), we assume that the household
has perfect consumption insurance. There exists a continuum of mass one of members who
are, at any point in time, either employed or unemployed. The fractions of employed and
unemployed workers are representative of the population at large. The household pools the
income of all the members together before choosing per capita consumption.2
2.1.1
Search and Matching
The representative firm posts a number of job vacancies, Vt , to attract unemployed workers,
Ut . Vacancies are filled via a constant returns to scale matching function:
G(Ut , Vt ) =
Ut Vt
(Utι
+ Vtι )1/ι
,
(1)
in which ι > 0. This matching function, originated from Den Haan, Ramey, and Watson
(2000), has the desirable property that matching probabilities fall between zero and one.
In particular, define θt ≡ Vt /Ut as the vacancy-unemployment (V /U) ratio. The probability for an unemployed worker to find a job per unit of time (the job finding rate) is
−1/ι
f (θt ) ≡ G(Ut , Vt )/Ut = 1 + θ−ι
. The probability for a vacancy to be filled per unit
t
of time (the vacancy filling rate) is q(θt ) ≡ G(Ut , Vt )/Vt = (1 + θιt )−1/ι . It follows that
f (θt ) = θt q(θt ) and q ′ (θt ) < 0, meaning that an increase in the scarcity of unemployed
workers relative to vacancies makes it harder to fill a vacancy. As such, θt is labor market
tightness from the firm’s perspective, and 1/q(θt ) is the average duration of vacancies.
2
It should be noted that relaxing the perfect consumption insurance might weaken disaster dynamics.
With imperfect insurance, unemployed workers would accept jobs at lower wages, reducing unemployment
and dampening disaster dynamics. However, Krusell, Mukoyama, and Sahin (2010) show that the search
model with imperfect consumption insurance behaves almost as in the representative agent model. Intuitively, individual self-insurance via asset accumulation is quite effective. Also, inelastic wages required for
the model to match labor market volatilities would counteract the dampening effect of imperfect insurance.
Sorting out these conflicting mechanisms is left for future work, however, due to computational complexity.
4
The representative firm incurs costs in posting vacancies. Following Pissarides (2009),
we assume that the unit costs per vacancy, denoted κt , contain two components, the proportional costs, κ0 , and the fixed costs, κ1 :
κt ≡ κ0 + κ1 q(θt ),
(2)
in which κ0 , κ1 > 0. The fixed costs capture matching costs, such as training, interviewing,
negotiation, and administrative setup costs of adding a worker to the payroll, costs paid after
a hired worker arrives but before wage bargaining takes place. The marginal costs of hiring
arising from the proportional costs, κ0 /q(θt ), increase with the mean duration of vacancies,
1/q(θt ). In contrast, the marginal costs from the fixed costs are “fixed” at κ1 , which is
independent of the duration of vacancies. The total marginal costs of hiring equal κ0 /q(θt ) +
κ1 . In expansions, the labor market is tighter for the firm (θt is higher), meaning that the
vacancy filling rate, q(θt ), is lower. As such, the marginal costs of hiring are procyclical.
Jobs are destroyed at a constant rate of s per period. Employment, Nt , evolves as:
Nt+1 = (1 − s)Nt + q(θt )Vt ,
(3)
in which q(θt )Vt is the number of new hires. Population is normalized to be one, Ut + Nt = 1,
meaning that Nt and Ut are also the rates of employment and unemployment, respectively.
2.1.2
The Representative Firm
The firm takes the aggregate productivity, Xt , as given. We specify xt ≡ log(Xt ) as:
xt+1 = ρxt + σǫt+1 ,
(4)
in which ρ ∈ (0, 1) is the persistence, σ > 0 is the conditional volatility, and ǫt+1 is an independently and identically distributed (i.i.d.) standard normal shock. The firm uses labor to
produce output, Yt , with a constant returns to scale production technology,
Yt = Xt Nt .
We abstract from capital, but include it later in a model extension (Section 3.2).
5
(5)
The dividends to the firm’s shareholders are given by:
Dt = Xt Nt − Wt Nt − κt Vt ,
(6)
in which Wt is the wage rate. Taking Wt , the household’s stochastic discount factor, Mt+1 ,
and the vacancy filling rate, q(θt ), as given, the firm posts the optimal number of vacancies
to maximize the cum-dividend market value of equity, St :
"∞
#
X
St ≡
max ∞ Et
Mt+τ (Xt+τ Nt+τ − Wt+τ Nt+τ − κt+τ Vt+τ ) ,
{Vt+τ ,Nt+τ +1 }τ =0
(7)
τ =0
subject to equation (3) and a nonnegativity constraint on vacancies, Vt ≥ 0.3
Because q(θt ) > 0, Vt ≥ 0 is equivalent to q(θt )Vt ≥ 0, meaning that the only source
of job destruction is the exogenous separation of workers from the firm. We abstract from
endogenous job destruction for parsimony. However, incorporating this feature is likely to
strengthen, rather than weaken our quantitative results. Intuitively, job destruction should
rise during recessions, amplifying disaster dynamics.4
Let λt denote the multiplier on the constraint q(θt )Vt ≥ 0. From the first-order conditions
with respect to Vt and Nt+1 , we obtain the intertemporal job creation condition:
κ0
κ0
+ κ1 − λt = Et Mt+1 Xt+1 − Wt+1 + (1 − s)
+ κ1 − λt+1
.
q(θt )
q(θt+1 )
(8)
Intuitively, the marginal costs of hiring at time t (with Vt ≥ 0 accounted for) equal the
marginal value of employment to the firm, which in turn equals the marginal benefits of
hiring at period t+1, discounted to t with the stochastic discount factor, Mt+1 . The marginal
benefits at t + 1 include the marginal product of labor, Xt+1 , net of the wage rate, Wt+1 , plus
the marginal value of employment, which equals the marginal costs of hiring at t + 1, net of
3
The Vt ≥ 0 constraint has been ignored so far in the macro labor literature. Using a globally nonlinear
algorithm, we find that the constraint is occasionally binding in the model’s simulations. Because a negative
vacancy does not make economic sense, we opt to impose the constraint to solve the model accurately.
However, the constraint is not a central ingredient of the model. In our benchmark calibration, for instance,
the constraint only binds for 0.21% of the time, which is extremely rare.
4
Den Haan, Ramey, and Watson (2000), for instance, show that endogenous job destruction amplifies
aggregate shocks in a business cycle model with labor market search frictions.
6
separation. Finally, the optimal vacancy policy also satisfies the Kuhn-Tucker conditions:
q(θt )Vt ≥ 0,
2.1.3
λt ≥ 0,
and λt q(θt )Vt = 0.
(9)
Equilibrium Wages
The wage rate is determined endogenously by applying the sharing rule per the outcome of a
generalized Nash bargaining process between employed workers and the firm. Let η ∈ (0, 1)
be the workers’ relative bargaining weight, and b the workers’ flow value of unemployment
activities. The equilibrium wage rate is given by (see the Online Appendix for derivations):
Wt = η (Xt + κt θt ) + (1 − η)b.
(10)
The wage rate is increasing in labor productivity, Xt , and the total vacancy costs per
unemployed worker, κt θt . Intuitively, the more productive the workers are, and the more
costly for the firm to fill a vacancy, the higher the wage rate will be for employed workers. In
addition, the flow value of unemployment activities, b, and the workers’ bargaining weight, η,
affect the wage elasticity to labor productivity. The lower η is, and the higher b is, the more
the wage rate will be tied with the constant b, inducing a lower wage elasticity to productivity.
2.1.4
Competitive Equilibrium
The competitive equilibrium consists of vacancy posting, Vt ≥ 0, multiplier, λt ≥ 0, and
consumption, Ct , such that (i) Vt and λt satisfy the intertemporal job creation condition
(22) and the Kuhn-Tucker conditions (9), while taking the stochastic discount factor Mt+1
and wages in equation (10) as given; and (ii) the goods market clears:
Ct + κt Vt = Xt Nt .
2.2
(11)
Solving for the Competitive Equilibrium
We use a projection algorithm to solve for the competitive equilibrium. The equilibrium is
not Pareto optimal. Intuitively, the firm in the decentralized economy does not take into
account the congestion effect of posting a new vacancy on the labor market when maximizing
7
its market value of equity, whereas the social planner does when maximizing social welfare
(Hosios 1990). As such, we must work with the optimality conditions directly.
The state space consists of employment and productivity, (Nt , xt ). The goal is to solve for
the optimal vacancy, V (Nt , xt ), and the multiplier, λ(Nt , xt ), from the functional equation
(8), while also satisfying the Kuhn-Tucker conditions (9). The standard projection method
would approximate the Vt and λt functions directly. With the Vt ≥ 0 constraint, the functions
are not smooth, making the approximations tricky and cumbersome. As such, we adapt
the Christiano and Fisher (2000) parameterized expectations method by approximating the
conditional expectation (the right-hand side) of equation (8), denoted Et ≡ E(Nt , xt ). We
then exploit a convenient mapping from Et to Vt and λt to eliminate the need to parameterize
λt separately.
In particular, after obtaining the parameterized Et , we first calculate
q̃(θt ) = κ0 / (Et − κ1 ) . If q̃(θ t ) < 1, Vt ≥ 0 is not binding, we set λt = 0 and q(θt ) = q̃(θt ). We
then solve θt = q −1 (q̃(θt )), in which q −1 (·) is the inverse function of q(θt ), and Vt = θt (1−Nt ).
If q̃(θt ) ≥ 1, Vt ≥ 0 is binding, we set Vt = 0, θ t = 0, q(θt ) = 1, and λt = κ0 + κ1 − Et .
2.3
Calibration and Basic Moments
Because we focus on rare disasters, we calibrate the model to a historical cross-country
dataset of output and consumption compiled by Barro and Ursua (2008). For labor market moments, because historical cross-country data for unemployment and vacancies are not
available, we calibrate the model to a historical U.S. sample. Finally, for variables such as
wages and net payouts, we calibrate the model to the (only available) postwar U.S. data.
Our calibration does not reply on steady state relations, which hold very poorly in
simulations due to the model’s nonlinearity. As such, we take care in reporting a wide range
of model moments to compare with data moments. Also, for parameters that are important
in driving our results, we perform extensive comparative statics to quantify their impact.
2.3.1
Data
We obtain the historical cross-country panel from Robert Barro’s Web site. The BarroUrsua dataset ends in 2009, and we extend it through 2013. The dataset contains annual
8
real consumption and output series for multiple countries. The starting points of the output
series range from 1790 for United States to 1911 for Korea and South Africa, and the starting points of the consumption series range from 1800 for Sweden to 1938 for Greece. We
drop countries with missing data as well as those with only postwar data. We impose this
restriction to ensure that estimates of disaster moments are (relatively) unbiased.
Table 1 reports descriptive statistics for output and consumption growth rates. From
Panel A, the output growth has a volatility of 5.63% per annum, a skewness of −1.02, and
a kurtosis of 11.87, all averaged across countries. The first-order autocorrelation is 0.16, but
high-order autocorrelations are largely zero. From Panel B, the consumption growth has a
volatility of 6.37%, a skewness of −0.55, and a kurtosis of 9.19, on average. The first-order
autocorrelation is 0.07, but high-order autocorrelations are again close to zero.5
2.3.2
Calibration
We set the time discount factor, β, to e−log(1.0573)/12 = 0.9954 to match the average discount
rate of 5.73% per annum in a historical cross-country panel of asset prices (Section 3.3).
Following Gertler and Trigari (2009), we set the persistence of the log productivity, ρ, to be
0.951/3 , and choose its conditional volatility, σ, to match the output growth volatility in the
data. This procedure yields a value of 0.01 for σ, which implies an output growth volatility
of 5.31% per annum in the model, which is close to 5.63% in the data.6
We calibrate the labor market parameters in the spirit of Hagedorn and Manovskii (2008).
The workers’ bargaining weight, η, is set to be 0.04, which implies a wage elasticity to labor
productivity of 0.51 in the model. This value is close to 0.47 in the postwar U.S. data.7
5
Consistent with Barro and Ursua (2008), the output growth volatility, 5.63%, is lower than the consumption growth volatility, 6.37%, in the cross-country panel. As pointed out by Barro and Ursua, government purchases tend to increase sharply during wartime. This expansion in government spending decreases consumption for a given level of output, thereby raising the consumption volatility relative to the output volatility.
6
While Cole and Ohanian (1999) emphasize productivity shocks in the Great Depression, monetary
shocks seem to play an important, if not a dominant role (Friedman and Schwartz 1963). We remain agnostic
about the origins of disasters, but shed light on the endogenous mechanism related to the labor market.
7
Following Hagedorn and Manovskii (2008), we measure wages as labor share times labor productivity
(real output per job) in the nonfarm business sector from Bureau of Labor Statistics. The sample is
quarterly from 1947 to 2013. We detrend wages and labor productivity with log deviations from the
Hodrick-Prescott (HP, 1997) trend with a smoothing parameter of 1,600.
9
Table 1 : Properties of Real Per Capita Output and Consumption Growth
Results are based on the Barro-Ursua (2008) historical cross-country panel extended through 2013.
σ Y and σ C denote volatilities (in percent), SY and SC skewness, KY and KC kurtosis, and ρYi and
ρC
i the ith-order autocorrelations of log annual output and consumption growth, respectively.
Panel A: Output growth
Argentina
Australia
Austria
Belgium
Brazil
Canada
Chile
China
Colombia
Denmark
Egypt
Finland
France
Germany
Iceland
India
Indonesia
Italy
Japan
Korea
Mexico
Netherlands
New Zealand
Norway
Peru
Portugal
Russia
South Africa
Spain
Sri Lanka
Sweden
Switzerland
Taiwan
Turkey
U.K.
U.S.
Uruguay
Venezuela
Average
Panel B: Consumption growth
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
6.62
6.33
8.33
6.97
4.84
5
5.77
7.04
2.22
3.51
4.67
4.63
6.16
10.13
4.93
4.79
5.69
4.64
6.22
6.93
4.19
6.47
4.56
3.65
4.81
4.12
8.3
4.8
4.48
4.48
4.12
4.58
8.77
8.11
2.86
4.28
7.76
8.34
−0.57
0.4
−6.17
1.31
−0.44
−0.78
−1.04
−0.78
−0.63
−0.82
0.92
−0.70
−0.81
−7.91
−0.44
−0.12
−2.3
−1.33
−2.24
−1.64
−1.33
1.1
0.43
−0.58
−1.33
0
−0.72
−1.66
−1.70
−0.18
−0.74
−0.24
−4.06
−0.6
−0.77
0
−0.54
0.44
4.48
5.99
61.39
21.44
4.17
5.06
5.47
3.53
4.16
7.5
7.2
6.36
9.89
86.02
4.5
4.67
12.74
13.49
15.35
9.59
6.61
29.05
5.04
5.92
5.88
4.32
6.27
14.61
12.24
3.79
5.5
4.32
31.39
5.3
5.03
5.15
3.43
4.05
0.02
−0.05
0.18
0.32
0.14
0.26
0.07
0.48
0.34
0
−0.08
0.21
0.07
0.3
0.15
−0.2
0.45
0.26
0.27
0.14
0
0.17
0.15
0.09
0.44
0.01
0.34
0.02
0.24
0.08
0
−0.01
0.35
0.13
0.3
0.25
−0.01
0.09
−0.11
−0.14
0.05
0.05
−0.05
0.11
−0.15
0.19
0.14
−0.12
−0.06
−0.13
−0.09
−0.04
0.08
0.02
0.22
−0.07
0.03
−0.1
0.21
−0.17
−0.17
−0.13
0.02
0.18
0.23
0.15
0.04
−0.01
−0.16
−0.23
0
0.13
−0.02
0.06
−0.05
0.15
0.04
0.13
−0.06
0
−0.03
−0.07
−0.07
0.16
−0.05
0.05
0.29
0.06
0.09
−0.11
0.04
0.3
0.11
−0.03
0.16
−0.03
0.04
−0.07
0
0.01
−0.16
0.02
0.16
0.04
0.04
0.19
−0.15
0.09
−0.04
−0.05
−0.15
−0.11
−0.14
0.09
−0.07
0.09
−0.15
0.03
0.03
−0.15
−0.07
0.19
−0.16
0.08
−0.01
−0.11
0.08
−0.16
−0.1
−0.1
−0.09
0.14
0.09
−0.03
−0.07
−0.04
0.19
0.07
−0.14
0.18
0.04
−0.11
0.01
−0.07
0.11
0
0.03
−0.09
−0.2
−0.16
−0.14
−0.08
5.63 −1.02 11.87
0.16
0
Argentina
Australia
Belgium
Brazil
Canada
Chile
Colombia
Denmark
Egypt
Finland
France
Germany
Greece
India
Italy
Japan
Korea
Mexico
Netherlands
New Zealand
Norway
Peru
Portugal
Russia
Spain
Sweden
Switzerland
Taiwan
Turkey
U.K.
U.S.
Venezuela
0.02 −0.02 Average
10
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
7.84
4.95
8.81
7.25
4.65
9.1
6.18
5.25
6.09
5.73
6.27
5.27
9.83
4.31
3.59
6.78
6.6
6.03
7.3
5.89
3.73
4.62
4.41
10.4
7.27
4.85
7.86
8.9
7.78
2.65
3.79
9.8
0.14
−1.02
−1.14
0.34
−1.03
−1.2
0.54
−0.68
0.3
−1.02
−0.99
−0.57
0.11
0.47
0.16
−1.54
−1.09
1.20
−0.79
−0.74
−0.27
−1.15
−0.48
−1.35
−2.81
0.04
0.43
−2.33
−1.02
−0.35
−0.06
0.18
4.21
8.25
12.97
4.14
6.19
5.26
7.18
10.34
6.3
7.94
12.94
7.55
12.04
5.97
7.76
20.71
6.26
11.66
22.68
7.98
9.81
6.32
3.21
12.27
25.14
4.94
5.39
16.2
6.69
8.48
3.49
3.93
−0.14
0.15
0.26
−0.26
0
0.08
−0.22
−0.10
−0.2
0.14
0.32
0.24
0.24
−0.06
0.38
0.21
0.18
−0.09
0.12
−0.13
0
0.39
0.22
0.37
−0.02
−0.09
−0.34
0.28
0.15
0.29
0.04
−0.03
0.04
0.06
0.19
−0.03
0.16
−0.13
0.04
−0.31
−0.15
−0.1
0.15
0.21
0.39
0.18
0.31
0.11
0.03
−0.04
0.14
−0.14
−0.31
0
0.23
0.08
−0.02
−0.12
−0.11
−0.02
0.07
0.01
0.07
0.11
−0.27
0.08
0
0.1
−0.16
−0.11
0.05
0.01
0.19
−0.01
−0.02
0.27
0.16
−0.07
0.1
0.18
−0.29
0.04
−0.2
0.08
0.14
−0.12
−0.01
0.2
−0.1
−0.17
0.04
0.05
0.01
−0.04
−0.07
−0.11
0.08
−0.06
−0.4
−0.06
−0.04
−0.12
−0.3
0.21
−0.01
−0.04
−0.27
−0.06
0.11
0.34
0.09
0.2
−0.07
−0.02
−0.18
0.07
−0.01
−0.01
0.1
−0.03
−0.06
0.13
−0.12
−0.03
−0.11
−0.04
−0.03
0.06
6.37 −0.55 9.19
0.07
0.03
0 −0.02
The flow value of unemployment activities, b, merits more discussion. Shimer (2005)
pins down b = 0.4 by assuming that the only benefit for unemployment is government unemployment insurance. However, Hagedorn and Manovskii (2008) argue that in a perfectly
competitive labor market, b should equal the flow value of employment. In particular, b
measures not only unemployment insurance, but also the total value of home production,
self-employment, disutility of work, and leisure. In the model, the marginal product of labor
is unity, to which b should be close. We set b to be 0.85, which is identical to that in Rudanko
(2011), and is far less extreme than 0.96 in Hagedorn and Manovskii.
Two additional remarks on b are in order. First, b = 0.85 implies a reasonable magnitude
of profits. The profits-to-output ratio in the model’s simulations is on average 8.58% (with a
cross-simulation standard deviation of 0.78%), which is relatively close to 9.12% in the U.S.
data from 1929 to 2013.8 In contrast, b = 0.96 would imply too small, and b = 0.4 as in
Shimer (2005) would give rise to unrealistically large profits (31%, Section 2.4.2).
Second, more important, we view our relatively high-b calibration only as a parsimonious
metaphor for inelastic wages. Prior studies show that inelastic wages are important to explain labor market volatilities (Hall 2005; Gertler and Trigari 2009) as well as business cycle
fluctuations (Christiano, Eichenbaum, and Evans 2005; Smets and Wouters 2007). Sticky
wages also play an important role in large declines in economic activity during the Great Depression (Bernanke and Carey 1996; Bordo, Erceg, and Evans 2000; Cole and Ohanion 2004).
We set the job separation rate, s, to be 4%, which is within the range of estimates from
Davis, Faberman, and Haltiwanger (2006). For the elasticity parameter in the matching function, ι, we set it to be 1.25, which is close to that in Den Haan, Ramey, and Watson (2000).
To pin down the proportional and fixed costs of vacancy, κ0 and κ1 , we first experiment
to keep the total unit costs of vacancy, κt , around 0.7 on average in simulations. This level
is necessary for the model to reproduce a realistic unemployment rate of 6.28%, which is not
far from the average unemployment rate of 6.98% in the long U.S. sample from April 1929 to
8
Output (gross domestic product) data are from National Income and Product Accounts (NIPA) Table
1.1.6, and profits data are from NIPA Table 1.12 row 13 (corporate profits with inventory valuation
adjustment and capital consumption adjustment). We use the implicit price deflator of gross domestic
product (NIPA Table 1.1.9) to deflate profits. The sample is annual from 1929 to 2013.
11
December 2013 from Global Financial Data. The evidence on the relative weights of the proportional and fixed costs of vacancy seems scarce. To pin down κ0 and κ1 separately, we target
the (quarterly) unemployment volatility of 21.76% in the long U.S. sample. This procedure
yields κ0 = κ1 = 0.5, which implies an unemployment volatility of 23.41% in the model.
Is the magnitude of the vacancy (hiring) costs in the model empirically plausible? As
noted, the marginal costs of vacancy posting in terms of labor productivity (output per
worker) equal 0.7 in the model, which is the average of κ0 + κ1 q(θt ) in simulations. The
marginal costs of hiring, κ0 /q(θt ) + κ1 , are on average 1.93. Merz and Yashiv (2007) estimate the marginal costs of hiring to be 1.48 times the average output per worker with a
standard error of 0.57. As such, albeit somewhat high, 1.93 is at least plausible. For the
total costs of vacancy, κt Vt , the average in the model is about 0.66% of annual wages. For
comparison, the estimated labor adjustment costs in Bloom (2009) imply hiring and firing
costs of “about 1.8% of annual wages” and high fixed costs of “around 2.1% of annual revenue (p. 663).” Both empirical studies are based on postwar U.S. data. However, because
the labor market tends to become less frictional over time, it seems reasonable to argue that
the labor market during the Great Depression is at least as frictional as that in the postwar
period. As such, our calibration of vacancy costs seems empirically plausible.
2.3.3
Basic Moments
From the baseline model’s stationary distribution (after 6,000 burn-in months), we repeatedly
simulate 10,000 samples, each with 1,836 months. On each sample, we time-aggregate the
monthly output into 153 annual observations and the first 1,656 monthly consumption
observations into 138 annual observations.
(Time-aggregation means that we add up
12 monthly observations within a given year, and treat the sum as the year’s annual
observation.) The sample lengths match the cross-country average sample lengths in Table
1. We then calculate the annual volatilities, skewness, kurtosis, and autocorrelations of log
consumption growth and log output growth. For each moment, we report the mean as well as
the 5 and 95 percentiles across the 10,000 simulations. We also report p-values that are the
frequencies with which a given model moment is larger than its data counterpart in Table 1.
12
Table 2 : Basic Moments in the Baseline Model
The data moments for output and consumption are based on the Barro-Ursua (2008) historical
cross-country panel extended through 2013, and the unemployment moments are based on the long
U.S. sample from April 1929 to December 2013. The model moments are from 10,000 simulated
samples. For each moment, we report the mean as well as the 5 and 95 percentiles across the
simulations. The p-values are the percentages with which a given model moment is larger than its
data counterpart. σ Y and σ C denote volatilities, SY and SC skewness, KY and KC kurtosis, and
ρYi and ρC
i the ith-order autocorrelations of log output and consumption growth rates, respectively.
E[U ], SU , and KU are the mean, skewness, and kurtosis of the monthly unemployment rate,
respectively, and σ U is its quarterly volatility. We take quarterly averages of monthly unemployment
rates, and detrend the quarterly series as HP-filtered proportional deviations from the mean with
a smoothing parameter of 1,600. σ Y , σ C , E[U ], and σ U are in percent.
Panel A: Log output growth
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
Panel B: Log consumption growth
data
mean
5%
95%
p-value
5.63
−1.02
11.87
0.16
0
0.02
−0.02
5.31
0.85
12.8
0.24
−0.11
−0.12
−0.11
2.90
−0.39
3.08
0.01
−0.31
−0.32
−0.31
11.56
3.35
34.99
0.63
0.23
0.08
0.08
0.31
0.99
0.41
0.58
0.16
0.1
0.23
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
data
mean
5%
95%
p-value
6.37
−0.55
9.19
0.07
0.03
0
−0.02
4.65
0.91
14.4
0.23
−0.12
−0.12
−0.11
2.12
−0.50
3.16
−0.02
−0.33
−0.34
−0.32
11.26
3.57
38.59
0.65
0.24
0.09
0.09
0.21
0.96
0.54
0.79
0.14
0.15
0.23
data
mean
5%
95%
p-value
2.02
21.76
3.52
23.41
1.49
5.45
5.82
53.49
0.85
0.43
Panel C: Unemployment
E[U ]
KU
data
mean
5%
95%
p-value
6.98
7.26
6.28
19.18
4.83
5.24
10.54
41.78
0.19
0.87
SU
σU
Table 2 shows that the output volatility is 5.31% per annum in the model. The data
moment, 5.63%, lies within the 90% confidence interval of the model’s bootstrapped distribution with a p-value of 0.31. The consumption volatility of 4.65% in the model is smaller
than the output volatility (due to consumption smoothing), but is substantially smaller than
6.37% in the data. This data moment still lies within the 90% confidence interval with a
p-value of 0.21. Also, both output and consumption growth rates are somewhat negatively
skewed in the data, with skewness coefficients of −1.02 and −0.55, respectively. However, the
output and consumption growth rates in the model are slightly positively skewed, with coefficients of 0.85 and 0.91, and the data moments have bootstrapped p-values of 0.99 and 0.96,
respectively. As such, the model has trouble in matching the negative skewness in the data.
13
The model fares better in matching the leptokurtic distributions of output and consumption growth in the data. The kurtosis of the output growth is 11.87, and that of the consumption growth is 9.19 in the data. Table 2 reports that the corresponding model moments
are 12.8 and 14.4, respectively, and both data moments lie well within the 90% confidence
intervals. Finally, both output and consumption growth rates are positively autocorrelated
at the first lag, and weakly negatively autocorrelated at longer lags in the model. All the
autocorrelations in the data are within the 90% confidence intervals of the model.
Panel C reports the unemployment moments both in the data and in the model. The
data moments are based on the long U.S. sample from April 1929 to December 2013 (1,017
months) obtained from Global Financial Data. The mean unemployment rate is 6.28% in the
model, and the data moment of 6.98% lies within the 90% confidence interval with a p-value
of 0.19. The model also reproduces the positively skewed and leptokurtic unemployment rate
distribution. The skewness and kurtosis are 3.52 and 19.18 in the model, respectively, and the
data moments of 2.02 and 7.26 are both within the 90% confidence intervals from the model.
We follow the macro labor literature in computing the unemployment volatility in the
model. We take quarterly averages of monthly unemployment and vacancy rates, and detrend with HP-filtered proportional deviations from the mean. (We do not take logs because
vacancies can hit zero in the model when the Vt ≥ 0 constraint is binding.) The unemployment volatility is 23.41% in the model, which is close to 21.76% in the data. The model also
implies a vacancy volatility of 14.32%, a labor market tightness volatility of 18.41%, and a
Beveridge curve with a negative unemployment-vacancy correlation of −0.46 (untabulated).
2.4
Endogenous Disasters
Most important, the baseline economy gives rise to endogenous disasters.
2.4.1
Quantitative Results
We simulate one million months from the model, and report the empirical cumulative distribution functions for unemployment, output, and consumption. Figure 1 shows that unemployment is positively skewed with a long right tail. The mean unemployment rate is 6.23%,
14
Figure 1 : Empirical Cumulative Distribution Functions, the Baseline Model
Results are based on one million months simulated from the baseline model’s stationary distribution.
Panel B: Output
Panel C: Consumption
1
1
0.8
0.8
0.8
0.6
0.4
0.2
0
0
Probability
1
Probability
Probability
Panel A: Unemployment
0.6
0.4
0.2
0.2
0.4
0.6
0.8
Unemployment
1
0
0
0.6
0.4
0.2
0.5
Output
1
1.5
0
0
0.5
1
Consumption
1.5
and the median 4.91%. The 2.5 percentile of the unemployment rate, 4.31%, is close to the
median, whereas the 97.5 percentile is far away, 16.95%. As a mirror image, employment is
negatively skewed with a long left tail. Consequently, output and consumption both exhibit
rare but deep disasters. With small probabilities, the economy falls off a cliff.
Do the rare disasters arising endogenously from the model resemble those in the data?
Barro and Ursúa (2008) apply a peak-to-trough method on their cross-country panel to
identify rare disasters, defined as cumulative fractional declines in per capita consumption
or output of at least 10%.9 We apply the peak-to-trough method on the extended crosscountry panel. Table 3 reports the estimates. For output, the disaster probability is 7.83%,
the average size of disasters 21.99%, and the average duration 3.72 years. For consumption,
the disaster probability is 8.57%, the average size 23.16%, and the duration 3.75 years.
Our disaster probability estimates, 7.83% and 8.57% for output and consumption, are
higher than those in Barro and Ursúa, 3.69% and 3.63%, respectively. The crux is that we
9
Suppose there are two states, normalcy and disaster. The disaster probability measures the likelihood
with which the economy shifts from normalcy to disaster in a given year. The number of disaster years
is the number of years in the interval between peak and trough for each disaster event. The number of
normalcy years is the total number of years in the sample minus the number of disaster years. Finally, the
disaster probability is the ratio of the number of disasters over the number of normalcy years.
15
Table 3 : Moments of Economic Disasters
The data moments are estimated from applying the Barro-Ursúa (2008) peak-to-trough method
on their cross-country panel extended through 2013. We adjust for trend growth in the data. The
model moments are from 10,000 simulations, each with 1,836 months. On each artificial sample,
we time-aggregate output and consumption into annual observations, and apply the peak-to-trough
method to identify disasters as cumulative fractional declines in output or consumption of at least
10%. We report the averages, 5 and 95 percentiles, and p-values across the simulations. The
disaster probabilities and average size are in percent, and the average duration is in terms of years.
Data
Model
Mean
5%
95%
p-value
2.24
12.7
3.2
8.57
46.24
6
0.09
0.33
0.79
0.71
11.26
3
5.83
62.13
7
0.00
0.36
0.81
Panel A: Output
Probability
Size
Duration
7.83
21.99
3.72
5.04
22.22
4.44
Panel B: Consumption
Probability
Size
Duration
8.57
23.16
3.75
2.86
25.64
4.91
adjust for trend growth in the data, 1.85% per annum for output and 1.74% for consumption,
to be consistent with our model with no growth. Ignoring trend growth in the data yields
the disaster probabilities of 3.47% for output and 4.15% for consumption in our extended
sample, estimates that are close to Barro and Ursúa’s. The average size and duration estimates are relatively unaffected. Clearly, adjusting for trend growth in the data raises the
hurdle for the model to match the disaster probabilities in the data.
We simulate 10,000 artificial samples from the model’s stationary distribution, each with
1,836 months. On each sample, we time-aggregate monthly output and consumption into
annual observations, and then apply the Barro-Ursua peak-to-trough measurement. The
output disaster probability in the model is 5.04%, which is lower than 7.83% in the data.
The cross-simulation standard deviation of this probability is 2.02%, meaning that the data
moment is within two standard deviations from the model (p-value = 0.09). The average
disaster size in the model, 22.21%, is close to the data moment, 21.99%. The average duration in the model of 4.44 years is longer than 3.72 years in the data, but the data duration
16
Figure 2 : Distributions of Output and Consumption Disasters by Size and Duration
from the Baseline Model’s Stationary Distribution
Yt disasters by size
Ct disasters by size
5
5
4
4
2
1
0
0
0.2
0.4 0.6 0.8
Cumulative decline
1
1.5
3
2
1
0
0
Ct disasters by
duration
0.2
0.4 0.6 0.8
Cumulative decline
1
0.5
0
1
1.5
Number of disasters
3
Yt disasters by
duration
Number of disasters
Number of disasters
Number of disasters
Results are based on 10,000 simulations. Yt is output, and Ct consumption.
1 2 3 4 5 6 7 8 9 10
Duration
1
0.5
0
1 2 3 4 5 6 7 8 9 10
Duration
lies within the 90% confidence interval from the model (p-value = 0.79).
For consumption disasters, Table 3 shows that the disaster probability is only 2.87% in
the model, which is substantially lower than 8.57% in the data, and the data probability lies
outside the 90% confidence interval from the model (p-value = 0.00). The average disaster
size is 25.58% in the model, which is close to 23.16% in the data. The average duration
is 4.91 years, which is longer than 3.75 years in the data, but the data duration again lies
within the 90% confidence interval from the model (p-value = 0.81).
Figure 2 reports the frequency distributions of output and consumption disasters by size
and duration averaged across 10,000 simulations from the model. The size and duration
distributions display similar patterns as those in the data (Barro and Ursua 2008, Figures 1
and 2). In particular, the size distributions seem to follow a power-law density.
2.4.2
Intuition: Comparative Statics
To shed light on the intuition behind the model’s disaster dynamics, we conduct six
comparative statics. (i, ii) We reduce the flow value of unemployment activities, b, from 0.85
in the benchmark calibration to 0.825, and then to 0.4. Because of its importance, we consider
17
Table 4 : Comparative Statics for the Disaster Moments in the Baseline Model
The first column reports the disaster moments from the benchmark calibration (Table 3), as well
as six comparative statics: (i, ii) b = 0.825 and b = 0.4 are for the flow value of unemployment
activities set to 0.825 and 0.4, respectively; (iii) s = 0.035 is for the job separation rate set to 0.035;
(iv) κt = 0.7 is for the proportional unit costs of vacancy κ0 = 0.7 and the fixed unit costs κ1 = 0;
(v) ι = 1.1 is for the elasticity of the matching function set to 1.1; and (vi) η = 0.05 is for the
workers’ bargaining weight set to 0.05. In each experiment, all the other parameters are identical
to those in the benchmark calibration. All the model moments are based on 10,000 simulations.
Baseline
b = 0.825
b = 0.4
s = 0.035
κt = 0.7
ι = 1.1
η = 0.05
4.05
18.2
4.51
5.29
21.97
4.41
5.57
22.69
4.4
1.85
20.19
5.1
3.04
25.05
4.88
3.59
25.21
4.78
Panel A: Output
Probability
Size
Duration
5.04
22.22
4.44
3.61
16.07
4.57
2.53
13.41
4.7
4.42
19.87
4.5
Panel B: Consumption
Probability
Size
Duration
2.86
25.64
4.91
1.62
16.31
5.19
1.32
12.35
5.2
2.43
22.25
4.97
two alternative values of b. (iii) We lower the job separation rate, s, from 0.04 to 0.035. (iv)
We adjust the unit costs of vacancy from κt = 0.5 + 0.5q(θt ) to κt = 0.7, i.e., κ0 = 0.7 and
κ1 = 0 (the average κt is 0.7 under the benchmark calibration). (v) We reduce the elasticity
of the matching function, ι, from 1.25 to 1.1. Finally, (vi) we raise the workers’ bargaining
power, η, from 0.04 to 0.05. In each experiment, all the other parameters remain unchanged.
Table 4 reports the results. With b = 0.825, the output disaster probability reduces from
5.04% to 3.61%, and the consumption disaster probability from 2.86% to 1.62%. The average
disaster size is lowered from 22.22% to 16.07% for output and from 25.64% to 16.31% for
consumption. The average duration of disasters increases slightly. In addition, the output
and consumption volatilities drop to 3.39% and 2.62%, and the mean and volatility of the
unemployment rate fall to 5.03% and 11%, respectively.
With the low flow value of unemployment activities b = 0.4, the disaster probability falls
to 2.53% for output and 1.32% for consumption. The average disaster size drops to 13.41%
and 12.35%, respectively. Also, the low value of b implies low wages, giving rise to an unrealistically high average profits-to-output ratio of 31%. The mean unemployment rate falls
18
to 3.97%, and the unemployment volatility to only 0.14%, consistent with Shimer (2005).
However, the presence of disaster dynamics even with b = 0.4 implies that these dynamics
are even more robust, quantitatively, to changes in b than the unemployment volatility.
Intuitively, with small profits, wages are inelastic to productivity. As noted, the wage
elasticity to productivity is 0.51 in the benchmark calibration. When productivity is low,
wages remain relatively high, shrinking the small profits further to stifle job creation flows.
In contrast, with large profits, wages are sensitive to shocks to productivity. With b = 0.4,
the wage elasticity to productivity increases to 0.66. As such, when productivity is low, employment falls, but wages drop as well, providing incentives for the firm to hire unemployed
workers to offset job destruction flows. Consequently, the disaster dynamics are dampened.
In the third experiment, reducing the separation rate lowers disaster probabilities and
size. With s = 0.035, the disaster probability is 4.42% for output and 2.43% for consumption,
both of which are still substantial. The disaster size is slightly decreased, and the duration
slightly increased. Intuitively, because jobs are destroyed at a lower rate, the economy can
create enough jobs to shore up employment in time to reduce disaster risks.
Without the fixed costs of vacancy, the disaster probability drops from 5.04% to 4.05%
for output, and from 2.86% to 1.85% for consumption. The disaster size falls from 22.22% to
18.2% for output, and from 25.64% to 20.19% for consumption. In addition, the output and
consumption volatilities drop to 4.09% and 3.3%, respectively. The mean unemployment
rate falls from 6.28% to 5.55%, and volatility from 0.23 to 0.15 (Pissarides 2009).
Reducing the elasticity of the matching function, ι, from 1.25 to 1.1 increases the disaster
probability slightly to 5.29% for output and 3.04% for consumption. The disaster size and
duration, as well as output, consumption, and unemployment volatilities all remain relatively unchanged. The mean unemployment rate increases from 6.28% to 6.68%. Intuitively,
a lower matching elasticity means that the labor market is more frictional in matching vacancies with unemployed workers. Because job creation flows are hampered, whereas job
destruction flows remain unchanged, the unemployment rate rises.
Finally, increasing the workers’ bargaining weight, η, from 0.04 to 0.05 increases the
19
Figure 3 : The Vacancy Filling Rate and the Marginal Costs of Hiring in the
Baseline Model, Small and Large Profits
Let x1 < x2 < . . . < x17 denote the x grid. In each panel, the blue solid line is for xt = x3 , the red
dashed line for xt = x9 , and the black dashdot line for xt = x15 .
Panel A: q(θt ),
b = 0.85
1
0.8
Panel B:
Panel C: q(θt ), b = 0.4
κ0 /q(θ t ) + κ1 , b = 0.85
Panel D:
κ0 /q(θ t ) + κ1 , b = 0.4
5
1
12
0.8
10
4
0.6
8
0.6
3
6
0.4
0.4
2
0.2
0
0
0.2
0.4 0.6
Employment
0.8
1
1
0
4
0.2
0.2
0.4 0.6
Employment
0.8
1
0
0.2
2
0.4
0.6
0.8
Employment
1
0
0.2
0.4
0.6
0.8
Employment
1
disaster probability to 5.57% for output and 3.59% for consumption. As workers gain more
bargaining power, the mean unemployment rate rises to 6.76%, causing the output volatility
to increase from 5.31% to 5.58%, and the consumption volatility from 4.65% to 5.03%.
However, the unemployment volatility is barely changed at 0.24.
2.4.3
Additional Intuition: Downward Rigidity in Marginal Hiring Costs
In addition to the elasticity (and magnitude) of wages, the other key determinant of the
firm’s hiring decisions is the marginal costs of hiring, κ0 /q(θt ) + κ1 . To illustrate the properties of the marginal costs, Figure 3 plots the vacancy filling rate, q(θt ), and the marginal
costs of hiring in the model with small profits (b = 0.85) and with large profits (b = 0.4).
Each panel has three lines, each corresponding to a different level of productivity.
Panel A shows that when productivity is low, the vacancy filling rate, q(θt ), is essentially
unity with b = 0.85. Intuitively, the labor market is populated by a large number of unemployed workers competing for a few vacancies. Filling a vacancy with an unemployed worker
is guaranteed, with no room for the vacancy filling rate to increase further. Accordingly, the
marginal costs of hiring equal the constant κ0 + κ1 , with no room to drop further, giving rise
20
to the downward rigidity in the marginal costs (Panel B). The rigid marginal costs suppress
the firm’s incentives of hiring, smothering job creation flows and giving rise to disasters.
Arising from the intrinsic nature of the matching process, this downward rigidity subsists even without the fixed costs of vacancy (κ1 = 0). By putting the constant κ1 into the
marginal costs of hiring, the fixed costs further restrict the ability of the marginal costs to
decline, fortifying the downward rigidity. This economic mechanism explains why removing
the fixed costs makes disasters less frequent and less severe in the baseline model (Table 4).
Panels C and D show that the downward rigidity in the marginal costs is absent with large
profits. The vacancy filling rate, q(θt ), is quite sensitive to employment when profits are large
(Panel C). Intuitively, with a low b of 0.4, vacancies are plentiful even when productivity is
low. The labor market is populated by a fair number of both vacancies and unemployed workers. As employment falls, q(θt ) keeps climbing, reducing the marginal costs of hiring (Panel
D). The falling marginal costs stimulate job creation flows, dampening disaster dynamics.
2.4.4
An Alternative Calibration
The baseline model is a textbook economic model, yet we opt to target disaster moments
from the Barro-Ursua (2008) dataset, which contains not only economic disasters such as
the Great Depression, but also wars and natural disasters such as earthquakes, floods, and
epidemics. This choice merits further discussion.
First, we calibrate the volatility of productivity shocks, σ, to match the output volatility
in the data. To the extent that wars and natural disasters are exogenous to the model economy, these impulses can, at least in principle, be captured by large negative productivity
shocks. The model then quantifies the impact of these impulses on endogenous variables such
as unemployment and output. Even for the Great Depression, the model does not identify
its origins. Second, empirically disentangling economic disasters from other types of disasters unavoidably requires judgement calls that might appear arbitrary. For instance, wars
could be endogenous responses to harsh economic conditions, which give rise to destructive
conflicts among rival nations. Conversely, the Great Depression has been argued to origi-
21
nate from the First World War (Temin 1989).10 Finally, to the extent that wars provide an
independent propagation mechanism, beyond the original impulses, the Barro-Ursua (2008)
disaster moments pose a very high hurdle for any economic model to match.
To supplement our main results, we have experimented with an alternative calibration.
Instead of matching the output volatility of 5.63% per annum in the cross-country panel, we
rescale the volatility of productivity shocks, σ, to match the output volatility of 4.28% in the
historical 1790–2013 U.S. sample. This target is conservative in that the output volatility is
higher, 4.89%, in the 1929–2013 U.S. sample from NIPA. We target the U.S. data because
the damage from world wars on its economy is relatively small compared with other nations.
In particular, we set σ to 0.00925, which implies a volatility of 4.25% for output and 3.61%
for consumption in the baseline model (the Online Appendix). The unemployment rate is
5.83% on average, and its volatility 18.71%. More important, the disaster dynamics remain
substantial. The disaster probability is 4%, size 19.62%, and duration 4.57 years for output,
and the disaster probability is 2.14%, size 22.29%, and duration 5.02 years for consumption.
3
Extensions
The disaster dynamics are robust to several extensions to the baseline model.
3.1
Home Production
In this subsection, we extend the baseline model to incorporate home production (Benhabib,
Rogerson, and Wright 1991). The household derives utility not only from the consumption
of market goods, Cmt , but also from the consumption of nonmarket, home produced goods,
Cht . We define the composite consumption bundle as:
e
e 1/e
Ct ≡ [aCmt
+ (1 − a)Cht
] ,
10
(12)
In particular, Temin (1989, p. 1) writes: “The origins of the Great Depression lie largely in the
disruptions of the First World War. Its spread owes much to the hostilities and continuing conflicts that
were created by the war and the Treaty of Versailles. And its effects—particularly the victory of National
Socialism in Germany—clearly extend to the Second World War.”
22
in which e ∈ (0, 1] and a ∈ [0, 1]. The elasticity of substitution between market and nonmarket goods is 1/(1 − e), and a is the relative weight of market goods over nonmarket goods.
The household has log utility over the composite consumption, log(Ct ). The marginal utility
e−1
of the market consumption is φt ≡ aCmt
/Cte , and the stochastic discount factor is:
Mt+1
φ
≡ β t+1 = β
φt
Cmt+1
Cmt
e−1 Ct
Ct+1
e
.
(13)
The home production technology is given by:
Cht = Xh Ut ,
(14)
in which Xh > 0 is a constant. This technology is nonstochastic. Aguiar, Hurst, and Karabarbounis (2013), for instance, find no evidence that suggests shocks to home production.
Finally, the equilibrium Nash-wage becomes (see the Online Appendix for derivations):
Wt = η(Xt + κt θt ) + (1 − η)zt ,
in which zt is the total flow value of unemployment activities, defined as:
1−e
1−a
Cmt
zt ≡ Xh
+ b.
a
Cht
(15)
(16)
The market clearing condition becomes: Cmt +κt Vt = Xt Nt . The rest of the model, including
the intertemporal job creation condition, remains identical to the baseline model.
3.1.1
Quantitative Results with Home Production
We set the value of b to 0.5, which is the benefit of unemployment activities other than home
production, such as unemployment insurance, disutility of work, and leisure. In the absence
of concrete evidence on home technology, we set its productivity parameter, Xh , to be unity,
which is the long-term mean of the market technology. We next calibrate the volatility of
market productivity, σ, the relative weight of the market goods, a, and the parameter governing the elasticity between market and home consumption, e, to match three data moments,
including the output volatility, as well as the mean and volatility of unemployment. To
facilitate comparison, all the other parameter values remain identical to the baseline model.
23
Table 5 : Quantitative Results, The Home Production Model
The model results are based on 10,000 simulations. σ Y and σ C are the volatilities, SY and SC
skewness, KY and KC kurtosis, and ρYi and ρC
i ith-order autocorrelations of log output and
consumption growth, respectively. ProbY , SizeY , and DurY , as well as ProbC , SizeC , and DurC are
the probability, size, and duration of output and consumption disasters, respectively. E[U ], SU ,
and KU are the mean, skewness, and kurtosis of monthly unemployment rates, respectively, and
σ U is the quarterly volatility. σ Y , σ C , E[U ], and σ U are in percent.
Data
σ
a
e
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
E[U ]
KU
Model
Data
0.01
0.8
0.85
0.014
0.8
0.85
0.014
0.8
0.9
0.014
0.85
0.85
σ
a
e
5.63
−1.02
11.87
0.16
0
0.02
−0.02
7.83
21.99
3.74
3.41
0.06
3.83
0.15
−0.13
−0.1
−0.08
5
15
4.32
5.29
0.15
4.92
0.16
−0.13
−0.11
−0.08
9.95
18.58
3.74
4.62
0.13
4.95
0.15
−0.13
−0.1
−0.08
8.2
17.21
3.88
3.9
0.01
3.42
0.14
−0.12
−0.1
−0.08
6.88
15.43
3.98
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
6.98
7.26
5.97
7.48
6.58
10.42
5.33
15.73
4.5
9.87
SU
σU
Model
0.01
0.8
0.85
0.014
0.8
0.85
0.014
0.8
0.9
0.014
0.85
0.85
6.37
−0.55
9.19
0.07
0.03
0
−0.02
8.57
23.16
3.75
2.91
0.09
4.22
0.15
−0.13
−0.1
−0.08
3.35
14.42
4.65
4.67
0.2
5.73
0.16
−0.14
−0.11
−0.09
7.52
18.06
3.99
3.74
0.2
5.97
0.15
−0.14
−0.11
−0.08
4.95
16.34
4.31
2.9
0.03
3.48
0.14
−0.12
−0.1
−0.08
3.43
13.81
4.59
2.02
21.76
1.86
10.75
2.44
19.53
3.06
15.3
2.1
4.07
This procedure yields σ = 0.014, a = 0.8, and e = 0.85. Table 5 reports the results. Together, these parameter values imply an output volatility of 5.29%, which is close to 5.31%
in the baseline model. The mean unemployment rate is 6.58%, which is between that in
the baseline model, 6.28%, and the data moment, 6.98%. The unemployment volatility is
19.53%, which is close to 21.76% in the data, but somewhat lower than 23.41% in the baseline
model. The (market) consumption volatility is 4.67%, which is close to 4.65% in the baseline
model. The model also reproduces the positively skewed and leptokurtic unemployment rate
distribution. The skewness and kurtosis are 2.44 and 10.42, which are smaller than 3.52 and
19.18 in the baseline model, but are closer to the data moments of 2.02 and 7.26, respectively.
More important, disaster dynamics are robust to home production. The disaster probability is 9.95% for output, which is even higher than 7.83% in the data and 5.04% in the
baseline model. The disaster probability is 7.52% for consumption, which is still lower than
24
8.57% in the data, but much higher than 2.87% in the baseline model. However, the disaster
size is somewhat smaller, 18.58% for output versus 21.99% in the data, and 18.06% for consumption versus 23.16% in the data. The disaster duration is close to the data, 3.74 versus
3.72 years for output, and 3.99 versus 3.75 for consumption.
Table 5 also reports three comparative statics. First, not surprisingly, reducing the volatility of market productivity from 0.014 to 0.01, which is the value in the baseline model, lowers
the output volatility to 3.41%. The mean unemployment rate falls to 5.97%, and its volatility
to 10.75%. Even with the low σ of 0.01, the disaster probability is still 5% for output, and
3.35% for consumption. However, the disaster size is smaller, 15% for output, and 14.42%
for consumption. Also, the duration rises to 4.32 years for output and 4.65 for consumption.
Increasing the e parameter governing the elasticity of substitution between market and
home goods from 0.85 to 0.9 weakens the disaster dynamics in the model. Both disaster probability and size decrease, and duration increases. The mean unemployment rate falls from
6.58% to 5.33%, and volatility from 19.53% to 15.3%. The output volatility also falls from
5.29% to 4.62%, and the consumption volatility from 4.67% to 3.74%. Intuitively, the magnitude of market consumption dominates that of home consumption (Cmt /Cht is 15.4 under the
benchmark home production calibration). As e increases from 0.85 to 0.9, the two types of
consumption become more substitutable. Home consumption loses its relative appeal, despite
its small size. As a result, the flow value of unemployment activities, zt , falls from 0.87 to
0.83, which in turn lowers unemployment and its volatility, dampening the disaster dynamics.
Increasing the relative weight of market goods, a, from 0.8 to 0.85 also dampens the
disaster dynamics. The disaster probability falls from 9.95% to 6.88% for output, and the
disaster size from 18.58% to 15.43%. The output volatility also drops from 5.29% to 3.9%.
More drastically, the mean unemployment rate falls from 6.58% to 4.5%, and the unemployment volatility from 19.53% to only 4.07%. Intuitively, as the relative weight of market goods
increases in the utility function, home production becomes less valuable to the household.
As a result, the flow value of unemployment activities, zt , falls from 0.87 to 0.77, which
greatly lowers unemployment and its volatility, weakening the disaster dynamics.
Finally, rescaling the volatility of market productivity, σ, to 0.012 matches the output
25
volatility of 4.28% in the historical U.S. sample (the Online Appendix). The mean unemployment rate falls to 6.23%, and the unemployment volatility 14.81%. The disaster dynamics remain substantial. The disaster probability is 7.52%, size 16.65%, and duration 3.98 years for
output, and the probability is 5.39%, size 16.11%, and duration 4.26 years for consumption.
3.2
Capital
In this subsection, we augment the baseline model in Section 2 with capital, which is standard in the business cycle literature (Kydland and Prescott 1982). The firm uses labor, Nt ,
and capital, Kt , to produce with a constant returns of scale technology:
Yt = Xt Ktα Nt1−α ,
(17)
in which Yt is output, and α ∈ (0, 1) is the capital’s weight. We specify xt = log(Xt ) as:
xt+1 = (1 − ρ)x̄ + ρxt + σǫt+1 ,
(18)
in which x̄ is the (non-zero) unconditional mean of xt . We rescale x̄ to ensure the average
marginal product of labor to be unity to facilitate comparison with the baseline model.
The firm incurs adjustment costs when investing. In particular, capital accumulates as:
Kt+1 = (1 − δ)Kt + Φ(It , Kt ),
in which δ is the capital depreciation rate, It is investment, and
"
1−1/ν #
It
a2
Kt ,
Φ(It , Kt ) ≡ a1 +
1 − 1/ν Kt
(19)
(20)
is the installation function (Boldrin, Christiano, and Fisher 2001; Kaltenbrunner and
Lochstoer 2010). In equation (20), ν > 0 is the supply elasticity of capital. We set
a1 = δ/(1 − ν) and a2 = δ 1/ν to ensure no adjustment costs in the deterministic steady state.
The investment Euler equation is given by:
!#
"
1/ν
1/ν
1
It
1
It+1
Yt+1
1 It+1
+
,
= Et Mt+1 α
(1 − δ + a1 ) +
a2 Kt
Kt+1 a2 Kt+1
ν − 1 Kt+1
26
(21)
and the intertemporal job creation condition becomes:
Yt+1
κ0
κ0
+ κ1 − λt = Et Mt+1 (1 − α)
− Wt+1 + (1 − s)
+ κ1 − λt+1
. (22)
q(θt )
Nt+1
q(θt+1 )
The equilibrium wage, Wt , which the firm takes as given, follows (see the Online Appendix):
Yt
Wt = η (1 − α)
+ κt θt + (1 − η)b.
(23)
Nt
Finally, the goods market clearing condition becomes Ct + It + κt Vt = Yt .
3.2.1
Quantitative Results with Capital
We calibrate the capital’s weight in production, α, to be 1/3. We calibrate the unconditional mean of log productivity, x̄ = −0.771, to ensure the long-term average of the marginal
product of labor, (1 − α)Yt /Nt , to be unity in simulations. For the capital elasticity, ν, we
vary it from 0.5 to 2, covering a wide range of empirically plausible values. To facilitate
comparison, all the other parameter values are identical to those in the baseline model.
The two model columns with σ = 0.01 in Table 6 show the smoothing effect of investment.
These columns add capital (with ν = 2) to the baseline model. The output volatility drops
from 5.31% in the baseline model to 3.35%, and the consumption volatility from 4.65% to
2.38%. The mean unemployment rate falls from 6.28% to 5.98%, and volatility from 23.41%
to 14%. The disaster probabilities and size, and the skewness and kurtosis of output and
consumption growth all fall. In particular, the output disaster size falls from 22.22% to
15.76%. However, its disaster probability is still 4.55%, which remains sizeable.
In the remaining model columns, we rescale the conditional volatility of productivity
shocks, σ, to 0.014, so that the output volatility is 5.11% (with ν = 2), which is close to 5.31%
in the baseline model. The consumption volatility is 3.74%, which is lower than 4.65% in the
baseline model, again indicating consumption smoothing. The mean unemployment rate is
7.46%, which is higher than 6.28% in the baseline model, and the unemployment volatility
is 22.51%, which is close to that in the baseline model. The investment growth volatility
is 6.98%, which, although much lower than 23.33% in the long U.S. sample, is relatively
27
Table 6 : Quantitative Results, The Capital Model
The model results are based on 10,000 simulations. σ Y , σ C , and σ I are the volatilities, SY , SC ,
I
and SI skewness, KY , KC , and KI kurtosis, and ρYi , ρC
i and ρi ith-order autocorrelations of log
output, consumption, and investment growth, respectively. ProbY , SizeY , and DurY , as well as
ProbC , SizeC , and DurC are the probability, size, and duration of output and consumption disasters,
respectively. E[U ], SU , and KU are the mean, skewness, and kurtosis of monthly unemployment
rates, respectively, and σ U is the quarterly volatility. σ Y , σ C , σ I , E[U ], and σ U are in percent.
Data
σ
ν
Model
Data
0.01
2
0.014
2
0.014
1.5
0.014
0.5
σ
ν
Model
0.01
2
0.014
2
0.014
1.5
0.014
0.5
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
5.63
−1.02
11.87
0.16
0
0.02
−0.02
7.83
21.99
3.72
3.35
0.1
4.11
0.18
−0.1
−0.08
−0.07
4.55
15.76
4.58
5.11
0.12
4.5
0.19
−0.09
−0.07
−0.06
9.45
18.97
3.89
5.1
0.11
4.49
0.19
−0.1
−0.08
−0.07
9.4
18.81
3.87
4.93
0.1
4.34
0.17
−0.12
−0.09
−0.08
9.07
18.08
3.8
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
6.37
−0.55
9.19
0.07
0.03
0
−0.02
8.57
23.16
3.75
2.38
0.08
4.67
0.21
−0.08
−0.07
−0.06
2.08
14.9
5.39
3.74
0.12
5.18
0.22
−0.07
−0.06
−0.06
5.31
17.69
4.51
4
0.14
5.1
0.2
−0.09
−0.08
−0.06
5.95
17.68
4.33
4.75
0.17
4.79
0.17
−0.12
−0.09
−0.08
8.18
17.98
3.9
σI
SI
KI
ρI1
ρI2
ρI3
ρI4
23.33
−0.79
8.72
0.22
−0.04
−0.54
−0.2
4.52
0.2
4.51
0.17
−0.12
−0.09
−0.08
6.98
0.2
4.94
0.17
−0.12
−0.09
−0.07
6.06
0.17
4.92
0.18
−0.11
−0.09
−0.07
2.88
0
4.66
0.19
−0.1
−0.08
−0.07
E[U ]
SU
KU
σU
6.98
2.02
7.26
21.76
5.98
2.51
11
14
7.46
2.55
11.09
22.51
7.45
2.55
11.12
22.57
6.92
2.64
11.65
22.27
close to 8.94% from 1951 onward.11 Consistent with data, investment growth in the model
is positively autocorrelated at the first lag, but negatively autocorrelated at longer lags.
More important, disaster dynamics are robust to the inclusion of capital. The disaster
probability is 9.45% for output, which is somewhat higher than 7.83% in the data and 5.04%
in the baseline model. The disaster probability is 5.31% for consumption, which is still lower
than 8.57% in the data, but higher than 2.86% in the baseline model. However, the disaster
size is smaller, 18.97% for output versus 21.99% in the data, and 17.69% for consumption
versus 23.16% in the data. The disaster duration is somewhat higher than those in the data,
11
We obtain the series for the real U.S. gross private domestic investment from NIPA Table 1.1.6. The
sample is annual from 1929 to 2013. Because a historical cross-country panel of investment is unavailable,
we do not estimate disaster moments for investment.
28
3.89 versus 3.72 years for output, and 4.51 versus 3.75 for consumption.
The rest of the model columns in Table 6 examines the impact of the supply elasticity of
capital, ν, on the quantitative results. Reducing ν from 2 to 1.5 and further to 0.5 dampens
investment dynamics, but amplifies consumption dynamics. The investment volatility drops
from 6.98% to 6.06% and further to 2.88%, whereas the consumption volatility rises from
3.74% to 4% and further to 4.75%. In addition, the consumption disaster probability goes
up from 5.31% to 5.95% and further to 8.18%, whereas the disaster size increases somewhat
from 17.69% to 17.98%, and duration decreases somewhat from 4.51 to 3.9 years. Intuitively,
ν governs the magnitude of capital adjustment costs. A falling ν means rising adjustment
costs, which dampen investment dynamics, but amplify consumption dynamics.
In contrast, varying ν shows only weak impact on output and unemployment dynamics.
As ν falls from 2 to 0.5, the output volatility falls from 5.11% only to 4.93%, the disaster
probability from 9.45% to 9.07%, and the disaster size and duration are relatively unaffected.
The mean unemployment rate falls from 7.46% to 6.92%, skewness from 2.55 to 2.64, and
kurtosis from 11.09 to 11.65. The unemployment volatility is virtually unchanged.
Finally, in the capital model with ν = 2, rescaling the volatility of productivity shocks, σ,
to 0.012 implies an output volatility of 4.25% per annum, which is close to 4.28% in the historical U.S. sample (the Online Appendix). In the rescaled model, the consumption volatility
is 3.06%, and the investment volatility 5.88%. The mean unemployment rate is 6.79%, and
the unemployment volatility 18.77%. More important, the disaster dynamics remain substantial. The disaster probability is 7.17%, size 17.45%, and duration 4.15 years for output,
and the disaster probability is 3.7%, size 16.31%, and duration 4.84 years for consumption.
3.3
Recursive Utility
In this subsection, we augment the baseline model in Section 2 with recursive preferences,
which are standard in the asset pricing literature (Bansal and Yaron 2004; Barro 2009).
Instead of log utility, the household maximizes recursive utility, denoted Jt , over consumption (Kreps and Porteus 1978; Epstein and Zin 1989) by trading risky shares issued by
29
the representative firm and a risk-free bond. The recursive utility function is given by:
Jt = (1 −
1− 1
β)Ct ψ
+β
1−γ Et Jt+1
1−1/ψ
1−γ
1
1−1/ψ
,
(24)
in which ψ is the elasticity of intertemporal substitution, and γ is the relative risk aversion.
The stochastic discount factor, Mt+1 , becomes:
Mt+1 ≡ β
Ct+1
Ct
− ψ1
Jt+1
1
1−γ 1−γ
Et [Jt+1
]
! ψ1 −γ
,
(25)
f
= 1/Et [Mt+1 ].
and the risk-free rate is given by Rt+1
The consumption Euler equation implies 1 = Et [Mt+1 Rt+1 ], in which Rt+1 ≡ St+1 /(St −
Dt ) is the stock return (St is the cum-dividend equity value). Under constant returns:
Rt+1 =
Xt+1 − Wt+1 + (1 − s) [κ0 /q(θt+1 ) + κ1 − λt+1 ]
.
κ0 /q(θt ) + κ1 − λt
(26)
Intuitively, the stock return is the tradeoff between the marginal benefits of hiring over period t+1 and the marginal costs of hiring in period t (Cochrane 1991). The rest of the model,
including the job creation condition and wages, remains identical to the baseline model.
3.3.1
Quantitative Results with Recursive Utility
Following Bansal and Yaron (2004), we set the risk aversion γ = 10, and the elasticity of intertemporal substitution ψ = 1.5. With the conditional volatility, σ, of 0.01 for productivity
shocks as in the baseline model, the recursive utility model implies an output volatility of
7.34%, which is higher than 5.63% in the data and 5.31% in the baseline model. As such, we
rescale σ to 0.0093, which implies an output volatility of 5.67%. We also report three comparative statics: (i) γ = 7.5; (ii) ψ = 1; and (iii) γ = ψ = 1 (log utility), all with σ = 0.0093.
Table 7 shows that disaster dynamics are robust to recursive utility. With γ = 10 and
ψ = 1.5, the disaster probability is 4.49%, the disaster size 23.92%, and duration 4.46 years
for output, and the disaster probability is 2.51%, size 28.86%, and duration 4.84 years for
consumption. Although the disaster probabilities are smaller than those in the data, the
disaster size is comparable. In addition, the disaster moments are not sensitive to utility pa30
Table 7 : Quantitative Results, Recursive Utility
The model results are based on 10,000 simulated samples. σ Y and σ C denote the volatilities,
SY and SC skewness, KY and KC kurtosis, and ρYi and ρC
i the ith-order autocorrelations of log
output and consumption growth, respectively. ProbY , SizeY , and DurY are the probability, size,
and duration of output disasters, respectively, and ProbC , SizeC , and DurC are analogously defined
for consumption disasters. E[U ], SU , and KU are the mean, skewness, and kurtosis of monthly
unemployment rates, respectively, and σ U is the quarterly unemployment volatility. σ Y , σ C , σ I ,
E[U ], and σ U are in percent. E[R − Rf ] is the equity premium, E[Rf ] the risk-free rate, σ R the
stock market volatility, and σ Rf the interest rate volatility, all in percent per annum.
Data
γ
ψ
Model
Data
10
1.5
7.5
1.5
10
1
1
1
γ
ψ
Model
10
1.5
7.5
1.5
10
1
1
1
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
5.63
−1.02
11.87
0.16
0
0.02
−0.02
7.83
21.99
3.72
5.67
0.87
15.47
0.21
−0.14
−0.13
−0.1
4.49
23.92
4.46
4.97
0.81
14.18
0.19
−0.14
−0.12
−0.1
4.03
22.17
4.56
4.99
0.76
12.36
0.23
−0.12
−0.12
−0.11
4.53
21.92
4.5
4.11
0.61
10.36
0.20
−0.12
−0.12
−0.1
5.03
22.25
4.45
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
6.37
−0.55
9.19
0.07
0.03
0
−0.02
8.57
23.16
3.75
5.05
0.88
17.09
0.19
−0.15
−0.13
−0.1
2.51
28.86
4.84
4.35
0.81
15.69
0.18
−0.15
−0.12
−0.09
2.12
26.51
5.01
4.35
0.81
14.2
0.22
−0.13
−0.13
−0.11
2.51
25.6
4.9
3.44
0.67
11.9
0.19
−0.14
−0.12
−0.1
2.84
25.7
4.93
E[U ]
SU
KU
σU
6.98
2.02
7.26
21.76
6.26
3.66
20.71
25.67
5.88
3.57
20.75
21.99
6.23
3.46
18.48
22.93
5.7
3.29
18
17.88
E[R − Rf ]
E[Rf ]
σR
σ Rf
4.69
1.04
20
12.32
4.45
2.58
15.79
1.64
1.1
2.87
15.15
1.39
4.97
2.6
15.73
1.98
0.22
2.93
14.5
1.54
rameters. Reducing γ to 7.5 lowers the disaster probability to 4.03% for output and 2.12% for
consumption, but the disaster size and duration are largely unaffected. Reducing ψ to unity
lowers the disaster size slightly to 21.92% for output and 25.6% for consumption, but disaster
probabilities are barely changed. Even with log utility (γ = ψ = 1), the disaster moments
are relatively close to those with γ = 10 and ψ = 1.5. The disaster probability rises somewhat to 5.03% for output and to 2.84% for consumption, and the average size falls slightly
to 22.25% for output and 25.7% for consumption. The durations are largely unaffected.
The utility parameters do affect output and consumption volatilities. Moving from recursive utility with γ = 10 and ψ = 1.5 to log utility, the output volatility falls from 5.67% to
4.11%, and the consumption volatility from 5.05% to 3.44%. For labor market moments, the
31
mean unemployment rate drops from 6.26% to 5.7%, and volatility from 25.67% to 17.88%.
However, the unemployment skewness and kurtosis are relatively unaffected.
For asset pricing moments, we compile a historical cross-country panel of real stock market returns and real interest rates, by drawing from Global Financial Data and the Dimson,
Marsh, and Staunton (2002) dataset updated through 2013. The panel contains 20 countries. The starting points of the series range from 1801 for the United Kingdom to 1901 for
several countries such as Canada and Switzerland. Table 7 reports that the equity premium
in the cross-country panel is 4.69% per annum, and the stock market volatility is 20%. Both
are adjusted for financial leverage. The interest rate is on average 1.04%, and its volatility
12.32%. The high interest rate volatility likely reflects sovereign default risk during disasters.
In contrast, the U.S. is relatively immune to sovereign default. In its historical 1836–2013
sample, the interest rate volatility is only 5.53%, and the mean interest rate is 1.78% (the
equity premium is 4.53%, and the stock market volatility 14.14%).12
The recursive utility model does a reasonable job in matching asset prices. With γ = 10
and ψ = 1.5, the model reproduces an equity premium of 4.45% and a stock market volatility
of 15.79%. However, the mean interest rate is 2.58%, which is somewhat high relative to
1.04% in the cross-country panel and 1.78% in the long U.S. sample. The model also implies
a low interest rate volatility of 1.64% (as we do not model sovereign default for parsimony).
This interest rate volatility is much lower than those in Jermann (1998) and Boldrin,
Christiano, and Fisher (2001). Both models feature internal habit, which implies that the
elasticity of intertemporal substitution, ψ, is close to zero. Intuitively, with a lower ψ, the
household cares more about consumption smoothing, and is less willing to rearrange consumption over time to interest rate changes. As such, it takes a larger interest rate change
to induce the household to accept a given change in consumption, raising the interest rate
volatility. In contrast, recursive utility allows a high intertemporal elasticity, which is separate from the high risk aversion, giving rise to a stable interest rate.
Table 7 shows that the equity premium is tightly linked to the relative risk aversion,
12
The asset pricing literature has overwhelmingly focused on the postwar U.S. data. In the 1951–2013
U.S. sample, the equity premium is 5.46% per annum, the stock market volatility 12.78%, both of which
are adjusted for financial leverage. The real interest rate 1%, and the interest rate volatility 2.2%.
32
γ. Lowering γ to 7.5 is sufficient to shrink the equity premium from 4.34% to only 1.1%,
although the stock market volatility falls only slightly. In contrast, quantity dynamics are
much less sensitive to the change in γ. In particular, the output volatility falls from 5.67%
only to 4.97%, and the output disaster probability from 4.49% to 4.03% (the disaster size
from 23.92% to 22.17%). For unemployment, the mean drops from 6.26% slightly to 5.88%,
and the volatility from 25.67% to 21.99%. Unlike the risk aversion, the equity premium is
robust to changes in the intertemporal elasticity of substitution, ψ. Lowering ψ from 1.5
to unity even raises the equity premium somewhat to 4.97%. Finally, with log utility, the
equity premium vanishes at 0.22%. Overall, echoing Tallarini (2000), although critical for
asset prices, the risk aversion seems much less important for quantity dynamics.
Finally, rescaling the volatility of productivity shocks, σ, to 0.00875 yields an output
volatility of 4.3% per annum, which is close to 4.28% in the historical U.S. sample (the
Online Appendix). The consumption volatility in the model falls to 3.68%, and the equity premium 1.78%. The mean unemployment rate falls to 5.7%, and the unemployment
volatility 19.41%. More important, the disaster dynamics remain substantial. The disaster probability is 3.38%, size 20.7%, and duration 4.65 years for output, and the disaster
probability 1.76%, size 24.54%, and duration 5.05 years for consumption.
3.3.2
Intuition: Dividends, Profits, and Inelastic Wages
It is well known that explaining the equity premium in general equilibrium production
economies is challenging (Rouwenhorst 1995). Dividends are often counterfactually countercyclical in these models. Intuitively, dividends equal profits minus investment, and profits
equal output minus wages. When the labor market is frictionless, wages equal the marginal
product of labor. Profits are proportional to, and as procyclical as output. Because investment is more procylical than output (and profits) due to consumption smoothing, dividends
(profits minus investment) must be countercyclical (Kaltenbrunner and Lochstoer 2010).
The search model avoids the pitfall of countercyclical dividends. The crux is that wages
are delinked from the marginal product of labor. The wage elasticity to labor productivity is
0.54 in the model, which is not far from 0.47 in the postwar U.S. data. Because of inelastic
33
wages, profits are more procyclical than output. Working as operating leverage, inelastic
wages magnify the procyclical dynamics of profits. This amplified procyclicality of profits is
sufficient to overcome the procyclicality of vacancy costs to turn dividends procyclical.13
The dividend and profit dynamics in the model are largely consistent with those in the
data. Dividends in production economies correspond to net payout (dividends plus stock
repurchases minus new equity issues) in the data. In the postwar U.S. data, the cyclical component of real net payout has a correlation of 0.54 with the cyclical components of both real
output and consumption.14 The relative volatility of net payout (the volatility of its leverageadjusted cyclical component divided by that of real output) is 20.18. In the model, the correlation between dividends and output is 0.59, and that between dividends and consumption
is 0.65. The relative volatility of dividends is 12.84 in the model with a standard deviation
of 3.01, which is low relative to 20.18 in the data. Finally, the relative volatility of profits to
output is 5.18 in the data.15 The model counterpart is 4.12, with a standard deviation of 1.14.
3.3.3
Time-varying Risk Premiums
With our globally nonlinear algorithm, we can study the model’s implications for timevarying risk premiums. Table 8 reports long-horizon regressions of stock market excess
returns and consumption growth on log price-to-consumption. We do not use log priceto-dividend because dividends can be negative in the model. Consistent with Beeler and
Campbell (2012), stock prices forecast excess returns, but not consumption growth, both in
the historical (1836–2013) annual U.S. sample and in the postwar quarterly U.S. sample.16
13
Danthine and Donaldson (2002) show that the priority status of wages magnifies dividend risks.
However, without uninsurable distribution risk, their model only produces an equity premium of about
1% per annum. Favilukis and Lin (2015) quantify the role of infrequent wage renegotiations in a general
equilibrium production economy with long run risks and labor adjustment costs. In contrast, we derive
equilibrium wages in the search framework, and obtain the equity premium via disaster risks.
14
Net payout is net dividends of nonfinancial corporate business minus net increase in corporate equities of
nonfinancial business from Financial Accounts of the Federal Reserve Board (Jermann and Quadrini 2012).
The sample is quarterly from the fourth quarter of 1951 to the fourth quarter of 2013. We use implicit price
deflator for gross domestic product (NIPA Table 1.1.9) to deflate net payout. We detrend real net payout,
gross domestic product, and consumption as HP-filtered proportional deviations from the mean with a
smoothing parameter of 1,600. We do not take logs because the net payout can be negative in the data.
15
Annual real profits and output are detrended as HP-filtered proportional deviations from the mean
with a smoothing parameter of 100.
16
Stock returns, interest rates, and S&P composite index series are from Global Financial Data. For the
historical sample, the real consumption data are from Barro and Ursua (2008) extended through 2013. For
34
Table 8 : Predicting Excess Returns and Consumption growth with the Log
Price-to-Consumption Ratio
P
f
The long-horizon predictive regression of excess returns is H
h=1 log(1 + Rt+h ) − log(1 + Rt+h ) =
f
αR + β R log(Pt /Ct ) + ǫR
t+h , in which H is the forecast horizon, Rt the real stock market return, R
the real interest rate, Pt the real stock market index, and Ct real consumption. Excess returns in the
data are adjusted for financial leverage. The long-horizon predictive regression of log consumption
P
C
growth is H
h=1 log(Ct+h /Ct ) = αC + β C log(Pt /Ct ) + ǫt+h . log(Pt /Ct ) is standardized to have
a mean of zero and a variance of unity. H ranges from one year (1y) to five years (5y), and
from one quarter (1q) to 20 quarters (20q). The t-statistics are Newey-West (1987) adjusted for
heteroscedasticity and autocorrelations of 2(H − 1) lags. The slopes and the R2 s are in percent.
The model moments are averaged across 10,000 simulations.
H
1y
2y
3y
4y
5y
1q
U.S. annual data, 1836–2013
βR
tR
2
RR
βC
tC
2
RC
−3.68
−4.54
8.1
0.46
1.69
1.48
−7.64
−5.49
17.09
0.21
0.43
0.15
−10.43
−5.2
23.65
0.05
0.07
0.01
−13.66
−5.57
31.31
−0.10
−0.10
0.02
4q
−1.91
−1.83
2.04
−0.67
−2.19
7.62
−3.44
−2.17
3.49
−1.68
−2.82
11.95
−4.71
−2.24
4.65
−2.69
−3.37
16.28
−5.75
−2.28
5.59
−3.62
−3.83
20.1
12q
16q
20q
U.S. quarterly data, 1947q2–2013q4
−16.65
−6.43
38.97
−0.15
−0.13
0.03
−0.74
−2.31
1.75
0.06
1.66
1.23
−3.39
−2.75
8.06
0.07
0.40
0.27
Recursive utility
βR
tR
2
RR
βC
tC
2
RC
8q
−6.78
−2.94
16.64
−0.05
−0.13
0.05
−9.69
−3.24
24.56
−0.16
−0.27
0.31
−12.18
−3.36
30.67
−0.24
−0.31
0.51
−15.03
−3.68
36.69
−0.34
−0.34
0.77
−8.20
−2.60
12.14
−3.41
−4.24
27.63
−9.49
−2.82
13.97
−4.19
−4.76
31.82
Recursive utility
−6.62
−2.31
6.35
−4.42
−4.19
23.32
−0.74
−1.65
1.17
−0.00
−1.11
2.56
−2.72
−2.04
4.14
−0.56
−2.14
8.54
−4.89
−2.19
7.35
−1.53
−2.98
16.09
−6.68
−2.38
9.96
−2.51
−3.65
22.43
In particular, in the historical sample, the negative slopes for the log price-to-consumption
ratio across 1- to 5-year horizon are all more than 4.5 standard errors from zero, and the R2
rises monotonically from 8.1% to 38.97%. The postwar evidence is similar, although the tstatistics for the slopes are somewhat smaller. In contrast, the consumption growth is largely
unpredictable. None of the slopes are significant at the 5% level. The R2 starts at 1.48% in
the 1-year horizon, but quickly fades to zero afterward. The postwar evidence is similar.
The model predicts time-varying risk premiums. The price-to-consumption slopes are all
negatively, and mostly significant, but the amount of predictability is weaker in the model
than in the data. In the annual frequency, the R2 rises from 2.04% to only 6.35% (38.97% in
the postwar sample, real consumption is real per capita nondurables plus services from NIPA Table 7.1.
35
the data), and in the quarterly frequency, the R2 goes from 1.17% to 13.97% (36.69% in the
data). More important, the model overstates the predictability of consumption growth. The
slopes are all negative, and mostly significant. The annual R2 ranges from 7.62% to 23.32%,
and the quarterly R2 from 2.56% to 31.82%. In contrast, R2 never exceeds 2% in the data.
Intuitively, sticky wages give rise to operating leverage. In bad times, output falls, but
sticky wages cause profits to drop disproportionately more than output. By dampening the
procyclical covariation of wages, sticky wages magnify the procyclical covariation (risk) of
dividends, causing the equity premium to rise. More important, the impact of sticky wages
is stronger in worse economic conditions, when the profits are even smaller because of the
lower labor productivity. This time-varying operating leverage magnifies the risk and risk
premiums, making the equity premium countercyclical.
3.3.4
The Timing Premium
Epstein, Farhi, and Strzalecki (2014) show that the household in the Bansal-Yaron (2004)
model would give up an implausibly high fraction, 31%, of its consumption stream (dubbed
the timing premium) for all the consumption risk to be resolved immediately. This timing
premium seems high because it is only about the early resolution of risk, and the household
cannot use this information to modify its risky consumption stream.17
Because we follow Bansal and Yaron’s (2004) calibration of preferences, it is natural to
ask what the timing premium is in our model with recursive utility. With the risk aversion
γ = 10 and the elasticity of intertemporal substitution ψ = 1.5, our timing premium is
only 17%. Intuitively, the long run risks model assumes high persistence of the consumption
growth, meaning that the risks are not resolved until much later. As such, the household
that prefers early resolution of risk would pay a high timing premium for the risks to be
resolved. In contrast, the expected consumption growth in our disaster risks model is much
less persistent. Not surprisingly, our timing premium is much smaller.18
Formally, the timing premium is π ≡ 1 − J0 /J0⋆ , in which J0 is the household’s utility with risk resolved
gradually, and J0⋆ is the utility with risk resolved instantly. In practice, we follow Epstein, Farhi, and
Strzalecki (2014) and calculate J0⋆ via Monte Carlo simulations with a fixed time horizon of T = 2, 500
months (and pasting J0 as the continuation value at T ).
18
We simulate one million months from the stationary distribution of our recursive utility model. Fitting
17
36
4
Conclusion
Our key insight is that the textbook Diamond-Mortensen-Pissarides search model of equilibrium unemployment, once solved accurately with a globally nonlinear projection algorithm,
gives rise endogenously to rare disasters. Intuitively, in bad times, sticky wages remain
relatively high, reducing profits. The marginal costs of hiring fail to decline, running into
downward rigidity, which is an inherent attribute of the matching process. The falling profits and rigid hiring costs combine to stifle job creation flows, depressing the economy into
disasters. Our new insight seems potentially important, as the search model of equilibrium
unemployment has been adopted throughout modern macroeconomics.
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40
For Online Publication Only
A
Data
We detail the data sources used in our study.
A.1
A.1.1
Historical Cross-country Panels
Output and Consumption
The Barro-Ursúa (2008) historical cross-country panel of real per capita output and consumption is obtained from Robert Barro’s Web site. Their panel ends in 2009, and we
extend it through 2013. In particular, we construct real per capita gross domestic product
(GDP) growth with two series from World Development Indicators (WDI) at the World Bank
(http://data.worldbank.org/data-catalog/world-development-indicators), real GDP growth
(code: NY.GDP.MKTP.KD.ZG), gGDP , and the population growth (code: SP.POP.GROW),
gPOP . Both growth rate series are in annual percent. We take 2006 as the base year and set
the GDP index, Y2006 , as 100. For year t after 2006, we calculate its GDP index as:
Qt
(1 + gGDP /100)
.
Yt = 100 × Q2006
t
(1
+
g
/100)
POP
2006
(A.1)
We construct real per capita consumption growth analogously from two WDI
series, annual percent growth of household final consumption expenditure (code:
NE.CON.PRVT.KD.ZG) and the population growth (code: SP.POP.GROW).
Finally, WDI seems to revise its published data retrospectively. To be consistent, we opt
to update the output and consumption series from 2006 onward, as opposed to 2009, when
the original Barro-Ursúa panel ends.
A.1.2
Asset Prices
We compile our cross-country panel of real stock market returns and real interest rates from
Global Financial Data (GFD) and the Dimson, Marsh, and Staunton (DMS, 2002) dataset
updated through 2013. We purchase the DMS dataset from Morningstar.
In constructing our cross-country panel, we search the GFD database for countries with
total return indexes going back to at least 1930s. We then supplement the GFD data with
the DMS database. For a given country, the starting year of the sample for real stock returns
41
often differs from that for real interest rates. In such cases, we take the common sample
period for stocks and bills. We only manage to verify dividend yields data for four countries
from GFD, but the series are not adjusted for inflation. As such, we opt not to use the
Barro-Ursúa (2009) data of asset prices. In any event, the basic asset pricing moments in
our cross-country panel are relatively close to those reported in Barro and Ursúa (2009).
Table A.1 reports descriptive statistics of real stock and bill returns across countries.
Both raw and leverage-adjusted financial moments are reported. Following Barro (2006), we
e −Rf ], in which ω
calculate the leverage-adjusted equity premium as E[R −Rf ] = (1 −ω)E[R
is leverage, defined as the market value of debt/(the market value of debt + the market value
f
e
of equity), and E[R−R
] is the equity premium without adjusting for leverage. The leverageet +(1−ω)Rtf . We set ω =
adjusted stock market volatility, σ R , is the standard deviation of ω R
0.29, which is the mean leverage ratio across 39 countries in Fan, Titman, and Twite (2012).
It should be noted that although standard in the literature (Barro 2006, Section III; see
also Kaltenbrunner and Lochstoer 2010), the leverage adjustment procedure is somewhat
crude. However, this simple procedure allows one to abstract from modeling sovereign default risk, which is an important topic on its own. However, its modeling would take us too
far afield. In any event, accounting for long-term (and defaultable) debt would reduce the
equity premium in the data. As such, our procedure raises the hurdle on the model to match
the equity premium. However, to the extent that bond returns are more volatile than bill
rates, our procedure likely understates the leverage-adjusted stock market volatility.
A.2
The Long U.S. Sample
Output (gross domestic product) data are from National Income and Product Accounts
(NIPA) Table 1.1.6, and profits data are from NIPA Table 1.12 row 13 (corporate profits
with inventory valuation adjustment and capital consumption adjustment). We use the implicit price deflator of gross domestic product (NIPA Table 1.1.9) to deflate profits. The
sample is annual from 1929 to 2013.
The series of monthly unemployment rates in the U.S. from April 1929 to December 2013
are from Global Financial Data.
We obtain the series for the real U.S. gross private domestic investment from NIPA Table
1.1.6. The sample is annual from 1929 to 2013.
42
Table A.1 : Properties of Asset Prices
Results are based on a cross-country panel of real stock market returns and real interest rates drawn
from GFD and the DMS dataset updated through 2013. “Start” is the starting year of the sample
e − Rf ] are the average
e σ e , and E[R
for a given country (all country samples end in 2013). E[R],
R
stock market returns, the stock market volatility, and the equity premium, respectively, without
adjusting for financial leverage. E[R − Rf ] and σ R are the equity premium and the stock market
volatility, respectively, after adjusting for leverage. E[Rf ] is the mean interest rate, and σ Rf is the
interest rate volatility. All moments are in annual percent.
Start
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
South Africa
Spain
Sweden
Switzerland
United Kingdom
United States
Average
A.3
1876
1901
1901
1901
1901
1901
1896
1870
1901
1901
1901
1901
1901
1901
1901
1901
1901
1901
1801
1836
e
E[R]
σ Re
E[Rf ]
σ Rf
9.81
6.36
6.34
7.16
8.15
9.57
6.34
5.26
7.45
5.57
9.1
8.14
8.32
8.47
9.78
6.92
8.36
7.3
6.32
8.16
16.68
38.04
26.21
17.79
23.05
30.65
26.18
31.9
25.53
28.78
30.24
24.8
25.74
30.04
29.78
28.25
21.49
20.9
16.39
19.3
1.24
−2.44
1.34
1.7
3.49
0.32
−1.58
−1.21
1.63
−0.85
−0.56
1.9
2.42
2.22
1.14
2.05
1.24
2.29
2.67
1.78
5.28
24.55
14.67
5.73
13.09
11.47
10.58
16.53
12.08
12.6
13.51
13.22
14.76
13.08
15.39
18.42
6.91
12.07
6.96
5.53
7.64
25.59
1.04
12.32
e − Rf ]
E[R
E[R − Rf ]
σR
8.57
8.79
5
5.46
4.66
9.25
7.92
6.47
5.82
6.42
9.66
6.24
5.9
6.25
8.63
4.86
7.12
5.01
3.66
6.38
6.08
6.24
3.55
3.88
3.31
6.57
5.62
4.59
4.13
4.56
6.86
4.43
4.19
4.44
6.13
3.45
5.06
3.56
2.6
4.53
12.34
31.19
20.82
13.14
18.28
23.01
19.54
25.56
20.13
21.76
23.36
19.48
21.21
23.31
24.09
23.69
15.71
16.57
12.6
14.14
6.6
4.69
20
Postwar U.S. Data
Following Hagedorn and Manovskii (2008), we measure wages as labor share times labor
productivity (real output per job) in the nonfarm business sector from Bureau of Labor
Statistics. The sample is quarterly from 1947 to 2013. We detrend wages and labor
productivity with log deviations from the Hodrick-Prescott (HP, 1997) trend with a
smoothing parameter of 1,600.
As in Jermann and Quadrini (2012), we measure the net payout as net dividends of
nonfinancial corporate business (Table F.102, series FA106121075.Q) minus net increase in
corporate equities of nonfinancial business (Table F.101, series FA103164103.Q).
43
B
B.1
Derivations
Wages
We derive equilibrium wages under recursive utility. The case with log utility is similar (and
more standard). Let η ∈ (0, 1) denote the relative bargaining weight of the worker, JN t the
marginal value of an employed worker to the representative family, JU t the marginal value
of an unemployed worker to the representative family, φt the marginal utility of the representative family, SN t the marginal value of an employed worker to the representative firm,
and SV t the marginal value of an unfilled vacancy to the representative firm.
A worker-firm match turns an unemployed worker into an employed worker for the representative household as well as an unfilled vacancy into a filled vacancy (an employed worker)
for the firm. As such, we can define the total surplus from the Nash bargain as:
Λt ≡
JN t − JU t
+ SN t − SV t .
φt
(B.1)
The wage equation (10) is determined via the Nash worker-firm bargain:
max
{Wt }
JN t − JU t
φt
η
(SN t − SV t )1−η ,
(B.2)
The outcome of maximizing equation (B.2) is the surplus-sharing rule:
JN t − JU t
= ηΛt = η
φt
JN t − JU t
+ SN t − SV t .
φt
(B.3)
As such, the worker receives a fraction of η of the total surplus from the wage bargain. In
what follows, we derive the wage equation (10) from the sharing rule in equation (B.3).
B.1.1
Workers
To derive JN t and JU t , we need to specify the details of the representative household’s probf
lem. Tradeable assets consist of risky shares and a risk-free bond. Let Rt+1
denote the
risk-free interest rate, which is known at the beginning of period t, Πt the household’s
financial wealth, χt the fraction of the household’s wealth invested in the risky shares,
f
Π
Rt+1
≡ χt Rt+1 + (1 − χt )Rt+1
the return on wealth, and Tt the taxes raised by the gov-
44
ernment. The household’s budget constraint is given by:
Πt+1
= Πt − Ct + Wt Nt + Ut b − Tt .
Π
Rt+1
(B.4)
The household’s dividends income, Dt , is included in the current financial wealth, Πt .
Let φt denote the Lagrange multiplier for the household’s budget constraint (B.4). The
household’s maximization problem is given by:
Jt = (1 −
1− 1
β)Ct ψ
1−γ
Jt+1
+ β Et
1−1/ψ
1−γ
1
1−1/ψ
− φt
Πt+1
− Πt + Ct − Wt Nt − Ut b + Tt ,
Π
Rt+1
(B.5)
The first-order condition for consumption yields:
φt = (1 −
−1
β)Ct ψ
(1 −
1− 1
β)Ct ψ
1−1/ψ
1−γ
1−γ
+ β Et Jt+1
1
1−1/ψ
−1
,
(B.6)
which gives the marginal utility of consumption.
Recalling Nt+1 = (1 − s)Nt + f (θt )Ut and Ut+1 = sNt + (1 − f (θt ))Ut , we differentiate Jt
in equation (B.5) with respect to Nt :
JN t
1
= φt Wt +
1−
×
1−
1
ψ
1−γ
1
ψ
(1 −
1− 1
β)Ct ψ
+ β Et
1−γ
Jt+1
1−1/ψ
1−γ
1
1−1/ψ
−1
1−1/ψ
−γ
1−γ 1−γ −1
Et (1 − γ)Jt+1
[(1 − s)JN t+1 + sJU t+1 ] . (B.7)
β Et Jt+1
Dividing both sides by φt :
JN t
= Wt +
φt
β
1
−ψ
(1 − β)Ct


1
1−γ
Et Jt+1
Dividing and multiplying by φt+1 :

− ψ1
1
1−γ
 ψ1 −γ


−γ
Et Jt+1
[(1 − s)JN t+1 + sJU t+1 ] .
 ψ1 −γ
Jt+1

1
1−γ
1−γ
Et Jt+1
JU t+1
JN t+1
+s
.
= Wt + Et Mt+1 (1 − s)
φt+1
φt+1
JN t

= Wt + Et β
φt
Ct+1
Ct

45
(1 − s)
JN t+1
φt+1
(B.8)

JU t+1 
+s

φt+1
(B.9)
Similarly, differentiating Jt in equation (B.5) with respect to Ut yields:
JU t
1
= φt b +
1−
×
1−
1
ψ
1−γ
1
ψ
(1 −
1− 1
β)Ct ψ
1−γ
Jt+1
+ β Et
1−1/ψ
1−γ
1
1−1/ψ
−1
1−1/ψ
−γ
1−γ 1−γ −1
Et (1 − γ)Jt+1
[f (θt )JN t+1 + (1 − f (θt ))JU t+1 ] (B.10)
.
β Et Jt+1
Dividing both sides by φt :
JU t
=b+
φt
β
1
−ψ
(1 − β)Ct


1
1
1−γ
1−γ
Et Jt+1
 ψ1 −γ

−γ
Et Jt+1
[f (θt )JN t+1 + (1 − f (θt ))JU t+1 ] .
(B.11)
Dividing and multiplying by φt+1 :

JU t

= b + Et β
φt
Ct+1
Ct
− ψ1


Jt+1
1−γ 1−γ
Jt+1
1
Et
 ψ1 −γ


JN t+1
JU t+1 
f (θt )
+ (1 − f (θt ))

φt+1
φt+1
JU t+1
JN t+1
+ (1 − f (θt ))
.
= b + Et Mt+1 f (θt )
φt+1
φt+1
B.1.2
(B.12)
The Firm
Start by rewriting the infinite-horizon value-maximization problem of the firm recursively as:
St = Xt Nt − Wt Nt − κt Vt + λt q(θt )Vt + Et [Mt+1 St+1 ],
(B.13)
subject to Nt+1 = (1 − s)Nt + q(θt )Vt . The first-order condition with respect to Vt says:
Equivalently,
SV t = −κt + λt q(θt ) + Et [Mt+1 SN t+1 q(θt )] = 0.
(B.14)
κ0
+ κ1 − λt = Et [Mt+1 SN t+1 ].
q(θt )
(B.15)
In addition, differentiating St with respect to Nt yields:
SN t = Xt − Wt + (1 − s)Et [Mt+1 SN t+1 ].
Combining the last two equations yields the job creation condition in equation (22).
46
(B.16)
B.1.3
The Wage Equation
From equations (B.9), (B.12), and (B.16), the total surplus of the worker-firm relationship is:
Λt
JU t+1
JN t+1
−b
+s
= Wt + Et Mt+1 (1 − s)
φt+1
φt+1
JN t+1
JU t+1
−Et Mt+1 f (θt )
+ (1 − f (θt ))
+ Xt − Wt + (1 − s)Et [Mt+1 SN t+1 ]
φt+1
φt+1
JN t+1 − JU t+1
JN t+1 − JU t+1
= Xt − b + (1 − s)Et Mt+1
+ SN t+1
− f (θt )Et Mt+1
φt+1
φt+1
= Xt − b + (1 − s)Et [Mt+1 Λt+1 ] − ηf (θt )Et [Mt+1 Λt+1 ] ,
(B.17)
in which the last equality follows from the definition of Λt and the surplus sharing rule (B.3).
The sharing rule implies SN t = (1 − η)Λt , which, combined with equation (B.16), yields:
(1 − η)Λt = Xt − Wt + (1 − η)(1 − s)Et [Mt+1 Λt+1 ] .
(B.18)
Combining equations (B.17) and (B.18) yields:
Xt − Wt + (1 − η)(1 − s)Et [Mt+1 Λt+1 ] = (1 − η)(Xt − b) + (1 − η)(1 − s)Et [Mt+1 Λt+1 ]
−(1 − η)ηf (θt )Et [Mt+1 Λt+1 ]
Xt − Wt = (1 − η)(Xt − b) − (1 − η)ηf (θt )Et [Mt+1 Λt+1 ]
Wt = ηXt + (1 − η)b + (1 − η)ηf (θt )Et [Mt+1 Λt+1 ] .
Using equations (B.3) and (B.15) to simplify further:
Wt = ηXt + (1 − η)b + ηf (θt )Et [Mt+1 SN t+1 ]
κ0
+ κ1 − λt .
= ηXt + (1 − η)b + ηf (θt )
q(θt )
(B.19)
(B.20)
Using the Kuhn-Tucker conditions, when Vt > 0, then λt = 0, and equation (B.20) reduces
to the wage equation (10) because f (θt ) = θt q(θt ). On the other hand, when the nonnegativity constraint is binding, λt > 0, but Vt = 0 means θt = 0 and f (θt ) = 0. Equation (B.20)
reduces to Wt = ηXt + (1 − η)b. Because θt = 0, the wage equation (10) continues to hold.
47
B.1.4
Acceptable Wages
More generally, any wages between the workers’ and firm’s reservation wages would be acceptable (Hall 2005). Workers would want to work when wages are no less than their reservation
wage, denoted, W t , which is the wage rate that sets the workers’ surplus to zero in the workerfirm match. In particular, setting JN t in equation (B.9) to JU t in equation (B.12) yields:
JU t+1 − JN t+1
.
W t = b + Et Mt+1 (1 − f (θt ) − s)
φt+1
(B.21)
On the other hand, a variable wage rate must also be lower than the highest wage that the
firm is willing to pay, W t , the firm’s reservation wage that sets the firm’s surplus to zero. In
particular, combining equations (B.15) and (B.16) and setting SN t to zero yield:
κ0
W t = Xt + (1 − s)
+ κ1 − λt .
q(θt )
B.1.5
(B.22)
Home Production
With home production, the marginal value of an unemployed worker to the household is:
JU t =
∂ log(Ct ) ∂Cht
+ φt b + βEt [f (θt )JN t+1 + (1 − f (θt ))JU t+1 ] .
∂Cht ∂Ut
(B.23)
It follows that:
JU t
= Xh
φt
1−a
a
Cmt
Cht
1−e
+ b + βEt [f (θt )JN t+1 + (1 − f (θt ))JU t+1 ] .
(B.24)
The wage equation (15) under home production then follows from the same derivations in
Appendix B.1.3, after redefining b as Xh ((1 − a)/a) (Cmt /Cht )1−e + b.
B.1.6
Capital
In the capital model, the derivation of wages is analogous. The firm’s problem becomes:
St = Yt − Wt Nt − It − κt Vt + λt q(θt )Vt + Et [Mt+1 St+1 ],
(B.25)
The marginal product of labor becomes (1 − α)Yt /Nt , and the rest of the proof follows.
48
B.2
Stock Returns
We prove equation (26) following the same logic in Liu, Whited, and Zhang (2009) in the
context of the q-theory of investment. Rewrite the equity value maximization problem as:

X
N
−
W
N
−
κ
V
t+τ
t+τ
t+τ
t+τ
t+τ
t+τ
∞


X
St =
max ∞ Et  Mt+τ  −µt+τ [Nt+τ +1 − (1 − s)Nt+τ
 ,
{Vt+τ ,Nt+τ +1 }τ =0
τ =0
−Vt+τ q(θt+τ )] + λt+τ q(θt+τ )Vt+τ


(B.26)
in which µt is the Lagrange multiplier on the employment accumulation equation, and λt is
the Lagrange multiplier on the V -constraint on job creation. The first-order conditions with
respect to Vt and Nt+1 in maximizing the market value of equity are given by, respectively:
κ0
+ κ 1 − λt ,
q(θt )
= Et Mt+1 Xt+1 − Wt+1 + (1 − s)µt+1 .
µt =
(B.27)
µt
(B.28)
The Kuhn-Tucker conditions are given by equation (9). Define dividends as Dt ≡ Xt Nt −
Wt Nt − κt Vt and the ex-dividend equity value as Pt ≡ St − Dt . Expanding St yields:
Pt + Xt Nt − Wt Nt − κt Vt = St = Xt Nt − Wt Nt − κt Vt + λt q(θt )Vt
−µt [Nt+1 − (1 − s)Nt − Vt q(θt )] + Et Mt+1 [Xt+1 Nt+1 − Wt+1 Nt+1 − κt+1 Vt+1
−µt+1 [Nt+2 − (1 − s)Nt+1 − Vt+1 q(θt+1 )] + λt+1 q(θt+1 )Vt+1 + . . .
(B.29)
Recursively substituting equations (B.27) and (B.28) yields:
Pt + Xt Nt − Wt Nt − κt Vt = Xt Nt − Wt Nt + µt (1 − s)Nt .
(B.30)
Using equation (B.27) to simplify further:
Pt = κt Vt + µt (1 − s)Nt = µt [(1 − s)Nt + q(θt )Vt ] + λt q(θt )Vt = µt Nt+1 ,
in which the last equality follows from equation (9).
49
(B.31)
To show equation (26), we expand the stock returns:
µ Nt+2 + Xt+1 Nt+1 − Wt+1 Nt+1 − κt+1 Vt+1
St+1
= t+1
S t − Dt
µt Nt+1
Xt+1 − Wt+1 − κt+1 Vt+1 /Nt+1 + µt+1 [(1 − s) + q(θt+1 )Vt+1 /Nt+1 ]
=
µt
Xt+1 − Wt+1 + (1 − s)µt+1 µt+1 q(θt+1 )Vt+1 − κt+1 Vt+1
=
+
µt
µt Nt+1
Xt+1 − Wt+1 + (1 − s)µt+1
,
(B.32)
=
µt
Rt+1 =
in which the last equality follows because the Kuhn-Tucker conditions imply:
(B.33)
µt+1 q(θt+1 )Vt+1 − κt+1 Vt+1 = −λt+1 q(θt+1 )Vt+1 = 0.
C
Computation
In this section, we provide computational details for solving different models.
C.1
The Baseline Model
We exploit a convenient mapping from the conditional expectation function, Et , defined as:
E(Nt , xt ) = Et Mt+1
κ0
Xt+1 − Wt+1 + (1 − s)
+ κ1 − λ(Nt+1 , xt+1 )
q(θt+1 )
,
(C.1)
to policy and multiplier functions to eliminate the need to parameterize the multiplier separately. After obtaining Et , we first calculate q̃(θ t ) = κ0 / (Et − κ1 ) . If q̃(θt ) < 1, the Vt ≥ 0
constraint is not binding, we set λt = 0 and q(θt ) = q̃(θt ). We then solve θt = q −1 (q̃(θt )),
in which q −1 (·) is the inverse function of q(θt ), and Vt = θt (1 − Nt ). If q̃(θ t ) ≥ 1, the Vt
nonnegativity constraint is binding, we set Vt = 0, θt = 0, q(θt ) = 1, and λt = κ0 + κ1 − Et .
We then perform the following set of substitutions:
Ut = 1 − Nt
(C.2)
Nt+1 = (1 − s)Nt + q(θt )V (Nt , xt )
(C.3)
xt+1 = ρxt + σǫt+1
(C.4)
C(Nt , xt ) = exp(xt )Nt − [κ0 + κ1 q(θt )] V (Nt , xt )
Wt = η [exp(xt ) + [κ0 + κ1 q(θt )] θ t ] + (1 − η)b
50
(C.5)
(C.6)
We approximate the xt process in equation (4) based on the discrete state space method
of Rouwenhorst (1995) with 17 grid points.19 This grid is large enough to cover the values
of xt within four unconditional standard deviations from its unconditional mean. We set
the minimum value of Nt to be 0.05 and the maximum value to be 0.99. We use cubic
splines with 75 basis functions on the N space to approximate E(Nt , xt ) on each grid point
of xt . We use extensively the approximation took kit in the CompEcon Toolbox in Matlab
of Miranda and Fackler (2002). To obtain an initial guess for the projection algorithm, we
use the second-order perturbation solution via Dynare.
Solving the nonlinear model takes a lot of care, otherwise the projection algorithm would
not converge. Unlike the value function, iterating on the first-order conditions is typically
not a contraction mapping. We use the idea of homotopy continuation methods (Judd 1998,
p. 179) extensively to ensure convergence for a wide range of parameter values.
When we solve the model with a new set of parameters, we set the lower bound of Nt to be
0.4 to alleviate the burden of nonlinearity on the nonlinear solver. After obtaining the model’s
solution, we then apply homotopy to gradually reduce the lower bound to 0.05. Time-wise,
solving the baseline model is relatively fast, but with trial-and-errors that come with homotopy, solving the extended models with a parametrization that admits strong nonlinearity
can take weeks. Indeed, we have encountered several specifications and parameterizations of
the extended models for which the projection solver has failed to converge at all.
Panel A of Figure C.1 reports the error in the E functional equation (C.1) (Et minus the
right-hand-side). The error, in the magnitude of 10−14 , is extremely small. As such, our
projection algorithm does an accurate job in approximating the competitive equilibrium.
C.2
Home Production
The conditional expectation function remains equation (C.1). In addition to equation (C.2)–
(C.4), we perform several substitutions that are specific to the home production model:
Cm (Nt , xt ) = exp(xt )Nt − [κ0 + κ1 q(θt )] V (Nt , xt )
(C.7)
Ch (Nt , xt ) = Xh (1 − Nt )
(C.8)
Wt = η [exp(xt ) + [κ0 + κ1 q(θt )] θt ]
#
" 1−e
Cmt
1−a
+b
+(1 − η) Xh
a
Cht
19
(C.9)
Kopecky and Suen (2010) show that the Rouwenhorst (1995) method is more reliable and accurate than
other methods in approximating highly persistent first-order autoregressive processes.
51
Figure C.1 : Expectational Errors, The Baseline and Home Production Models
Panel B: Home production
Panel A: The baseline model
x 10
−14
x 10
5
5
0
0
−5
1
Employment
−5
1
0.5
0.5
0.5
0.5
0
0 −0.5
−15
Employment
Log productivity
0
0 −0.5
Productivity
We continue to discretize the xt process with 17 grid points per Rouwenhorst (1995).
We use cubic splines with 50 basis functions on the Nt space, [0.35, 0.99]. We obtain an
initial guess from the second-order perturbation solution via Dynare on a smaller grid of Nt ,
[0.8, 0.99], and then gradually reduce the minimum Nt to 0.35 via homotopy. Panel B of
Figure C.1 shows that our projection solution to the home production model is accurate.
C.3
Capital
In the extended model with capital, the state space consists of employment, capital, and productivity, (Nt , Kt , xt ). The goal is to solve for the optimal vacancy function, V (Nt , Kt , xt ),
the multiplier function, λ(Nt , Kt , xt ), and the optimal investment function, I(Nt , Kt , xt ),
from the following two functional equations:
1/ν
I(Nt , Kt , xt )
Y (Nt+1 , Kt+1 , xt+1 )
= Et Mt+1 α
Kt
K(Nt+1 , Kt+1 , xt+1 )
1/ν
##
1
I(Nt+1 , Kt+1 , xt+1 )
I(Nt+1 , Kt+1 , xt+1 )
1
+
(C.10)
(1 − δ + a1 ) +
a2 K(Nt+1 , Kt+1 , xt+1 )
ν − 1 K(Nt+1 , Kt+1 , xt+1 )
κt
Y (Nt+1 , Kt+1 , xt+1 )
− λ(Nt , Kt , xt ) = Et Mt+1 (1 − α)
− Wt+1
q(θt )
Nt+1
κt+1
− λ(Nt+1 , Kt+1 , xt+1 )
,
(C.11)
+ (1 − s)
q(θt+1 )
1
a2
Also, V (Nt , Kt , xt ) and λ(Nt , Kt , xt ) must satisfy the Kuhn-Tucker conditions (9).
An advantage of the installation function in equation (20) is that optimal investment
52
Figure C.2 : Errors, The Capital Model
Panel A: The Et error, capital
x 10
Panel B: The Et error, capital
−15
−15
x 10
5
5
0
0
−5
−5
0.9
0.8
1
0
Capital
Productivity
Panel C: The investment Euler
equation error, capital
x 10
−15
x 10
2
0
0
−2
0.8
Employment
1
0
Productivity
−15
−2
40
0.5
0.7
0.5
0 0
Panel D: The investment Euler
equation error, capital
2
0.9
1
20
0.5
0.7
Employment
−10
40
Capital
Productivity
1
20
0.5
0 0
Productivity
is always positive. When investment goes to zero, the marginal benefit of investment,
∂Φ(It , Kt )/∂It = a2 (It /Kt )−1/ν , goes to infinity. As such, there is no need to impose the
It ≥ 0 constraint, and we approximate I(Nt , Kt , xt ) directly. For the intertemporal job
creation condition in equation (C.11), we use the Christiano-Fisher (2000) parameterized
expectations method by approximating the conditional expectation (the right-hand-side).
We discretize the xt process via the Rouwenhorst (1995) method with 17 grid points. We
use cubic splines with 10 basis functions on the Nt space, [0.65, 0.99], and 10 basis functions
on the Kt space, [15, 40]. We approximate I(Nt , Kt , xt ) and E(Nt , Kt , xt ) on each grid point
of xt . (The deterministic steady state capital is 32.5.) Figure C.2 shows that the errors of
the two functional equations from the projection algorithm are extremely small.
53
Figure C.3 : Errors, The Recursive Utility Model
Panel A: The Jt error, recursive
utility
Panel B: The Et error, recursive
utility
−16
−14
x 10
x 10
10
5
5
0
0
−5
1
0.5
0.5
0 −0.5
Employment
Productivity
0.5
0.5
0
Employment
C.4
−5
1
0
0 −0.5
Productivity
Recursive Utility
In this extended model, the state space consists of employment and productivity, (Nt , xt ).
The goal is to solve for the optimal vacancy function, V (Nt , xt ), the multiplier function,
λ(Nt , xt ), and an indirect utility function, J(Nt , xt ), from two functional equations:
1
1− ψ
1−γ
1−1/ψ
1−γ
1
1−1/ψ
+ β Et J(Nt+1 , xt+1 )
(C.12)
(1 − β)C(Nt , xt )
κ0
κ0
+ κ1 − λ(Nt , xt ) = Et Mt+1 Xt+1 − Wt+1 + (1 − s)
+ κ1 − λ(Nt+1 , xt+1 )
,
q(θt )
q(θt+1 )
(C.13)
J(Nt , xt ) =
in which
Mt+1
C(Nt+1 , xt+1 )
=β
C(Nt , xt )
− ψ1 "
J(Nt+1 , xt+1 )
1
Et [J(Nt+1 , xt+1 )1−γ ] 1−γ
# ψ1 −γ
.
(C.14)
Also, V (Nt , xt ) and λ(Nt , xt ) must satisfy the Kuhn-Tucker conditions (9).
We use the Christiano-Fisher (2000) parameterized expectations method by
approximating the right-hand-side of equation (C.13). We discretize the xt process via the
Rouwenhorst (1995) method with 17 grid points. We use cubic splines with 50 basis functions
on the Nt space, [0.05, 0.99], to approximate J(Nt , xt ) and E(Nt , xt ) on each grid point of
xt . Figure C.3 shows that the errors from the projection algorithm are extremely small.
54
C.4.1
Calculating the Timing Premium
To reduce computational time,
we work with a sparse Nt -xt grid.
The xt
grid is the same as before with 17 grid points, but the Nt grid contains only
N(1) , N(5) , N(10) , N(15) , N(20) , N(25) , N(30) , N(35) , N(40) , N(45) , and N(50) , in which N(i) is the ith
grid point on the original Nt grid with 50 grid points. The household’s indirect utility with
risk resolved gradually, J0 , on the sparse grid is available from our projection algorithm.
We calculate the utility with risk resolved instantly, J0⋆ , on each grid point on the sparse
grid. Following Epstein, Fahri, and Strzalecki (2014), we simulate in total 10,000 sample
paths, each with T = 2, 500 months, using the grid point in question as the initial condition
of Nt and xt . On the nth path, we compute:
1−1/ψ
⋆
J0(n)
=
T −1
X
1− 1
1− 1
β t−1 (1 − β)Ct(n)ψ + β T −1 JT (n)ψ ,
(C.15)
t=1
in which Ct(n) is the time-t consumption, and JT (n) is the time-T utility with risk resolved
gradually on the nth sample path. We then calculate J0⋆ as the cross-simulation averaged
⋆
J0(n)
, and the timing premium as 1 − J0 /J0⋆ on the grid point.
Once the timing premiums are calculated on all the grid points on the sparse grid, we
simulate one million months from the sparse grid, using cubic splines to interpolate the timing premiums on observations that are not directly on the sparse grid. The timing premium
for the recursive model is then the time-series average of the simulated timing premiums.
Because of our model’s nonlinearity, we have opted to use the cumbersome multi-step
simulations (that take about 15 hours to run on a 32-CPU workstation) to calculate the
timing premium. In contrast, Epstein, Fahri, and Strzalecki (2014) only work with the
stochastic steady state. For robustness, we have also computed the timing premium in our
recursive utility model at its stochastic steady state, with employment Nt = 0.9374 and log
productivity xt = 0. For only one initial condition, we can afford to simulate 100,000 sample
paths as in Epstein et al., each with 2,500 months. We find the timing premium is 17.2%
in our model’s stochastic steady state, which is close to 17% calculated from the whole grid.
With 10,000 simulated sample paths, the timing premium remains unchanged at 17.2%.
55
D
D.1
Supplementary Results
An Alternative Calibration
Table A.2 reports the detailed results from the alternative calibration, in which we rescale
the volatility of productivity shocks, σ, in the baseline model and its extentions to target
the real output growth volatility of 4.28% per annum in the historical U.S. sample from
1790 to 2013. All the other model parameters are unchanged. The table shows that despite
the lower output volatility than 5.63% in the Barro-Ursua (2008) cross-country panel, the
disaster dynamics that arise endogenously within the models remain substantial.
D.2
D.2.1
Additional Results from the Recursive Utility Model
Comparative Statics
Table 7 in the main text reports comparative statics for the recursive utility model by varying utility parameters. We report additional comparative statics by varying labor market
parameters in Table A.3. We observe that labor market dynamics and disaster dynamics are
tightly connected with asset pricing dynamics. In particular, reducing the value of unemployment volatilities, b, from 0.85 to 0.825 lowers the unemployment volatility from 25.67%
to 8.16%, and the equity premium from 4.45% per annum to 0.64%. Such sensitivities of
the unemployment volatility to b are also present in Hagedorn and Manovskii (2008). The
output disaster probability falls from 4.49% to 2.58%, and size from 23.92% to 15.39%.
Lowering the separation rate and removing the fixed costs of vacancy both serve to reduce
the unemployment volatility, disaster probabilities and size, as well as the equity premium.
Intuitively, a lower separation rate means that jobs are destructed at a lower rate, and all
else equal, the economy can create enough jobs to dampen disaster dynamics, reducing the
equity premium. Removing the fixed costs of vacancy weaken the downward rigidity of the
marginal costs of hiring, allowing the economy to create more jobs in bad times. As such,
disaster dynamics are dampened, and the equity premium lowered. Reducing the elasticity
of the matching function, ι, makes the labor market more frictional in matching vacancies
with unemployed workers. Because job creation flows are hampered, the disaster dynamics
are strengthened, and the equity premium increased. Finally, increasing the workers’ bargaining power, η, from 0.04 to 0.05 raises the unemployment rate as well as the output and
consumption volatilities. The disaster dynamics are also strengthened. However, the equity
premium falls slightly, as the wage elasticity of productivity rises from 0.54 to 0.6.
56
Table A.2 : Quantitative Results, An Alternative Calibration
In model columns, BL is the baseline model, HP the home production model, CA the capital
model, and RU the recursive utility model. We rescale the volatility of productivity shocks, σ, to
be 0.00925, 0.012, 0.012, and 0.00875 in BL, HP, CA, and RU, respectively, to match the output
growth volatility of 4.28% per annum in the 1790–2013 U.S. sample. All the other parameters are
identical to those reported in the main text. All the model results are based on 10,000 simulations.
σ Y , σ C , and σ I are the volatilities, SY , SC , and SI skewness, KY , KC , and KI kurtosis, and ρYi , ρC
i
and ρIi ith-order autocorrelations of log output, consumption, and investment growth, respectively.
ProbY , SizeY , and DurY and ProbC , SizeC , and DurC are the probability, size, and duration of
output and consumption disasters, respectively. E[U ], SU , and KU are the mean, skewness, and
kurtosis of monthly unemployment rates, respectively, and σ U is the quarterly volatility. σ Y , σ C , σ I ,
E[U ], and σ U are in percent. E[R − Rf ] is the equity premium, E[Rf ] the risk-free rate, σ R the
stock market volatility, and σ Rf the interest rate volatility, all in percent per annum.
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
σI
SI
KI
ρI1
ρI2
ρI3
ρI4
BL
HP
CA
RU
4.25
0.65
10.77
0.2
−0.12
−0.12
−0.1
4
19.62
4.57
4.28
0.1
4.26
0.15
−0.13
−0.1
−0.08
7.52
16.65
3.98
4.25
0.11
4.38
0.19
−0.09
−0.07
−0.06
7.17
17.45
4.15
4.3
0.69
12.62
0.17
−0.14
−0.12
−0.09
3.38
20.7
4.65
5.88
0.23
4.94
0.17
−0.11
−0.09
−0.08
57
BL
HP
CA
RU
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
3.61
0.69
12.29
0.2
−0.13
−0.12
−0.1
2.14
22.29
5.02
3.71
0.14
4.86
0.16
−0.13
−0.1
−0.08
5.39
16.11
4.26
3.06
0.09
5
0.21
−0.07
−0.06
−0.06
3.7
16.31
4.84
3.68
0.72
14.26
0.16
−0.15
−0.12
−0.09
1.76
24.54
5.05
E[U ]
SU
KU
σU
E[R − Rf ]
E[Rf ]
σR
σ Rf
5.83
3.29
17.75
18.71
6.23
2.18
9.05
14.81
6.79
2.58
11.3
18.77
5.7
3.46
19.92
19.41
1.78
2.82
14.32
1.22
Table A.3 : Additional Comparative Statics, Recursive Utility
RU is the benchmark calibration. The remaining columns report six comparative statics: (i, ii) b = 0.825 and b = 0.4 are for the value
of unemployment activities set to 0.825 and 0.4, respectively; (iii) s = 0.035 is for the job separation rate set to 0.035; (iv) κt = 0.7 is for
the proportional unit costs of vacancy κ0 = 0.7 and the fixed unit costs κ1 = 0; (v) ι = 1.1 is for the elasticity of the matching function
set to 1.1; and (vi) η = 0.05 is for the workers’ bargaining weight set to 0.05. In each experiment, all the other parameters are identical
to those in the benchmark calibration. The results are based on 10,000 simulated samples. σ Y and σ C denote the volatilities, SY and
SC skewness, KY and KC kurtosis, and ρYi and ρC
i the ith-order autocorrelations of log output and consumption growth, respectively.
ProbY , SizeY , and DurY are the probability, size, and duration of output disasters, respectively, and ProbC , SizeC , and DurC are
analogously defined for consumption disasters. E[U ], SU , and KU are the mean, skewness, and kurtosis of monthly unemployment
rates, respectively, and σ U is the quarterly unemployment volatility. σ Y , σ C , σ I , E[U ], and σ U are in percent. E[R − Rf ] is the equity
premium, E[Rf ] the risk-free rate, σ R the stock market volatility, and σ Rf the interest rate volatility, all in percent per annum.
58
RU b = 0.825 b = 0.4 s = 0.035 κt = 0.7 ι = 1.1 η = 0.05
RU b = 0.825 b = 0.4 s = 0.035 κt = 0.7 ι = 1.1 η = 0.05
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
5.67
0.87
15.47
0.21
−0.14
−0.13
−0.1
4.49
23.92
4.46
2.98
2.47
0.17
0
5.53
3.36
0.14
0.14
−0.13 −0.12
−0.1 −0.1
−0.08 −0.08
2.58
1.82
15.39 13.35
4.8
4.93
4.67
0.8
14.07
0.17
−0.15
−0.12
−0.09
3.68
21.48
4.6
3.77
5.77
0.48
0.88
9.85
15.2
0.14
0.21
−0.14 −0.14
−0.11 −0.13
−0.08 −0.1
3.14
4.83
18.19 23.71
4.68
4.42
5.86
0.84
14.85
0.23
−0.13
−0.13
−0.11
4.95
23.83
4.41
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
E[U ]
SU
KU
σU
6.26
3.66
20.71
25.67
4.86
2.63
14.58
8.16
5.06
3.76
22.91
23.24
5.39
3.12
18.99
13.8
6.66
3.47
18.65
25.81
E[R − Rf ] 4.45
E[Rf ]
2.58
σR
15.79
1.64
σ Rf
3.97
0.3
2.84
0.12
6.7
3.48
19.03
24.99
5.05
0.88
17.09
0.19
−0.15
−0.13
−0.1
2.51
28.86
4.84
2.24
2.04
0.2
0
6.25
3.36
0.14
0.14
−0.13 −0.12
−0.1 −0.1
−0.08 −0.08
1.06
1.03
15.32 12.14
5.45
5.35
4.08
0.8
15.52
0.16
−0.15
−0.12
−0.09
1.94
25.72
5.02
3.01
5.14
0.51
0.89
10.94 16.83
0.13
0.2
−0.15 −0.15
−0.11 −0.13
−0.08 −0.1
1.35
2.73
20.84 28.21
5.26
4.81
5.34
0.85
16
0.22
−0.14
−0.13
−0.11
3.13
27.32
4.74
0.64
2.88
13.66
0.63
2.67
2.75
15.94
1.3
1.32
2.87
16.97
0.98
4.33
2.5
14.59
1.65
0.23
2.86
4.68
0.16
4.57
2.59
15.61
1.67
D.2.2
Time-varying Volatilities
Table A.4 reports long-horizon regressions of volatilities of excess returns and consumption growth on the log price-to-consumption ratio. Neither volatility appears predictable in
the historical data, but the consumption volatility seems at least somewhat predictable in
the postwar data. The slopes are all negative, and significant in short horizons up to four
quarters. The R2 rises from 2.18% at the 1-quarter to 9.83% at the 20-quarter horizon.
The model is largely consistent with the lack of predictability for the excess return
volatility. However, the model overstates the predictability of the consumption volatility. In
the quarterly frequency, in particular, the slopes are all negative, and mostly significant. The
R2 rises from 3.11% at the 1-quarter to 18.74% at the 20-quarter horizon (9.83% in the data).
D.3
Leisure in the Utility Function
In this subsection, we modify the baseline model by incorporating leisure into the utility function. In particular, the household’s utility becomes log(Ct +hUt ), in which h > 0 is a constant.
This utility is the logarithmic form of the general Greenwood, Hercowitz, and Huffman (1988)
preferences. The stochastic discount factor becomes Mt+1 = β((Ct + hUt )/(Ct+1 + hUt+1 )).
Going through similar steps as in Section B.1 yields the equilibrium wage:
Wt = η(Xt + κt θt ) + (1 − η)z,
(D.1)
in which z ≡ b+h is the flow value of unemployment activities. The rest of the leisure model,
including the intertemporal job creation condition, remains identical to the baseline model.
We set b = 0.5 and h = 0.35, or z = 0.85, to ease comparison with the baseline model. We
also rescale the volatility of productivity shocks, σ, to 0.0095 to align the output volatility in
the leisure model, 5.5% per annum, with that in the data, 5.63%, as well as that in the baseline model, 5.31%. Table A.5 shows that the results from the leisure model are largely comparable with those from the baseline model. The output disaster probability is 4.37%, which is
somewhat lower than 5.04% in the baseline model, but the average size and duration are both
close. For consumption disasters, the probability is 2.41%, which is somewhat lower than
2.87% in the baseline model, but the average size is 27%, which is slightly higher than 25.58%
in the baseline model. We also report two comparative statics, (i) b = 0.45 and h = 0.4; as
well as (ii) b = 0.4 and h = 0.45. The results seem insensitive to the parameter changes.
59
Table A.4 : Predicting Volatilities of Stock Market Excess Returns and Consumption
Growth with the Log Price-to-Consumption Ratio
For a given H, the excess return volatility is σ R
t,t+H−1 =
PH−1
h=0
R
|ǫR
t+h |, in which ǫt+h is the h-period-
f
ahead residual from the first-order autoregression on excess returns, log(1 + Rt+1 ) − log(1 + Rt+1
).
Excess returns in the data are adjusted for financial leverage. The long-horizon predictive regression
R
of the excess return volatility is log σ R
t+1,t+H = αR +β R log(Pt /Ct )+ut+h . The consumption growth
P
H−1 C
C
volatility is σ C
h=0 |ǫt+h |, in which ǫt+h is the h-period-ahead residual from the firstt,t+H−1 =
order autoregression on consumption growth, log(Ct+1 /Ct ). The long-horizon predictive regression
C
of the consumption growth volatility is log σ C
t+1,t+H = αC + β C log(Pt /Ct ) + ut+h . log(Pt /Ct ) is
standardized to have a mean of zero and a variance of unity. For annual data, H ranges from one
(1y) to five years (5y), and for quarterly data, from one (1q) to 20 quarters (20q). The t-statistics
are Newey-West (1987) adjusted for heteroscedasticity and autocorrelations of 2(H − 1) lags. The
slopes and the R2 s are in percent. The model moments are averaged across 10,000 simulations.
H
1y
2y
3y
4y
5y
1q
U.S. annual data, 1836–2013
βR
tR
2
RR
βC
tC
2
RC
−8.36
−0.96
0.44
−6.48
−0.72
0.32
0.09
0.01
0.00
−7.85
−0.96
0.86
−1.12
−0.24
0.04
−6.88
−0.82
0.98
−1.31
−0.31
0.07
−5.75
−0.63
0.88
4q
8q
12q
16q
20q
U.S. quarterly data, 1947q2–2013q4
−0.13
−0.03
0.00
−4.79
−0.50
0.70
−10.79 −1.23
3.19
5.08
5.55
5.56
−1.74 −0.21
0.53
0.92
1.06
1.11
0.93
0.05
0.55
2.07
3.45
4.47
−15.87 −14.10 −13.04 −12.41 −11.25 −10.17
−2.51 −2.16 −1.68 −1.50 −1.35 −1.24
2.18
6.19
8.33
9.33
9.49
9.83
Recursive utility
βR
−3.05 −3.28 −3.10 −2.86 −2.63
tR
−0.40 −0.57 −0.61 −0.61 −0.59
2
RR
0.61
1.02
1.56
2.06
2.46
β C −34.78 −33.27 −31.51 −29.80 −28.16
tC
−3.52 −4.01 −3.71 −3.51 −3.34
2
RC
8.05
14.85
18.24
19.56
19.70
Recursive utility
−2.16 −3.52 −3.55 −3.27 −2.94 −2.62
−0.27 −0.43 −0.54 −0.58 −0.58 −0.57
0.87
1.26
2.60
4.14
5.27
6.09
−29.42 −24.86 −22.45 −20.75 −19.15 −17.66
−2.33 −1.81 −2.19 −2.41 −2.47 −2.47
3.11
5.35
11.00
16.07
18.21
18.74
60
Table A.5 : Quantitative Results, Leisure in the Utility
BL is the baseline model. The other columns report the results from the leisure model, in which
we rescale the volatility of productivity shocks, σ, to be 0.0095. We vary b from 0.5, 0.45, to 0.4,
and correspondingly h from 0.35, 0.4, to 0.45. All the other parameters are identical to those in
the baseline model. σ Y and σ C are the volatilities, SY and SC , KY and KC kurtosis, and ρYi
and ρC
i ith-order autocorrelations of log output and consumption growth, respectively. ProbY ,
SizeY , and DurY and ProbC , SizeC , and DurC are the probability, size, and duration of output
and consumption disasters, respectively. E[U ], SU , and KU are the mean, skewness, and kurtosis
of monthly unemployment rates, respectively, and σ U is the quarterly volatility. σ Y , σ C , σ I , E[U ],
and σ U are in percent. The results are based on 10,000 simulations.
BL
b
h
σ
σY
SY
KY
ρY1
ρY2
ρY3
ρY4
ProbY
SizeY
DurY
E[U ]
SU
Leisure
BL
0.85
0
0.01
0.5
0.35
0.0095
0.45
0.4
0.0095
0.4
0.45
0.0095
b
h
σ
5.31
0.85
12.8
0.24
−0.11
−0.12
−0.11
5.04
22.22
4.44
5.5
0.85
15.83
0.18
−0.16
−0.13
−0.1
4.37
22.7
4.47
5.53
0.83
16
0.17
−0.16
−0.13
−0.09
4.38
22.98
4.46
5.68
0.81
16.61
0.16
−0.16
−0.13
−0.09
4.39
23.18
4.45
6.28
3.52
6
3.63
5.97
3.66
6
3.7
61
Leisure
0.85
0
0.01
0.5
0.35
0.0095
0.45
0.4
0.0095
0.4
0.45
0.0095
σC
SC
KC
ρC
1
ρC
2
ρC
3
ρC
4
ProbC
SizeC
DurC
4.65
0.91
14.4
0.23
−0.12
−0.12
−0.11
2.86
25.64
4.91
4.91
0.86
17.02
0.16
−0.18
−0.13
−0.09
2.41
27.01
4.88
4.95
0.85
17.31
0.16
−0.17
−0.13
−0.09
2.42
27.51
4.86
5.11
0.82
17.81
0.15
−0.17
−0.13
−0.09
2.43
27.83
4.84
KU
σU
19.18
23.41
21.09
24.15
21.59
24.51
22.01
25.38
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