In this paper, I investigate the scope of a model with exogenous habit formation - or `catching up with the Joneses`, see Abel (1990) - to generate the observed equity premium as well as other key macroeconomic facts. Along the way, I derive restrictions for four out of eight parameters for a rather general preference specification of habit formation by imposing consistency with long-run growth, the leisure share, the aggregate Frisch elasticity of labor supply, the observed risk-free rate, and the observed Sharpe ratio. I show that a DSGE model with (exogenous and lagged) habits in both leisure and consumption, but not necessarily with additional persistence, is well capable of matching the observed asset market facts as well as macro facts, provided one allows for moderate real wage stickiness and provided one allows for sufficient curvature on preferences, as dictated by the asset market observations. Without wage stickiness, delivery on both the asset pricing implications as well as the macroeconomic implications seems to be much harder.asset pricing, wage rigidity, habit formation, Frisch elasticity, Sharpe ratio, log-linear approximation

Harald Uhlig

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
SFB 649 Discussion Paper 2007-003 
Explaining Asset Prices 
with External Habits 
and Wage Rigidities in a 
DSGE Model. 
 
Harald Uhlig* 
* Humboldt Universität zu Berlin, Germany 
Deutsche Bundesbank, CentER and CEPR 
This research was supported by the Deutsche 
Forschungsgemeinschaft through the SFB 649 "Economic Risk". 
 
http://sfb649.wiwi.hu-berlin.de 
ISSN 1860-5664 
 
SFB 649, Humboldt-Universität zu Berlin 
Spandauer Straße 1, D-10178 Berlin 
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 Explaining Asset Prices with External Habits
and Wage Rigidities in a DSGE Model.∗
Harald Uhlig
Humboldt Universit¨ at zu Berlin
Deutsche Bundesbank, CentER and CEPR
This version: January 15, 2007
∗This was written for the 2006 AEA session E3 “Habit Persistence and the Macroe-
conomy”, chaired by Martin Uribe and Stephanie Schmitt-Grohe. The discussant
was Fatih Guvenen. Address: Prof. Harald Uhlig, Humboldt Universit¨ at zu Berlin,
Wirtschaftswissenschaftliche Fakult¨ at, Spandauer Str. 1, 10178 Berlin, GERMANY. e-
mail: uhlig@wiwi.hu-berlin.de, phone: +49-30-2093 5626, fax: +49-30-2093 5934, home
page http://www.wiwi.hu-berlin.de/wpol/. This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 “Economic Risk”. I am grateful to Fatih
Guvenen for useful feedback.Explaining Asset Prices with External Habits and Wage
Rigidities in a DSGE Model.1
This paper is a progress report on understanding the relationship between
prices and allocations of risks on ﬁnancial markets versus macroeconomic
choices and allocations. In this paper, I investigate the scope of a model
with exogenous habit formation – or “catching up with the Joneses”, see
Abel (1990) – to generate the observed equity premium as well as other key
macroeconomic facts. Along the way, I derive restrictions for four out of eight
parameters for a rather general preference speciﬁcation of habit formation by
imposing consistency with long-run growth, the leisure share, the aggregate
Frisch elasticity of labor supply, the observed risk-free rate, and the observed
Sharpe ratio.
The high reward for holding risk on ﬁnancial markets implies that the
stochastic discount factor of the marginal investor shows large ﬂuctuations,
which are highly correlated with aggregate risk. Models with a representative
agent - to which this paper belongs - therefore require, that risk aversion,
measured appropriately, must be high. That alone, however is not enough:
when economic choices are endogenous, agents typically have the possibility
to insulate the risk-sensitive dimensions of their preferences against aggregate
risk.
Thus, the literature on generating both asset pricing facts as well as
macroeconomic facts within one model has increasingly pointed to labor mar-
ket frictions as possibly important for a joint explanation. Endogenous labor
1This research was supported by the Deutsche Forschungsgemeinschaft through the
SFB 649 “Economic Risk”. I am grateful to Fatih Guvenen for useful feedback.
1supply decisions on a frictionless labor market provide agents with an insur-
ance device against ﬂuctuations in consumption. This insurance possibility
then renders these models incapable of generating high Sharpe ratios or eq-
uity premia, unless additional frictions on labor markets such as separated
labor markets or wage rigidities are introduced, see e.g. Lettau and Uh-
lig (2000), Boldrin, Christiano and Fisher (2001), Guvenen (2003) or Uhlig
(2006). Furthermore, wage rigidities have recently been emphasized as key
to understanding aggregate ﬂuctuations more generally, see e.g. Hall (2005),
Shimer (2005) and Blanchard and Gali (2005). I shall therefore pay partic-
ular attention to the role of wage rigidities here.
1 The model
I shall use small letters to denote the choices of an individual agent, and
capital letters to denote economy-wide averages per agent. Production is
organized by ﬁrms, hiring labor and renting capital from households at a
market wage Wt and a market dividend Dt. I assume that production is
competitive and follows a Cobb-Douglas production function,
Yt = K
θ
t−1 (e
ztNt)
1−θ (1)
and therefore
Wt = (1 − θ)Yt/Nt, Dt = θYt/Kt−1 (2)
Technology zt evolves according to a random walk with drift,
zt = γzt−1 + ǫt (3)
where I shall assume in the linearized version of the model, that ǫt is normally
iid with standard deviation σǫ.
2A representative agent has preferences given by
U = E
" ∞ X
t=0
β
t((ct − Ht)(A + (lt − Ft)ν))
1−η − 1
1 − η
#
(4)
where ct and lt denote consumption and leisure, and β,ν,η,A are parameters,
satisfying ν > 0,η > ν/(ν + 1) to assure monotonicity and concavity on
the domain. She is endowed with initial capital k−1 and one unit of time
per period, which can be used as labor or leisure. She maximizes these
preferences over choices of investment, consumption and labor, taking as
given the “exogenous habits” of Ht for consumption and Ft for leisure as
well as real wages Wt for labor and dividends Dt for providing capital to
ﬁrms,
1 = nt + lt (5)
ct + xt = Dtkt−1 + Wtnt (6)
kt =
Ã
1 − δ + g
Ã
xt
kt−1
!!
kt−1 (7)
Deﬁne
˜ δ = δ + e
γ − 1 (8)
I assume that the adjustment cost function g(·) satisﬁes
g(˜ δ) = ˜ δ, g
′(˜ δ) = 1, g
′′(˜ δ) = −
1
ξ
(9)
for some ξ > 0, see Jermann (1998). The exogenous habits evolve according
to
Ht = e
γ ((1 − ζ)χCt−1 + ζHt−1) (10)
Ft = (1 − φ)ψLt−1 + φFt−1 (11)
where Ct and Lt are aggregate average levels of consumption and leisure.
There is an extra constant γ in the speciﬁcation for Ht which is tied to the
3productivity growth rate in (3). I do this for algebraic simplicity. Exoge-
nous habits give rise to externalities, which may be corrected with taxes, see
Ljungqvist and Uhlig (2000): I shall ignore this issue here.
I assume that labor markets are demand constrained. The usual ﬁrst-
order condition of the agent for supplying labor would be
W
f
t =
∂u/∂l
∂u/∂c
= ν
ct − Ht
A(lt − Ft)1−ν + lt − Ft
(12)
at the friction-free wage W
f
t . Due to some unmodelled friction, I assume
that not all labor supply reaches the market. I assume that the steady state
supply of labor is ﬁxed at some exogenously imposed level below the level
of the frictionless economy. Locally around the steady state, households are
therefore willing to supply labor at the going market wage, assumed to be
governed by
Wt = (e
γWt−1)
  ³
W
f
t
´1− 
(13)
This is a real wage ridity as recently postulated by e.g. Hall (2005) and
Shimer (2005). The particular speciﬁcation here follows Blanchard and Gali
(2005). Note that the frictionless scenario is included as a special case for
  = 0.
An equilibrium is deﬁned in the usual way, except for imposing that
labor markets are demand-constrained. In particular, individual choices will
coincide with aggregate choices, e.g. ct = Ct, and I shall now drop the
distinction between the two. One can show that there is a steady state in
the productivity-detrended variables ˜ Ct = Ct/exp(zt−1) and similarly ˜ kt, ˜ Ht,
˜ Wt, ˜ W
f
t , ˜ Yt and the stationary variables Lt,Ft,Nt. I denote this steady state
with bars.
Assuming that t counts quarters, I shall set δ = 0.015, γ = 0.0075,
θ = 0.33, as is common, implying a nonstochastic growth rate of 3% per
4year. I shall also impose on my choice of preferences parameters, that they
are consistent with a nonstochastic return of ¯ R = 1.01 and a share of time
spent as leisure as ¯ L = 2/3. With this, compute e.g.
¯ X
¯ Y
=
˜ δθ
¯ R − 1 + δ
= 0.30 and κ =
1 − ¯ L
(1 − θ)¯ L
Ã
1 −
¯ X
¯ Y
!
= 0.52 (14)
which will be useful below. The eight preference parameters (A,β,η,ν,χ,ζ,ψ,φ)
shall now be constrained by two assumptions already made and by two fur-
ther observations: the Frisch elasticity of labor supply and the Sharpe ratio.
2 Constraints on preferences
2.1 Macroeconomic constraints
Consider a more general preference speciﬁcation with a per-period felicity
function u(ct−Ht,lt−Ft). Impose that leisure is constant along the balanced
growth path. This implies that
u((ct − Ht,lt − Ft) =
((ct − Ht)v(lt − Ft))
1−η − 1
1 − η
(15)
for some function v(·) and up to the intercept and scaling. For a log-
linear approximation, the derivatives of logv(·) and logv′(·) around the
steady state characterize this function suﬃciently. In my speciﬁcation, I
set v(lt − Ft) = A + (lt − Ft)ν. Thus, my habit preference speciﬁcation is
general up to a second-order approximation, subject to obeying the balanced
growth condition for preferences of the form (15). For algebraic convenience,
deﬁne
α = A(1 − ψ)
−ν ¯ L
−ν, ˜ η =
η
1 − χ
, ˜ ν =
ν(1 − η)
(1 + α)(1 − ψ)
, (16)
5Comparing the marginal utility of leisure with its marginal opportunity
costs yields
1 − ψ
1 − χ
1 + α
ν
≥
¯ C
¯ W f ¯ L
= κ (17)
where the inequality ought to be strict in order to induce the labor market to
be demand-constrained, and where the latter equality derives from the ﬁrst-
order conditions of the ﬁrm and steady state substitutions. This equation
delivers my ﬁrst constraint
1
η
≥ 1 + κ
˜ ν
˜ η
(18)
as well as, equivalently,
α ≥ κν
1 − χ
1 − ψ
− 1 (19)
Let τ be the Frisch elasticity of labor supply, which I shall treat here as the
elasticity of desired labor supply with respect to a change in the frictionless
wage, holding marginal utility of consumption constant. Given preference
parameters χ and η, deﬁne
Υ(χ,η) =
1 − ¯ N
τ ¯ N
−
Ã
2 −
1
η
!
1
(1 − χ)κ
(20)
After some calculation, I obtain the second constraint
ν = 1 − (1 − ψ)Υ(χ,η) (21)
I will use this equation to calculate the implied value for ν. Note that ν > 0
for all ψ ∈ [0,1), χ ∈ [0,1) and η > 1, if τ > τ = (1 − ¯ N)κ/( ¯ N(κ + 1)).
2.2 Asset pricing constraints
Let Rt+1 be the return on some asset between period t and t+1. The Lucas
asset pricing equation is
1 = βEt
"
λt+1
λt
Rt+1
#
(22)
6where
λt = uc,t = (ct − Ht)
−η(A + (lt − Ft)
ν)
1−η (23)
For the nonstochastic growth path (although not for the average risk-free
rate in the stochastic economy), this implies the third constraint
β = e
ηγ ¯ R
−1 (24)
Deﬁne the news,
ǫλ,t+1 = log(λt+1) − Et[log(λt+1)] (25)
and deﬁne similarly ǫR,t+1,ǫc,t+1 and ǫl,t+1. Let σλ, σR, ρλ,R be the standard
deviations of ǫλ,t+1 and ǫR,t+1 and their correlation. Deﬁne similarly σc, σl
and ρc,l. I assume homoskedasticity throughout. Let r
f
t be the logarithm of
the risk-free return from t to t + 1. Deﬁne the Sharpe ratio
S R =
logEt[Rt+1] − r
f
t
σR
(26)
Assuming joint normality in (22) delivers
S R = −ρλ,rσλ (27)
see Lettau-Uhlig (2002) for a detailed derivation. The maximally possible
Sharpe ratio therefore is
S R = σλ (28)
In asset pricing, distributional assumptions and the choice of numerical
approximation methods are not innocuous, see e.g. Judd and Guo (2001) and
Weitzmann (2005). Also, nonlinearities are key for Campbell and Cochrane
(1999) to explain a number of facts jointly. Finally, the linear habit preference
speciﬁcation in (4) can generate an ill-deﬁned maximization problem, unless
7the shock process is suﬃciently restricted. Resolving these issues is beyond
the scope of this paper. Rather, these remarks shall serve as a caveat for the
log-linear approach pursued here. This approach has the advantage of being
well-understood and imposing a tight discipline on the exercise.
To a log-linear approximation,
ǫλ,t+1 = −˜ ηǫc,t+1 + ˜ νǫl,t+1 (29)
Equation (28) therefore implies a quadratic equation in ˜ η and ˜ ν,
Ã
(1 − χ)S R
ησc
!2
= 1 − ρc,l
σl
σc
˜ ν
˜ η
+
Ã
σl
σc
˜ ν
˜ η
!2
(30)
which is my forth constraint. Note that ˜ ν = 0, if ˜ ησc = S R, which is the
benchmark case of no inﬂuence of leisure on asset pricing, and holds for
separable preferences, η = 1.
3 Numerical strategy and results
3.1 Parameterization
The observations above constrain the preference parameters (α,β,η,ν,χ,ψ),
leaving two degrees of freedom. I use (χ,ψ) to parameterize this solution
manifold. I assume that the demand-constraint on labor markets is moderate
and I shall therefore treat (18) as equality. Replace ˜ ν/˜ η in (30) with (18),
imposing equality there. Given χ, equation (30) is a quadratic equation in η,
which generally has two solutions. I shall use the solution to the right of the
minimum. If χ is suﬃciently small – which means χ ≤ 0.97 in the numerical
calculations – this results in η > 1. Calculate Υ(η,χ) and therefore ν in
(21). Find α from (19) with equality. Find β from (24), where I allow β > 1.
Check ν > 0, η > ν/(ν + 1) and α > −1 or start over with a new (χ,ψ).
8For equation (30) I shall use S R = 0.15 as the quarterly Sharpe ratio. On
an annual basis, this implies a Sharpe ratio of approximately 0.3, which is
lower than the usual 0.5 ratio quoted in the literature, but appropriate here,
given the deﬁnition in term of log-returns. I use σc = 0.67%, σl = 0.45% and
ρc,l = −0.33, calculated from taking ﬁrst-diﬀerences of the log-series rather
than innovation standard deviations, which would be more appropriate. For
the Frisch elasticity of labor supply, I use a value of 3.
The remaining free parameters are thus (χ,ψ,ζ,φ,ξ, ,σǫ). I always
rescale σǫ so that the HP-ﬁltered volatility of ouput equals 2. For the other
parameters, I conducted a hyperparameter search, using a grid for each of
these six parameters. The goal is to ﬁnd parameter constellations which en-
dogenously deliver a quarterly Sharpe ratio of 0.15, a risk-free rate volatility
of 1.7%, a ratio of consumption volatility to output volatility of 0.47, a ra-
tio of investment volatility to output volatility of 3.95 and a ratio of labor
volatility to output volatility of 1.03. I minimize a criterion function, impos-
ing a weight of 1 on squared deviations for all values, except using a weight of
50 for the Sharpe ratio. I exclude solutions with explosive behaviour (which
may happen with β > 1) or other numerical problems.
For the grid, I use 0,.3,.6,.8,.9,.95,.97 for both habit level parameters χ
and ψ. I use 0,.1,.2,.3,.5,.7,.9 for both habit persistence parameters φ and
ζ. I set ξ−1 = 0,.05,0.1,0.15,0.2,0.5,1. Finally, I use   = .2,.3,.35,.4,.6,.8.
The overall minimum has been found at ψ = 0.97,χ = 0.8,ζ = φ = 0,ξ−1 =
0.1,  = 0.35 implying η = 7,ν = 1.5,β = 1.04,α = 4.3, while the minimum
without sticky wages was found at ψ = 0.9,χ = 0.97,ζ = 0,φ = 0.7,ξ−1 =
0.2. Apparently, habit persistence is not needed (i.e. ζ = φ = 0), if sticky
wages are allowed for. The required degree of wage stickiness is moderate.
When wages are restricted to be ﬂexible, it seems important to allow for
9considerable persistence in leisure habit.
At the overall minimum, the model simulations deliver S R = 0.25 on
an annualized basis, σr = 1.84, σc/σy = 0.85, σn/σy = 0.81, σx/σy = 2.22,
σǫ/0.172 = 1.63, corr(c,y)= 0.91, corr(n,y) = 0.73 and corr(x,y) = 0.84,where
e.g. σc now denotes the volatility of HP-ﬁltered consumption. These numbers
are close to the data. The model delivers both the observed Sharpe ratio as
well as a moderate risk-free rate volatility. Consumption ﬂuctuates more
in the model than what is observed in the data, while investment ﬂuctuates
less. Also, I need the ﬂuctuations in the productivity innovations to be about
60% higher than what is traditionally assumed. This may simply reﬂect the
necessity for shocks that have not been included here yet.
At the ﬂexible wage minimum, the model simulations deliver S R = 0.08
on an annualized basis, σr = 2.12, σc/σy = 0.31, σn/σy = 0.66, σx/σy = 2.69,
σǫ/0.172 = 1.37, corr(c,y)= 0.72, corr(n,y) = 0.94 and corr(x,y) = 0.62. Now,
the Sharpe ratio is just a quarter of what it ought to be, despite giving this
particular target a high weight in the criterion function. Apparently, it is
hard to match the observed Sharpe ratio without giving up considerably on
other quantitative implications of the model.
In summary, a DSGE model with (exogenous and lagged) habits in both
leisure and consumption, but not necessarily with additional persistence, is
well capable of matching the observed asset market facts as well as macro
facts, provided one allows for moderate real wage stickiness and provided one
allows for suﬃcient curvature on preferences, as dictated by the asset market
observations. Without wage stickiness, delivery on both the asset pricing
implications as well as the macroeconomic implications seems to be much
harder.
10References
[1] Abel, Andrew. ”Asset Prices under Habit Formation and Catching Up
with the Joneses,” American Economic Review, American Economic
Association,1990. 80(2), pp. 38-42.
[2] Blanchard, Olivier and Gali, Jordi. “Real Wage Rigidities and the New
Keynesian Model,”. Working paper 05-27, MIT, Department of Eco-
nomics, 2005.
[3] Boldrin, Michele, Christiano, Lawrence J. and Fisher, Jones D.M.
“Habit Persistence, Asset Returns, and the Business Cycle.” American
Economic Review, 2001. 91(1), pp. 149-66.
[4] Campbell, John Y. and John H. Cochrane (1999). “By Force of Habit: A
Consumption-Based Explanation of Aggregate Stock Market Behavior.”
Journal of Political Economy, 1999. 107(2), pp. 205-51.
[5] Guvenen, Fatih. “A Parsimonious Macroeconomic Model for Asset Pric-
ing: Habit Formation or Cross-Sectional Heterogeneity?”. Draft, Univer-
sity of Texas-Austin (2003).
[6] Hall, Robert F. “Employment Fluctuations with Equilibrium Wage
Stickiness.” American Economic Review, 2005. 95(1), pp. 50-65.
[7] Jermann, U.J. “Asset pricing in production economies.” Journal of Mon-
etary Economics, 1998. 41, pp. 257-275.
[8] Judd, Kenneth L. and Guu, Sy-Ming. “Asymptotic methods for asset
market equilibrium analysis.” Economic Theory, 2001. 18, pp. 127-157.
11[9] Lettau, Martin and Uhlig, Harald. “Can Habit Formation Be Reconciled
with Business Cycle Facts?” Review of Economic Dynamics, 2000. 3(1),
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[10] Lettau, Martin and Uhlig, Harald. “The Sharpe Ratio and Preferences:
A Parametric Approach.” Macroeconomic Dynamics, 2002. 6(2), pp.
242-65.
[11] Ljungqvist, Lars and Uhlig, Harald. “Tax Policy and Aggregate Demand
Management under Catching Up with the Joneses.” American Economic
Review, 2000. 90(3), pp. 356-366.
[12] Shimer, Robert. “The Cyclical Behavior of Equilibrium Unemployment,
Vacancies and Wages: Evidence and Theory.” American Economic Re-
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[13] Uhlig, Harald (2006), “Macroeconomics and Asset Markets: some Mu-
tual Implications.” Draft, Humboldt Universit¨ at zu Berlin, 2005.
[14] Weitzmann, Martin L. “Risk, Uncertainty, and Asset-Pricing ‘Puzzles’.”
Draft, Harvard University, 2005.
12 
SFB 649 Discussion Paper Series 2007 
 
For a complete list of Discussion Papers published by the SFB 649, 
please visit http://sfb649.wiwi.hu-berlin.de. 
 
001 "Trade  Liberalisation,  Process and Product Innovation, and Relative Skill 
  Demand" by Sebastian Braun, January 2007. 
002  "Robust Risk Management. Accounting for Nonstationarity and Heavy 
  Tails" by Ying Chen and Vladimir Spokoiny, January 2007. 
003  "Explaining Asset Prices with External Habits and Wage Rigidities in a 
  DSGE Model." by Harald Uhlig, January 2007. 
 
SFB 649, Spandauer Straße 1, D-10178 Berlin 
http://sfb649.wiwi.hu-berlin.de 
 
This research was supported by the Deutsche 
Forschungsgemeinschaft through the SFB 649 "Economic Risk". 

Explaining Asset Prices with External Habits and Wage Rigidities in a DSGE Model.

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Explaining Asset Prices with External Habits and Wage Rigidities in a DSGE Model.

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