Macroeconometric Analysis Chapter 4. DSGE Models: Three Examples Chetan Dave David N. DeJong
Macroeconometric Analysis
Chapter 4. DSGE Models: Three Examples
Chetan Dave David N. DeJong
Chapters 2 and 3 provided background for preparing structural models for empirical
analysis. The �rst step of the preparation stage is the construction of a linear approximation
of the structural model under investigation, which takes the form
Axt+1 = Bxt + C�t+1 +D�t+1:
The purpose of this chapter is to demonstrate the completion of this �rst step for three
prototypical model environments that will serve as examples throughout the remainder of
the text. This will set the stage for Part II, which outlines and demonstrates alternative
approaches to pursuing empirical analysis.
The �rst environment is an example of a simple real business cycle (RBC) framework,
patterned after that of Kydland and Prescott (1982). The foundation of models in the RBC
tradition is a neoclassical growth environment, augmented with two key features: a labor-
leisure trade-o¤ that confronts decision makers; and uncertainty regarding the evolution
of technological progress. The empirical question Kydland and Prescott (1982) sought to
address was the extent to which such a model, bereft of market imperfections and featuring
fully �exible prices, could account for observed patterns of business cycle activity while
capturing salient features of economic growth. This question continues to serve as a central
focus of this active literature; an overview is available in the collection of papers presented
in Cooley (1995).
Viewed through the lens of an RBC model, business cycle activity is interpretable as re-
�ecting optimal responses to stochastic movements in the evolution of technological progress.
Such interpretations are not without controversy. Alternative interpretations cite the exis-
1
tence of market imperfections, costs associated with the adjustment of prices, and other
nominal and real frictions as potentially playing important roles in in�uencing business cy-
cle behavior, and giving rise to additional sources of business cycle �uctuations. Initial
skepticism of this nature was voiced by Summers (1986); and the collection of papers con-
tained in Mankiw and Romer (1991) provide an overview of DSGE models that highlight the
role of, e.g., market imperfections in in�uencing aggregate economic behavior. As a com-
plement to the RBC environment, the second environment presented here (that of Ireland,
2004) provides an example of a model within this neo-Keynesian tradition. Its empirical
purpose is to simultaneously evaluate the role of cost, demand and productivity shocks in
driving business cycle �uctuations. Textbook references for models within this tradition are
Benassy (2002) and Woodford (2003).
The realm of empirical applications pursued through the use of DSGE models extends
well beyond the study of business cycles. The third environment serves as an example of
this point: it is a model of asset-pricing behavior adopted from Lucas (1978). The model
represents �nancial assets as tools used by households to optimize intertemporal patterns of
consumption in the face of exogenous stochastic movements in income and dividends earned
from asset holdings. Viewed through the lens of this model, two particular features of asset-
pricing behavior have proven exceptionally di¢ cult to explain. First, Shiller (1981) used a
version of the model to underscore the puzzling volatility of prices associated with broad
indexes of assets (such as the Standard & Poor�s 500), highlighting what has come to be
known as the �volatility puzzle�. Second, Mehra and Prescott (1985) used a version of the
model to highlight the puzzling dual phenomenon of a large gap observed between aggregate
returns on risky and risk-free assets. When coupled with exceptionally low returns yielded
2
by risk-free assets, this feature came to be known as the �equity premium�puzzle. The texts
of Shiller (1989) and Cochrane (2001) provide overviews of literatures devoted to analyses
of these puzzles.
1 Model I: A Real Business Cycle Model
1.1 Environment
The economy consists of a large number of identical households; aggregate economic
activity is analyzed by focusing on a representative household. The household�s objective
is to maximize U , the expected discounted �ow of utility arising from chosen streams of
consumption and leisure:
maxct;lt
U = E0
1Xt=0
�tu(ct; lt): (1)
In (1), E0 is the expectations operator conditional on information available at time 0, � 2
(0; 1) is the household�s subjective discount factor, u(�) is an instantaneous utility function,
and ct and lt denote levels of consumption and leisure chosen at time t.
The household is equipped with a production technology that can be used to produce a
single good yt. The production technology is represented by
yt = ztf(kt; nt); (2)
where kt and nt denote quantities of physical capital and labor assigned by the household
to the production process, and zt denotes a random disturbance to the productivity of these
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inputs to production (that is, a productivity or technology shock).
Within a period, the household has one unit of time available for division between labor
and leisure activities:
1 = nt + lt: (3)
In addition, output generated at time t can either be consumed or used to augment the stock
of physical capital available for use in the production process in period t+1. That is, output
can either be consumed or invested:
yt = ct + it; (4)
where it denotes the quantity of investment. Finally, the stock of physical capital evolves
according to
kt+1 = it + (1� �)kt; (5)
where � denotes the depreciation rate. The household�s problem is to maximize (1) subject
to (2)-(5), taking k0 and z0 as given.
Implicit in the speci�cation of the household�s problem are two sets of trade-o¤s. One is a
consumption/savings trade-o¤: from (4), higher consumption today implies lower investment
(savings), and thus from (5), less capital available for production tomorrow. The other is
a labor/leisure trade-o¤: from (3), higher leisure today implies lower labor today and thus
lower output today.
In order to explore quantitative implications of the model, it is necessary to specify
explicit functional forms for u(�) and f(�), and to characterize the stochastic behavior of
4
the productivity shock zt. We pause before doing so to make some general comments. As
noted, an explicit goal of the RBC literature is to begin with a model speci�ed to capture
important characteristics of economic growth, and then to judge the ability of the model to
capture key components of business cycle activity. From the model builder�s perspective,
the former requirement serves as a constraint on choices regarding the speci�cations for
u(�); f(�) and the stochastic process of zt. Three key aspects of economic growth serve as
constraints in this context: over long time horizons the growth rates of fct; it; yt; ktg are
roughly equal (balanced growth), the marginal productivity of capital and labor (re�ected
by relative factor payments) are roughly constant over time, and flt; ntg show no tendencies
for long-term growth.
Beyond satisfying this constraint, functional forms chosen for u(�) are typically strictly
increasing in both arguments, twice continuously di¤erentiable, strictly concave and satisfy
limc!0
@u(ct; lt)
@ct= lim
l!0
@u(ct; lt)
@lt=1: (6)
Functional forms chosen for f(�) typically feature constant returns to scale and satisfy similar
limit conditions.
Finally, we note that the inclusion of a single source of uncertainty in this framework,
via the productivity shock zt, implies that the model carries nontrivial implications for
the stochastic behavior of a single corresponding observable variable. For the purposes of
this chapter, this limitation is not important; however, it will motivate the introduction of
extensions of this basic model in Part II of the text.
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1.1.1 Functional Forms
The functional forms presented here enjoy prominent roles in the macroeconomics lit-
erature. Instantaneous utility is of the constant relative risk aversion (CRRA) form:
u(ct; lt) =
�c't l
1�'t
1� �
�1��; (7)
where � > 0 determines two attributes: it is the coe¢ cient of relative risk aversion, and also
determines the intertemporal elasticity of substitution, given by 1�(for textbook discussions,
see e.g., Blanchard and Fischer, 1998; or Romer, 2001).1 Note that the larger is �, the
more intense is the household�s interest in maintaining a smooth consumption/leisure pro�le.
Also, ' 2 (0; 1) indicates the importance of consumption relative to leisure in determining
instantaneous utility.
Next, the production function is of the Cobb-Douglas variety:
yt = ztk�t n
1��t ; (8)
where � 2 (0; 1) represents capital�s share of output. Finally, the log of the technology shock
is assumed to follow a �rst-order autoregressive, or AR(1), process:
log zt = (1� �) log(z) + � log zt�1 + "t (9)
"t � NID(0; �2); � 2 (�1; 1): (10)
1When � = 1; u(:) = log(:):
6
The solution to the household�s problem may be obtained via standard application of the
theory of dynamic programming (e.g., as described in Stokey and Lucas, 1989). Necessary
conditions associated with the household�s problem expressed in general terms are given by
@u(ct; lt)
@lt=
�@u(ct; lt)
@ct
���@f(kt; nt)
@nt
�(11)
@u(ct; lt)
@ct= �Et
�@u(ct+1; lt+1)
@ct+1��@f(kt+1; nt+1)
@kt+1+ (1� �)
��: (12)
The intratemporal optimality condition (11) equates the marginal bene�t of an additional
unit of leisure time with its opportunity cost: the marginal value of the foregone output
resulting from the corresponding reduction in labor time. The intertemporal optimality con-
dition (12) equates the marginal bene�t of an additional unit of consumption today with its
opportunity cost: the discounted expected value of the additional utility tomorrow that the
corresponding reduction in savings would have generated (higher output plus undepreciated
capital).
Consider the qualitative implications of (11) and (12) for the impact of a positive produc-
tivity shock on the household�s labor/leisure and consumption/savings decisions. From (11),
higher labor productivity implies a higher opportunity cost of leisure, prompting a reduction
in leisure time in favor of labor time. From (12), the curvature in the household�s utility
function carries with it a consumption-smoothing objective. A positive productivity shock
serves to increase output, thus a¤ording an increase in consumption; but since the marginal
utility of consumption is decreasing in consumption, this drives down the opportunity cost of
savings. The greater is the curvature of u(:); the more intense is the consumption-smoothing
objective, and thus the greater will be the intertemporal reallocation of resources in the face
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of a productivity shock.
Dividing (11) by the expression for the marginal utility of consumption, and employing
the functional forms introduced above, these conditions can be written as
�1� '
'
�ctlt
= (1� �)zt
�ktnt
��(13)
c'(1��)�1t l
(1�')(1��)t = �Et
(c'(1��)�1t+1 l
(1�')(1��)t+1
"�zt+1
�nt+1kt+1
�1��+ (1� �)
#): (14)
1.2 The Nonlinear System
Collecting components, the system of nonlinear stochastic di¤erence equations that
comprise the model is given by
�1� '
'
�ctlt
= (1� �)zt
�ktnt
��(15)
c�t l�t = �Et
(c�t+1l
�t+1
"�zt+1
�nt+1kt+1
�1��+ (1� �)
#)(16)
yt = ztk�t n
1��t (17)
yt = ct + it (18)
kt+1 = it + (1� �)kt (19)
1 = nt + lt (20)
log zt = (1� �) log(z) + � log zt�1 + "t; (21)
where � = '(1��)�1 and � = (1�')(1��): Steady states of the variables fyt; ct; it; nt; lt; kt; ztg
may be computed analytically from this system. These are derived by holding zt to its steady
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state value z, which we set to 1:
y
n= �;
c
n= � � ��;
i
n= ��; n =
1
1 +�
11��� �
1�''
� �1� ��1��
� ; l = 1� n;k
n= �;
(22)
where
� =
��
1=� � 1 + �
� 11��
� = ��:
Note that in steady state the variables fyt; ct; it; ktg do not grow over time. Implicitly,
these variables are represented in the model in terms of deviations from trend, and steady
state values indicate the relative heights of trend lines. To incorporate growth explicitly,
consider an alternative speci�cation of zt:
zt = z0(1 + g)te!t ; (23)
!t = �!t�1 + "t: (24)
Note that, absent shocks, the growth rate of zt is given by g; and that removal of the
trend component (1 + g)t from zt yields the speci�cation for log zt given by (21). Further,
the reader is invited to verify that under this speci�cation for zt, fct; it; yt; ktg will have a
common growth rate given by g1�� :Thus the model is consistent with the balanced-growth
requirement, and as speci�ed, all variables are interpreted as being measured in terms of
deviations from their common trend.
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There is one subtlety associated with the issue of trend removal that arises in dealing with
the dynamic equations of the system. Consider the law of motion for capital (19). Trend
removal here involves division of both sides by�1 + g
1���t; however, the trend component
associated with kt+1 is�1 + g
1���t+1
; so the speci�cation in terms of detrended variables is
�1 +
g
1� �
�kt+1 = it + (1� �)kt: (25)
Likewise, there will be a residual trend factor associated with ct+1 in the intertemporal
optimality condition (16). Since ct+1 is raised to the power � = '(1 � �) � 1; the residual
factor is given by�1 + g
1����:
c�t l�t = �Et
(�1 +
g
1� �
��c�t+1l
�t+1
"�zt+1
�nt+1kt+1
�1��+ (1� �)
#): (26)
With � negative (insured by 1�< 1; i.e., an inelastic intertemporal elasticity of substitution
speci�cation), the presence of g provides an incentive to shift resources away from (t + 1)
towards t:
Exercise 1 Rederive the steady state expressions (22) by replacing (19) with (25), and (16)
with (26). Interpret the intuition behind the impact of g on the expressions you derive.
1.3 Linearization
The linearization step involves taking a log-linear approximation of the model at steady
state values. In this case, the objective is to map (15)-(21) into the linearized system
Axt+1 = Bxt + C�t+1 + D�t+1 for eventual empirical evaluation. Regarding D, dropping
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Et from the Euler equation (16) introduces an expectations error in the model�s second
equation, therefore D = [0 1 0 0 0 0 0]0. Likewise, the presence of the productivity shock in
the model�s seventh equation (21) implies C = [0 0 0 0 0 0 1]0.
Regarding A and B, using the solution methodology discussed in Chapter 2, these can
be constructed by introducing the following system of equations into a gradient procedure
(where time subscripts are dropped so that, e.g., y = yt and y0 = yt+1):
0 = log(1� '
') + log c0 � log l0 � log(1� �)� log z0 � � log k + � log n0 (27)
0 = � log c+ � log l � log � � � log c0 � � log l0 � log�� exp(log z0)
exp [(1� �) log n0]
exp [(1� �) log k0]+ (1� �)
�(28)
0 = log y0 � log z0 � � log k � (1� �) log n0 (29)
0 = log y0 � log fexp [log (c0)] + exp [log (i0)]g (30)
0 = log k0 � log fexp [log (i0)] + (1� �) exp [log (k)]g (31)
0 = � log fexp [log (n0)] + exp [log (l0)]g (32)
0 = log z0 � � log z: (33)
The mapping from (15)-(21) to (27)-(33) involves four steps. First, logs of both sides of each
equation are taken; second, all variables not converted into logs in the �rst step are converted
using the fact, e.g., that y = exp(log(y)); third, all terms are collected on the right-hand side
of each equation; fourth, all equations are multiplied by �1: Derivatives taken with respect
to log y0; etc. evaluated at steady state values yield A; and derivatives taken with respect
to log y; etc. yield �B: Note that capital installed at time t is not productive until period
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t+ 1; thus, e.g., k rather than k0 appears in (29).
Having obtained A;B;C and D, the system can be solved using any of the solution
methods outlined in Chapter 2 to obtain a system of the form xt+1 = F (�)xt + et+1. This
system can then be evaluated empirically using any of the methods described in Part II.
Exercise 2 With xt given by
xt =
�log
yty; log
ctc; log
it
i; log
ntn; log
lt
l; log
kt
k; log
ztz
�0
and
� = [� � � ' � � �]0 = [0:33 0:975 2 0:5 0:06 0:9]0;
show that the steady state values of the model are y = 0:9; c = 0:7; i = 0:2; n =
0:47; l = 0:53 and k = 3:5 (and take as granted z = 1): Next, use a numerical gradient
procedure to derive
A =
266666666666666666666664
0 1 0 0:33 �1 0 �1
0 1:5 0 �0:12 0:5 0 �0:17
1 0 0 �0:67 0 0 �1
1 �0:77 �0:23 0 0 0 0
0 0 �0:18 0 0 1 0
0 0 0 �0:47 �0:53 0 0
0 0 0 0 0 0 1
377777777777777777777775
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B =
266666666666666666666664
0 0 0 0 0 0:33 0
0 1:5 0 0 0:5 �0:9 0
0 0 0 0 0 0:33 0
0 0 0 0 0 0 0
0 0 0 0 0 0:77 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0:9
377777777777777777777775
:
Exercise 3 Rederive the matrices A and B given the explicit incorporation of growth in the
model. That is, derive A and B using the steady state expressions obtained in Exercise 1,
and using (25) and (26) in place of (19) and (16).
2 Model II: Monopolistic Competition and Monetary
Policy
This section outlines a model of imperfect competition featuring �sticky�prices. The
model includes three sources of aggregate uncertainty: shocks to demand, technology and the
competitive structure of the economy. The model is due to Ireland (2004), who designed it to
determine how the apparent role of technology shocks in driving business-cycle �uctuations
is in�uenced by the inclusion of these additional sources of uncertainty.
From a pedological perspective, the model di¤ers in two interesting ways relative to the
RBC model outlined above. While the linearized RBC model is a �rst-order system of
di¤erence equations, the linearized version of this model is a second-order system. However,
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as we demonstrate, it is possible to represent a system of arbitrary order into the �rst-
order form taken by Axt+1 = Bxt + C�t+1 + D�t+1, given appropriate speci�cation of the
elements of xt: Second, the problem of mapping implications carried by a stationary model
into the behavior of non-stationary data is revisited from an alternative perspective than that
adopted in the discussion of the RBC model. Rather than assuming the actual data follow
stationary deviations around deterministic trends, here the data are modelled as following
drifting random walks; stationarity is induced via di¤erencing rather than detrending.2
2.1 Environment
The economy once again consists of a continuum of identical households. Here, there
are two distinct production sectors: an intermediate-goods sector and a �nal-goods sector.
The former is imperfectly competitive: it consists of a continuum of �rms that produce
di¤erentiated products which serve as factors of production in the �nal-goods sector. While
�rms in this sector have the ability to set prices, they face a friction in doing so. Finally,
there is a central bank.
2Nelson and Plosser (1982) initiated an intense debate regarding the nature of non-stationarity thatcharacterizes macroeconomic variables. The issue has proven di¢ cult to resolve, thus the use of alterna-tive detrending methods remains common practice. For an overview of the debate, see, e.g., DeJong andWhiteman (1993) and Stock (1994).
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2.1.1 Households
The representative household maximizes lifetime utility de�ned over consumption, money
holdings, and labor:
maxct;mt;nt
U = E0
1X0
�t
(at log ct + log
mt
pt� n�t
�
)(34)
s:t: ptct +btrt+mt = mt�1 + bt�1 + � t + wtnt + dt; (35)
where � 2 (0; 1) and � � 1: According to the budget constraint (35), the household divides
its wealth between holdings of bonds bt and money mt; bonds mature at the gross nominal
rate rt between time periods. The household also receives transfers � t from the monetary
authority and works nt hours in order to earn wages wt to �nance its expenditures. Finally,
the household owns an intermediate-goods �rm, from which it receives a dividend payment
dt. Note from (34) that the household is subject to an exogenous demand shock at that
a¤ects its consumption decision.
Recognizing that the instantaneous marginal utility derived from consumption is given by
atct; the �rst-order conditions associated with the household�s choices of labor, bond holdings
and money holdings are given by
�wtpt
��atct
�= n��1t (36)
�Et
��1
pt+1
��at+1ct+1
��=
�1
rtpt
��atct
�(37)�
mt
pt
��1+ �Et
��1
pt+1
��at+1ct+1
��=
�1
pt
��atct
�: (38)
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Exercise 4 Interpret how (36)-(38) represent the optimal balancing of trade-o¤s associated
with the household�s choices of n, b and m.
2.1.2 Firms
There are two types of �rms, one that produces a �nal consumption good yt; which sells
at price pt, and a continuum of intermediate-goods �rms that supply inputs to the �nal-good
�rm. The output of the ith intermediate good is given by yit; which sells at price pit: The
intermediate goods combine to produce the �nal good via a constant elasticity of substitution
(CES) production function. The �nal-good �rm operates in a competitive environment and
pursues the following objective:
maxyit
�Ft = ptyt �1Z0
pityitdi (39)
s:t: yt =
8<:1Z0
y�t�1�t
it di
9=;�t
�t�1
: (40)
The solution to this problem yields a standard demand for intermediate inputs and a price
aggregator:
yit = yt
�pitpt
���t(41)
pt =
8<:1Z0
p1��tit di
9=;1
1��t
: (42)
Notice that �t is the markup of price above marginal cost; randomness in �t provides the
notion of a cost-push shock in this environment.
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Intermediate-goods �rms are monopolistically competitive. Since the output of each �rm
enters the �nal-good production function symmetrically, the focus is on a representative �rm.
The �rm is owned by the representative household, thus its objectives are aligned with the
household�s. It manipulates the sales price of its good in pursuit of these objectives, subject
to a quadratic adjustment cost:
maxpit
�Iit = E0
1X0
�t�atct
��dtpt
�; (43)
s:t: yit = ztnit (44)
yit = yt
�pitpt
���t(45)
c(pit; pit�1) =�
2
�pit
�pit�1� 1�2yt; � > 0; (46)
where � is the gross in�ation rate targeted by the monetary authority (described below),
and the real value of dividends in (43) is given by
dtpt=
�pityit � wtnt
pt� c(pit; pit�1)
�: (47)
The associated �rst-order condition may be written as
(�t � 1)�pitpt
���t ytpt= �t
�pitpt
���t�1 wtpt
ytzt
1
pt
���
�pit
�pit�1� 1�
yt�pit�1
� ��Et
�at+1at
ctct+1
�pit+1�pit
� 1�yt+1pit+1�p2it
��: (48)
The left-hand side of (48) re�ects the marginal revenue to the �rm generated by an increase
in price; the right-hand side re�ects associated marginal costs. Under perfect price �exibility
17
(� = 0) there is no dynamic component to the �rm�s problem; the price-setting rule reduces
to pit = �t�t�1
wtzt; which is a standard markup of price over marginal cost wt
zt. Under �sticky
prices�(� > 0) the marginal cost of an increase in price has two additional components: the
direct cost of a price adjustment, and an expected discounted cost of a price change adjusted
by the marginal utility to the households of conducting such a change. Empirically, the
estimation of � is of particular interest: this parameter plays a central role in distinguishing
this model from its counterparts in the RBC literature.
2.1.3 The Monetary Authority
The monetary authority chooses the nominal interest rate according to a Taylor Rule.
With all variables expressed in terms of logged deviations from steady state values, the rule
is given by
ert = �rert�1 + ��e�t + �gegt + �oeot + "rt; "rt � IIDN(0; �2r); (49)
where e�t is the gross in�ation rate, egt is the gross growth rate of output, and eot is the outputgap (de�ned below). The �i parameters denote elasticities. The inclusion of ert�1 as an inputinto the Taylor Rule allows for the gradual adjustment of policy to demand and technology
shocks, e.g., as in Clarida, Gali, and Gertler (2000).
The output gap is the logarithm of the ratio of actual output yt to capacity output byt.Capacity output is de�ned to be the �e¢ cient� level of output, which is equivalent to the
18
level of output chosen by a benevolent social planner who solves:
maxbyt;nit US = E0
1X0
�t
8><>:at log byt � 1�0@ 1Z
0
nitdi
1A�9>=>; (50)
s:t: byt = zt
0@ 1Z0
n�t�1�tit di
1A�t
�t�1
: (51)
The solution to this problem is simply
byt = a1�
t zt: (52)
2.1.4 Stochastic Speci�cation
In addition to the monetary policy shock "rt introduced in (49), the model features a
demand shock at, a technology shock zt, and a cost-push shock �t. The former is IID; the
latter three evolve according to
log(at) = (1� �a) log(a) + �a log(at�1) + "at; a > 1 (53)
log(zt) = log(z) + log(zt�1) + "zt; z > 1 (54)
log(�t) = (1� ��) log(�) + �� log(�t�1) + "�t; � > 1; (55)
with j�ij < 1; i = a; �: Note that the technology shock is non-stationary: it evolves as a
drifting random walk. This induces similar behavior in the model�s endogenous variables,
and necessitates the use of an alternative to the detrending method discussed above in the
context of the RBC model. Here, stationarity is induced by normalizing model variables
19
by zt. For the corresponding observable variables, stationarity is induced by di¤erencing
rather than detrending: the observables are measured as deviations of growth rates (logged
di¤erences of levels) from sample averages. Details are provided in the linearization step
discussed below.
The model is closed through two additional steps. The �rst is the imposition of symmetry
among the intermediate-goods �rms:
yit = yt; nit = nt; pit = pt; dit = dt: (56)
The second is the requirement that the money and bond markets clear:
mt = mt�1 + � t (57)
bt = bt�1 = 0: (58)
2.2 The Non-Linear System
In its current form, the model consists of twelve equations: the household�s �rst-order
conditions and budget constraint; the aggregate production function; the aggregate real div-
idends paid to the household by its intermediate-goods �rm; the intermediate-goods �rm�s
�rst-order condition; the stochastic speci�cations for the structural shocks; and the expres-
sion for capacity output. Ireland�s (2004) empirical implementation focused on a linearized
reduction to an eight-equation system consisting of an IS curve; a Phillips curve; the Taylor
Rule (speci�ed in linearized form in (49)); the three exogenous shock speci�cations; and
de�nitions for the growth rate of output and the output gap.
20
The reduced system is recast in terms of the following normalized variables:
::yt =
ytzt;
::ct =
ctzt;
::byt = bytzt; �t =
ptpt�1
;
::
dt =(dt=pt)
zt;
::wt =
(wt=pt)
zt;
::mt =
(mt=pt)
zt;
::zt =
ztzt�1
:
Using the expression for real dividends given by (47), the household�s budget constraint is
rewritten as
::yt =
::ct +
�
2
��t�� 1�2 ::yt: (59)
Next, the household�s �rst-order condition (37) is written in normalized form as
at::ct= �rtEt
�at+1::ct+1
� 1::zt+1
� 1
�t+1
�: (60)
Next, the household�s remaining �rst-order conditions, the expression for the real dividend
payment (47) it receives, and the aggregate production function can be combined to elim-
inate wages, money, labor, dividends and capacity output from the system. This serves to
introduce the expression for the output gap into the system:
ot �ytbyt =
::yt
a1�
t
: (61)
Finally, normalizing the �rst-order condition of the intermediate-goods �rm and the stochas-
21
tic speci�cations leads to the following non-linear system:
::yt =
::ct +
�
2
��t�� 1�2 ::yt (62)
at::ct
= �rtEt
�at+1::ct+1
� 1::zt+1
� 1
�t+1
�(63)
0 = 1� �t + �t
::ctat
::y��1t � �
��t�� 1� �t�+ ��Et
� ::ctat+1::ct+1at
��t+1�
� 1� �t+1
�
::yt+1::yt
�(64)
gt =
::zt::yt
::yt�1
(65)
ot =ytbyt =
::yt
a1�
t
(66)
log(at) = (1� �a) log(a) + �a log(at�1) + "at (67)
log(�t) = (1� ��) log(�) + �� log(�t�1) + "�t (68)
log(::zt) = log(z) + "zt (69)
Along with the Taylor Rule, this is the system to be linearized.
2.3 Linearization
Log-linearization proceeds with the calculation of steady state values of the endogenous
variables:
r =z
��; c = y =
�a� � 1�
� 1�
; o =
�� � 1�
� 1�
; (70)
(62)-(69) are then log-linearized around these values. As with Model I, this can be accom-
plished through the use of a numerical gradient procedure. However, as an alternative to this
approach, here we follow Ireland (2004) and demonstrate the use of a more analytically ori-
ented procedure. In the process, it helps to be mindful of the re-con�guration Ireland worked
22
with: an IS curve; a Phillips curve; the Taylor Rule; the shock processes; and de�nitions of
the growth rate of output and the output gap.
As a �rst step, the variables appearing in (62)-(69) are written in logged form. Log-
linearization of (62) then yields eyt � log� ::yty
�= ect, since the partial derivative of eyt with
respect to e�t (evaluated at steady state) is zero.3 Hence upon linearization, this equation iseliminated from the system, and ect is replaced by eyt in the remaining equations.Next, recalling that Etezt+1 = 0; log-linearization of (63) yields
0 = ert � Ete�t+1 � (Eteyt+1 � eyt) + Eteat+1 � eat: (71)
Relating output and the output gap via the log-linearization of (66),
eyt = 1
�eat + eot; (72)
the term Eteyt+1 � eyt may be substituted out of (71), yielding the IS curve:eot = Eteot+1 � (rt � Ete�t+1) + �1� ��1
�(1� �a)eat: (73)
Similarly, log-linearizing (64) and eliminating eyt using (72) yields the Phillips curve:e�t = �Ete�t+1 + eot � eet; (74)
where = �(��1)�
and eet = 1�e�t. This latter equality is a normalization of the cost-push
3Recall our notational convention: tildes denote logged deviations of variables from steady state values.
23
shock; like the cost-push shock itself, the normalized shock follows an AR(1) process with
persistence parameter �� = �e; and innovation standard deviation �e =1���.
The resulting IS and Phillips curves are forward looking: they include the one-step-ahead
expectations operator. However, prior to empirical implementation, Ireland augmented these
equations to include lagged variables of the output gap and in�ation in order to enhance the
empirical coherence of the model. This �nal step yields the system he analyzed. Dropping
time subscripts and denoting, e.g., eot�1 as eo�; the system is given by
eo = �oeo� + (1� �o)Eteo0 � (r � Ete�0) + �1� ��1�(1� �a)ea (75)
e� = ���e�� + �(1� ��)Ete�0 + eo� ee (76)
eg0 = ey0 � ey + ez0 (77)
eo0 = ey0 � ��1ea0 (78)
er0 = �rer + ��e�0 + �geg0 + �oeo0 + "0r (79)
ea0 = �aea+ "0a (80)
ee0 = �eee+ "0e (81)
ez0 = "0z (82)
where the structural shocks �t = f"rt; "at; "et; "ztg are IIDN with diagonal covariance matrix
�. The additional parameters introduced are �o 2 [0; 1] and �� 2 [0; 1]; setting �o = �� = 0
yields the original microfoundations.
The augmentation of the IS and Phillips curves with lagged values of the output gap and
in�ation converts the model from a �rst- to a second-order system. Thus a �nal step is re-
24
quired in mapping this system into the �rst-order speci�cation Axt+1 = Bxt+C�t+1+D�t+1:
This is accomplished by augmenting the vector xt to include not only contemporaneous ob-
servations of the variables of the system, but also to include lagged values of the output gap
and in�ation:
xt � [eot eot�1 e�t e�t�1 eyt ert egt eat eet ezt]0:This also requires the introduction of two additional equations into the system: e�0 = e�0 andeo0 = eo0. Specifying these as the �nal two equations of the system, the corresponding matricesA and B are given by
A =
266666666666666666666666666666666664
�(1� �0) 1 �1 0 0 0 0 0 0 0
0 � ��(1� ��) 1 0 0 0 0 0 0
0 0 0 0 �1 0 1 0 0 �1
1 0 0 0 �1 0 0 ��1 0 0
��0 0 ��� 0 0 1 ��g 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
377777777777777777777777777777777775
(83)
25
B =
266666666666666666666666666666666664
0 �o 0 0 0 �1 0 (1� ��1)(1� �a)�a 0 0
0 0 0 ��� 0 0 0 0 �1 0
0 0 0 0 �1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 �r 0 0 0 0
0 0 0 0 0 0 0 �a 0 0
0 0 0 0 0 0 0 0 �e 0
0 0 0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0
377777777777777777777777777777777775
: (84)
Further, de�ning �t = [�1t �2t]0; where �1t+1 = Eteot+1 � eot+1 and �2t+1 = Ete�t+1 � e�t+1;
the matrices C and D are given by
C =
26666666666666666664
04x4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
02x4
37777777777777777775
; D =
266666641� �0 1
0 �(1� ��)
02x8
37777775 (85)
The �nal step needed for empirical implementation is to identify the observable variables
of the system. For Ireland, these are the gross growth rate of output gt; the gross in�ation
rate �t; and the nominal interest rate rt (all measured as logged ratios of sample averages).
26
Under the assumption that output and aggregate prices follow drifting random walks, gt and
�t are stationary; the additional assumption of stationarity for rt is all that is necessary to
proceed with the analysis.
Exercise 5 Consider a pth� order di¤erence equation for yt of the form
yt = �1yt�1 + �2yt�2 + :::+ �pyt�p + "t; "t � IID:
Construct vectors (xt; et) and matrices (�;�) so that the model may be re-cast in �rst-order
form as
xt+1 = �xt +�et+1:
Exercise 6 Solve the linearized system (75)-(82) using any of the methods outlined in Chap-
ter 2. Note that the vector of deep parameters is now given by:
� = [z � � ! � � �x �� �r �� �g �x �a �� �a �� �z �r]0:
Exercise 7 Consider the following CRRA form for the instantaneous utility function for
Model II:
u(ct;mt
pt; nt) = at
c�t�+ log
mt
pt� n�t
�:
1. Derive the non-linear system under this speci�cation.
2. Sketch the linearization of the system via a numerical gradient procedure.
27
3 Model III: Asset Pricing
The �nal model is an adaptation of Lucas�(1978) one-tree model of asset-pricing behav-
ior. Alternative versions of the model have played a prominent role in two important strands
of the empirical �nance literature. The �rst, launched by Shiller (1981) in the context of a
single-asset version of the model, concerns the puzzling degree of volatility exhibited by prices
associated with aggregate stock indexes. The second, launched by Mehra and Prescott (1985)
in the context of a multi-asset version of the model, concerns the puzzling coincidence of a
large gap observed between the returns of risky and risk-free assets, and a low average risk-
free return. Resolutions to both puzzles have been investigated using alternate preference
speci�cations. After outlining single- and multi-asset versions of the model given a generic
speci�cation of preferences, alternative functional forms are introduced. Overviews of the
role of preferences in the equity-premium literature are provided by Kocherlakota (1996) and
Cochrane (2001); and in the stock-price volatility literature by DeJong and Ripoll (2004).
3.1 Single-Asset Environment
The model features a continuum of identical households and a single risky asset. Shares
held during period (t � 1), st�1; yield a dividend payment dt at time t; time-t share prices
are pt. Households maximize expected lifetime utility by �nancing consumption ct from
an exogenous stochastic dividend stream, proceeds from sales of shares, and an exogenous
stochastic endowment qt. The utility maximization problem of the representative household
is given by
maxct
U = E0
1Xt=0
�tu(ct); (86)
28
where � 2 (0; 1) again denotes the discount rate, and optimization is subject to
ct + pt(st � st�1) = dtst�1 + qt: (87)
Since households are identical, equilibrium requires st = st�1 for all t, and thus ct = dtst+qt =
dt + qt (hereafter, st is normalized to 1). Combining this equilibrium condition with the
household�s necessary condition for a maximum yields the pricing equation
pt = �Etu0(dt+1 + qt+1)
u0(dt + qt)(dt+1 + pt+1): (88)
From (88), following a shock to either dt or qt, the response of pt depends in part upon
the variation of the marginal rate of substitution between t and t+ 1. This in turn depends
upon the instantaneous utility function u(�). The puzzle identi�ed by Shiller (1981) is that
pt is far more volatile than what (88) would imply, given the observed volatility of dt:
The model is closed by specifying stochastic processes for (dt; qt). These are given by
log dt = (1� �d) log(d) + �d log(dt�1) + "dt (89)
log qt = (1� �q) log(q) + �q log(qt�1) + "qt; (90)
with j�ij < 1; i = d; q; and 2664 "dt
"qt
3775 � IIDN(0;�): (91)
29
3.2 Multi-Asset Environment
An n-asset extension of the environment leaves the household�s objective function intact,
but modi�es its budget constraint to incorporate the potential for holding n assets. As a
special case, Mehra and Prescott (1985) studied a two-asset speci�cation, including a risk-
free asset (ownership of government bonds) and risky asset (ownership of equity). In this
case, the household�s budget constraint is given by
ct + pet(set � set�1) + pft s
ft = dts
et�1 + sft�1 + qt; (92)
where pet denotes the price of the risky asset, set represents the number of shares held in the
asset during period t� 1; and pft and sft are analogous for the risk-free asset. The risk-free
asset pays one unit of the consumption good at time t if held at time t�1 (hence the loading
factor of 1 associated with sft�1 on the right-hand-side of the budget constraint).
First-order conditions associated with the choice of the assets are analogous to the pricing
equation (88) established in the single-asset speci�cation. Rearranging slightly:
�Etu0(ct+1)
u0(ct)�pet+1 + dt
pet= 1 (93)
�Etu0(ct+1)
u0(ct)� 1
pft= 1: (94)
De�ning gross returns associated with the assets as
ret+1 =pet+1 + dt
pet
rft+1 =1
pft;
30
Mehra and Prescott�s identi�cation of the equity premium puzzle centers on
�Etu0(ct+1)
u0(ct)rft+1 = 1 (95)
Etu0(ct+1)
u0(ct)
hret+1 � rft+1
i= 0; (96)
where (96) is derived by subtracting (94) from (93). The equity premium puzzle has two
components. First, taking fctg as given, the average value of re � rf is quite large: given
CRRA preferences, implausibly large values of the risk-aversion parameter are needed to
account for the average di¤erence observed in returns. Second, given a speci�cation of u(c)
that accounts for (96), and again taking prices as given, the average value observed for rf is
far too low to reconcile with (95). This second component is the risk-free rate puzzle.
3.3 Alternative Preference Speci�cations
As noted, alternative preference speci�cations have been considered for their potential in
resolving both puzzles. Here, in the context of the single-asset environment, three forms for
the instantaneous utility function are presented in anticipation of the empirical applications
to be presented in Part II of the text: CRRA preferences; habit/durability preferences; and
self control preferences. The presentation follows that of DeJong and Ripoll (2004), who
sought to evaluate empirically the ability of these preference speci�cations to make headway
in resolving the stock-price volatility puzzle.
31
3.3.1 CRRA
Once again, CRRA preferences are parameterized as
u(ct) =c1� t
1� ; (97)
thus > 0 measures the degree of relative risk aversion, and 1= the intertemporal elasticity
of substitution. The equilibrium pricing equation is given by
pt = �Et(dt+1 + qt+1)
�
(dt + qt)� (dt+1 + pt+1): (98)
Notice that, ceteris paribus, a relatively large value of will increase the volatility of price
responses to exogenous shocks, at the cost of decreasing the correlation between pt and dt
(due to the heightened role assigned to qt in driving price �uctuations). Since fdtg and fqtg
are exogenous, their steady states d and q are simply parameters. Normalizing d to 1 and
de�ning � = q
d, so that � = q; the steady state value of consumption (derived from the
budget constraint) is c = 1 + �. And from the pricing equation,
p =�
1� �d =
�
1� �: (99)
Letting � = 1=(1+r), where r denotes the household�s discount rate, (99) implies p=d = 1=r.
Thus as the household�s discount rate increases, its asset demand decreases, driving down
the steady state price level. Empirically, the average price/dividend ratio observed in the
data serves to pin down � under this speci�cation of preferences.
32
Exercise 8 Linearize the pricing equation (98) around the model�s steady state values.
3.3.2 Habit/Durability
Following Ferson and Constantinides (1991) and Heaton (1995), an alternative speci�-
cation of preferences that introduces habit and durability into the speci�cation of preferences
is parameterized as
u(ht) =h1� t
1� ; (100)
with
ht = hdt � �hht ; (101)
where � 2 (0; 1), hdt is the household�s durability stock, and hht its habit stock. The stocks
are de�ned by,
hdt =1Xj=0
�jct�j (102)
hht = (1� �)1Xj=0
�jhdt�1�j = (1� �)1Xj=0
�j1Xi=0
�ict�1�i (103)
where � 2 (0; 1) and � 2 (0; 1). Thus the durability stock represents the �ow of services
from past consumption, which depreciates at rate �. This parameter also represents the
degree of intertemporal substitutability of consumption. The habit stock can be interpreted
as a weighted average of the durability stock, where the weights sum to one. Notice that
more recent durability stocks, or more recent �ows of consumption, are weighted relatively
heavily; thus the presence of habit captures intertemporal consumption complementarity.
The variable ht represents the current level of durable services net of the average of past
services; the parameter � measures the fraction of the average of past services that is netted
33
out. Notice that if � = 0, there would only be habit persistence, while if � = 0 only durability
survives. Finally, when � = 0, the habit stock includes only one lag. Thus estimates of these
parameters are of particular interest empirically.
Using the de�nitions of durability and habit stocks, ht becomes
ht = ct +
1Xj=1
"�j � �(1� �)
j�1Xi=0
�i�j�i�1
#ct�j �
1Xj=0
�jct�j; (104)
where �0 � 1. Thus for these preferences, the pricing equation is given by
pt = �Et
1Pj=0
�j�j
� 1Pi=0
�ict+1+j�i
�� 1Pj=0
�j�j
� 1Pi=0
�ict+j�i
�� (dt+1 + pt+1) ; (105)
where as before ct = dt + qt in equilibrium.
To see how the presence of habit and durability can potentially in�uence the volatility of
the prices, rewrite the pricing equation as
pt = �Et(ct+1 + �1ct + �2ct�1 + :::)� + ��1(ct+2 + �1ct+1 + �2ct + :::)� + :::
(ct + �1ct�1 + �2ct�2 + :::)� + ��1(ct+1 + �1ct + �2ct�1 + :::)� + :::(dt+1 + pt+1) :
(106)
When there is a positive shock to say qt, ct increases by the amount of the shock, say
�q. Given (89)-(90), ct+1 would increase by �q�q, ct+2 would increase by �2q�q, etc. Now,
examine the �rst term in parenthesis both in the numerator and the denominator. First,
in the denominator ct will grow by �q. Second, in the numerator ct+1 + �1ct goes up by��q + �1
��q 7 �q. Thus, whether the share price pt increases by more than in the standard
CRRA case depends ultimately on whether �q+�1 7 1. Notice that if �j = 0 for j > 0, the
34
equation above reduces to the standard CRRA utility case. If we had only habit and not
durability, then �1 < 0, and thus the response of prices would be greater than in the CRRA
case. This result is intuitive: habit captures intertemporal complementarity in consumption,
which strengthens the smoothing motive relative to the time-separable CRRA case.
Alternatively, if there was only durability and not habit, then 0 < �1 < 1, but one still
would not know whether � + �1 7 1. Thus with only durability, we cannot judge how the
volatility of pt would be a¤ected: this will depend upon the sizes of � and �1. Finally, we
also face indeterminacy under a combination of both durability and habit: if � is large and
� is small enough to make � + �1 < 1, then we would get increased price volatility. Thus
this issue is fundamentally quantitative. Finally, with respect to the steady state price, note
from (106) that it is identical to the CRRA case.
Exercise 9 Given that the pricing equation under Habit/Durability involves an in�nite num-
ber of lags, truncate the lags to 3 and linearize the pricing equation (106) around its steady
state.
3.3.3 Self-Control Preferences
Consider next a household that every period faces a temptation to consume all of its
wealth. Resisting this temptation imposes a self-control utility cost. To model these prefer-
ences we follow Gul and Pesendorfer (2004), who identi�ed a class of dynamic self-control
preferences. In this case, the problem of the household can be formulated recursively as
W (s; P ) = maxs0fu(c) + v(c) + �EW (s0; P 0)g �maxes0 v(ec); (107)
35
where P = (p; d; e); u(:) and v(:) are Von Neuman-Morgenstern utility functions; � 2 (0; 1);
ec represents temptation consumption; and s0 denotes share holdings next period. While u(:)is the momentary utility function, v(:) represents temptation. The problem is subject to the
following budget constraints:
c = ds+ e� p(s0 � s) (108)
ec = ds+ e� p(es0 � s): (109)
In the speci�cation above, v(c) � maxes0 v(ec) � 0 represents the disutility of self-control
given that the agent has chosen c. With v(c) speci�ed as strictly increasing, the solution for
maxes0 v(ec) is simply to drive ec to the maximum allowed by the constraint ec = ds+e�p(es0�s),which is attained by setting es0 = 0. Thus the problem is written as
W (s; P ) = maxs0fu(c) + v(c) + �EW (s0; P 0)g � v(ds+ e+ ps) (110)
subject to
c = ds+ e� p(s0 � s): (111)
The optimality condition reads
[u0(c) + v0(c)] p = �EW 0(s0; P 0); (112)
and since
W 0(s; P ) = [u0(c) + v0(c)] (d+ p)� v0(ds+ e+ ps)(d+ p); (113)
36
the optimality condition becomes
[u0(c) + v0(c)] p = �E [u0(c0) + v0(c0)� v0(d0s0 + e0 + p0s0)] (d0 + p0): (114)
Combining this expression with the equilibrium conditions s = s0 = 1 and c = d+ e yields
p = �E (d0 + p0)
�u0(d0 + e0) + v0(d0 + e0)� v0(d0 + e0 + p0)
u0(d+ e) + v0(d+ e)
�: (115)
Notice that when v(:) = 0, there is no temptation, and the pricing equation reduces to
the standard case. Otherwise, the term u0(d0 + e0) + v0(d0 + e0) � v0(d0 + e0 + p0) represents
tomorrow�s utility bene�t from saving today. This corresponds to the standard marginal
utility of wealth tomorrow u0(d0 + e0), plus the term v0(d0 + e0) � v0(d0 + e0 + p0) which
represents the derivative of the utility cost of self-control with respect to wealth.
DeJong and Ripoll (2004) assume the following functional forms for the momentary and
temptation utility functions:
u(c) =c1�
1� (116)
v(c) = �c�
�; (117)
with � > 0, which imply the following pricing equation:
p = �E [d0 + p0]
�(d0 + e0)� + �(d0 + e0)��1 � �(d0 + e0 + p0)��1
(d+ e)� + �(d+ e)��1
�: (118)
The concavity/convexity of v(:) plays an important role in determining implications of
this preference speci�cation for the stock-price volatility issue. To understand why, rewrite
37
(118) as
p = �E [d0 + p0]
24 (d0+e0)�
(d+e)� + �(d+ e) �(d0 + e0)��1 � (d0 + e0 + p0)��1
�1 + �(d+ e)��1+
35 : (119)
Suppose � > 1; so that v(:) is convex, and consider the impact on p of a positive endowment
shock. This increases the denominator, while decreasing the term
�(d+ e) �(d0 + e0)��1 � (d0 + e0 + p0)��1
�
in the numerator. Both e¤ects imply that relative to the CRRA case, in which � = 0, this
speci�cation reduces price volatility in the face of an endowment shock, which is precisely
the opposite of what one would like to achieve in seeking to resolve the stock-price volatility
puzzle.
The mechanism behind this reduction in price volatility is as follows: a positive shock to d
or e increases the household�s wealth today, which has three e¤ects. The �rst (�smoothing�)
captures the standard intertemporal motive: the household would like to increase saving,
which drives up the share price. Second, there is a �temptation�e¤ect: with more wealth
today, the feasible budget set for the household increases, which represents more temptation
to consume, and less willingness to save. This e¤ect works opposite to the �rst, and reduces
price volatility with respect to the standard case. Third, there is the �self-control�e¤ect: due
to the assumed convexity of v(:), marginal self-control costs also increase, which reinforces
the second e¤ect. As shown above, the last two e¤ects dominate the �rst, and thus under
convexity of v(:) the volatility is reduced relative to the CRRA case.
38
In contrast, price volatility would not necessarily be reduced if v(:) is concave, and thus
0 < � < 1. In this case, when d or e increases, the term
�(d+ e) �(d0 + e0)��1 � (d0 + e0 + p0)��1
�
increases. On the other hand, if � � 1 + > 0, i.e., if the risk-aversion parameter > 1,
the denominator also increases. If the increase in the numerator dominates that in the
denominator, then higher price volatility can be observed than in the CRRA case.
To understand this e¤ect, note that the derivative of the utility cost of self-control with
respect to wealth is positive if v(:) is concave: v0(d0 + e0)� v0(d0 + e0 + p0) > 0. This means
that as agents get wealthier, self-control costs become lower. This explains why it might
be possible to get higher price volatility in this case. The mechanism behind this result
still involves the three e¤ects discussed above: smoothing, temptation, and self-control. The
di¤erence is on the latter e¤ect: under concavity, self-control costs are decreasing in wealth.
This gives the agent an incentive to save more rather than less. If this self-control e¤ect
dominates the temptation e¤ect, then these preferences will produce higher price volatility.
Notice that when v(:) is concave, conditions need to be imposed to guarantee that W (:)
is strictly concave, so that the solution corresponds to a maximum (e.g., see Stokey and
Lucas, 1989). In particular, the second derivative of W (:) must be negative:
� (d+ e)� �1 + �(�� 1)�(d+ e)��2 � (d+ e+ p)��2
�< 0 (120)
which holds for any d, e, and p > 0, and for > 0, � > 0, and 0 < � < 1. The empirical
39
implementation in Part II of the text proceeds under this set of parameter restrictions.
Finally, from the optimality conditions under self-control preferences, steady-state temp-
tation consumption is ec = 1+ � + p. From (118), the steady-state price in this case is given
by
p = � (1 + p)
"(1 + �)� + � (1 + �)��1 � � (1 + � + p)��1
(1 + �)� + � (1 + �)��1
#: (121)
Regarding (121), the left-hand-side is a 45-degree line. The right-hand side is strictly con-
cave in p, has a positive intercept, and a positive slope that is less than one at the intercept.
Thus (121) yields a unique positive solution for p� for any admissible parameterization of
the model. (In practice, (121) can be solved numerically, e.g., using GAUSS�s quasi-Newton
algorithm NLSYS; see Judd, 1998, for a presentation of alternative solution algorithms.) An
increase in � causes the function of p on the right-hand-side of (121) to shift down and �at-
ten, thus p is decreasing in �. The intuition for this is again straightforward: an increase in
� represents an intensi�cation of the household�s temptation to liquidate its asset holdings.
This drives down its demand for asset shares, and thus p. Note the parallel between this
e¤ect and that generated by an increase in r, or a decrease in �, which operates analogously
in both (99) and (121).
Exercise 10 Solve for p in (121) using � = 0:96; = 2; � = 0:01; � = 10; � = 0:4:
Linearize the asset-pricing equation (119) using the steady state values for�p; d; q
�implied
by these parameter values.
40
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