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 inuencing 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 inuencing 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 & Poors 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 premiumpuzzle. 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 households 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 households 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

3

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 households problem is to maximize (1) subject

to (2)-(5), taking k0 and z0 as given.

Implicit in the specication of the households 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 specied 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 builders perspective,

the former requirement serves as a constraint on choices regarding the specications 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 (reected

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.

5

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 households interest in maintaining a smooth consumption/leisure prole.

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 capitals 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 households 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 households 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 benet 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 benet 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 households 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 households 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

7

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

8

state value z, which we set to 1:

y

n= �;

c

n= � � ��; i

n= ��; n =

1

1 +�

11��� �

1�''

� �1� ��1��

� ; l = 1� n; kn = �;(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 specication 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 specication for log zt given by (21). Further,

the reader is invited to verify that under this specication 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 specied, all variables are interpreted as being measured in terms of

deviations from their common trend.

9

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 specication 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

specication), 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

10

Et from the Euler equation (16) introduces an expectations error in the models second

equation, therefore D = [0 1 0 0 0 0 0]0. Likewise, the presence of the productivity shock in

the models 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

11

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

12

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 stickyprices. 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 inuenced 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,

13

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 specication 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).

14

2.1.1 Households

The representative household maximizes lifetime utility dened over consumption, money

holdings, and labor:

maxct;mt;nt

U = E0

1X0

�t

(at log ct + log

mtpt� 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 households 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)�

mtpt

��1+ �Et

��1

pt+1

��at+1ct+1

��=

�1

pt

��atct

�: (38)

15

Exercise 4 Interpret how (36)-(38) represent the optimal balancing of trade-o¤s associated

with the households 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

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

households. 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 ination 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) reects the marginal revenue to the rm generated by an increase

in price; the right-hand side reects associated marginal costs. Under perfect price exibility

17

(� = 0) there is no dynamic component to the rms problem; the price-setting rule reduces

to pit = �t�t�1wtzt; 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 ination rate, egt is the gross growth rate of output, and eot is the outputgap (dened 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 dened 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 = E01X0

�t

8>:at log byt � 1�0@ 1Z

0

nitdi

1A�9>=>; (50)

s:t: byt = zt0@ 1Z

0

n�t�1�tit di

1A�t

�t�1

: (51)

The solution to this problem is simply

byt = a 1�t zt: (52)2.1.4 Stochastic Specication

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 models 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 households 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 rms

rst-order condition; the stochastic specications for the structural shocks; and the expres-

sion for capacity output. Irelands (2004) empirical implementation focused on a linearized

reduction to an eight-equation system consisting of an IS curve; a Phillips curve; the Taylor

Rule (specied in linearized form in (49)); the three exogenous shock specications; and

denitions 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 households budget constraint is

rewritten as

::yt =

::ct +

�

2

��t�� 1�2 ::yt: (59)

Next, the households 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 households 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 specications 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-conguration Ireland worked

22

with: an IS curve; a Phillips curve; the Taylor Rule; the shock processes; and denitions 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 withrespect 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-push3Recall 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 ination 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 byeo = �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

ination 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 specication 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 ination:

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, dening �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 ination

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;mtpt; nt) = at

c�t�+ log

mtpt� n

�t

�:

1. Derive the non-linear system under this specication.

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

specications. After outlining single- and multi-asset versions of the model given a generic

specication 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

households 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 identied 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 households objective function intact,

but modies its budget constraint to incorporate the potential for holding n assets. As a

special case, Mehra and Prescott (1985) studied a two-asset specication, including a risk-

free asset (ownership of government bonds) and risky asset (ownership of equity). In this

case, the households budget constraint is given by

ct + pet(s

et � set�1) + p

ft sft = dts

et�1 + s

ft�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 specication. Rearranging slightly:

�Etu0(ct+1)

u0(ct)�pet+1 + dt

pet= 1 (93)

�Etu0(ct+1)

u0(ct)� 1pft

= 1: (94)

Dening gross returns associated with the assets as

ret+1 =pet+1 + dt

pet

rft+1 =1

pft;

30

Mehra and Prescotts identication of the equity premium puzzle centers on

�Etu0(ct+1)

u0(ct)rft+1 = 1 (95)

Etu0(ct+1)

u0(ct)

hret+1 � r

ft+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 specication 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 Specications

As noted, alternative preference specications 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 specications 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�t1� ; (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

dening � = qd, 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 households discount rate, (99) implies p=d = 1=r.

Thus as the households 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 specication of preferences.

32

Exercise 8 Linearize the pricing equation (98) around the models 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 specication of preferences

is parameterized as

u(ht) =h1�t1� ; (100)

with

ht = hdt � �hht ; (101)

where � 2 (0; 1), hdt is the households durability stock, and hht its habit stock. The stocks

are dened 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 denitions 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 inuence 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 innite 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 identied 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 specication above, v(c) � maxes0 v(ec) � 0 represents the disutility of self-control

given that the agent has chosen c. With v(c) specied 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

tomorrows utility benet 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 specication 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

specication 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 households 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 temptatione¤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-controle¤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 givenby

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 GAUSSs 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 intensication of the households 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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