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%Date: Thu, 11 Feb 1999 14:24:53 -0600 (CST)

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%TCIDATA{Created=Fri Oct 18 13:38:19 1996}
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\begin{document}

\author{Jahangir Aziz and Luc Leruth\thanks{%
The views expressed are those of the authors and do not necessarily
represent those of the Fund. An earlier version of the paper was issued as a 
\textit{Working Paper of the International Monetary Fund No. WP/97/19}. It
is available at www.imf.org.} \\
%EndAName
International Monetary Fund\\
Washington D.C. 20431\\
and \\
University of Liege, Belgium}
\date{February 1999}
\title{Cyclical Effects of the Composition of Government Purchases}
\maketitle

\begin{abstract}
This paper constructs a general equilibrium model with monopolistically
competitive firms and endogenous markups where government spending consists
of both consumption and investment goods. It is shown that when markups are
countercyclical, an increase in the share of investment goods in total
public expenditure, raises output, employment, and capital stock in the
long-run leading to increases in welfare and productivity. However, this
also raises the short run cyclical variability of the economy. In
particular, variance of output and employment arising from technological and
aggregate demand shocks increase as the long run share of government
investment goes up implying a trade-off between greater long-run efficiency
and higher short-run volatility.

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\tableofcontents

\newpage

\section{Introduction}

Since the mid-1980s, many of the OECD countries have undertaken significant
budgetary changes. Much of the subsequent analyses on the implications of
these changes have focussed primarily on the effects of reducing the levels
of overall expenditure fiscal deficits. However, along with reductions in
overall spending and fiscal deficits, in many of these countries the
composition of government spending has also undergone substantial changes.
In particular, over the last ten years or so, except in Japan, governments
have generally shifted away from investment spending. While the shift has
been most dramatic in the smaller OECD countries such as Australia, Ireland,
and New Zealand, among many of the larger countries, such as France,
Germany, and the United States, the compositional change has also been quite
pronounced (see Figure 1). This paper focuses on the effects of
compositional changes in government expenditure and shows how changes in the
relative shares of public consumption and investment goods can affect both
the long run efficiency as well as short run fluctuations of an economy even
when public spending is assumed to be unproductive.

Using a model with horizontally differentiated goods and monopolistic
competition we find that when markups are countercyclical, a reduction in
the share of investment goods in total public expenditure, lowers output,
employment, and capital stock in the steady state. While this leads to
declines in welfare and productivity in the long run, it also lowers the
short run cyclical variability of the economy. In particular, variance of
output and employment arising from technological and aggregate demand shocks
fall as the long run share of government investment goes down. Quantitative
estimates using the model calibrated, to the postwar U.S. and other selected
OECD countries, show that the effects of changing the composition of public
spending can be quite significant. These effects are, however, reversed if
markups are procyclical.

The link between the composition of public expenditure and growth has been
the focus of endogenous growth models such as those of Barro (1990) and
Devarajan et. al (1996), where a part or all of government expenditure is
assumed to be directly productive. Some authors have also explored the
impact of aggregate government purchases on short run output and employment
when public spending is not directly productive. In perfectly competitive
economies, an increase in government spending raises households' willingness
to supply more labor in response to the corresponding fall in wealth arising
from financing additional government spending and through an increase in
real interest rate induced by the higher government outlay (see, for
example, Aiyagari et. al (1989) and the references therein). In a model of
oligopolistic price setting, Rotemberg and Woodford (1992) showed that
higher government spending can result in increased output and employment
without any shift in households' consumption-leisure choices. However, these
studies have concentrated on the effect of aggregate public spending rather
than on its composition.

The economy we consider is one where final goods have two different end uses
--- consumption and investment. Under the assumption that firms cannot
discriminate between the two sets of buyers, the price elasticity of
aggregate demand is given by the average of the elasticities of consumption
and investment, weighted by the shares of the two end uses in total demand.
This implies that the composition of aggregate demand will determine the
final price elasticity faced by the producers of the commodity. In a world
where firms have some market power, the markup charged by a firm over its
marginal cost is then determined by this average elasticity. Consequently,
whenever the composition of the demand changes between the two end uses so
does the elasticity and in turn the markup. Since decisions regarding the
use of different inputs by firms depend on the markup, i.e., in equilibrium $%
MPK_{t}=\mu _{t}r_{t}$, where $MPK_{t}$ is the marginal product of capital, $%
r_{t}$ its rental price, and $\mu _{t}$ the markup, and $MPL_{t}=\mu
_{t}w_{t}$, where $MPL_{t}$ is the marginal product of labor and $w_{t}$ the
real wage rate, changes in the composition of aggregate output affect both
investment and employment in the economy.

The model is based on a framework originally developed by Gali (1994). Goods
are horizontally differentiated with each good being produced by a single
firm in a monopolistic market The price elasticity of demand for each good
is different for households --- who use it for consumption, and firms ---
who use it for investment. In this environment, we show that the government
can affect the efficiency and cyclical behavior of the economy by changing
composition of aggregate demand by altering the composition of its own
purchases. In order to isolate the effect of compositional changes in public
expenditure on the economy, we assume throughout the analysis that
government purchases neither enter households' utility nor the production
process.

When markups are countercyclical, which is a largely accepted feature of the
U.S. and OECD economies (Woodford and Rotemberg (1990), Bils (1987), Martins
et. al (1996)), decreasing the share of public investment while keeping
government expenditure unchanged, reduces output, employment, capital stock,
and welfare in the steady state. In this model, for markups to be
countercyclical, the price elasticity of investment needs to be larger than
that of consumption. When this happens, an decrease in the share of
investment lower the average elasticity, raises markups charged by profit
maximizing firms, leading to lower output, employment, investment, and
finally welfare. Moreover, the rise in the markup pushes the economy farther
away from the perfectly competitive equilibrium such that there are
efficiency losses, in terms of both labor and total factor productivity.
However, if the elasticity of investment is lower than that of consumption
so that markups become procyclical, decreasing the share of public
investment has the opposite effect.

Although, when the price elasticity of investment is larger than that of
consumption, decreases in steady state public investment lowers steady state
output and welfare by increasing the markup, the reduction in elasticity
which leads to the rise in the markup, also makes the aggregate demand less
elastic. Consequently, both exogenous demand and supply shocks lead to a
lower adjustment in quantity rather than prices, so that output and
employment variability are also reduced. Consequently, as the economy is
moved farther away from the perfectly competitive allocation, the cyclical
variability is lowered. With lower cyclical variations, depending on the
level of risk aversion, the increase in welfare can overcompensate the
losses in welfare arising from the lower efficiency so that decreases in
public investment can increase overall welfare. Thus, changes in the long
run composition of public expenditure entail a trade-off between lower long
run efficiency and lesser short run volatility.

The mechanism by which shocks affect markups in this model differs
significantly from the processes relied upon in Basu (1995) and Rotemberg
and Woodford (1992). In Basu, each producer faces a state dependent ``menu
cost'' for changing prices. This generates price rigidity over some range of
aggregate shocks such that markups become countercyclical. Rotemberg and
Woodford, uses a structure based on Rotemberg and Saloner (1986) where firms
within an oligopoly collude to keep prices above marginal cost with the
collusion being supported by the threat to revert to the competitive price
in future in response to any deviation. An exogenous increase in demand
raises the gains from undercutting the industry set price. To prevent a
breakdown of the collusion, the industry reverts to a lower price and hence
to a lower markup. In contrast to these mechanisms, in our model, all prices
are flexible and there is no collusion. The importance of markup variations
in understanding business cycle fluctuations dates back to Kalecki (1939)
and Keynes (1938). And in this paper, we add to this literature by showing
how permanent changes in the composition of public expenditure can affect
the cyclical response of the economy by changing the elasticity of aggregate
demand. Although, Bils (1989) provides a setup where the price elasticity
changes in response to demand shocks, the paper does not make a persuasive
case as to why this can occur. In contrast, in our model this link is made
explicit.

Among others, Hall (1988) and Evans (1992) have shown that empirically
productivity shocks can be correlated with variables, such as changes to
government expenditure and monetary impulses, although in principle these
innovations should be orthogonal to the technology. We show that, government
expenditure shocks can be positively correlated to productivity (as in Basu
1995), measured as the standard Solow residual, when markups are
countercyclical. As a result, the standard measures of the Solow residual
will tend to overestimate the contribution of technological shocks. It is
also shown that when markups are countercyclical if the average elasticity
is sufficiently large, then increases in public investment \textit{crowds in}
private investment. As public investment is not used in production, the
crowding in of private investment occurs because of gains in efficiency,
which increases the marginal revenue product of capital despite a fall in
the markup.

The original Gali (1994) framework was constructed to show how in the
absence of exogenous shocks, self-fulfilling revisions of expectations or
sunspots can generate cyclical fluctuations that are similar to those in
real economies. Although this property of the model is retained in this
paper, we find that in light of the more recent evidence on econometric
estimates of markups it is difficult to justify sunspot equilibria to be
reasonable mechanisms by which cyclical fluctuations occur. In particular,
when markups are procyclical, the existence of sunspots requires that, at
the steady state, the markup has to be larger than $2.$ Recent studies such
as those by Domowitz et. al (1988), Morrison (1990), and Martins et. al
(1996) typically estimate markups to be less than $1.7$.

The rest of the paper is organized as follows. Section II presents the
environment including the assumptions made with respect to the behavior of
government expenditure. Section III characterizes the equilibrium while
Section IV discusses the properties of the steady state. In Section V the
long run effects of changing the composition of public spending is
discussed. Issues involving the plausibility of sunspot equilibria are
discussed in section VI. Section VII shows the effects of changing the
composition of government expenditure on the economy when exogenous demand
and technological shocks are present. Section VIII concludes the
paper.\medskip

\section{The Model}

\subsection{Households}

We assume that there is a large number of identical consumers denoted by $%
i=1,2,...N$ whose preferences are described by

\[
E_{0}\sum_{t=0}^{\infty }\beta ^{t}\left[ u(c_{t}^{i})-v(l_{t}^{i})\right] 
\]
where $E_{0}$ is the expectation operator at time $t=0$ and $\beta \in \left[
0,1\right] $ is the discount factor. The utility $u(c_{t}^{i})$ derived by
consumer $i$ at time $t$ is given by

\[
u(c_{t}^{i})=M^{\frac{1}{1-\sigma }}\left[ \sum_{s=1}^{M}\left(
c_{st}^{i}\right) ^{\frac{\sigma -1}{\sigma }}\right] ^{\frac{\sigma }{%
\sigma -1}}\text{ } 
\]
where $\sigma >0$ and $c_{st}^{i}$ is the quantity of good $s$ consumed by $%
i $ at $t$ . This is a constant elasticity of substitution utility function
used previously in the literature on monopolistic competition (Spence 1976,
Dixit and Stiglitz 1977). The disutility from working $l_{t}^{i}$ hours is
given by

\[
v(l_{t}^{i})=\frac{\zeta }{1+\zeta }\left( l_{t}^{i}\right) ^{\frac{1+\zeta 
}{\zeta }}. 
\]

where $\zeta >0$. Normalizing each period wage to unity, denoting the total
income of consumer $i$ prior to paying taxes as $a_{t}^{i}$ and the lump-sum
tax as $T_{t}^{i}$, letting $p_{t}^{s}$ denote the price paid by consumers
for a unit of good $s$, and using $P_{t}$ as the consumer price index
corresponding to the composite consumption good purchased by the consumers
which is defined as 
\[
P_{t}=\left[ \frac{1}{M}\left( \sum_{s=1}^{M}\left( p_{t}^{s}\right)
^{1-\sigma }\right) ^{\frac{1}{1-\sigma }}\right] 
\]
the following are the consumption and labor supply functions (derived in the
appendix) of the $ith$ household:

\begin{equation}
\begin{array}{c}
c_{st}^{i}=\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{%
a_{t}^{i}-T_{t}^{i}}{P_{t}M}\right) ,\text{ }s=1,2,...M,\forall \text{ }%
i=1,2,...N \\ 
\\ 
l_{t}^{i}=P_{t}^{-\zeta },\text{ }\forall \text{ }i=1,2,...N
\end{array}
\end{equation}

\subsection{Firms}

There are $M$ firms, each producing a horizontally differentiated commodity
indexed by $j=1,2,...M$. Without loss of generality assume that $N=M$. The
technology to produce these goods is identical across firms and requires
both labor and capital with a Cobb-Douglas production function given by

\begin{equation}
y_{t}^{j}=z_{t}\left( k_{t}^{j}\right) ^{\alpha }\left( l_{t}^{j}\right)
^{1-\alpha }  \label{1}
\end{equation}
with $0<\alpha <1$ and where $y_{t}^{j}$, $k_{t}^{j},$ and $l_{t}^{j},$
denote the $jth$ firm's output, capital stock and labor input respectively. $%
z_{t}$ denotes the productivity shock and follows

\[
z_{t+1}=\phi ^{z}z_{t}+\varepsilon _{t+1}^{z} 
\]
where $\phi ^{z}<1$, measures the persistence in the shock and $\varepsilon
_{t}^{z}$ is an $i.i.d$. disturbance with mean $\bar{\varepsilon}$ and a
finite positive support $[\varepsilon ^{-},\varepsilon ^{+}]$. Capital
accumulation requires the use of all $M$ commodities and the law of motion
of capital is given by the following equation

\begin{equation}  \label{2}
k_{t+1}^{j}=(1-\delta )k_{t}^{j}+f(i_{t}^{j}),
\end{equation}
where

\begin{equation}
f(i_{t}^{j})=\left( \frac{1}{M}\right) ^{\frac{1}{\eta -1}}\left[
\sum_{h=1}^{M}\left( i_{ht}^{j}\right) ^{\frac{\eta -1}{\eta }}\right] ^{%
\frac{\eta }{\eta -1}}\text{ }  \label{3}
\end{equation}

$\eta >0$ and where $\delta \in \left[ 0,1\right] $ represents the rate of
depreciation. Thus the capital stock of each firm is increased by purchasing
goods from the other firms that are mixed together (through the same
function for all firms) to produce a composite investment good $f(i_{t}^{j})$%
. Basu (1995) uses a technology that is similar in some respects. There
output is produced using a constant returns to scale technology in labor and
intermediate goods, $Q_{t}^{j}=\left( L_{t}^{j}\right) ^{\alpha }\left(
I_{t}^{j}\right) ^{1-\alpha }$, where $I_{t}^{j}=\left(
\int_{0}^{1}I_{ki}^{1-\phi }dk\right) ^{\frac{1}{1-\phi }}$. However, $\phi $
which measures the elasticity of substitution among different goods in
production is also the same among the goods in consumption so that the price
elasticities of firms and households are identical. Consequently, in Basu
the transmission mechanism by which shocks filter through to firms is
different from the process relied upon in this model.

If we denote, as above, by $p_{t}^{h}$, $h=1,2,...M$, the price of good $h$,
the cost of the composite investment good purchased at each period (i.e. the
total investment expenditure of the firm) is given by $\nu
_{t}^{j}=\sum_{h=1}^{M}p_{t}^{h}i_{ht}^{j}$. Implicitly, this framework
requires that firms are not able to discriminate between types of buyers.
When a buyer purchases one unit of the good sold by a firm, that buyer can
be a consumer purchasing a part of his consumption composite, a firm
purchasing a part of its investment composite, or the government buying
either. We assume here that in none of these cases can the firm discriminate
and extract a different margin.

As generally followed in this literature, the firm's decision problem is
solved in two steps. In the first step, for a given level of capital
accumulated the firm solves the \textit{static} allocation problem of
choosing the optimal mix among $M$ commodities to produce the intermediate
capital goods and the number of labor hours and then in the second step,
using this solution, the \textit{intertemporal} problem of accumulating
capital is addressed. This results in the following decision rules for the
firm:

\begin{equation}
\begin{array}{c}
i_{ht}^{j}=\left( \frac{p_{t}^{h}}{\Pi _{t}}\right) ^{-\eta }\left( \frac{%
\nu _{t}^{j}}{\Pi _{t}M}\right) ,\text{ }h=1,2,...M \\ 
f(i_{t}^{j})=\frac{\nu _{t}^{j}}{\Pi _{t}}
\end{array}
\label{firmstatic}
\end{equation}
determine the optimal choice among the $M$ investment goods where 
\begin{equation}
\Pi _{t}=\left[ \left( \frac{1}{M}\right) \sum_{h=1}^{M}\left(
p_{t}^{h}\right) ^{1-\eta }\right] ^{\frac{1}{1-\eta }}
\end{equation}
is the price index of the composite investment good.

\begin{equation}
p_{t}^{j}=\mu _{t}^{j}\omega _{t}^{j}
\end{equation}
where $\mu _{t}^{j}=\frac{\xi _{t}^{j}}{\xi _{t}^{j}-1}$ is the mark-up, $%
\xi _{t}^{j}$ is the price elasticity of the $jth$ firm's final product, and

\[
\omega _{t}^{j}=\frac{1}{1-\alpha }z_{t}^{-\frac{1}{1-\alpha }}\left( \frac{%
y_{t}^{j}}{k_{t}^{j}}\right) ^{\frac{\alpha }{1-\alpha }} 
\]
is the marginal cost. The optimal capital accumulation policy function,
obtained by taking the first order condition in $k_{t+1}$ is given by

\begin{equation}
\frac{\Pi _{t}}{P_{t}}=\beta E_{t}\frac{1}{P_{t+1}}\left[ \frac{\alpha }{%
1-\alpha }z_{t+1}^{-\frac{1}{1-\alpha }}\left( \frac{y_{t+1}^{j}}{k_{t+1}^{j}%
}\right) ^{\frac{1}{1-\alpha }}+(1-\delta )\Pi _{t+1}\right]  \label{9}
\end{equation}

\subsection{The Government}

As discussed previously, we assume that the government collects lump sum
taxes from consumers for a total of $T_{t}$, with $T_{t}=%
\sum_{i=1}^{N}T_{t}^{i}$. The government spends the entire amount on
purchasing consumption and investment goods i.e., $G_{t}=T_{t}$. Once $T_{t}$
is determined, the budget allocation is carried out in a two step process.
First, the overall sectoral expenditure limits are determined. We denote
them by $\theta _{t}G_{t}$ for public investment and $(1-\theta _{t})G_{t}$
for public consumption, with $\theta _{t}\in [0,1]$. Alternatively, the
budget constraint of the government can be written as

\begin{equation}
\begin{array}{c}
\sum p_{t}^{s}c_{gt}^{s}\leq (1-\theta _{t})G_{t} \\ 
\\ 
\sum p_{t}^{h}i_{gt}^{h}\leq \theta _{t}G_{t}
\end{array}
\label{10}
\end{equation}

Two assumptions regarding the behavior of government purchases are made
next. First, $\theta _{t}=\theta $ for all $t$, and second we assume that
for any fixed amount earmarked for public consumption (public investment),
the government allocates the money among different goods exactly in the same
way as it would be done by the private sector. This would in effect occur if
the aggregator function for the government is a linear combination of $%
u(c_{t}^{i})$ and $f(i_{t}^{j})$ i.e., the government's allocation problem is

\[
\max \text{ }\omega M^{\frac{1}{1-\sigma }}\left[ \sum_{s=1}^{M}\left(
c_{st}^{i}\right) ^{\frac{\sigma -1}{\sigma }}\right] ^{\frac{\sigma }{%
\sigma -1}}+\left( 1-\omega \right) \left( \frac{1}{M}\right) ^{\frac{1}{%
\eta -1}}\left[ \sum_{h=1}^{M}\left( i_{ht}^{j}\right) ^{\frac{\eta -1}{\eta 
}}\right] ^{\frac{\eta }{\eta -1}} 
\]
subject to 
\[
\begin{array}{c}
\sum p_{t}^{s}c_{gt}^{s}\leq (1-\theta _{t})G_{t} \\ 
\\ 
\sum p_{t}^{h}i_{gt}^{h}\leq \theta _{t}G_{t}
\end{array}
\]

where $\theta _{t}$ and $G_{t}$ is given and the weight $\omega $ $\in \left[
0,1\right] $\footnote{%
In particular, note that if an overall budget constraint is only imposed,
i.e., $\sum p_{t}^{s}c_{gt}^{s}+\sum p_{t}^{h}i_{gt}^{h}\leq G_{t}$, then $%
\theta _{t}=\omega $.}. The following two equations are obtained as the
solution to the spending patterns of the government\footnote{%
Note that the elasticities of government's consumption goods and investment
goods expenditures are the same as for individual agents i.e., the
elasticities are $\sigma $ and $\eta $.}.

\begin{equation}
\begin{array}{c}
c_{gt}^{s}=\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{%
(1-\theta )G_{t}}{P_{t}}\right) ,\text{ }s=1,2,...M \\ 
\\ 
i_{gt}^{h}=\left( \frac{p_{t}^{h}}{\Pi _{t}}\right) ^{-\eta }\left( \frac{%
\theta G_{t}}{\Pi _{t}}\right) ,\text{ }h=1,2,...M
\end{array}
\label{govfoc}
\end{equation}

Finally, assume that government expenditure is stochastic in the sense that $%
g_{t}\equiv \frac{G_{t}}{Y_{t}}=(1-\phi ^{g})\bar{g}+\phi
^{g}g_{t-1}+\varepsilon _{t}^{g}$, where $\varepsilon _{t}^{g}$ has a finite
support $[g^{-},g^{+}]$ containing $\bar{g}$ and has an $i.i.d$.
distribution. This constitutes the demand shock to the system as opposed to $%
z_{t}$ which is the supply shock.\bigskip

\section{Symmetric Market Equilibrium}

The optimal decision rules derived in the previous two sections and the
overall feasibility conditions, and the price indices describe the market
equilibrium. However, before proceeding further with the characterization of
the equilibrium it is useful to discuss the aggregate demand faced by firms.
Note that the demand faced by the firm $j$ comprising of the demand by
consumers, other firms and the government is given by

\[
\begin{array}{c}
y_{t}^{j}=\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{%
a_{t}-G_{t}}{P_{t}}\right) +\left( \frac{p_{t}^{h}}{\Pi _{t}}\right) ^{-\eta
}\left( \frac{\nu _{t}}{\Pi _{t}}\right) + \\ 
\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{(1-\theta
)\Upsilon _{t}}{P_{t}}\right) +\left( \frac{p_{t}^{h}}{\Pi _{t}}\right)
^{-\eta }\left( \frac{\theta \Upsilon _{t}}{\Pi _{t}}\right)
\end{array}
\]

\begin{equation}
y_{t}^{j}=\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{%
a_{t}-\theta G_{t}}{P_{t}}\right) +\left( \frac{p_{t}^{h}}{\Pi _{t}}\right)
^{-\eta }\left( \frac{\nu _{t}+\theta G_{t}}{\Pi _{t}}\right)  \label{12}
\end{equation}

As is evident from (14), if $\theta =0$, then aggregate demand is not
affected at all by government expenditure.

Now let $\lambda _{t}^{j}$ denote the share of investment demand. Then the
price elasticity of the $jth$ good is given by

\begin{equation}  \label{13}
\xi _{t}^{j}=(1-\lambda _{t}^{j})\sigma +\lambda _{t}^{j}\eta
\end{equation}

The price elasticity $\xi _{t}^{j}$ is a weighted sum of the consumers and
investors elasticities. And therefore, the markup will depend on the weight, 
$\lambda _{t}^{j}$. Furthermore, note that if the two elasticities are the
same then $\mu _{t}=\sigma $ , a constant.

Focusing our attention to only symmetric equilibria where all firms produce
the same quantity and use the same amount of inputs. Moreover, all consumers
consume and save the same amount. Furthermore, since firms cannot
discriminate between consumers and investors price charged for any
particular commodity will be the same, regardless of its end use. As a
result $P_{t}=\Pi _{t}$. With the assumption of symmetry, all firm and
household specific indices are dropped.

The demand for labor by firms can be written as $l_{t}=\left( \frac{x_{t}}{%
z_{t}}\right) ^{\frac{1}{1-\alpha }}k_{t}$ where, $x_{t}\equiv \frac{y_{t}}{%
k_{t}}$, the output-capital ratio. Using this and equation (11) one derives
the $P_{t}$ as

\begin{equation}
P_{t}=\left( \frac{x_{t}}{z_{t}}\right) ^{-\frac{1}{\zeta (1-\alpha )}%
}k_{t}^{-\frac{1}{\zeta }}  \label{14}
\end{equation}
The real wage in the economy $w_{t}$ is then given by

\[
w_{t}=\frac{1}{P_{t}} 
\]
Replacing (16) in (8) provides a closed form expression for the
output-capital ratio

\begin{equation}  \label{15}
x_{t}=A_{x}z_{t}^{\frac{1+\zeta }{1+\alpha \zeta }}\mu _{t}^{-\frac{\zeta
(1-\alpha )}{1+\alpha \zeta }}k_{t}^{-\frac{(1-\alpha )}{1+\alpha \zeta }},%
\text{ }A_{x}=(1-\alpha )^{\frac{\zeta (1-\alpha )}{1+\alpha \zeta }}
\end{equation}
Using (17) one can subsequently derive

\begin{equation}  \label{16}
y_{t}=A_{x}z_{t}^{\frac{1+\zeta }{1+\alpha \zeta }}\mu _{t}^{\frac{-\zeta
(1-\alpha )}{1+\alpha \zeta }}k_{t}^{\frac{\alpha (1+\zeta )}{1+\alpha \zeta 
}}
\end{equation}

\begin{equation}  \label{17}
l_{t}=A_{l}z_{t}^{\frac{\zeta }{1+\alpha \zeta }}\mu _{t}^{\frac{-\zeta }{%
1+\alpha \zeta }}k_{t}^{\frac{\alpha \zeta }{1+\alpha \zeta }},\text{ }%
A_{l}=\left( 1-\alpha \right) ^{\frac{\zeta }{1+\alpha \zeta }}
\end{equation}

Note that both output and labor are functions of capital stock $k_{t}$ and $%
\lambda _{t}$. One can use (18) to find solutions for consumption, $c_{t},$
and investment, $i_{t}$. However, $\lambda _{t}$ is an endogenous variable
and needs to be solved. We next turn out attention to solving for $\lambda
_{t}$. To do so, consider equation (11). Under the assumption that $%
P_{t}=\Pi _{t}$, $\left( \ref{9}\right) $ becomes 
\begin{equation}
\rho +\delta =E_{t}\frac{1}{P_{t}}\left[ \frac{\alpha }{1-\alpha }z_{t+1}^{-%
\frac{1}{1-\alpha }}\left( \frac{y_{t+1}}{k_{t+1}}\right) ^{\frac{1}{%
1-\alpha }}\right] \qquad \text{ }\rho =\frac{1}{\beta }-1
\label{multiplier}
\end{equation}
Using the fact that at period $t$, $k_{t+1}$ is known the above can be
written as

\begin{equation}
k_{t+1}=A_{k}\left( E_{t}\left[ \left( \frac{z_{t+1}}{\mu _{t+1}}\right) ^{%
\frac{1+\zeta }{1+\alpha \zeta }}\right] \right) ^{\frac{1+\alpha \zeta }{%
1-\alpha }}\text{ \qquad }A_{k}=\left( \frac{\alpha }{\rho +\delta }\right)
^{\frac{1+\alpha \zeta }{1-\alpha }}\left( 1-\alpha \right) ^{\zeta \text{ }}
\label{18}
\end{equation}
Note that $\mu _{t+1}$ is a function of $\lambda _{t+1}$. Therefore, if $%
\lambda _{t+1}$ is perceived to increase and $\mu ^{^{\prime }}(.)\neq 0$,
then private investors will increase or decrease $k_{t+1}$. If, for example, 
$\mu ^{^{\prime }}(.)<0$, then $k_{t+1}$ will increase. However, this will
imply that $\lambda _{t}$ will also rise. The law of motion of capital is
given by $\left( \ref{2}\right) $

\begin{equation}  \label{19}
k_{t+1}=\left[ (1-\delta )+(\lambda _{t}-\theta g_{t})x_{t}\right] k_{t}
\end{equation}
Rearranging the above yields,

\[
(\lambda _{t}-\theta g_{t})x_{t}=\frac{k_{t+1}}{k_{t}}-(1-\delta ) 
\]
equivalently,

\begin{equation}
(\lambda _{t}-\theta g_{t})x_{t}=\frac{A_{k}}{k_{t}}\left( E_{t}\left[
\left( \frac{z_{t+1}}{\mu _{t+1}}\right) ^{\frac{1+\zeta }{1+\alpha \zeta }}%
\right] \right) ^{\frac{1+\alpha \zeta }{1-\alpha }}-(1-\delta )
\end{equation}

\noindent

\noindent Using the derived first order conditions a symmetric market
equilibrium for this economy can be defined as:

\textit{Given }$\{k_{0},z_{0}\}$\textit{, a sequence of \{z}$_{t}$\textit{,g}%
$_{t}$\textit{\}, and government policy described by }$\{\theta ,\tau _{t}\}$%
\textit{, asymmetric equilibrium for this economy is a sequence }$%
\{y_{t},c_{t},l_{t},k_{t+1},\lambda _{t},\mu _{t},P_{t}\}$\textit{\ such
that:}

\begin{enumerate}
\item[(1)]  $x_{t}=A_{x}z_{t}^{\frac{1+\varsigma }{1+\alpha \varsigma }}\mu
_{t}^{-\frac{\varsigma (1-\alpha )}{1+\alpha \varsigma }}k_{t}^{-\frac{%
(1-\alpha )}{1+\alpha \varsigma }}$

\item[(2)]  $y_{t}=A_{x}z_{t}^{\frac{1+\varsigma }{1+\alpha \varsigma }}\mu
_{t}^{-\frac{\varsigma (1-\alpha )}{1+\alpha \varsigma }}k_{t}^{\frac{\alpha
(1+\varsigma )}{1+\alpha \varsigma }}$

\item[(3)]  $l_{t}=A_{l}z_{t}^{\frac{\varsigma }{1+\alpha \varsigma }}\mu
_{t}^{-\frac{\varsigma }{1+\alpha \varsigma }}k_{t}^{-\frac{\alpha \varsigma 
}{1+\alpha \varsigma }}$

\item[(4)]  $P_{t}=\left( \frac{x_{t}}{z_{t}}\right) ^{-\frac{1}{\varsigma
(1-\alpha )}}k_{t}^{-\frac{1}{\varsigma }}$

\item[(5)]  $\varpi _{t}=\frac{1}{P_{t}}$

\item[(6)]  $\mu _{t}=\frac{\varepsilon _{t}}{\varepsilon _{t}-1}$

\item[(7)]  $\xi _{t}=(1-\lambda _{t})\sigma +\lambda _{t}\eta $

\item[(8)]  $\tau _{t}=\frac{\Upsilon _{t}}{y_{t}}$

\item[(9)]  $(\lambda _{t}-\theta g_{t})x_{t}=\frac{k_{t+1}}{k_{t}}%
-(1-\delta )$

\item[(10)]  $k_{t+1}=A_{k}\left( E_{t}\left[ \left( \frac{z_{t+1}}{\mu
_{t+1}}\right) ^{\frac{1+\varsigma }{1+\alpha \varsigma }}\right] \right) ^{%
\frac{1+\alpha \varsigma }{1-\alpha }}$

\item[(11)]  $g_{t}=\tau _{t}$
\end{enumerate}

\section{Steady State and Long Run Policy}

Conditions $(1)-(11)$ completely characterize the competitive equilibrium.
The only intertemporal equations are (9)-(11). Assuming that $\tau _{t}=\tau 
$ and $\bar{z}$ and $\bar{g}$ are the mean values of $z_{t}$ and $g_{t}$ and
letting $\psi \equiv \theta \bar{g}$, in a \textit{deterministic steady state%
} i.e., $z_{t}=\bar{z}$ and $g_{t}=\bar{g}$, the steady state values of
capital, share of investment, and mark-up are given by

\begin{equation}
(\bar{\lambda}-\psi )\bar{\mu}=\frac{\alpha \delta }{\rho +\delta }
\label{share}
\end{equation}

\smallskip

\begin{equation}
\bar{k}=A_{k}\bar{z}^{\frac{1+\zeta }{1-\alpha }}\left( \frac{1}{\bar{\mu}}%
\right) ^{\frac{1+\zeta }{1-\alpha }}\emph{\ }  \label{capital}
\end{equation}

\begin{equation}
\bar{\mu}=\left( \frac{(\eta -\sigma )\bar{\lambda}+\sigma }{(\eta -\sigma )%
\bar{\lambda}-(1-\sigma )}\right)  \label{markup}
\end{equation}

\noindent

Note that from $\left( \ref{markup}\right) $ it follows that when $\frac{%
\partial \mu }{\partial \lambda }$ $>0$ as long as $\sigma >\rho $. When $%
\lambda =\psi $, the $RHS$ of $\left( \ref{share}\right) $ is $0$ and
monotonically increasing if $\sigma >\rho $. Therefore, as long as $\frac{%
\alpha \delta }{\rho +\delta }<(1-\psi )\left( \frac{\eta }{\eta -1}\right) $
there will exist some $0\leq \lambda <1$ for which $\left( \ref{share}%
\right) $ is satisfied and a steady state exists. In particular, given the
monotonicity of $(\bar{\lambda}-\psi )\bar{\mu}$ the steady state will also
be unique. However, if $\sigma <\rho $ then $\frac{\partial \mu }{\partial
\lambda }$ $<0$ and in which case multiple steady states can exist (Gali,
1994).

\smallskip From $\left( \ref{capital}\right) $ it follows that when $\bar{\mu%
}$ is high, $\bar{k}$ is low. Since steady state output is given by 
\[
\bar{y}=A_{x}\bar{z}^{\tfrac{1+\zeta }{1+\alpha \zeta }}\bar{\mu}^{\tfrac{%
-\zeta (1-\alpha )}{1+\alpha \zeta }}\bar{k}^{\tfrac{\alpha (1+\zeta )}{%
1+\alpha \zeta }} 
\]
\noindent a fall in the steady state markup unambiguously increases steady
state output. Similarly, when the markup falls both employment, 
\[
\bar{l}=A_{l}\bar{z}^{\tfrac{\zeta }{1+\alpha \zeta }}\bar{\mu}^{\tfrac{%
-\zeta }{1+\alpha \zeta }}\bar{k}^{\tfrac{\alpha \zeta }{1+\alpha \zeta }} 
\]
and labor productivity, 
\[
\frac{\bar{y}}{\bar{l}}=const%
%TCIMACRO{
%\TeXButton{Times}{\times %
%}}%
%BeginExpansion
\times %
%
%EndExpansion
\bar{z}^{\tfrac{1}{1-\alpha }}\mathit{\ }\bar{\mu}^{-\tfrac{\alpha }{%
1-\alpha }} 
\]
\noindent also increase.

Recall that $y_{t}=z_{t}k_{t}^{\alpha }l_{t}^{1-\alpha }$ or equivalently, $%
\ln y_{t}=\ln z_{t}+\alpha \ln k_{t}+\left( 1-\alpha \right) \ln l_{t}$. One
method of computing total factor productivity, $TFP$, is to define $%
TFP_{t}=\ln y_{t}-\alpha \ln k_{t}-\left( 1-\alpha \right) \ln l_{t}$. In
particular, this formulation has been used to derive the contribution of
productivity shock $z_{t}$ in growth and cyclical variability. In computing
the total factor productivity, following Solow (1957), researchers have used
the share of labor income in national income to approximate $(1-\alpha )$.
In the presence of perfect competition the share of labor income
approximates $(1-\alpha )$ quite well. However, under imperfect competition
i.e., $\mu >1$, the share of labor income is $\tfrac{1-\alpha }{\mu }$ and
that of capital $\tfrac{\alpha }{\mu }$. If these shares are used to compute
productivity then

\[
\frac{\partial TFP}{\partial \mu }=\frac{\partial \ln y}{\partial \mu }%
-\left( \frac{\alpha }{\mu }\right) \frac{\partial \ln k}{\partial \mu }%
-\left( \frac{1-\alpha }{\mu }\right) \frac{\partial \ln l}{\partial \mu } 
\]
It turns out that

\begin{equation}
\frac{\partial TFP}{\partial \mu }=-\left( \zeta (1-\alpha )^{2}+\alpha
(1+\zeta )^{2}\right) \left( 1-\frac{1}{\mu }\right)  \label{tfp}
\end{equation}

\noindent For $\mu >1$, $\frac{\partial TFP}{\partial \mu }<0$ rather than $%
0 $ if $\alpha $, the true labor elasticity, was used. Consequently, if the
steady state markup increases, not only does output increase as shown before
but also total factor productivity. This, however, should not be surprising.
The presence of the markup causes the equilibrium to be suboptimal in the
first place so that when the markup falls there are efficiency gains
reflected in the increase in total factor productivity. But, more
importantly, it also implies that factors that can change the markup, such
as changes in government expenditure (of a permanent kind for the steady
state analysis) will be correlated with the Solow residual, i.e., the
measured factor productivity. In particular, since $\mu =\left( 1-\frac{1}{%
\xi }\right) ^{-1}$, where $\xi =(\eta -\sigma )\lambda +\sigma $, if $%
\lambda $ increases and $\eta -\sigma >0$, then $\mu $ falls as a higher
elasticity implies a lower markup as predicted by standard microeconomic
theory. If $\eta -\sigma <0$ the opposite holds true. We summarize these
findings in the following proposition.

\vspace{0.25in}

\begin{proposition}
\textit{A decrease in the steady state markup is associated with higher
steady state levels of capital stock, employment and output and also higher
labor and total factor productivity. Moreover, if }$\left( \eta -\sigma
\right) $\textit{\ is positive(negative) then the share of investment in
output is positively(negatively) correlated with total factor productivity
in the long run.}\bigskip
\end{proposition}

\subsection{Public Expenditure Policy --- The Long Run Effects}

Proposition 1 makes it clear that changes in the equilibrium markup can have
important consequences for the economy in general. In this section we look
at a particular way of changing the markup, namely by altering the share of
public investment, $\psi $. To see how government policies reflected in the
composition of public expenditure can affect the markup of producers
consider the expression for the weighted elasticity $\xi _{t}=(1-\lambda
_{t})\sigma +\lambda _{t}\eta $. Denote by $\lambda _{t}^{p}$, the share of
investment in total private spending, then $\lambda _{t}=(1-g_{t})\lambda
_{t}^{p}+\theta _{t}g_{t}$ and

\[
\xi _{t}=\left( \eta -\sigma \right) \left[ (1-g_{t})\lambda _{t}^{p}+\theta
_{t}g_{t})\right] +\sigma 
\]

\noindent Government expenditure, $g_{t}$, by itself can only have a
second-order effect on the elasticity since if $\theta _{t}=0$ then $\xi
_{t}=\left( \eta -\sigma \right) (1-g_{t})\lambda _{t}^{p}+\sigma $ and $%
g_{t}$ can have an effect only if $\lambda _{t}^{p}$ changes, i.e., if
changes in government spending affects the way private agents allocate their
income between consumption and investment. In this model since investment is
determined by firms, who care only for the overall breakdown between
investment and consumption, and not who the end-users are ( by assumption
all end-users for the same functional purposes have the same elasticity)
changes in $g_{t}$ will be completely offset by changes in $\lambda _{t}^{p}$
such that $(1-g_{t})\lambda _{t}^{p}$ remains constant. If in contrast,
households decided on the level of investment then $\lambda _{t}^{p}$ would
probably not move enough to completely offset the change in $g_{t}$ such
that $(1-g_{t})\lambda _{t}^{p}$ would change leading to a change in the
elasticity. In particular it would depend on how government expenditure
changes both disposable income of households and the interest rate in the
economy and the reaction of private investment to such changes. However, if $%
\theta _{t}\neq 0$ then changes in both $\theta _{t}$ and $g_{t}$ matter in
equilibrium. When $g_{t}$ or $\theta _{t}$ is altered, the overall
composition of output changes causing the elasticity to change which in turn
leads to a change in the markup.

In the steady state, $(\bar{\lambda}-\psi )\bar{\mu}=\frac{\alpha \delta }{%
\rho +\delta }$ and $\bar{\mu}=\left( \frac{(\eta -\sigma )\bar{\lambda}%
+\sigma }{(\eta -\sigma )\bar{\lambda}-(1-\sigma )}\right) $. Substituting
the second equation in the first we have $(\lambda -\psi )\left( \frac{(\eta
-\sigma )\lambda +\sigma }{(\eta -\sigma )\lambda -(1-\sigma )}\right) -%
\frac{\alpha \delta }{\rho +\delta }=0$. Since $\frac{(\eta -\sigma )\bar{%
\lambda}+\sigma }{(\eta -\sigma )\bar{\lambda}-(1-\sigma )}\neq 0$ for $%
0\leq \lambda \leq 1$ and $\eta \geq 0$, the steady state relationship
between $\lambda $ and $\psi $ can be expressed as $\lambda =f(\psi )$.

\begin{lemma}
\textit{If }$\eta <\sigma $\textit{\ then }$0<\frac{df}{d\psi }<1$\textit{.
Otherwise, }$\frac{df}{d\psi }>1$\textit{\ if }$\xi (\xi -1)>(\lambda -\psi
)(\eta -\sigma )$\textit{\ and }$\frac{df}{d\psi }<0$\textit{\ if }$\xi (\xi
-1)<(\lambda -\psi )(\eta -\sigma )$\textit{.}
\end{lemma}

%TCIMACRO{
%\TeXButton{Proof}{\proof %
%}}%
%BeginExpansion
\proof %
%
%EndExpansion
Follows directly from differentiating $(f(\psi )-\psi )\left( \frac{(\eta
-\sigma )f(\psi )+\sigma }{(\eta -\sigma )f(\psi )-(1-\sigma )}\right) -%
\frac{\alpha \delta }{\rho +\delta }=0$.%
%TCIMACRO{
%\TeXButton{End Proof}{\endproof %
%}}%
%BeginExpansion
\endproof %
%
%EndExpansion

\noindent From the above lemma it follows that when $\eta <\sigma $, if
public investment increases, although share of total investment goes up --- $%
\frac{d\lambda }{d\psi }>0$ --- the increase is less than the amount by
which public investment is raised --- $\frac{d\lambda }{d\psi }<1$. This
implies that the share of private investment must fall. Since $\lambda $
increases when $\psi $ is raised and $\eta -\sigma <0$, it follows from $\xi
=(\eta -\sigma )\lambda +\sigma $ and $\mu =\left( 1-\frac{1}{\xi }\right)
^{-1}$ that the markup also increases. With $\mu $ being raised steady state
output falls (Proposition 1) so that not only does the share of private
investment decrease but also its level. Permanent increases in public
investment, in this case, \textit{crowds out} private investment. However,
if $\eta -\sigma >0$ and the price elasticity of aggregate demand is
sufficiently high, i.e., $\xi (\xi -1)>(\lambda -\psi )(\eta -\sigma )$, not
only does the share of total investment increase but also induces an
increase in the share of private investment --- $\frac{d\lambda }{d\psi }>1$%
. Given that $\eta -\sigma >0$, the steady state markup falls and output
also increases (again due to Proposition 1) so that the level of private
investment also increases. In contrast to the previous case, public
investment \textit{crowds in} private investment. Since increasing $\theta $
or $g$ while keeping the other unchanged in equivalent to increasing $\psi $
we have the following proposition.

\begin{proposition}
\textit{If }$\theta $\textit{\ is increased keeping }$g$\textit{\ constant
or }$g$\textit{\ is increased keeping }$\theta $\textit{\ constant and if }$%
\eta -\sigma >0$\textit{\ and }$\xi (\xi -1)>(\lambda -\psi )(\eta -\sigma )$%
\textit{\ then the steady state levels of output, private investment, and
employment are increased along with labor productivity and total factor
productivity. The opposite occurs if }$\eta -\sigma <0$\textit{.}
\end{proposition}

%TCIMACRO{
%\TeXButton{Proof}{\proof %
%}}%
%BeginExpansion
\proof %
%
%EndExpansion
Follows from proposition 1 and lemma 1.%
%TCIMACRO{
%\TeXButton{End Proof}{\endproof %
%}}%
%BeginExpansion
\endproof %
%
%EndExpansion

The long run effects of changing government expenditure and its composition
depend on whether investor elasticity is larger than consumer elasticity and
the weighted average of the two elasticities --- the elasticity of aggregate
demand. As we argue later in the paper, it is more likely that $\eta -\sigma
>0$ for the post war U.S. and other OECD economies. In which case,
increasing the share of investment in government expenditure does in fact
increase output, employment and productivity. The same effects are also
obtained if the share of government spending is increased keeping its
composition unchanged. To ascertain the short run effects we need to compute
the temporal equilibria of the economy. However, given the nature of the
environment closed form solutions cannot be obtained. To get around the
problem, we construct a linear approximation to the equilibria and study the
short run impact of government spending using the approximate solution.

\section{Sunspot Equilibria}

In a recent paper, Gali (1994) used this environment to show how equilibria
that are driven by self-fulfilling revisions of expectations can generate
time series with properties that are similar to those observed in U.S.
postwar business cycles. The belief that sunspot equilibria can generate
empirically reasonable time series properties have been shared by others
such as Farmer and Guo (1994) and Woodford (1991) among others. However,
unlike in Gali, the other models rely on some form of increasing returns or
externalities to deliver the sunspot equilibria. Although Gali's approach is
appealing since it eliminates the need to rely on exogenous shocks to
generate cycles, we find that for the U.S. postwar economy it is not a
plausible mechanism to use in studying cyclical fluctuations.

\subsection{Existence of sunspots}

To see how sunspot equilibria may be generated in this economy and why they
are not plausible we turn our attention to studying the model's cyclical
behavior. First, equilibrium conditions $(1),(6),(9),$ and $(10)$ are
linearized around the steady state, implicitly given by $\left( \ref{share}%
\right) $ and $\left( \ref{capital}\right) $ to obtain approximate closed
form solutions. Let $\bar{x}=\{\bar{\lambda},\bar{k},\bar{z}\}$ denote the
steady state of the system. The linearized equations are

\begin{equation}
\hat{x}_{t}=a_{1}\hat{z}_{t}-a_{2}\hat{\mu}_{t}-a_{3}\hat{k}_{t}  \label{eq1}
\end{equation}
\begin{equation}
\hat{\mu}_{t}=b_{1}\hat{\lambda}_{t}  \label{eq2}
\end{equation}
\begin{equation}
\hat{\lambda}_{t}=\frac{1}{c_{1}}\hat{k}_{t+1}-\frac{1}{c_{1}}\hat{k}_{t}-%
\frac{\delta }{c_{1}}\hat{x}_{t}+c_{2}\hat{g}_{t}  \label{eq3}
\end{equation}
\begin{equation}
\hat{k}_{t+1}=d_{1}E_{t}\left[ \hat{z}_{t+1}-\hat{\mu}_{t+1}\right]
\label{eq4}
\end{equation}

\noindent where the coefficients are the partial derivatives evaluated at
their steady state values (\textit{see the appendix for details}). By using $%
\left( \ref{eq1}\right) $ , $\left( \ref{eq2}\right) $ , $\left( \ref{eq3}%
\right) ,$ $\left( \ref{eq4}\right) $ becomes

\begin{equation}
\hat{k}_{t+1}=E_{t}\left[ e_{1}\hat{z}_{t+1}+e_{2}\hat{k}_{t+2}+e_{3}\hat{g}%
_{t+1}\right]  \label{capdiff}
\end{equation}

Now suppose $\mid e_{2}\mid >1$. Consider first the perfect foresight
equilibrium i.e., $\hat{z}_{t+1}=0$ and $\hat{g}_{t}=0$ $\forall
t=0,...\infty $. The first equation in this system is $\hat{k}_{1}=e_{2}\hat{%
k}_{2}$. Consequently, there is no restriction on $k_{1}$ imposed by the
initial capital stock $k_{0}$ i.e., the system of equations, $\hat{k}%
_{t+1}=e_{2}\hat{k}_{t+2}$ $\forall t=0,...\infty $, are unrestricted by the
initial stock of capital. There is an unique steady state given by $\hat{k}%
_{t+1}=0$ $\forall t=0,...\infty $. Moreover, the perfect foresight
equilibrium is stable. Any sequence of $\left\{ \hat{k}_{t+1}\right\} $
starting from an arbitrary $k_{1}$ will converge to the steady state $\hat{k}%
_{t+1}=0$. Since, $k_{0}$ does not restrict the choice of $k_{1}$, there is
an indeterminate number of perfect foresight equilibria. However, of these
equilibria there is one that is stationary namely, $\left\{ \hat{k}%
_{t+1}\right\} =0$ $\forall t=0,...\infty $ which is the steady state
equilibrium. But this is not the only stationary equilibrium. Around a
sufficiently small neighborhood of the steady state there may be more than
one such equilibria (Woodford 1991, Gali 1994). To see why the steady state
is not the only stationary equilibrium note that $E_{t}\left[ \hat{k}%
_{t+1}-e_{2}\hat{k}_{t+2}\right] =0$ is the version of $\left( \ref{capdiff}%
\right) $ when there is no intrinsic uncertainty ($\varepsilon _{t}^{z}=0$
and $\varepsilon _{t}^{g}=0$) which is the optimal capital accumulation rule
followed by investors. Lagged one period, the rule is given by $%
E_{t-1}\left( \hat{k}_{t+1}-\frac{1}{e_{2}}\hat{k}_{t}\right) =0$. Now
suppose investors' believe that investment follows $\hat{k}_{t+1}=\frac{1}{%
e_{2}}\hat{k}_{t}+s_{t}$, where the random variable $s_{t}$ is a realization
of an $i.i.d.$ probability distribution with $E_{t-1}s_{t}=0$ and described
over a closed support $\left[ a,b\right] $ containing the steady state.
Although this rule still implies that $E_{t-1}\left( \hat{k}_{t+1}-\frac{1}{%
e_{2}}\hat{k}_{t}\right) =0$ which is consistent with the optimal policy $%
\left( \ref{capdiff}\right) $, the extrinsic uncertainty $s_{t}$ can
generate investment paths that depend solely on it. If $\mid e_{2}\mid <1$
then the solution to $\hat{k}_{t+1}=\frac{1}{e_{2}}\hat{k}_{t}+s_{t}$ is
given by $\hat{k}_{t+1}=E_{t}\sum_{j=0}\left( e_{2}\right) ^{j}s_{t+j}$ $%
\forall t=1,...\infty $. As $E_{t}s_{t+j}=0$, $\hat{k}_{t+1}=0$ $\forall
t=1,...\infty $ is the unique stationary solution. But if $\mid e_{2}\mid >1$%
, then the solution is 
\begin{equation}
\hat{k}_{t+1}=\sum_{j=0}\left( \frac{1}{e_{2}}\right) ^{j}s_{t-j}\text{ }%
\forall t=0,...\infty
\end{equation}
with $\hat{k}_{1}=e_{2}s_{0}$ and the equilibrium path of capital stock is
determined entirely by current and past realizations of the sunspot variable 
$s_{t}$\footnote{%
The reason why such sunspot equilibria can exist is identical to that in
Woodford (1991). Without loss of generality assume that in period $t$ the
realization of the sunspot is a positive $s_{t}$. Furthermore, suppose all
investors believe that the choice of $\hat{k}_{t+2}$ will follow the rule $%
\hat{k}_{t+2}=\sum_{j=0}\left( \frac{1}{e_{2}}\right) ^{j}s_{t-j+1}$. Then
the positive realization of $s_{t}$ increases the expected value of $E_{t}%
\hat{k}_{t+2}$. However, this also implies that investment in $t+1$ will
need to be higher such that $E_{t}\hat{y}_{t+1}$ will be increased. In order
to support a higher equilibrium output in $t+1$ investors will then, as an
optimal response, increase their desired capital stock in period $t$ i.e., $%
\hat{k}_{t+1}$. Consequently, the seemingly inconsistent behavior of $\hat{k}%
_{t+1}=\frac{1}{e_{2}}\hat{k}_{t}+s_{t}$ instead of $\hat{k}_{t+1}=\frac{1}{%
e_{2}}\hat{k}_{t}$ -- as determined by the fundamentals of the environment
-- is justified.}. However, if agents do not believe that the sunspot
matters for the capital stock i.e., $\hat{k}_{t+1}\neq \frac{1}{e_{2}}\hat{k}%
_{t}+s_{t}$ but $\hat{k}_{t+1}=\frac{1}{e_{2}}\hat{k}_{t}$, then the only
stationary equilibrium possible is the one where $\{\hat{k}_{t+1}\}=0$ $%
\forall t=0,...\infty $.

Since the existence of sunspots depend on the parameter $e_{2}$ we next
provide its characterization.

\begin{lemma}
$\mid e_{2}\mid <1$\textit{\ if and only if }$1-\frac{\delta \left( 1-\alpha
\right) }{1+\zeta }\left[ 1+\left( \frac{\bar{\lambda}}{\bar{\lambda}-\bar{%
\psi}}\right) \frac{1}{\bar{\varepsilon}_{\mu }}\right] >1$\footnote{%
This lemma is identical to the result in Gali (1994) except that it is
modified to include $\psi $.}\textit{.}
\end{lemma}

%TCIMACRO{
%\TeXButton{Proof}{\proof %
%} }%
%BeginExpansion
\proof %
%
%EndExpansion
See appendix.%
%TCIMACRO{
%\TeXButton{End Proof}{\endproof %
%}}%
%BeginExpansion
\endproof %
%
%EndExpansion

While the theoretical possibility of sunspots existing have been established
it still remains to be shown whether empirically such equilibria should be
considered seriously. To this end, in the next section we use a suitably
parameterized economy to provide the answer.

\subsection{The Likelihood of Sunspots}

In this section, parameters of the model are calibrated to be compatible
with the post-war U.S. economy using long-run and micro-level data and
econometric studies. Following, Kydland and Prescott (1990) set $\bar{\lambda%
}=0.21$, the share of investment in aggregate output. Depending on whether
investment in defense equipment is included $\bar{\psi}=0.023$ or $0.03$.
The depreciation rate is set at $\delta =0.016$ per quarter such that the
steady state capital-output ratio is around $10$. This depreciation rate is
different from $0.025$ which is normally used approximating an annual rate
of $10$ percent. Using the average risk-free interest rate per quarter in
the postwar U.S. economy $\rho =0.01$ or $\beta =0.99$. Typically,
equilibrium business cycle models assume extremely large labor supply
elasticities. However, estimated elasticities have been in general very low
(close to zero for adult white males) and have rarely been over 2
(Killingsworth and Heckman 1986). In this model $\varsigma =1$. This leaves
behind unresolved the values for $\mu ,\alpha ,\eta $ and $\sigma $. Note
that the steady state condition $(\bar{\lambda}-\psi )\mu =\frac{\alpha
\delta }{\rho +\delta }$ implies that for any value of $\mu $ there is an
unique $\alpha $ consistent with it.

Using industry data, Hall (1987, 1989) estimated $\mu $ of over $1.8$ for
all the one-digit industries. Subsequent work by Domowitz, Hubbard and
Petersen (1988) and Morrison (1990) have shown that estimates of markups
that range from $1.2$ to $1.7$ are more reasonable. This range of values for 
$\mu $ seem more reasonable also on the ground that for the postwar U.S.
economy profit margins have been rather small. More recently, Martins,
Scarpetta and Pilat (1996) estimated the average markup for the U.S. economy
to be around $1.15$. On sectoral basis, they found that markups for most
industries were between $1.05$ and $1.54$. However, Gali (1994) assumes a
steady state value of $\mu \in [2.0,3.03]$ using evidence from Hall (1988).
These two ranges i.e., $[2.0,3.03]$ and $[1.15,1.7]$ have significant
behavioral differences. To see these differences in a more transparent
manner consider the data in Table 1. It describes the values of $\{\eta
,\sigma \}$ consistent with a given $\mu $. Table 1 also lists the
corresponding sign of $e_{2}$ and whether it is greater or less than $1$.
Recall that if $e_{2}$ is greater than $1$, then sunspots are possible.

\vspace{0.25in}

\begin{center}
\[
\stackunder{
\begin{array}{c}
\begin{tabular}{ccccc|c}
\hline\hline
$\mu $ & $\eta $ & $\sigma $ & $\alpha $ & $e_{2}$ & $sunspot$ \\ 
\hline\hline
$1.2$ & $0.00-5.15$ & $7.50-6.21$ & $0.32$ & $+$ & $yes$ \\ 
$1.2$ & $5.18-5.60$ & $6.20-6.10$ & $0.32$ & $-$ & $yes$ \\ 
$1.2$ & $5.61-5.98$ & $6.09-6.00$ & $0.32$ & $-$ & $no$ \\ 
$1.2$ & $6.00-30.0$ & $5.99-0.00$ & $0.32$ & $+$ & $no$ \\ \hline
$1.6$ & $0.00-2.58$ & $3.33-2.7$ & $0.47$ & $+$ & $yes$ \\ 
$1.6$ & $2.59-2.61$ & $2.69-2.68$ & $0.47$ & $-$ & $yes$ \\ 
$1.6$ & $2.62-2.65$ & $2.68-2.67$ & $0.47$ & $-$ & $no$ \\ 
$1.6$ & $2.66-13.33$ & $2.66-0.00$ & $0.47$ & $+$ & $no$ \\ \hline
$2.8$ & $0.00-1.53$ & $1.94-1.56$ & $0.74$ & $+$ & $yes$ \\ 
$2.8$ & $1.53-155$ & $\simeq 1.55$ & $0.74$ & $-$ & $yes$ \\ 
$2.8$ & $1.56-5.05$ & $1.54-0.68$ & $0.74$ & $+$ & $no$ \\ 
$2.8$ & $5.06-7.78$ & $0.67-0.00$ & $0.74$ & $+$ & $yes$ \\ \hline\hline
\end{tabular}
\end{array}
}{
\begin{array}{c}
\text{\textrm{Table\ 1.\ Sunspot\ Zones\ and\ Markups}}
\end{array}
} 
\]

\vspace{0.25in}
\end{center}

For every value of $\mu $ considered, for some region of $\left\{ \eta
,\sigma \right\} $, $\mid e_{2}\mid >1$ such that it is possible that
sunspot equilibria may exist for a wide range of markups. Moreover, as can
be seen in Table 1, the intervals of $\left\{ \eta ,\sigma \right\} $ for
which sunspots exist are also quite large. Consequently, the existence of
such equilibria in the environment discussed in this paper does not depend
on very specific values of parameters. Note, that for values of $\{\eta
,\sigma \}$ close to one another $\mid e_{2}\mid <1$ and for sufficiently
apart values $\mid e_{2}\mid >1$. This implies, as described by Gali, for
sunspots to exist the two elasticities need to be sufficiently far apart. If 
$\eta $ and $\sigma $ are close to one another $\bar{\varepsilon}_{\mu }$ is
extremely small such that $\frac{1}{\bar{\varepsilon}_{\mu }}$ is very large
which in turn implies that $\mid e_{2}\mid =\left\{ 1-\dfrac{\delta \left(
1-\alpha \right) }{1+\zeta }\left[ 1+\left( \dfrac{\bar{\lambda}}{\bar{%
\lambda}-\psi }\right) \dfrac{1}{\bar{\varepsilon}_{\mu }}\right] \right\}
^{-1}$ will be small and sunspot equilibria cannot exist.

However, when $\mu $ is small e.g., $1.2$ or $1.5$, sunspots exist only in
the region $\eta <\sigma $. For larger values of the markup e.g., for values
considered by Gali i.e., $\mu \in [2,3.03]$, $\mid e_{2}\mid >1$ in regions
where $\eta <\sigma $ as well as $\eta >\sigma $. The difference in the
behavior of the economy for larger values of the steady state $\mu $ turns
out to be quite significant. In the region $\eta >\sigma $ , the elasticity
of the markup to $\lambda $, $\bar{\varepsilon}_{\mu }=\dfrac{(\sigma -\rho )%
\bar{\lambda}}{\bar{\xi}\left( \bar{\xi}-1\right) }$ is negative. At the
empirical level, the U.S. economy displays two strong regularities

\begin{enumerate}
\item 
\begin{enumerate}
\item  a strongly \emph{procyclical} investment share\quotesinglbase

\item  a strongly \emph{countercyclical} markup\footnote{%
See for example, Rotemberg and Woodford 1990, Bils, 1987.}.
\end{enumerate}
\end{enumerate}

If these two features are to hold simultaneously then when output increases, 
$\lambda $ should increase and $\mu $ should fall. This implies that $%
\varepsilon _{\mu }<0$. Therefore, if the model is to be consistent with the
regularities displayed by the U.S. economy it follows that $\eta >\sigma $
since $\bar{\varepsilon}_{\mu }=\frac{(\sigma -\eta )\bar{\lambda}}{\bar{\xi}%
\left( \bar{\xi}-1\right) }$. While for parameter values consistent with a
steady state markup $\mu >2.0$, sunspots can exist ($\mid e_{2}\mid >1$)
when $\eta >\sigma $, in the range $\mu \in [1,2)$ this is not true. In fact
for $\mu \leq 2$, sunspots fail to exist for $\eta >\sigma $.

\begin{proposition}
\textit{If }$\eta >0$\textit{, }$\sigma >0$\textit{, and }$\eta >\sigma $%
\textit{, then at the steady state }$\mu >2$ \textit{for sunspot equilibria
to exist.\noindent }
\end{proposition}

%TCIMACRO{
%\TeXButton{Proof}{\proof %
%} }%
%BeginExpansion
\proof %
%
%EndExpansion
See appendix.%
%TCIMACRO{
%\TeXButton{End Proof}{\endproof %
%}}%
%BeginExpansion
\endproof %
%
%EndExpansion

Note that in the proof of Proposition 3, we did not use any specific
parameter values except for restricting the parameters to be positive and
the markup to be greater than $1$. Consequently, the proposition holds quite
generally and for this class of economies if the share of investment is to
be procyclical and the optimal markup countercyclical, at the steady state
the markup has to be over $100$ percent. Given the more recent studies
discussed previously, it is unlikely that the long-run markup for the U.S.
economy can be that large. In fact, the Martins, Scarpetta and Pilat (1996)
study shows that for most OECD countries markups are substantially lower
than $2$.

Simplifying the parameters $e_{1}$ and $e_{3}$, $\left( \ref{capdiff}\right) 
$ becomes 
\[
\hat{k}_{t+1}=E_{t}\left[ d_{1}(1-e_{2})\hat{z}_{t+1}+e_{2}\hat{k}%
_{t+2}+\left( \frac{\rho +\delta }{\alpha }\right) \psi ^{2}e_{2}\hat{g}%
_{t+1}\right] 
\]

\noindent This implies that if $e_{2}>1$ then $\hat{k}_{t+1}$ is negatively
correlated with technological shocks as $d_{1}(1-e_{2})<0$. In turn, $%
e_{2}>1 $ also implies a negative correlation for investment which we find
to be counter intuitive. Furthermore, if $\mu \in [2,3.03]$, the values used
by Gali, $\alpha \in [0.53,0.8]$ and the share of wages in national income
lies between $[0.24,0.07]$. This value is significantly different from the
observed share of around $0.64$ (Prescott, 1986). Consequently, given the
preponderance of both empirical evidence and the implied inconsistencies
with empirical regularities it seems unlikely that the version of this model
with sunspots will be a reasonable approximation of the real economy.

\section{Equilibrium with Exogenous Fluctuations}

In keeping with the restrictions derived in the previous section we follow
studies that have used a similar framework such as Rotemberg and Woodford
(1992) and Basu (1995) to restrict steady state markups to the interval $%
[1.2,1.7]$ that matches the more recent econometric estimates. As shown in
proposition 3, this restriction forces $\left| e_{2}\right| <1$, so that
sunspot equilibria are ruled out. Moreover, in the rest of this section
unless otherwise noted it will be assumed that $\eta >\sigma $ so that $%
\varepsilon _{\mu }<0$. Therefore, in order to generate cyclical
fluctuations the model will depend exclusively on the two exogenous shocks : 
$(i)$ the technological shock $\hat{z}_{t}$ and $(ii)$ the demand shock $%
\hat{g}_{t}$. Consider first the perfect foresight equilibrium i.e., $\hat{z}%
_{t}=0$ and $\hat{g}_{t}=0,$ $\forall t=0,...\infty $. The first equation in
this system is $\hat{k}_{1}=e_{2}\hat{k}_{2}$. Consequently, there is no
restriction on $k_{1}$ imposed by the initial capital stock $k_{0}$. The
unique steady state of the system is $\hat{k}=0$ and since $\mid e_{2}\mid
<1 $ it is unstable\textit{, } such that the only rational expectations
equilibrium is the one where $\{\hat{k}_{t+1}\}=0$ $\forall t=0,...\infty $.
Therefore, given any initial capital stock $k_{0}$, there is an unique
perfect foresight equilibrium which is also stationary. The instability of
the steady state allows the stochastic version of the difference equation $%
\left( \ref{capdiff}\right) $ to be forward stable such that there is an
unique rational expectations equilibrium given by

\begin{equation}
\hat{k}_{t+1}=e_{1}E_{t}\left[ \sum_{j=0}e_{2}^{j}\hat{z}_{t+j+1}\right]
+e_{3}E_{t}\left[ \sum_{j=0}e_{2}^{j}\hat{g}_{t+j+1}\right]
\end{equation}

\noindent Since $\hat{z}_{t+1}=\phi ^{z}\hat{z}_{t}+\hat{\varepsilon}%
_{t+1}^{z}$ and $E_{t}\left[ \hat{\varepsilon}_{t+1}\right] =0$ and $\hat{g}%
_{t+1}=\phi ^{g}\hat{g}_{t}+\hat{\varepsilon}_{t+1}^{g}$ and $E_{t}\left[ 
\hat{\varepsilon}_{t+1}^{g}\right] =0$ we have,

\begin{equation}
\hat{k}_{t+1}=\phi ^{z}e_{1}E_{t}\left[ \sum_{j=0}\left( \phi
^{z}e_{2}\right) ^{j}\hat{z}_{t}\right] +\phi ^{g}e_{3}E_{t}\left[
\sum_{j=0}\left( \phi ^{g}e_{2}\right) ^{j}\hat{g}_{t}\right]
\end{equation}
which can be expressed as

\begin{equation}
\hat{k}_{t+1}=\frac{\phi ^{z}e_{1}}{1-\phi ^{z}e_{2}}\hat{z}_{t}+\frac{\phi
^{g}e_{3}}{1-\phi ^{g}e_{2}}\hat{g}_{t}
\end{equation}

\noindent This is the capital accumulation law of motion generated by the
equilibrium conditions of the economy and using the linearized system of
equations $\left( \ref{eq1}\right) -\left( \ref{eql4}\right) $ all other
variables can be expressed in terms of $\hat{z}_{t}$ and $\hat{g}_{t}$. In
particular\footnote{%
The appendix lists the complete expressions for the different coefficients.},

\begin{equation}
\hat{\lambda}_{t}=\lambda ^{z}\hat{z}_{t}-\lambda ^{k}\hat{k}_{t}+\lambda
^{g}\hat{g}_{t}
\end{equation}

\begin{equation}
\hat{\mu}_{t}=b_{1}\lambda ^{z}\hat{z}_{t}-b_{1}\lambda ^{k}\hat{k}%
_{t}+b_{1}\lambda ^{g}\hat{g}_{t}
\end{equation}

\begin{equation}
\hat{y}_{t}=y^{z}\hat{z}_{t}+y^{k}\hat{k}_{t}+y^{g}\hat{g}_{t}
\label{eqlout}
\end{equation}

\begin{equation}
\hat{l}_{t}=l^{z}\hat{z}_{t}+l^{g}\hat{g}_{t}+l^{k}\hat{k}_{t}
\end{equation}

We investigate the effects of changing the long run share of investment
goods in overall government purchases, $\theta $, on the cyclical behavior
of aggregate output $y_{t}$, investment $i_{t}$, consumption $c_{t}$ and
labor $l_{t}$ keeping the long run level of overall government expenditure $%
\bar{g}$ constant. Using the expression for equilibrium output $(\ref{eqlout}%
)$\noindent derived in the appendix and ignoring the variations in the stock
of capital which are generally small,

\[
var(\hat{y})=\left( y_{1}^{a}\right) ^{2}\sigma _{z}^{2}+\left(
y_{3}^{a}\right) ^{2}\sigma _{g}^{2} 
\]

\noindent since $Cov(\hat{z},\hat{k})=0,$ $Cov(\hat{z},\hat{g})=0$, $Cov(%
\hat{g},\hat{k})=0$. Consequently, the way output responds to the exogenous
shocks, $\hat{z}_{t}$ and $\hat{g}_{t}$, depends on the parameters $y^{z}$
and $y^{g}$. These two parameters are affected by $\theta $. As a result,
when the long-run composition of government outlay changes, it affects the
way the economy reacts to both supply and demand shocks. The cyclical
variability of output can therefore, be accentuated or reduced by the choice
of $\theta $.

As noted in the discussion on the long run effects of changes in the share
of public investment, when $\theta $ changes the price elasticity of the
commodity also changes at the steady state also changes since 
\[
\bar{\xi}=\left( \eta -\sigma \right) \left[ (1-\bar{g})\bar{\lambda}%
^{p}+\theta \bar{g}\right] +\sigma 
\]
In particular, when $\left( \eta -\sigma \right) >0$, then an increase in $%
\theta $ will increase the elasticity as long as $\bar{\xi}(\bar{\xi}-1)>(%
\bar{\lambda}-\theta \bar{g})(\eta -\sigma )$ (see lemma 1). This will lead
to a reduction in the markup. Apart from increasing welfare by reducing the
markup and increasing productivity (Proposition 1) an increase in $\theta $
also impinges on the way the variability of output and variables such as
employment, consumption and investment. This occurs since an increase in the
elasticity results in a ``flattening'' of the aggregate demand function.
With a flatter demand function, exogenous shocks have a larger effect on
quantity than prices (see Figure 3). Consequently, the variability in
aggregate output for the same exogenous shocks will be larger resulting in a
larger short-run volatility leading to the following situation. On one hand,
increased government investment enhances productivity and welfare while on
the other it also raises the volatility in income. Depending on the risk
aversion of households, the loss in welfare due to increased volatility
could, in principle, overshadow the welfare gains due to increased
productivity leading a to fall in the total welfare.

\FRAME{ftbphF}{301.125pt}{200.75pt}{0pt}{}{}{Figure 3}{\special{language
"Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "T";width
301.125pt;height 200.75pt;depth 0pt;original-width 331.25pt;original-height
255.9375pt;cropleft "0";croptop "1";cropright "1";cropbottom
"0";tempfilename 'govtcyl1.wmf';tempfile-properties "XNP";}}

\section{Growth and Cyclical Effects of Changing Public Investment}

Although it is possible to derive closed form solutions expressing all the
variables in terms of the exogenous shocks (see the appendix) evaluating the
effects of increased $\theta $ analytically is still too cumbersome.
Instead, we use the parametric restrictions derived in the previous section
to simulate the effects first for the U.S. economy and then for other OECD
countries.

\subsection{Simulation Results for the U.S. Economy}

For the U.S. economy, in the experiments we keep $g$ constant at the long
run average for government purchases which is around $0.21$. The steady
state value of $\theta $ is around $0.15$ which includes investment goods
for defense. This implies that government investment is about $3.1$ percent
of GDP.

\vspace{0.25in}

\[
\stackunder{}{
\begin{array}{c}
\text{\textrm{Table\ 2.\ \ Parameters\ used\ in\ Simulations}} \\ 
\\ 
\begin{tabular}{l|ll}
\hline\hline
\textrm{Parameter description} & \textrm{Value} &  \\ \hline\hline
$\beta -$\textrm{\ household's discount factor} & $\mathrm{0.99}$ &  \\ 
$\varsigma -$\textrm{\ wage elasticity of labor} & $\mathrm{1}$ &  \\ 
$\alpha -$\textrm{\ output elasticity of capital} & $\mathrm{0.467}$ &  \\ 
$\delta -$\textrm{\ rate of depreciation of capital stock} & $\mathrm{0.016}$
&  \\ 
$\bar{\lambda}-$\textrm{\ steady state share of total investment in GDP} & $%
\mathrm{0.21}$ &  \\ 
$\bar{\mu}-$\textrm{\ steady state markup} & $\mathrm{1.6}$ &  \\ 
$\bar{\theta}-$\textrm{\ share of investment in government expenditure} & $%
\mathrm{0.15}$ &  \\ 
$\bar{g}-$\textrm{\ steady state share of government expenditure in GDP} & $%
\mathrm{0.21}$ &  \\ 
$\phi ^{z}-$\textrm{\ serial correlation of technology shock} & $\mathrm{0.95%
}$ &  \\ 
$\phi ^{g}-$\textrm{\ serial correlation of government expenditure shock} & $%
\mathrm{0.82}$ &  \\ 
$\sigma _{z}-$\textrm{\ standard deviation of technology shock} & $\mathrm{%
0.00763}$ &  \\ 
$\sigma _{g}-$\textrm{\ standard deviation of government expenditure shock}
& $\mathrm{0.0241}$ &  \\ \hline\hline
\end{tabular}
\end{array}
} 
\]

\vspace{0.25in}

We take as the benchmark the steady state implied by the parameter values
enumerated in Table 2 . The unresolved parameters are $\eta $ and $\sigma $.
However, given the procyclicality of markups as discussed in the previous
section we know that $\eta >\sigma $. In this region we consider two sets of
values for these variables $\{\eta =2.75,\sigma =2.65\}$ and $\left\{ \eta
=3.5,\sigma =2.46\right\} $. In the first set the difference between the two
elasticities is small and in the second relatively large. For each set of
parameters we vary the $\theta $ keeping $g$ constant at $0.21$. A five
percentage point increase (decrease) in $\theta $ implies a one percentage
point increase(decrease) in share of government investment in GDP. The
steady state share of public investment, $\psi $, is around $0.03.$ Tables
3a and 3b summarizes the effects of changing this share by $one$ percentage
point or equivalently changing $\theta $ from its steady state value of $%
0.15 $ by $five$ percentage points.

\vspace{0.2in}

\[
\stackunder{
\begin{array}{c}
\begin{tabular}{lll|l}
\hline\hline
&  & $\{\eta =2.75,\sigma =2.65\}$ & $\{\eta =3.5,\sigma =2.46\}$ \\ 
\hline\hline
\textrm{Percentage change in }$\bar{\lambda}$ &  & \multicolumn{1}{|l|}{%
\textrm{-1.054}} & \textrm{-1.096} \\ 
\textrm{Percentage change in }$\bar{\mu}$ &  & \multicolumn{1}{|l|}{\textrm{%
0.038}} & \textrm{0.41} \\ 
\textrm{Percent change in }$\bar{k}$ &  & \multicolumn{1}{|l|}{\textrm{-0.089%
}} & \textrm{-0.95} \\ 
\textrm{Percent change in }$\bar{l}$ &  & \multicolumn{1}{|l|}{\textrm{-0.044%
}} & \textrm{-0.476} \\ 
\textrm{Percent change in }$\bar{c}$ &  & \multicolumn{1}{|l|}{\textrm{-0.058%
}} & \textrm{-0.625} \\ 
\textrm{Percent change in }$\bar{y}$ &  & \multicolumn{1}{|l|}{\textrm{-0.065%
}} & \textrm{-0.702} \\ 
\textrm{Percent change in welfare} &  & \multicolumn{1}{|l|}{\textrm{-0.82}}
& \textrm{-0.502} \\ \hline\hline
\end{tabular}
\end{array}
}{
\begin{array}{c}
\begin{array}{c}
\text{\textrm{Table\ 3a. Steady\ State\ Effects\ of\ a One Percent\ Decrease}%
}\mathrm{\ } \\ 
\text{\textrm{in\ Public\ Investment-to-GDP ratio}}
\end{array}
\end{array}
} 
\]

\vspace{0.2in}

When $\psi $ is lowered by $1$ percentage point, the share of total
investment $\lambda ,$ falls by more than $1$ percentage for both cases
i.e., $\eta $ and $\sigma $ are close to one another (Table 3a, column A)
and $\eta $ and $\sigma $ relatively far apart (Table 3a, column B).
However, the multiplier effect is somewhat larger when the two elasticities
are further apart. On the other hand, when $\eta $ and $\sigma $ are close
to one another, the effect on the steady state markup is very small. The
markup, $\mu $ remains almost unchanged. This occurs because the average
elasticity, which determines the level of the markup, given by $\left( \eta
-\sigma \right) \lambda +\sigma $ is approximately constant at $\sigma $
when $\eta -\sigma $ is small. When the difference between $\eta $ and $%
\sigma $ is large the change in the markup is by almost half a percentage
point. In a similar vein, capital stock, labor, output and private
consumption rise in both cases although the increase is sharper when the two
elasticities are further apart.

When $\eta =3.5$ and $\sigma =2.46$, the effect of percentage decrease in
public investment leads to almost half a percent decrease in employment. If $%
\theta $ was held constant at $0.15$, to achieve the same effect government
expenditure as a percentage of output would have had to increase by $5$
percentage points, i.e., from $0.21$ to about $0.26$. Thus keeping the
composition of public expenditure constant, permanent decreases in
government spending permanently decreases employment and output. In models
with constant markup, although such effects are present, the causality works
through the labor supply decisions of households and depends almost entirely
on the income effect on leisure. A decrease in government expenditure leads
to an increase in private wealth and lowers real interest rates. Both these
have the effect of lowering the marginal utility of wealth which, in turn,
causes labor supply to fall (Barro 1981). However, in this environment the
causality works from the demand side. With a decrease in the share of
investment, the price elasticity of a firm's output increases causing the
markup to rise and consequently leading to a decrease in labor demand (for a
related analysis see Rotemberg and Woodford 1992).

\vspace{0.2in}

\[
\stackunder{
\begin{array}{c}
\begin{tabular}{ll|l|l}
\hline\hline
&  & $\left\{ \eta =2.75,\sigma =2.65\right\} $ & $\{\eta =3.5,\sigma
=2.46\} $ \\ \hline\hline
\textrm{Percent change in }$\sigma _{y}$\textrm{\ due to }$\hat{z}$ &  & 
\textrm{-0.927} & \textrm{-3.52} \\ 
\textrm{Percent change in }$\sigma _{y}$\textrm{\ due to }$\hat{g}$ &  & 
\textrm{-85.16} & \textrm{-100.41} \\ 
\textrm{Percent change in }$\sigma _{l}$\textrm{\ due to }$\hat{z}$ &  & 
\textrm{-2.26} & \textrm{-4.72} \\ 
\textrm{Percent change in }$\sigma _{l}$\textrm{\ due to }$\hat{g}$ &  & 
\textrm{-85.16} & \textrm{-100.41} \\ 
\textrm{Percent change in welfare} &  & \textrm{0.03} & \textrm{0.05} \\ 
\hline\hline
\end{tabular}
\end{array}
}{
\begin{array}{c}
\begin{array}{c}
\text{\textrm{Table\ 3b.\ Cyclical\ Effects\ of\ a One\ Percentage\ Decrease}%
}\mathrm{\ } \\ 
\text{\textrm{in\ Public\ Investment-to-GDP ratio}}
\end{array}
\end{array}
} 
\]

\vspace{0.2in}

The short run effects of permanently changing $\theta $ are quite
significant, even when the difference between $\eta $ and $\sigma $ is small
(column A, Table 3b). While the percent decrease in the variance of output
from the technology shock is by almost $1$ percent, for labor it is by over $%
2$ percent. The effect is more dramatic for the variance caused by demand
shocks. For both employment and output the variance falls by over $85$
percent and by over $100$ percent when the elasticities are further apart.
This reduction in the variance of employment and output leads to a rise in
households' welfare. Welfare increases by around $0.03$ percent. As
discussed previously, a decrease in the public investment reduces the
elasticity and increases the markup. While this leads to an decline in
welfare due to efficiency losses, it also makes aggregate demand less
elastic. With a more inelastic demand, adjustments in quantity (output and
therefore employment) are smaller than in prices. This fall in volatility
acts to increase the level of welfare.

\subsubsection{Correlation of Total Factor Productivity with Demand Shocks}

Assume in this section that $\hat{z}_{t}=0$, so that the only exogenous
shock to the system is $\hat{g}_{t}$. Letting $\hat{x}_{t}\equiv \tfrac{%
x_{t}-\bar{x}}{\bar{x}}\simeq \ln \left( \tfrac{x_{t}}{\bar{x}}\right) $,
total factor productivity is given by 
\[
\hat{q}_{t}=y^{k}\left( 1-\frac{1}{\bar{\mu}}\right) \hat{k}_{t}+y^{g}\left(
1-\frac{1}{\bar{\mu}}\right) \hat{g}_{t} 
\]
where $y^{g}=$ $-\frac{\varsigma (1-\alpha )\xi _{\mu }\lambda ^{g}}{%
1+\alpha \zeta }$. As long as the elasticity of $\lambda $ with respect to
markup is negative, a condition satisfied if $\eta >\sigma $ --- which is
the same requirement to ensure that the markup is procyclical --- the Solow
residual, $\hat{q}_{t}$, will be positively correlated to $\hat{g}_{t}$,
namely the aggregate demand shock. Consequently, in this economy if Solow
residuals are measured in the standard way i.e., the difference between
actual output and the share weighted contribution of capital and labor, it
will be biased upwards as shocks such as $\hat{g}_{t}$ that are exogenous to
the true technological shock, will be picked up. While, Hall (1988) and
Evans (1992) have empirically shown that productivity is correlated with
variables such as government expenditure shocks, Basu (1995) provides an
argument why this may take place in a model where final good production
requires intermediate goods. In this environment however, the driving force
behind the result is the wedge between the elasticities of firms and
households for the same goods.\bigskip

\subsection{Simulation Results for Other OECD Economies}

This section is based largely on a study by Royen (1998), who uses a similar
environment to estimate the effects of changing the composition of
government purchases on output, employment, capital stock and private
consumption in OECD\ member countries

\subsubsection{Calibration Issues}

\strut The depreciation rate and the elasticity of labor supply are assumed
to be common across the $14$ OECD member countries used in this study. The
specific values for the two parameters $\delta =0.10$ and $\varsigma =1$.
Estimates of mark-ups for OECD countries are available from Martins,
Scarpetta, and Pilat, who calculated markup ratios for $14$ countries for
the period $1970-1992$. Markups, $\mu $ , are available for $36$
manufacturing sectors. The average markup ratios range from $1.13$ in
Belgium and Finland to $1.26$ for Japan. In this paper, for the US, the
markup has been estimated to $1.18$ for the 1970's period and to $1.15$ for
the 1980's - substantially lower than the $1.6$ used in our simulations.
Furthermore, Martins, Scarpetta, and Pilat, provide evidence of the
countercyclical markups in these countries. As before, for the model
economy, this implies that investment elasticity is higher than consumption
elasticity i.e., $\eta >\sigma $. Royen uses two sets of specific values for
these elasticities which are consistent with the estimated OECD markups. In
the first set, the distance between the two elasticities is set
approximately at $1.5$, while in the other it is $3$. Given the
countercyclical mark-ups and their low levels, this results in high values
for $\eta $ and $\sigma .$ The elasticity of capital in production, $\alpha
, $ is given for each country by the steady state relation $(\overline{%
\lambda }-\psi )\overline{\mu }=\frac{\alpha \delta }{\rho +\delta }$, where 
$\rho $ is given by the average risk-free interest rate in each country%
\footnote{%
The average three month treasury bill rate for the period considered is used.%
}. Country-specific values for the long-run share of investment in
government expenditure, the share of government expenditure in GDP, and
share of investment in GDP were used.

\subsubsection{Steady State Effects}

In the 1970s, the impact of an increase in $\bar{\theta}$ by five percentage
points, with $\bar{g}$ remaining constant, on the long-run share of
investment in GDP is the largest in the UK and Denmark of around $1.7$
percentage points, while it has a relatively low impact Japan and Germany of
about $0.7$ percentage. The OECD average is around $1.3$ percentage point
change. In the 1980's, the impact on share of investment increases for
almost all countries. The average increase for OECD countries is about $1.6$
percentage points, with the highest impact in France, Sweden, and Denmark of
over $2$ percentage points. This result does not alter when the difference
between the two elasticities is altered. The effect on the markup ratios,
however, increases, as one would expect, when the gap between investment and
consumption elasticity increases. When the difference in the two
elasticities is around $1.5$, markups fall by about $0.1$ percent in these
countries, while the decline is about $0.25$ percent when the difference is
doubled. The 1970's, with the exception of Japan and Norway, the effect of a 
$5$ percentage change in $\bar{\theta}$ has little impact on output, ranging
from $0.07$ percent increase in Belgium to $0.5$ percent in the Netherlands.
In Norway, however, a significantly different pattern emerges. Changing $%
\bar{\theta}$ by five percentage point increases output by more than $1.6$
percent when the distance between the two elasticities is about $3$. For the
1980's, the differences among countries are less pronounced. Changing $%
\theta $ has a relatively larger impact in France, Italy, Australia, Norway
and Finland, where it increase output by almost $0.7$ percent.

\section{Concluding Remarks}

That aggregate demand affects the cyclical behavior of an economy by
changing the markup is well established in the literature with markups
behaving in a countercyclical manner as increases in demand raise the
elasticity of demand faced by monopolistically competitive firms such that
profit maximization leads to a reduction in their markups. Building on the
framework developed by Gali (1994), who showed that the composition of
aggregate demand can have similar impact, we found that changes in
government purchases can have efficiency effects as well as affect the short
run volatility of macroeconomic variables like output and employment. In
particular, when the price elasticities of investment is larger than that of
consumption, markups behave countercyclically. In such a situation if the
share of investment in total public expenditure is increased aggregate
demand becomes more elastic and consequently, optimal markup is reduced.
With a fall in the markup the economy is pushed towards the competitive
equilibrium with the resulting increase in productivity raising the marginal
revenue products of both capital and labor, such that in equilibrium, firms
hire more workers and accumulate more capital, which in turn increases the
steady output, employment and capital stock. However, the increase in the
elasticity of aggregate demand implies that in response to exogenous shifts
in demand or supply quantity adjustment will be more than price adjustment
thereby increasing the cyclical variability of output and employment. This
trade-off between welfare gains due to increased efficiency and welfare
losses due to greater variability is the critical decision that policy
makers need to make when changing the composition of public expenditure.

For the postwar U.S. economy, a one percentage increase in the share of
public investment, keeping the share of total government spending constant,
raises steady state output by almost $0.07$ percent and welfare by $0.8$
even when the elasticities of consumption and investment are close to one
another. For larger differences in the two elasticities while the increase
in output is larger the welfare gains are less since there is an offsetting
increase in disutility from increased hours of work. However, this also
increases the short run variability of the economy. The variance of output
from the same technology shock increases by almost one percent while the
variance of employment is raised by about $2$ percent. The resulting loss in
welfare from this increased uncertainty is about $0.03$ percent.

We conclude the paper by discussing briefly a particular shortcoming of the
model. Typically government expenditure is made of consumption and
investment goods and labor, i.e., public sector employees. If a government
changes the number of its employees, the economy can also be affected.
However, this effect will be through the supply side. By changing its level
of employment, the government can affect the amount of labor input available
to the private sector. i.e., the effective labor supply faced by firms in
the private sector. The effect of such changes could be significant both
quantitatively as well as in terms of its direction. One cannot draw a
strict analogy between changes in the composition of government purchases
and public sector employment since the latter will affect the economy
primarily by changing the composition of available inputs and their relative
prices. In this paper, however, we did not consider the effects of changes
in the public sector employment.

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\section{References}

\begin{enumerate}
\item  \noindent Aiyagari, S. Rao, Lawrence Christiano, and Martin
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\item  \noindent Barro, Robert J., ``Government Spending in a Simple Model
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\item  \noindent Barro, Robert J., ``Output Effects of Government
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\item  \noindent Bils, Mark, ``Pricing in a Customer Market,'' \textit{%
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\item  \noindent Devarajan, Shantayanan, Vinaya Swaroop, and Heng-fu Zou,
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\item  \noindent Domowitz Ian, R. Glenn Hubbard, and Bruce C. Petersen,
``Market Structure and Cyclical Fluctuations in U.S. Manufacturing,'' 
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\item  \noindent Evans, Charles L., ``Productivity Shocks and Real Business
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\item  \noindent Farmer, Roger E.A., and Jang-Ting Guo, ``Real Business
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\item  \noindent Gal\'{i}, Jordi, ``Monopolistic Competition, Business
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\item  \noindent Hall, Robert E., ``The Relation Between Price and Marginal
Cost in U.S. Industry,'' \textit{Journal of Political Economy} (Chicago),
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\item  \noindent Kalecki, Michael, ``The Determinants of Distribution of the
National Income,'' \textit{Econometrica}, Vol. 6, (April 1938), pp. 97-112.

\item  \noindent Keynes, John Maynard, ``Relative Movements of Real Wages
and Output,'' \textit{Economic Journal}, Vol. 49, (March 1939), pp. 34-51.

\item  \noindent Killingsworth, Mark R., and James J. Heckman, ``Female
Labor Supply: A Survey,'' in \textit{Handbook of Labor Economics} (Orley
Ashenfelter and Richard Layard eds.,), Vol. 1, Amsterdam: North-Holland,
(1986).

\item  \noindent Kydland, Finn E. and Edward C. Prescott, `` Business
Cycles: Real Facts and a Monetary Myth,'' 
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(Winter 1990), pp. 3-18.

\item  \noindent Martins, Joaquim O., Stefano Scarpetta, and Dirk Pilat,
``Mark-up Ratios in Manufacturing Industries: Estimates for 14 OECD
Countries,'' \textit{OECD Working Paper No. 162}, Paris, (1996).

\item  \noindent Morrison, Catherine J., ``Market Power, Economic
Profitability and Productivity Growth Measurement: An Integrated Structural
Approach,'' \textit{NBER Working Paper No. 3355}, Cambridge, Mass., (May
1990).

\item  \noindent Rotemberg, Julio J., and Garth Saloner, ``A
Supergame-Theoritic Model of Price Wars During Booms,'' \textit{American
Economic Review}, Vol. 76, (June 1986), pp. 390-407.

\item  \noindent Rotemberg, Julio J., and Michael Woodford, ``Oligopolistic
Pricing and the Effects of Aggregate Demand on Economic Activity,'' \textit{%
Journal of Political Economy} (Chicago), Vol. 100, No. 6, (1992), pp.
1153-1207.

\item  \noindent Rotemberg, Julio J., and Michael Woodford, ``Cyclical
Markups: Theories and Evidence,'' \textit{NBER Working Paper No. 3534},
Cambridge, Mass., (December 1990).

\item  Royen, Marie-Christine, ``Impact of a Change in Public Spending
Policies on Margins and Business Cycles in OECD Countries,'' GEMME Working
Paper, University of Liege, (October 1998).

\item  \noindent Solow, Robert M., ``Technical Change and the Aggregate
Production,'' 
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Vol. 39 (August 1957), pp. 312-20.

\item  \noindent Woodford, Michael, ``Self-fulfilling Expectations and
Fluctuations in Aggregate Demand,'' in \textit{New Keynesian Economics}
(N.Gregory Mankiw and David Romer, Eds.), Vol. 2, MIT Press, Cambridge, MA,
(1991).
\end{enumerate}

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\section{Appendix}

\subsection{Solution to the Household's Problem}

The objective function of the household is given by

\[
E_{0}\sum_{t=0}^{\infty }\beta ^{t}\left[ u(c_{t}^{i})-v(l_{t}^{i})\right] 
\]
where

\[
u(c_{t}^{i})=M^{\frac{1}{1-\sigma }}\left[ \sum_{s=1}^{M}\left(
c_{st}^{i}\right) ^{\frac{\sigma -1}{\sigma }}\right] ^{\frac{\sigma }{%
\sigma -1}},\text{ }\sigma >0 
\]
and

\[
v(l_t^i)=\frac \zeta {1+\zeta }\left( l_t^i\right) ^{\frac{1+\zeta }\zeta },%
\text{ }\zeta >0. 
\]

Denoting by $q_{t}^{h}$ the price of the shares in firm $h$, we can write
the budget constraint of consumer $i$ as a function of his income, the
return on the shares that he owns and the taxes he pays to the government.
Thus, if $s_{ht}^{i}$ is the number of shares he owns in firm $h$ at time $t$
and $d_{t}^{h}$ the dividend he receives from these shares, the budget
constraint is given by

\[
\sum_{s=1}^{M}p_{t}^{s}c_{st}^{i}\leq l_{t}^{i}+\sum_{h=1}^{M}\left(
d_{t}^{h}+q_{t}^{h}\right)
s_{ht}^{i}-\sum_{h=1}^{M}q_{t}^{h}s_{h,t+1}^{i}-T_{t}^{i} 
\]
where, $T_{t}^{i}$ is the tax paid by the $ith$ household.

If we define the income of consumer $i$ net of taxes as $a_{t}^{i}$, with

\[
a_{t}^{i}\equiv l_{t}^{i}+\sum_{h=1}^{M}\left( d_{t}^{h}+q_{t}^{h}\right)
s_{ht}^{i}-\sum_{h=1}^{M}q_{t}^{h}s_{h,t+1}^{i} 
\]
and assume that all shares are distributed equally between consumers at time 
$t=0$, the consumer's problem can be written as

\begin{equation}
\begin{array}{c}
\max \text{ }E_{0}\sum_{t=0}^{\infty }\beta ^{t}\left[
u(c_{t}^{i})-v(l_{t}^{i})\right] \\ 
\\ 
\text{s.t. }\sum_{s=1}^{M}p_{t}^{s}c_{st}^{i}\leq a_{t}^{i}-T_{t}^{i} \\ 
\\ 
s_{h0}^{i}=\frac{1}{N}
\end{array}
\label{house}
\end{equation}
As discussed in the text let $P_{t}$ be the consumer price index defined as 
\[
P_{t}=\left[ \frac{1}{M}\left( \sum_{s=1}^{M}\left( p_{t}^{s}\right)
^{1-\sigma }\right) ^{\frac{1}{1-\sigma }}\right] 
\]
then the solution to the consumer's problem is 
\begin{equation}
\begin{array}{c}
c_{st}^{i}=\left( \frac{p_{t}^{s}}{P_{t}}\right) ^{-\sigma }\left( \frac{%
a_{t}^{i}-T_{t}^{i}}{P_{t}M}\right) ,\text{ }s=1,2,...M,\forall \text{ }%
i=1,2,...N \\ 
\\ 
l_{t}^{i}=P_{t}^{-\zeta },\text{ }\forall \text{ }i=1,2,...N \\ 
\\ 
\frac{q_{t}^{h}}{P_{t}}=\beta E_{t}\frac{1}{P_{t+1}}\left(
d_{t+1}^{h}+q_{t+1}^{h}\right) ,\text{ }h=1,2,...M,\text{ }\forall \text{ }%
i=1,2,...N \\ 
\\ 
\lim \inf_{T\rightarrow \infty }E\beta ^{T}\sum_{h=1}^{M}\left( \frac{%
q_{T}^{h}}{P_{T}}\right) s_{hT}^{i}=0,\text{ }h=1,2,...M,\text{ }\forall 
\text{ }i=1,2,...N
\end{array}
\text{ }  \label{housefoc}
\end{equation}

The impact of the taxes clearly appears in the first equation characterizing
the demand for the consumption goods. The second equation is the labor
supply as a function of the composite price index with an elasticity of $%
\zeta $ as defined above. The third is the Euler condition and corresponds
to the familiar capital asset pricing equation. The fourth equation is a
limit condition that guarantees that the value function is well
defined.\medskip

\subsection{Solution to the Firm's Problem}

Using the solutions to the static allocation problem faced by the firm from
equation $\left( \ref{firmstatic}\right) $, the \textit{intertemporal}
problem of the $jth$ firm is given by

\begin{equation}
\begin{array}{c}
\max \text{ }E_{0}\sum_{t=0}^{\infty }\text{ }\beta ^{t}\frac{d_{t}^{j}}{%
P_{t}} \\ 
\\ 
\text{s.t.\qquad }d_{t}^{j}=p_{t}^{j}y_{t}^{j}-l_{t}^{j}-\nu _{t}^{j} \\ 
\\ 
y_{t}^{j}=z_{t}\left( k_{t}^{j}\right) ^{\alpha }\left( l_{t}^{j}\right)
^{1-\alpha } \\ 
\\ 
k_{t+1}^{j}=(1-\delta )k_{t}^{j}+\frac{\nu _{t}^{j}}{\Pi _{t}}
\end{array}
\label{6}
\end{equation}

\noindent where $\beta $ is the discount factor, $d_{t}^{j}$ is the dividend
paid by firm $j$ at time $t$, $P_{t}$ the price index for the consumption
goods defined similarly to $\Pi _{t}$, and $E_{0}$ is the expectation
operator at time $t=0$. Since the intertemporal problem is also identical
across firms, we can drop the subscript $j$.

One can thus summarize the problem in $\left( \ref{6}\right) $ by

\begin{equation}
\max \text{ }E_{0}\sum_{t=0}^{\infty }\text{ }\beta ^{t}\frac{1}{P_{t}}\left[
p_{t}y_{t}-z_{t}^{-\frac{1}{1-\alpha }}y_{t}^{\frac{1}{1-\alpha }}k_{t}^{-%
\frac{\alpha }{1-\alpha }}-\Pi _{t}\left( k_{t+1}-(1-\delta )k_{t}\right) %
\right]  \label{7}
\end{equation}
where $z_{t}^{-\frac{1}{1-\alpha }}y_{t}^{\frac{1}{1-\alpha }}k_{t}^{-\frac{%
\alpha }{1-\alpha }}$ is the variable cost at time $t$. Solving $\left( \ref
{7}\right) $ by choosing $\left\{ y_{t},k_{t+1}\right\} $ one gets

\[
p_{t}^{^{\prime }}y_{t}+p_{t}=\frac{1}{1-\alpha }z_{t}^{-\frac{1}{1-\alpha }%
}\left( \frac{y_{t}}{k_{t}}\right) ^{\frac{\alpha }{1-\alpha }} 
\]
where $p_{t}^{^{\prime }}$ represents the derivative of $p_{t}$ with respect
to $y_{t}$. The above equation can be rearranged as

\[
p_{t}\left( \frac{p_{t}^{^{\prime }}y_{t}}{p_{t}}+1\right) =\frac{1}{%
1-\alpha }z_{t}^{-\frac{1}{1-\alpha }}\left( \frac{y_{t}}{k_{t}}\right) ^{%
\frac{\alpha }{1-\alpha }} 
\]
or,

\[
p_{t}\left( 1-\frac{1}{\xi _{t}}\right) =\frac{1}{1-\alpha }z_{t}^{-\frac{1}{%
1-\alpha }}\left( \frac{y_{t}}{k_{t}}\right) ^{\frac{\alpha }{1-\alpha }} 
\]
where $\xi _{t}$, is the price elasticity of the commodity and $\frac{1}{%
1-\alpha }z_{t}^{-\frac{1}{1-\alpha }}\left( \frac{y_{t}}{k_{t}}\right) ^{%
\frac{\alpha }{1-\alpha }}$ is the marginal cost. This gives us the standard
price setting rule in the case of monopolistic competition namely, that the
price is set equal to a mark-up over the marginal cost, i.e.,

\begin{equation}  \label{8}
p_{t}=\mu _{t}\omega _{t}
\end{equation}
where $\mu _{t}=\frac{\xi _{t}}{\xi _{t}-1}$ is the mark-up and

\[
\omega _{t}=\frac{1}{1-\alpha }z_{t}^{-\frac{1}{1-\alpha }}\left( \frac{y_{t}%
}{k_{t}}\right) ^{\frac{\alpha }{1-\alpha }} 
\]
is the marginal cost. After rearranging, the optimal capital accumulation
policy function, obtained by taking the first order condition in $k_{t+1}$
is given by

\begin{equation}
\frac{\Pi _{t}}{P_{t}}=\beta E_{t}\frac{1}{P_{t+1}}\left[ \frac{\alpha }{%
1-\alpha }z_{t+1}^{-\frac{1}{1-\alpha }}\left( \frac{y_{t+1}}{k_{t+1}}%
\right) ^{\frac{1}{1-\alpha }}+(1-\delta )\Pi _{t+1}\right]  \label{9}
\end{equation}
where $\frac{\alpha }{1-\alpha }z_{t+1}^{-\frac{1}{1-\alpha }}\left( \frac{%
y_{t+1}}{k_{t+1}}\right) ^{\frac{1}{1-\alpha }}$ corresponds to the decline
in the variable or labor cost due to a unit increase in the capital stock.
Equations $\left( \ref{4}\right) $, $\left( \ref{8}\right) $, and $\left( 
\ref{9}\right) $ characterize the optimal policy functions of the firms in
terms of output at time $t$ and capital stock at time $t+1$, thus fully
characterizing the solution to the firm's problem.

\subsection{The Linear Approximations}

The linearization of the following equations

\begin{equation}  \label{eql1}
x_{t}-A_{x}z_{t}^{\frac{1+\zeta }{1+\alpha \zeta }}\mu _{t}^{-\frac{\zeta
(1-\alpha )}{1+\alpha \zeta }}k_{t}^{-\frac{(1-\alpha )}{1+\alpha \zeta }}=0
\end{equation}
\begin{equation}  \label{eql2}
\mu _{t}-\frac{(\eta -\sigma )\lambda _{t}+\sigma }{(\eta -\sigma )\lambda
_{t}-(1-\sigma )}=0
\end{equation}

\begin{equation}  \label{eql3}
(\lambda _{t}-\theta g_{t})x_{t}-\frac{k_{t+1}}{k_{t}}+(1-\delta )=0
\end{equation}

\begin{equation}
k_{t+1}-A_{k}\left\{ E_{t}\left[ \left( \frac{z_{t+1}}{\mu _{t+1}}\right) ^{%
\frac{1+\zeta }{1+\alpha \zeta }}\right] \right\} ^{\frac{1+\alpha \zeta }{%
1-\alpha }}=0  \label{eql4}
\end{equation}
resulted in

\begin{equation}  \label{eqll1}
\hat{x}_{t}=a_{1}\hat{z}_{t}-a_{2}\hat{\mu}_{t}-a_{3}\hat{k}_{t}
\end{equation}
\begin{equation}  \label{eqll3}
\hat{\mu}_{t}=b_{1}\hat{\lambda}_{t}
\end{equation}

\begin{equation}
\hat{\lambda}_{t}=\frac{1}{c_{1}}\hat{k}_{t+1}-\frac{1}{c_{1}}\hat{k}_{t}-%
\frac{\delta }{c_{1}}\hat{x}_{t}+c_{2}\hat{g}_{t}  \label{eqll2}
\end{equation}
\begin{equation}
\hat{k}_{t+1}=d_{1}E_{t}\left[ \hat{z}_{t+1}-\hat{\mu}_{t+1}\right]
\label{eqll4}
\end{equation}
where,

\noindent $
\begin{array}{l}
a_{1}=\left( \frac{1+\zeta }{1+\alpha \zeta }\right) \text{,}
\end{array}
\begin{array}{l}
a_{2}=\frac{\zeta (1-\alpha )}{1+\alpha \zeta }\text{,}
\end{array}
\begin{array}{l}
a_{3}=\frac{1-\alpha }{1+\alpha \zeta }\text{,}
\end{array}
\begin{array}{l}
b_{1}=\left( \frac{\alpha \delta }{\rho +\delta }\right) \left( \frac{%
(\sigma -\rho )}{\left( (\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}%
\right) \frac{\bar{\lambda}}{\left( \bar{\lambda}-\psi \right) }\text{,}
\end{array}
$

\noindent $
\begin{array}{l}
c_{1}=\delta \frac{\bar{\lambda}}{\left( \bar{\lambda}-\psi \right) }\text{,}
\end{array}
\begin{array}{l}
c_{2}=\left( \frac{\psi }{\bar{\lambda}}\right) \text{,}
\end{array}
\begin{array}{l}
d_{1}=\left( \frac{1+\zeta }{1-\alpha }\right) \text{,}
\end{array}
$and$
\begin{array}{l}
\hat{\omega}_{t}=\frac{\omega _{t}-\bar{\omega}}{\bar{\omega}}\text{ for }%
\omega =x,\lambda ,k,z,v,g\text{.}
\end{array}
$ Note that at the steady state $(\bar{\lambda}-\psi )\left( \frac{(\eta
-\sigma )\bar{\lambda}+\sigma }{(\eta -\sigma )\bar{\lambda}-(1-\sigma )}%
\right) =\frac{\alpha \delta }{\rho +\delta }$. Consequently, 
\[
\left( \frac{\alpha \delta }{\rho +\delta }\right) \left( \frac{(\sigma
-\eta )}{\left( (\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}\right) =(%
\bar{\lambda}-\psi )\left( \frac{(\sigma -\eta )}{\left( (\eta -\sigma )\bar{%
\lambda}-(1-\sigma )\right) \left( (\eta -\sigma )\bar{\lambda}+\sigma
\right) }\right) 
\]
Since $\bar{\xi}=(\eta -\sigma )\bar{\lambda}+\sigma $, the $b_{1}$ becomes $%
\bar{\lambda}\left( \frac{(\sigma -\rho )}{\bar{\xi}\left( \bar{\xi}%
-1\right) }\right) $. From the definition of the elasticity of the markup
with respect to $\lambda $, we know that $\bar{\varepsilon}_{\mu }=\dfrac{%
(\sigma -\eta )\bar{\lambda}}{\bar{\xi}\left( \bar{\xi}-1\right) }$.
Consequently, $b_{1}=\bar{\varepsilon}_{\mu }$. Replacing $\left( \ref{eqll1}%
\right) $ and $\left( \ref{eqll3}\right) $ in $\left( \ref{eqll2}\right) $
we have under the assumption that $c_{1}-\delta a_{2}b_{1}\neq 0$

\begin{equation}
\hat{\lambda}_{t}=\frac{1}{c_{1}-\delta a_{2}b_{1}}\hat{k}_{t+1}-\frac{%
\left( 1-\delta a_{3}\right) }{c_{1}-\delta a_{2}b_{1}}\hat{k}_{t}-\frac{%
\delta a_{1}}{c_{1}-\delta a_{2}b_{1}}\hat{z}_{t}+\frac{c_{1}c_{2}}{%
c_{1}-\delta a_{2}b_{1}}\hat{g}_{t}
\end{equation}
Using the above and leading $\left( \ref{eqll3}\right) $ by one period

\[
\hat{\mu}_{t+1}=b_{1}\left( \frac{1}{c_{1}-\delta a_{2}b_{1}}\hat{k}_{t+2}-%
\frac{\left( 1-\delta a_{3}\right) }{c_{1}-\delta a_{2}b_{1}}\hat{k}_{t+1}-%
\frac{\delta a_{1}}{c_{1}-\delta a_{2}b_{1}}\hat{z}_{t+1}+\frac{c_{1}c_{2}}{%
c_{1}-\delta a_{2}b_{1}}\hat{g}_{t+1}\right) 
\]
Note that $c_{1}c_{2}=\frac{\delta \psi }{\lambda -\psi }$ . Replacing the
above in $\left( \ref{eqll4}\right) $ we have, provided $c_{1}-b_{1}\left(
d_{1}-\delta \right) \neq 0$,

\begin{equation}
\hat{k}_{t+1}=E_{t}\left[ e_{1}\hat{z}_{t+1}+e_{2}\hat{k}_{t+2}+e_{3}\hat{g}%
_{t+1}\right]  \label{diff}
\end{equation}
where,

$
\begin{array}{l}
e_{1}=\frac{d_{1}\left[ c_{1}+b_{1}\delta \left( a_{1}-a_{2}\right) \right] 
}{c_{1}-b_{1}\left( d_{1}-\delta \right) } \\ 
e_{2}=-\frac{d_{1}b_{1}}{c_{1}-b_{1}\left( d_{1}-\delta \right) }=-\frac{%
d_{1}\bar{\varepsilon}_{\mu }}{c_{1}-\bar{\varepsilon}_{\mu }\left(
d_{1}-\delta \right) } \\ 
e_{3}=-\frac{d_{1}c_{1}c_{2}b_{1}}{c_{1}-b_{1}\left( d_{1}-\delta \right) }=-%
\frac{d_{1}c_{1}c_{2}\bar{\varepsilon}_{\mu }}{c_{1}-\bar{\varepsilon}_{\mu
}\left( d_{1}-\delta \right) }
\end{array}
$

\medskip \noindent Next, we list below some of the other coefficients of the
linearized system of equations:\noindent 
\[
k^{z}=\frac{\phi ^{z}e_{1}}{1-\phi ^{z}e_{2}},k^{g}=\frac{\phi ^{g}e_{3}}{%
1-\phi ^{g}e_{2}},\lambda ^{z}=\frac{k^{z}-\delta a_{1}}{c_{1}-\delta
a_{2}b_{1}},\lambda ^{k}=\frac{\left( 1-\delta a_{3}\right) }{c_{1}-\delta
a_{2}b_{1}},\lambda ^{g}=\frac{k^{g}+c_{1}c_{2}}{c_{1}-\delta a_{2}b_{1}} 
\]
\noindent 
\[
y^{z}=\left( \frac{\left( 1+\zeta \right) -\zeta (1-\alpha )b_{1}\lambda ^{z}%
}{1+\alpha \zeta }\right) ,y^{k}=\left( \frac{\alpha (1+\zeta )+\zeta
(1-\alpha )b_{1}\lambda ^{k}}{1+\alpha \zeta }\right) ,y^{g}=-\frac{%
\varsigma (1-\alpha )b_{1}\lambda ^{g}}{1+\alpha \zeta } 
\]
\[
l^{z}=\frac{\zeta }{1+\alpha \zeta }(1-b_{1}\lambda ^{z}),l^{g}=-\frac{\zeta
b_{1}\lambda ^{g}}{1+\alpha \zeta },l^{k}=\frac{\zeta }{1+\alpha \zeta }%
(\alpha +b_{1}\lambda ^{k}) 
\]
\medskip \bigskip

\subsection{Proof of Lemma 2}

Using the expression for $d_{1},$ $c_{1}$and $b_{1}$ one gets

\[
e_{2}=-\left\{ \frac{\left( \frac{\alpha \delta }{\rho +\delta }\right)
\left( \frac{(\sigma -\eta )}{\left( (\eta -\sigma )\bar{\lambda}+\sigma
\right) ^{2}}\right) }{\delta \left( \frac{1-\alpha }{1+\zeta }\right) \left[
1-\left( \frac{1+\zeta }{\delta \left( 1-\alpha \right) }-1\right) \left( 
\frac{\alpha \delta }{\rho +\delta }\right) \left( \frac{(\sigma -\eta )}{%
\left( (\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}\right) \right] }%
\right\} 
\]

\[
\frac{1}{e_{2}}=-\left\{ \frac{\delta \left( \frac{1-\alpha }{1+\zeta }%
\right) \left[ 1-\left( \frac{1+\zeta }{\delta \left( 1-\alpha \right) }%
-1\right) \left( \frac{\alpha \delta }{\rho +\delta }\right) \left( \frac{%
(\sigma -\eta )}{\left( (\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}%
\right) \right] }{\left( \frac{1+\zeta }{1-\alpha }\right) \left( \frac{%
\alpha \delta }{\rho +\delta }\right) \left( \frac{(\sigma -\rho )}{\left(
(\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}\right) }\right\} 
\]
\[
=-\delta \left( \frac{1-\alpha }{1+\zeta }\right) \left[ \frac{1}{\left( 
\frac{1+\zeta }{1-\alpha }\right) \left( \frac{\alpha \delta }{\rho +\delta }%
\right) \left( \frac{(\sigma -\eta )}{\left( (\eta -\sigma )\bar{\lambda}%
+\sigma \right) ^{2}}\right) }+1-\left( \frac{1+\zeta }{\delta \left(
1-\alpha \right) }\right) \right] 
\]
\[
=-\left\{ \frac{\delta \left( 1-\alpha \right) }{1+\zeta }\left[ 1+\frac{1}{%
\left( \frac{\alpha \delta }{\rho +\delta }\right) \left( \frac{(\sigma
-\eta )}{\left( (\eta -\sigma )\bar{\lambda}+\sigma \right) ^{2}}\right) }%
\right] -1\right\} 
\]
Using that fact that the steady state $\left( \frac{\alpha \delta }{\rho
+\delta }\right) \left( \frac{(\sigma -\eta )}{\left( (\eta -\sigma )\bar{%
\lambda}+\sigma \right) ^{2}}\right) =\left( \frac{\bar{\lambda}-\psi }{\bar{%
\lambda}}\right) \bar{\varepsilon}_{\mu }$, we have

\[
\frac{1}{e_{2}}=-\left\{ \frac{\delta \left( 1-\alpha \right) }{1+\zeta }%
\left[ 1+\left( \frac{\bar{\lambda}}{\bar{\lambda}-\psi }\right) \frac{1}{%
\bar{\varepsilon}_{\mu }}\right] -1\right\} 
\]
Taking modulus, $\frac{1}{\mid e_{2}\mid }$ $=1-\frac{\delta \left( 1-\alpha
\right) }{1+\zeta }\left[ 1+\left( \frac{\bar{\lambda}}{\bar{\lambda}-\psi }%
\right) \frac{1}{\bar{\varepsilon}_{\mu }}\right] $. If $\mid e_{2}\mid <1$,
then $\mid \frac{1}{e_{2}}\mid >1$ i.e., $1-\frac{\delta \left( 1-\alpha
\right) }{1+\zeta }\left[ 1+\left( \frac{\bar{\lambda}}{\bar{\lambda}-\psi }%
\right) \frac{1}{\bar{\varepsilon}_{\mu }}\right] >1$.\medskip \bigskip

\subsection{Proof of Proposition 3}

Consider the expression for $e_{2}=\left\{ 1-\dfrac{\delta \left( 1-\alpha
\right) }{1+\zeta }\left[ 1+\left( \dfrac{\bar{\lambda}}{\bar{\lambda}-\psi }%
\right) \dfrac{1}{\bar{\varepsilon}_{\mu }}\right] \right\} ^{-1}$. Since $%
\bar{\varepsilon}_{\mu }=\frac{(\sigma -\eta )\bar{\lambda}}{\bar{\xi}\left( 
\bar{\xi}-1\right) }$ and $\bar{\xi}=(\eta -\sigma )\bar{\lambda}+\sigma $,
using the steady state condition $(\bar{\lambda}-\psi )\mu =\frac{\alpha
\delta }{\rho +\delta }$, we get $e_{2}=C_{2}\left[ \frac{C_{1}^{2}}{\eta
-\sigma }+C_{3}\right] ^{-1}$ where $C_{1}=(\eta -\sigma )\lambda +\sigma >0$%
, $C_{2}=\frac{\alpha (1+\zeta )}{(\rho +\delta )(1-\alpha )}>0$ and $%
C_{3}=C_{2}-\frac{\alpha \delta }{(\rho +\delta )}>0$. Now if $\mid
e_{2}\mid >1$ when $\eta -\sigma >0$ then $C_{2}>\left[ \frac{C_{1}^{2}}{%
\eta -\sigma }+C_{3}\right] $. This implies that $C_{2}-C_{3}>\frac{C_{1}^{2}%
}{\eta -\sigma }$, i.e. $\frac{\alpha \delta }{(\rho +\delta )}>\frac{%
C_{1}^{2}}{\eta -\sigma }$ which in turn results in $\eta -\sigma
>C_{1}^{2}\left( \frac{\alpha \delta }{\rho +\delta }\right) ^{-1}$. Since, $%
\eta -\sigma =\frac{C_{1}}{\lambda }-\sigma $, the inequality can be
rearranged as $\frac{C_{1}}{\lambda }-\sigma >C_{1}^{2}\left( \frac{\alpha
\delta }{\rho +\delta }\right) ^{-1}$. Again using the steady state
condition $(\bar{\lambda}-\psi )\mu =\frac{\alpha \delta }{\rho +\delta }$
and rearranging the expression for optimal markup such that $C_{1}=(1-\frac{1%
}{\mu })^{-1}$, the inequality becomes $\mu C_{1}-C_{1}^{2}\left( \frac{%
\lambda }{\lambda -\psi }\right) >\mu \sigma $. Now if $\sigma >0$ then $\mu
-C_{1}\left( \frac{\lambda }{\lambda -\psi }\right) >0$ i.e., $\mu -\left( 
\frac{\mu }{\mu -1}\right) \left( \frac{\lambda }{\lambda -\psi }\right) >0$%
. This implies that $\mu >1+\frac{\lambda }{\lambda -\psi }$. Since $\frac{%
\lambda }{\lambda -\psi }>1$ it implies that $\mu >2$.

\end{document}