Optimal Fiscal Policy in a Small Open Economy and the Structure of International Financial Markets

Banco de M´
exico
Documentos de Investigaci´
on
Banco de M´
exico
Working Papers
N◦ 2007-07
Optimal Fiscal Policy in a Small Open Economy and
the Structure of International Financial Markets
Josu´
e Fernando Cort´
es Espada
Banco de M´
exico
March 2007
La serie de Documentos de Investigaci´
on del Banco de M´
exico divulga resultados preliminares de
trabajos de investigaci´
on econ´
omica realizados en el Banco de M´
exico con la finalidad de propiciar
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on, as´ı como las
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necesariamente las del Banco de M´
exico.
The Working Papers series of Banco de M´
exico disseminates preliminary results of economic
research conducted at Banco de M´
exico in order to promote the exchange and debate of ideas. The
views and conclusions presented in the Working Papers are exclusively the responsibility of the
authors and do not necessarily reflect those of Banco de M´
exico.

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Documento de Investigaci´
on
Working Paper
2007-07
2007-07
Optimal Fiscal Policy in a Small Open Economy and the
Structure of International Financial Markets
Josu´
e Fernando Cort´
es Espada†
Banco de M´
exico
Abstract This paper characterizes the behavior of debt and tax rates in a small open
economy under both complete and incomplete markets. First, I show that when the govern-
ment follows an optimal fiscal policy and agents have access to complete markets, the value
of the government’s debt portfolio is negatively correlated with government spending, and
positively correlated with productivity and output, while output, labor, consumption and
the tax rate are uncorrelated with government spending shocks. The stochastic processes
followed by these variables inherit the serial-correlation properties of the stochastic process
of the productivity shock. Second, I show that if agents can only buy and sell one-period
risk-free bonds, public debt shows more persistence than other variables, and it is nega-
tively correlated with productivity and output, and positively correlated with government
spending. Moreover, the tax rate is positively correlated with government spending, while
consumption is negatively correlated.
Keywords: Complete markets, Incomplete markets, Optimal fiscal policy.
JEL Classification: E60, F34, F41, G15 and H21.
Resumen El presente art´ıculo caracteriza el comportamiento de la deuda p´
ublica y de la
tasa impositiva bajo mercados completos e incompletos en una econom´ıa peque˜
na y abierta.
En primer lugar, se demuestra que cuando el gobierno sigue una pol´ıtica fiscal ´
optima y
los agentes tienen acceso a mercados completos, el valor del portafolio de deuda del gobier-
no est´
a negativamente correlacionado con el gasto p´
ublico y positivamente correlacionado
con la productividad y el producto, mientras que el producto, el trabajo, el consumo y la
tasa impositiva no est´
an correlacionados con el gasto p´
ublico. Los procesos estoc´
asticos que
siguen estas variables heredan las propiedades estad´ısticas del proceso estoc´
astico que sigue
la productividad. En segundo lugar, se demuestra que si los agentes tienen acceso a mercados
incompletos, la deuda p´
ublica es m´
as persistente que la otras variables y est´
a negativamente
correlacionada con la productividad y el producto, y positivamente correlacionada con el
gasto p´
ublico. Adicionalmente, la tasa impositiva est´
a positivamente correlacionada con el
gasto p´
ublico, mientras que el consumo est´
a negativamente correlacionado.
Palabras Clave: Mercados completos, Mercados incompletos, Pol´ıtica fiscal ´
optima.
† Direcci´on General de Investigaci´on Econ´omica. Email: jfcortes@banxico.org.mx.

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1
Introduction
In this paper, I characterize the behavior of debt and tax rates in a small open economy
under both complete and incomplete markets using Ramsey’s approach to optimal taxation.
Since the seminal work of Lucas and Stokey (1983), an extensive literature characterizing
optimal …scal policy based on Ramsey’s approach to dynamic optimal taxation has emerged.
Most of the existing work, however has limited attention to closed economy environments.
This paper instead studies optimal …scal policy in a small open economy under both incom-
plete and complete markets. I follow the Ramsey approach in characterizing the optimal
…scal policy. In this approach the Ramsey planner chooses an allocation that maximizes the
household’s utility subject to the condition that this allocation be implementable as a com-
petitive equilibrium. I also abstract, as it is standard in the literature of optimal taxation,
from issues of time inconsistency.
The main contributions of the paper are the following: First, I show that when the gov-
ernment in a small open economy follows an optimal …scal policy and agents have access
to international complete asset markets, the value of the government’s debt portfolio is a
time invariant function of the underlying shocks. As a consequence, the value of the govern-
ment’s debt portfolio inherits the serial correlation structure of the shocks. Moreover, under
complete markets the value of the government’s debt portfolio is negatively correlated with
government spending, and positively correlated with productivity and output, while output,
labor, consumption and the tax rate are uncorrelated with government spending shocks.
The stochastic processes followed by output, labor, consumption and the tax rate inherit
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the serial- correlation properties of the stochastic process of the productivity shock. The
Ramsey planner …nances all innovations to government spending with state-contingent pay-
ments from the rest of the world. Second, I show that if agents in a small open economy
can only buy and sell one-period risk-free bonds, public debt shows more persistence than
other variables, and it is negatively correlated with productivity and output, and positively
correlated with government spending, since the government uses debt to smooth tax distor-
tions over time. Additionally, the tax rate is positively correlated with government spending,
while consumption is negatively correlated. The negative correlation between consumption
and government spending illustrates the limited insurance role played by non-contingent
debt.
The paper proceeds as follows. Section 2 discusses some of the literature on optimal …scal
policy. Section 3 presents the complete markets model and analyzes the dynamic properties
of the optimal …scal policy under complete markets. Section 4 presents the incomplete mar-
kets model and analyzes the dynamic properties of the optimal …scal policy under incomplete
markets . Section 5 concludes.
2
Related Literature
This paper is related to several studies about optimal …scal policy. An extensive literature
on optimal …scal policy has emerged since the seminal work of Lucas and Stokey (1983).
Most of the existing work, however has limited attention to closed economy environments.
This paper instead studies optimal …scal policy in a small open economy with incomplete
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markets.
In a closed economy environment with complete markets, Lucas and Stokey (1983) used
the Ramsey approach of optimal taxation to study the properties of optimal …scal policy.
They found that it is optimal to respond to …scal shocks by appropriately altering the state-
contingent return on government debt and keeping the tax rate roughly constant, so state-
contingent debt serves as an instrument to smooth tax distortions over time and states of
nature. They also show that tax rates and debt inherit the serial correlation structure of the
underlying shocks. Chari, Christiano and Kehoe (1994) analyzed the quantitative features
of optimal …scal policy in a standard real business cycle model with complete markets as
in Lucas and Stokey (1983). They showed that another way to keep tax rates stable over
the business cycle is to have non-state contingent debt with taxes on interest income that
vary with the shocks, in this case state-contingent taxes on interest income should be used
to provide insurance against adverse shocks. They found that in calibrated models to the
U.S., the standard deviation of optimal income taxes is close to zero while taxes on interest
income are highly volatile and serially uncorrelated.
Aiyagari et al (2002) restricted the government to issue only one-period non-contingent
debt. They showed that optimal …scal policy under this environment imposes a near random
walk behavior on taxes and debt irrespective of the degree of autocorrelation of the under-
lying shocks. They also found that the level of debt permanently increases after a …scal
shock, and that the response of the tax rate is a weighted average of a random walk and
a serially uncorrelated process. Their results a¢ rm partially the random walk hypothesis
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of Barro (1979) . Angeletos (2002), and Buera and Nicolini (2002) considered governments
restricted to trading non-contingent real debt of di¤erent maturities. They showed that
governments could use the maturity structure of non-contingent public debt to replicate the
complete markets optimal allocation. However, Buera and Nicolini showed that the gov-
ernment might need to take extremely large long and short positions in debt of di¤erent
maturities. Marcet and Scott (2000) compare the empirical implications of the model with
complete markets, the model with just one period risk free debt and US data. They show
that the one-period risk-free bond economy replicates the qualitative features of the data
better.
In an open economy setting, Riascos and Vegh (2004) consider an environment in which
government spending is determined endogenously. They show that when markets are com-
plete, the correlation between public consumption and output is zero, while if markets are
incomplete, the correlation between public consumption and output is large and positive.
In terms of the existing literature, this paper is closest to Riascos and Vegh (2004). Like
them, I study optimal …scal policy in a small open economy. However, this paper di¤ers in
two key respects from their paper. First, the goal of the present paper is to characterize
the behavior of optimal tax rates and government debt under both complete and incomplete
markets in a small open economy, while the goal of Riascos and Vegh (2004) is to analyze
the procyclicality of …scal policy in developing countries, so they do not analyze the optimal
behavior of public debt under complete and incomplete markets. Second, these authors
consider an endowment economy, while I consider a production economy with an elastic
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labor supply, so movements in the tax rate a¤ect the labor supply and output.
3
The Complete Markets Model
Consider a small open economy populated by an in…nite number of identical, in…nitely
lived consumers. In each period t = 0; 1; :: the economy experiences one of …netely many
events st 2 S = (1;2;::::N): We denote by st = (s0;:::::;st) the history of events up to and
including period t: The probability as of period 0, of any particular history st is
(st) : The
initial realization s0 is given. Asset markets are complete, both the government and private
agents have access to a complete set of Arrow-Debreu securities traded in world capital
markets. The government …nances an exogenous and stochastic sequence of unproductive
public consumption by issuing state-contingent debt and by taxing income at the rate
(st).
3.1
Households
Each household has preferences de…ned over consumption ct and labor ht: The representative
agent´s lifetime utility is given by:
1
XX t st U c st ;h st
(1)
t=0
st
where 0 <
< 1 denotes the subjective discount factor, c (st) and h (st) denote consump-
tion and labor conditional on the history of events st, and the single-period utility function
U is strictly increasing in consumption, decreasing in labor, strictly concave, and satis…es
the Inada conditions.
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Each period t, households have access to a set of N one-period state-contingent bonds
d (st; st+1), which pay one unit of consumption in a particular state of period t + 1. The
variable D (st) = (d (st; st+1))
denotes the portfolio of bonds of the representative
st+12S
agent at time t; conditional on history st. Let q (st+1 j st) be the period t price conditional
on history st of an asset that promises to pay one unit of consumption in period t + 1 in
the event that st+1 is realized. The value at t of the portfolio of state-contingent bonds
purchased in period t conditional on history st is:
vd st = Xq st+1 j st d st;st+1
st+1
The variable P q(st
st+1
j st+1) is the period t price of an asset that pays one unit of con-
sumption in every state in period t + 1, therefore, this variable represents the inverse of the
risk-free gross real interest rate. Letting R (st) denote the gross risk-free real interest rate,
we have
1
R st = P q(s
s
t+1
t+1
j st)
In each period t; households have access to a concave technology to transform labor into
output. The period-by-period budget constraint is given by
c st + Xq st+1 j st d st;st+1 d st + 1 st z st f h st
(2)
st+1
where
denotes the income tax rate imposed by the government, f the production function,
and z (st) a technology shock. The production function is increasing in labor, concave and
homogeneous of degree
< 1:
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In addition to this budget constraint, the household is subject to the following borrowing
constraint that prevents it from engaging in Ponzi schemes
lim
Qt st+j d st+j
0 f or all t; st
(3)
j!1 sX
t+j jst
where Qt (st+j) is the price of an asset that promises to pay one unit of consumption in
period t + j conditional on history st+j being realized, the Arrow-Debreu price Qt (st+j) is
denominated in units of the date t history st consumption good.
Qt st+j
= q st+1 j st q st+2 j st+1 ::::::q st+j j st+j 1
Qt st
= 1
The assumptions on the utility function imply that households will always choose allo-
cations such that constraints (2) and (3) hold with equality. These two constraints holding
with equality imply that the following intertemporal budget constraint must hold
1
1
d 1 + XXQ st 1 st z st f h st = XXQ st c st
(4)
t=0 st
t=0 st
where Q (st) is the price of an asset that promises to pay one unit of consumption in period
t conditional on history st being realized. The Arrow-Debreu price Q (st) is denominated in
units of the date zero consumption good.
Q st
= q (s1 j s0)q s2 j s1 ::::::q st j st 1
Q (s0) = 1
Expression (4) states that total wealth in period zero, which consists of the sum of initial
…nancial wealth and the present discounted value of after-tax income must equal the present
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discounted value of consumption. Given the stochastic processes fz (st); (st);Q(st)g1t=0
and the initial condition d 1, the household chooses state-contingent sequences fc(st);h(st)g1t=0
to maximize (1) subject to (4) : The …rst-order conditions associated with the household’s
maximization problem are (4), and:
tuc st
st = Q st
(5)
where
is the multiplier on the intertemporal budget constraint, and uc is the marginal
utility of consumption
uh (st) = 1
st
z st f 0 h st
(6)
uc (st)
First-order condition (6) shows that the tax rate introduces a wedge between the consumption-
leisure marginal rate of substitution and the marginal product of labor.
3.2
The Government
The government sets the tax rate on income and issues one-period state-contingent bonds
to …nance the exogenous sequence of government consumption, which is stochastic and
unproductive. In each period t, the government issues one-period state-contingent bonds
b (st; st+1), which pay one unit of consumption in a particular state of period t + 1. The
variable B (st) = (b (st; st+1))
denotes the debt portfolio of the government at time t;
st+12S
conditional on history st. The value of the government’s debt portfolio in period t conditional
on history st is:
vb st = Xq st+1 j st b st;st+1
st+1
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