Ramsey Optimal Fiscal Policy in Emerging
Countries: Is it Procyclical?
∗
Subrata Sarker
May, 2008
Draft, please do not cite without permission
Abstract
The purpose of this paper is to investigate the role of nancial market imperfections in explaining the procyclical scal policy in many of
the emerging developing countries. I examine whether, in the presence
of distortionary taxation, procyclical scal policy can be an optimal
outcome for a Ramsey planner in a one good, representative agent
small open economy model with only friction being an upward sloping supply curve for loans from an incomplete international nancial
market. The paper shows that the observed procyclicality of the scal variables may indeed be an optimal outcome as debt nancing of
decit becomes expensive during a recession due to the upward slope
of the loan supply function.
Procyclical scal policy, Ramsey optimal policy
Codes: E62, F41
Keywords:
JEL
PhD Student, Department of Economics, University of British Columbia, 997-1873
East Mall, Vancouver, BC, V6T 1Z1 Email: ssarker@interchange.ubc.ca.
∗
1
Introduction
In developed countries, scal policy is either countercyclical, i.e. they are
targeted to smooth the cyclical uctuations in income and employment, or
they are, at least, acyclical. However, in many of the emerging developing
countries, it appears that scal policy is procyclical. Procyclicality implies
that business cycle booms and recessions are accentuated by the government
policies. This may take place in several ways: tax rates fall in good times and
rise in bad times, government consumption rises in good times and falls in
bad times. These phenomena are often considered as signicant contributors
to the macroeconomic instability in emerging market economies.
These facts are contrary to the conventional wisdom in both the major
schools in macroeconomics. The Keynesian framework suggests that the
scal policy should be countercyclical i.e. the government should cut taxes
and increase expenditures during a recession and do the opposite during a
boom. In contrast, the neo-classical approach calls for acyclical scal policy
in the sense that optimal tax rates should remain largely constant over the
business cycles. This raises an important question: are policy procyclicalities
simply the result of misguided macroeconomic policies or are they the results
of specic circumstances in which the policy making authorities in these
countries operate? Understanding these puzzling issues is an important step
in macroeconomic stabilization in developing countries.
The purpose of this paper is to investigate the role of nancial market
imperfections in explaining the procyclical scal policy. I examine whether,
in the presence of distortionary taxation, procyclical scal policy can be an
optimal outcome for a Ramsey planner in a one good, representative agent
small open economy model with only friction being an incomplete international nancial market where the rate of interest depends negatively on the
net worth of the economy. The paper shows that the observed procyclicality
of the scal variables may indeed be an optimal outcome as debt nancing
of decit becomes expensive during a recession due to higher interest cost
of borrowing. The empirical motivation for such shape of the loan supply
function comes from the well-documented fact in Uribe and Yue (2006) and
Neymeyer and Perri (2005) that the interest rate in emerging countries is
negatively correlated with output whereas the correlation is either positive
1
or zero in developed countries.
The paper proceeds as follows. Section 2 briey discusses some of the stylized facts from the existing empirical studies. Section 3 discusses the two
main lines of arguments given to explain procyclical scal policy and, in that
context, provides justication for the exercise undertaken in this paper. In
section 4, I outline the baseline model with exogenous public expenditure,
parametrize it and discuss the results in response to a negative shock in productivity. Section 5 extends the baseline model by incorporating endogenous
public expenditure and discusses its implications. Section 6 provides the
concluding remarks.
2
Stylized Facts
Gavin and Perotti (1997), Braun (2001), Kaminski, Reinhart and Végh
(2004), Alesina, Campante and Tabellini (2007), Végh and Ilzetzki (2008)
provide ample evidences in support of procyclical scal policies. Braun (2001)
nds that while in OECD countries a one percentage point increase in GDP
is associated with a reduction of 0.37 percentage point in the ratio of government expenditures to GDP; in developing countries this ratio remains
unchanged. This implies, in developing economies, government expenditures
increase by the same proportion during economic expansions and decline by
the same proportion during recessions.
Kaminski, Reinhart and Végh (2004) is by far the most comprehensive
empirical exercise in this regard. They document some key stylized features of
the business cycle variation in scal policies in the developing and developed
countries using a dataset of 104 countries and covering a period between
1960-2003. I reproduce some of the key ndings from their study to motivate
the discussion further.
Table 1 presents the correlation of the cyclical components of GDP with
the general government expenditures and the ination tax rate. For the
middle income and the low income countries, general government expenditure
and the ination tax rates are clearly pro-cyclical whereas, for the OECD
countries, expenditure is acyclical and the ination tax rate is countercyclical.
2
π
Ination tax rate, which is dened as 1+π
, (where π is the rate of ination)
is used as an imperfect proxy for the actual tax rates.
Table 1: Correlation between Fiscal Policy and Real GDP
Countries
General
Government
Expenditure
HP Band-Pass
Filter
Filter
-0.06
-0.02
0.43*
0.44*
0.20*
0.23*
0.37*
0.34*
Ination
Tax
HP Band-Pass
Filter
Filter
0.16*
0.15*
-0.15*
-0.13*
-0.09*
-0.10*
-0.20*
-0.16*
OECD
Middle-High Income
Middle-Low Income
Low Income
Notes :An asterisk denotes statistical signicance at the 10 percent level
Source :Reproduced from Kaminski, Reinhart and Végh (2004).
Unfortunately, data on cyclical variation in actual tax rates are extremely
dicult to come by. There are, however, strong casual evidences suggesting
that tax rates are procyclical in developing countries. In a detail case study
on Uruguay Mailhos and Sosa (2000) found that over the period between 1975
to 1999 almost all the tax rates such as the value added tax rates, income tax
rates, wage tax rates and certain commodity tax rates were procyclical. They
also found that the taris charged by the state owned rms for public utilities, which often constitutes a form of implicit taxation, were signicantly
negatively correlated with the GDP. In addition, the taxes charged by the
social welfare authorities on dierent pension funds and housing or construction funds were also procyclical. Talvi and Végh (2005) reports that in 1995,
in the midst of a severe recession, both Argentina and Mexico implemented
major scal adjustments involving large increases in tax rates along with cuts
in public spending. In Mexico, the value-added tax rate was increased by ve
percentage points. In contrast, tax rates were reduced in Argentina during
the economic boom in 1991 to 1994.
To get some insights on the capital market access, I reproduce Table 2
from the Kaminski, Reinhart and Végh (2004). It is evident that the middleincome countries tend to borrow more during good times and lend during bad
3
times i.e. net capital ow seems to be pro-cyclical. Kaminski, Reinhart and
Végh (2004) further documents that there is almost no dierence in credit
ratings for the OECD countries during good and bad times whereas ratings
are clearly pro-cyclical for the middle income countries1 . These evidences,
coupled with the negative correlation between interest rate and output documented in Uribe and Yue (2006) and Neymeyer and Perri (2005),suggest that
these countries face a positive premium on interest rate when their output is
low. This makes dicult for the government to run a scal policy aimed at
smoothening the business cycle.
Table 2: Fluctuations in Capital Flow
Countries
Net Capital Inows/GDP
Good Times Bad Times
Amplitude
(1)
(2)
(1)-(2)
0.5
0.4
0.1
4.4
3
1.4
4.2
3
1.2
3.9
3.6
0.3
OECD
Middle-High Income
Middle-Low Income
Low Income
Note : Good (bad) times are dened as those years in which GDP growth is
above (below) the median.
Source : Reproduced from Kaminski, Reinhart and Végh (2004).
3
Existing Explanations
Existing theoretical explanations of procyclical scal policy have followed
under two main strands:
• Political-economic explanations by Talvi and Végh (2005) or Alesina,
Campante and Tabellini (2007) and others
• Explanations based on international credit market imperfections or in-
completeness as in Gavin and Perotti (1997), Aizenman, Gavin and
Hausmann (1996) or more recently by Riascos and Végh (2003).
1 There
is almost no variation in capital ow or credit ratings for the low income countries but that may be because of the fact that at such low ratings they are already shut
out of the international credit markets.
4
3.1
Political-economic Explanations
Talvi and Végh (2005) argue that budget surpluses create political pressure
for increased public spending from dierent interest groups. Government may
nd it optimal to lower taxes in good times to fend o spending pressures.
However, lowering taxes in good times introduces distortion. So optimal
choice is somewhere between choosing the levels of these two distortions, i.e.
during good times the governments increase public expenditure a bit while
at the same time lower the tax rates a bit. And it is the inability of the
governments to maintain primary surplus during booms, to pay o debt,
that forces them to lower public expenditures and raise tax rates during
recessions. The authors argue that since the uctuation in the tax base is
high in developing countries as compared to the developed countries, the
surplus is higher during booms and consequently, the spending pressure is
also higher, leading to procyclical policies. There is, however, an endogeniety
problem with this line of argument. Higher uctuation in the tax base in the
developing countries may, as well, be a consequence rather than a cause of the
pro-cyclical policies. Their reduced form modeling where public expenditure
is an increasing (and, in fact, at an increasing rate) function of the primary
surplus does not address this problem.
In a recent paper, Alesina, Campante and Tabellini (2007) argued that
in corrupt democracies voters minimize rent extraction by demanding procyclical policy. When a positive income shock hits the economy, voters demand immediate benets in the form of tax cuts or increases in productive
government spending or transfers. They fear that otherwise the available extra resources would be wasted in rents. Consequently governments can not
accumulate reserves in good times to run a countercyclial policy. They show
that in the data covering 83 countries over the period of 1960 to 2003, procyclicality is highly correlated with corruption and the correlation is stronger
in democracies. However, the causality is dicult to establish since corruption is highly correlated with credit rating in the data. Moreover Thornton
(2008) provided a contrasting evidence from 37 low-income African countries during 1960-2004 that the procyclicality is rather relatively weaker in
democracies.
5
3.2
Credit Market based Explanations
The main argument is that total or partial loss of access or costly access
to international credit in dicult times forces developing countries to contract government spending and raise taxes in bad times. Gevin and Perotti
(1997) made this claim based on empirical observations from Latin American
countries.
Riascos and Végh (2003) provide a formalization of the credit channel.
They consider a setting where government consumption provides direct utility to the representative agent in an endowment economy model and they
show that if developing countries have access to only risk free bonds from the
international market then government expenditure will be pro-cyclical under
the Ramsey optimal policy. In contrast, under complete market both private
and government consumption is constant across the states of the economy.
While this paper is an important rst step, there are several directions in
which further examination is called for. Firstly, there is no production in
their model and consequently no consumption-leisure choice. If labour is
introduced in their model2 then consumption (both public and private) will
co-move positively with output under both complete and incomplete market.
Merely an incomplete credit market, therefore, is not sucient to capture
the distinguishing characteristic of the credit market faced by the emerging
countries. Endogenizing public expenditure either by introducing it in the
utility function as a consumption good or by introducing it in the production
funcition as a ow of input (or stock of public capital) will trivially always
lead to optimal procyclical expenditure policy. The challenge then is rather
to show whether the other instrument of public policy (i.e. the tax rate)
is also moving procyclically3 to or not. Riascos and Végh (2003),however,
predict a counter-cyclical tax policy. This happens because the capital ow
is counter-cyclical in their model. Both the private agent and the government can borrow more during bad times as the interest rate remains constant. So the government does not need to resort to distortionary taxes to
generate revenue during bad times. However, counter-cyclical capital ow
is completely counter factual for the emerging countries as documented by
2 Except
when the utility function is additively separable between consumption and
leisure.
3 The terminology is a bit confusing here. A policy is pro-cyclical when it accentuate
the business cycle. In case of tax rate, a negative co-movement with output will be an
example of a pro-cyclical policy
6
Kaminski, Reinhart and Végh (2004).
This paper is in a sense an extension of the Riascos and Végh (2003) to
derive the optimal pro-cyclical scal policy result in a model that at least
qualitatively matches the basic features of the business cycle data in developing countries such as procyclical capital ow and negative comovement
between output and interest rates. Due to the complexity in calculating optimal policy, typically the literature has taken a normative approach with
tractable models4 . This paper makes a half-way journey toward using optimal policy models with a positive approach to match some observations
in the data qualitatively. Full journey requires quantitatively matching the
moments in the data with a much more complicated model. However, it
is extremely dicult to calculate optimal policy in an open economy under
incomplete credit market with such a model and therefore we make the rst
step with a simpler version.
4
Benchmark Model
Consider a small open economy with three agents: households, rms and the
government. The credit market faced by the economy is incomplete in the
sense that both the households and the governments have access only to one
period real discounted bond. The interest rate on that bond is negatively
related to the deviation of the real net-worth of the economy from its steady
state level. The net worth is dened as the value of current output adjusted
for the current level of total debt obligation. Households and the government
face higher interest rates in international capital market as they need to
borrow more from the international market. I assume the following functional
form for the interest rate5 :
(y−d−b)−(yt −dt −bt ))
(
rt = r + ψ e
−1
where yt is the level of current output and dt and bt are the real private and
public debt holdings at time t and y , d and b are their steady state levels
4 See
for example Gali and Monacelli (2008) in the open economy context or Siu (2004),
Schmitt-Grohe and Uribe (2004) in the closed economy context.
5 This is following Schmitt-Grohe and Uribe (2003). In their case, however, the risk
premium does not depend on the current level of output.
7
respectively. Note r is the rate of interest without the risk premium. It is
assumed that β(1 + r) = 1 where β ∈ (0, 1) denotes the subjective discount
factor. This formulation of interest rate essentially implies that the risk premium is a function of the productivity shock. Recently similar approach
has been adopted in Neumeyer and Perri (2003), Garcia-Cicco,Pancrazi and
Uribe (2006) and Aguir and Gopinath (2007). The idea is based on models of endogenous default by Eaton and Garsovitz (1981) or Arellano (2003)
in which default probabilities are high when productivity is low and consequently the risk premium is high. This helps to generate a negative correlation between the output and the interest rate in our model. Our reduced
form approach is subject to the usual critiques but our goal here is not to
provide a theory of the risk premium in the international credit market but
to show that how it's presence aects the optimal formulation of scal policies in emerging countries. Calculation of optimal scal policy in a model of
endogenous default is beyond the scope of this paper.
4.1
Households
The lifetime utility of the innitely lived representative household is given
by:
"
#
E
∞
X
β t U (ct , ht )
(1)
t=0
where c and h are the private consumption and hours worked respectively.
U (.) is increasing in consumption, decreasing in hours, strictly concave and
twice continuously dierentiable. The representative household trades in the
international bond market and receives after tax wage income from labour,
rents from capital and spends that on consumption, investment and debt
repayment. The household's budget constraint is given by:
dt+1 + wt (1 − τt )ht + zt kt = ct + It + (1 + rt )dt + ζ(dt+1 ) + κt
(2)
where w. zt and τ denotes the wage rate, rental rate and the tax rate respectively. ζ(dt+1 ) is the loan adjustment cost faced by the household which
is introduced with the sole purpose of introducing stationarity in the model.
κt denotes capital adjustment which is convex on investment and is dened
in the following way:
κt = κ(It , kt ), kI > 0, kII > 0.
8
Lastly,
It = kt+1 − (1 − δ)kt
(3)
In addition to the budget constraint, the household is also subject to the
usual no-Ponzi game conditions. Households' problem can be described as
maximization of (1) subject to (2) and (3) and the no-Ponzi conditions. Note
households do not internalize the fact that they face an upward sloping supply
curve for loan and they take interest rate as given. First order conditions
with respect to consumption, labour supply, investment and foreign bonds
imply:
−
uh (ct , ht )
= (1 − τt )wt
uc (ct , ht )
0
uc (ct , ht )(1 − ζ (dt+1 )) = Et β [(1 + rt+1 ) uc (ct+1 , ht+1 )]
dκt+1
dκt
) = Et β zt+1 + 1 − δ −
uc (ct , ht )(1 +
uc (ct+1 , ht+1 )
dkt+1
dkt+1
(4)
(5)
(6)
First-order condition (4) implies that the tax rate introduces a wedge
between the wage rate and the marginal rate of substitution between consumption and leisure. The higher the tax rate, the lower will be the labour
supply given a wage rate. The consumption Euler equation (5) indicates that
the utility derived by investing one unit of consumption today in the foreign
bond, to consume tomorrow, should be equal to the utility forgone by not
consuming it today. This is another way of saying that, at the optimum,
marginal benet of an additional unit of debt must equal its marginal cost.
Similar interpretation goes for equation (5) which is the Euler equation with
respect to capital.
4.2
Firms
Firms are owned by the households and they produce the nal good using a
constant return to scale technology in labour and capital:
Yt = At F (kt , ht ) = At ktη ht1−η
(7)
where At is a productivity parameter. The law of motion of At is given by
the following AR(1) process:
ln At+1 = ρ ln At + t+1
9
(8)
where ρ ∈ (0, 1) and t+1 ∼ N IID(0, σ2 ). The static prot maximization
problem of the representative rm can be expressed as follows:
max At ktη ht1−η − wt ht − zt kt
ht ,kt
This gives us the standard rst order condition which equals the marginal
benet of using a labour and capital at any period t to its marginal cost:
4.3
wt = (1 − η)At ktη h−η
t
(9)
zt = ηAt ktη−1 ht1−η
(10)
The Government
In the benchmark model, government expenditure is exogenously specied
at gt = ḡ, ∀t. Government nances its expenditure either by taxing wage income of the households or by borrowing from the international credit market
where it has access to one period real non-contingent bonds bt . Unlike the
household, government, however, internalizes the fact that the interest rate
faced in the international capital market depends on the real net worth of
the country. It is also assumed that a commitment technology exists and the
government can bind itself to a particular sequence of scal policy variables.
The budget constraint of the government is given by:
bt + τt wt ht = gt + (1 + rt−1 )bt−1
(11)
The government is also subject to a no-Ponzi game condition of the following
form:
#
"
!
lim Et
k→∞
4.4
t+k−1
Y
j=0
1
1 + rj
bt+j ≤ 0
(12)
Characterizing Equilibrium
A competitive equilibrium is dened in the usual way:
Denition 1 Given the household's and the government's initial real debt
d−1 and b−1 and the exogenous stochastic processes {At }∞
t=0 and exogenous
government expenditure ḡ , an equilibrium is an allocation {ct , kt , ht , dt }∞
t=0 ,
∞
∞
price system {wt , rt }t=0 and government policy {bt , τt }t=0 such that:
10
1. {ct , kt , ht , dt }∞
t=0 solve the household maximization problem subject to
the sequence of household budget constraints and the no-Ponzi constraint.
2. {kt , ht , wt , zt }∞
t=0 solve the rm's problem.
3. {bt , τt }∞
t=0 satisfy the sequence of government budget constraints and
the government's no-Ponzi constraint.
4. The factor and the goods markets are cleared
Competitive equilibrium is obtained subject to the condition that (2)-(6),
(9)-(11) and the no-Ponzi conditions are satised. To simplify the analysis
of Ramsey optimal policy, I characterize the equilibrium in primal form.
This involves restating the equilibrium conditions in terms of real allocations
alone. Given those allocations, it is possible to recover the equilibrium values
for the price and the policy variables. The following proposition presents the
primal form of the competitive equilibrium:
Proposition 2 Given the household's and the government's initial real debt
d0 and b0 and the exogenous stochastic processes {At , gt }∞
t=0 , the allocation
∞
{ct , kt , ht , dt , bt }t=0 satisfy (5),(6) and
dt+1 −
uh (ct , ht )
ht +ηAt F (kt , ht ) = ct +kt+1 −(1−δ)kt +(1+rt )dt +ζ(dt+1 )+κt
uc (ct , ht )
(13)
bt+1 +dt+1 +At F (kt , ht ) = ct +gt +(1+rt )(bt +dt )kt+1 −(1−δ)kt +ζ(dt+1 )+κt
if and only if they satisfy (2)-(6), (9)-(11).
(14)
Proof. See Appendix.
This implies once we have {ct , kt , ht , dt , bt }∞
t=0 satisfying (5), (6), (13)and
(14), the wage rate wt and the policy variable τt can be recovered from (9)
and (4) respectively.
4.5
Ramsey Problem
Ramsey optimal policy problem can be dened as the maximization of the
lifetime expected utility of the household (1) with respect to the real allocations subject to (5), (6), (13)and (14) and the no-Ponzi conditions. In a small
11
open economy with incomplete market, it is not possible to derive a date-zero
implementability constraint as in Lucas and Stokey (1983) and Chari et al.
(1991). Instead we need to use the sequential constraints mentioned above.
Note that the constraint (5) and (6), the consumption Euler equations of the
households, involve expectations of future variables. This makes the Ramsey
planner's problem non-recursive. The government needs to take account of
the future expectations of the private households therefore the optimal choice
at time t is not a time invariant function of the state variables {dt , bt , At } at
time t.
To solve the Ramsey problem, we need to reconstruct the government's
problem in a recursive framework. Marcet and Marimon (1998) show that in
such cases an equivalent recursive saddle point maximization problem can be
constructed from the original non-recursive problem by expanding the state
space by including a new state variable that summarizes the inuence of the
past events on the choice of current allocations. Following their methodology,
as a rst step, the original problem is rewritten as:
min
max
∞
{γt ,νt }∞
t=0 {ct ,kt+1 ,ht ,dt+1 ,bt+1 }t=0
E
∞
X
β t [U (ct , ht )
t=0
n
o
0
+νt uc (ct , ht )(1 − ζ (dt+1 )) − Et β [(1 + rt+1 ) uc (ct+1 , ht+1 )]
dκt
dκt+1
γt uc (ct , ht )(1 +
) − Et β At Fkt + 1 − δ −
uc (ct+1 , ht+1 )] ]
dkt+1
dkt+1
subject to (13)and (14) and taking as given d0 and b0 . νt , γt are costate variables which are the same as the Lagrange multipliers associated
with the constraint (5) and (6) of the original problem. This problem is still
non-recursive. However, using the law of iterated expectations, it can be
easily shown that the above problem is equivalent to the following recursive
problem:
min∞
max
{νt ,γt }t=0 {ct ,kt+1 ,ht ,dt+1 ,bt+1 }∞
t=0
0
+νt uc (ct , ht )(1 − ζ (dt+1 )) −
+γt uc (ct , ht )(1 +
E
∞
X
β t [U (ct , ht )
t=0
Ztν (1
+ rt )uc (ct , ht )
dκt
dκt
) − Ztγ (At Fkt + 1 − δ −
)uc (ct , ht )]
dkt+1
dkt
12
where Ztν = νt−1 and Ztγ = γt−1 . The enlarged state space of the problem
is now composed by the vectors At , kt , Ztν , Ztγ . Ztν and Ztγ track along the
dynamic the value to the planner of committing to the pre-announced policy
plan. The Lagrangian for this problem is given by:
L=
min
max
∞
{νt ,γt ,λt ,φt }∞
t=0 {ct ,kt+1 ,ht ,dt+1 ,bt+1 }t=0
E
∞
X
β t [U (ct , ht )
t=0
0
+νt uc (ct , ht )(1 − ζ (dt+1 )) − νt−1 (1 + rt )uc (ct , ht )
+γt uc (ct , ht )(1 +
+λt (dt+1 −
dκt
dκt
) − γt−1 (ηAt ktη−1 h1−η
+1−δ−
)uc (ct , ht )
t
dkt+1
dkt
uh (ct , ht )
ht + ηAt F (kt , ht ) − ct − kt+1 + (1 − δ)kt
uc (ct , ht )
−(1 + rt )dt − ζ(dt+1 ) − κt )
+φt (bt+1 + dt+1 + At F (kt , ht ) − ct − gt − (1 + rt )(bt + dt )
−kt+1 + (1 − δ)kt − ζ(dt+1 ) − κt )]
with d0 and b0 given. Following Khan, King and Wolman (2003), SchmittGrohe and Uribe (2004) and Faia and Monacelli (2007), ν−1 , γ−1 are set at
the steady state values implicit in the system of Ramsey rst order conditions. This way we ignore the optimal policy problem at time zero. Implicit
assumption is that the policy has been optimal in the past.
Ramsey rst order conditions are given by:
0
uct + νt uct ct (1 − ζ (dt+1 )) − νt−1 uct ct (1 + rt ) + γt uct ct (1 +
dκt
d
−γt−1 uct ct (At Fkt + 1 − δ −
) − λt [1 +
dkt
dct
0
uht + νt uct ht (1 − ζ (dt+1 )) − νt−1 (uct ht (1 + rt ) + uct
−γt−1 (uct ht (At Fkt + 1 − δ −
d
−λt [
dht
uht ht
u ct
+ ηAt Fht − dt
dκt
)
dkt+1
uht ht
] − φt = 0
uct
(15)
drt
dκt
) + γt uct ht (1 +
)
dht
dkt+1
dκt
) + uct At Fkt ,ht )
dkt
drt
drt
] − φt (At Fht − (dt + bt )
) = 0 (16)
dht
dht
13
d2 κt
dκt
dκt
drt+1
d
γt uc 2 + γt−1 uc
− (1 +
)(λt + φt ) + βEt [−νt uct+1
dkt+1
dkt+1 dkt
dkt+1
dkt+1
d
dκt+1
d
dκt+1
+γt+1 uct+1
− γt uct+1 (At+1 Fkkt+1 −
dkt+1 dkt+2
dkt+1 dkt+1
dκt+1 drt+1
−
dt+1 )
dkt+1 dkt+1
dκt+1 drt+1
−
(dt+1 + bt+1 ))] = 0
(17)
+φt+1 (At+1 Fkt+1 + 1 − δ −
dkt+1 dkt+1
drt+1
00
0
νt uct ζ (dt+1 ) + (1 − ζ (dt+1 ))(λt + φt ) − βEt νt uct+1
ddt+1
drt+1
drt+1
)+φt+1 (1+rt+1 +(dt+1 +bt+1 )
) = 0 (18)
−βEt [λt+1 (1+rt+1 +dt+1
ddt+1
ddt+1
drt+1
φt − βEt νt uct+1
dbt+1
drt+1
drt+1
−βEt [λt+1 dt+1
+ φt+1 (1 + rt+1 + (dt+1 + bt+1 )
)=0
(19)
dbt+1
dbt+1
λt+1 (ηAt+1 Fkt+1 + 1 − δ −
The Ramsey equilibrium is dened by the system of equations (5), (6),
(13), (14) and (15) to (19). Since it is not possible to solve this system
analytically, I present quantitative results based on log-linear approximation
of the rst order conditions.
4.6
Parametrization
The utility function is assumed to follow the following specication:
1−σ
U (ct , ht ) =
(cαt (1 − ht )1−α )
1−σ
−1
The parameter σ is the coecient of relative risk aversion and it set equal
to 5 following Mendoza and Uribe (2000) and Reinhart and Végh (1994). The
value for α is set so that one third of the time endowment is spent on working
in the steady state. The β is chosen so that the steady state risk-free rate
of interest is 0.04 per quarter. η is the share of capital in production which
equals 0.33.
I choose values of b̄ and d¯ such that the average public and private debt to
GDP ratios are 17 % and 13 % respectively. These are the average values for
14
Mexico during the last two decade. ḡ is chosen so that the average share of
the government expenditure to GDP is 19 % which again reects the average
value for the Mexican economy.
The debt adjustment cost function for the household debt is given by:
¯ 2 . The parameter ζ equals 0.0007 as in Schmitt-Grohe and
ζ(dt ) = ζ2 (dt − d)
Uribe (2004). The
capital
adjustment cost κ takes the following functional
2
φk
It −δkt
form: κt = 2 kt kt
. We choose a value of φk to match the average
quarterly volatility of investment to output in the Mexican data. The depreciation rate δ equals 0.05 which is following the values used for the Mexican
economy in Mendoza and Uribe (2000) and Reinhart and Végh (1994).
The most crucial parameter for this model is the risk-premium elasticity
parameter ψ . Unfortunately, estimate of this parameter that can be used in
the context of this model is not available in the literature. I use a value of
0.0005 so that the correlation of real interest rate with output generated by
this model is approximately equal to 0.49 which is the value for Mexico as
reported in Neumeyer and Perri (2005). However, estimating true value of
this parameter for a particular country and then calibrating the whole model
to that economy is an agenda for the future research. The persistence parameter ρ in the productivity shock is 0.95 as reported in Aguiar and Gopinath
(2007). We choose the standard deviation of the error term in the shock
process to match the average volatility of output in the Mexican data.
4.7
Results
Figure 1 depicts the impulse responses of the key variables in response to a
one percent negative productivity shock in the economy. Since the tax-base
declines and the wage-taxation introduces distortion and thereby further reduces labour supply during a period of recession, the optimal response of the
government should be to increase debt to meet the given expenditure target.
Incurring more debt is, however, costly given that the interest rate is higher
during the bad time and more borrowing by the government will increase the
interest rate further and thereby make it dicult for the private household
to smooth consumption. In our model this trade-o leads to an increase in
the tax rate while the government debt rather goes down and thereby osets
15
some of the increase in interest rate by increasing the net worth. Private
households, in contrast, have two motives to borrow. Following a bad shock
in the economy, their income goes down and they would like to borrow more
during that time to smooth consumption. However, following a bad shock,
investment falls sharply and that reduces the borrowing needs of the private
household for investment purposes. In equilibrium, the investment eect
dominates, at least during the more recent quarters after a shock, and this
allows them to reduce their debt burden. In sum, we get a procyclical scal
policy and capital ow as a welfare maximizing outcome.
5
A Model with Endogenous Government Expenditure
In the preceding section we have seen that the optimal tax rate can be procyclical in the presence of a loan supply function that relates interest rate to
the level of net-worth of the economy. However, in that model government
expenditure is exogenously specied at a constant level. In contrast, in the
date we know that the government expenditure is strongly pro-cyclical. This
calls for endogenously modeling the government expenditure. As we have
mentioned before that the standard ways of introducing public expenditure,
as a ow of production or consumption goods or as a stock of productive capital, trivially leads to procyclical expenditure. We, therefore, take the simple
approach of introducing it as a ow of government provided consumption
goods which are exogenous to households but are optimally chosen by the
government.
We keep the basic set up of the benchmark model intact. However, the
Ramsey Planner now optimizes the following utility function:
1−σ
U (ct , ht .gt ) =
(cαt (1 − ht )1−α )
1−σ
−1
+ θlog(gt )
Ramsey planner has new choice variable gt and consequently an additional
rst order condition. We solve the model as before using the log-linear
approximation of the optimality conditions around a non-stochastic steady
state. The new parameter θ is calibrated to give steady state public expenditure to GDP ratio of 19% in the steady state.
16
5.1
Results
Figure 2 depicts the impulse responses to a one percent negative productivity
shock in this economy. Public expenditure now declines following a bad
shock. This is expected given the incompleteness of the credit market though
we have a separable utility specication for government expenditure. Tax
base declines since output falls and the government is, as before, faced with
the trade-o of increasing taxes or raising debt. However, now the planner
can reduce the expenditure and thereby minimize the tax-distortion of the
consumption-leisure choice. As the impulse response functions show, the
optimal choice is a combination of reduction in expenditure and increase in
taxes. However, the negative co-movement between the output and the tax
rate is now much weaker. The behavior of the other variables remain broadly
unchanged.
6
Conclusion
The analysis above suggests that it is possible to explain the observed procyclicalities in scal policy variables with a standard neoclassical model provided frictions in the credit market are taken into account. This is not to
say that the political economic explanations have no merit of their own. At
the policy level, dealing with these two types of frictions requires building
up two dierent types of institutions as discussed in Talvi and Végh (2005).
The purpose is just to emphasize that one should not ignore the need of
developing mechanism to strengthen the access to the international credit
market for the developing countries.
This paper is still a preliminary attempt in its approach to modeling the
behavior of the government and the international creditors in the optimal
policy context. While there are several examples in the literature of endogenously generating the risk premium by introducing the probability of
sovereign default, we have rather taken a reduced form approach. Using
those models in the optimal policy context will be extremely dicult. Moreover, we don't intend to provide a theory of the risk premium. Rather the
objective is to show that, given what we observe in the credit market, the
pro-cyclical scal policy is optimal. In that context, it will be more useful
to take up larger small open economy models that match the data well and
17
then conduct the optimal policy exercise in those models. It would also be
useful to gather data on country-specic risk premium and tax rate variability to calibrate the model in a proper fashion. Moreover, the uctuations
in dierent components of the public expenditure needs to be examined in
detail so that the public expenditure can be endogenized in a more realistic
fashion.
18
References
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A
Proof of proposition 2
Proof. First I show that {ct , kt , ht , dt , bt }∞
t=0 satisfying (2)-(6) and (9)-(11)
also satisfy (5), (6), (13) and (14). To derive (13), substitute (3)and (4) in
(2). (14) is obtained by adding (2) and (11) and substituting for wt and zt
from (9) and (10).
Now we need to check that if the allocations {ct , kt , ht , dt , bt }∞
t=0 satisfy (5),
(6), (13) and (14), then they also satisfy (2)-(6),(9)-(11). Set wt and τt such
that (9) and (4) hold so that they are satised by construction. Use the
denition of wt and τt in (13) to get back (2). Again use the denition of wt
and τt and subtract (2) from (14) to recover (11).
21
Figure 1: Benchmark Model: Impulse Responses to 1% negative Productivity
Shock
22
Figure 2: Model with Endogenous Public Expenditure: Impulse Responses
to 1% negative Productivity Shock
23