The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
DEBT AND DEFICIT CEILINGS, AND SUSTAINABILITY OF FISCAL POLICIES:
AN INTERTEMPORAL ANALYSIS
by
Merih Uctum and Michael Wickens
Federal Reserve Bank of New York
Research Paper No. 9615
June 1996
This paper is being circulated for purposes of discussion and comment only.
The contents should be regarded as preliminary and not for citation or quotation without
permission of the author. The views expressed are those of the author and do not necessarily
reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045
DEBT AND DEFICIT CEILINGS, AND SUSTAINABILITY OF FISCAL POLICIES:
AN INTERTEMPORAL ANALYSIS
by
MERIH UCTUM* AND MICHAEL WICKENS**
May 1996
• Economist, Federal Reserve Bank of New York.
**Professor of Economics, Department of Economics, University of York, United Kingdom.
We wquld like to thank Deborah Roseveare, Michael Woodford for very helpful comments and Sandra Viana
for excellent assistance. The opinions expressed in this paper are solely those of the authors and do not
reflect the opinions of the Federal Reserve Bank of New York or the Federal Reserve System.
Abstract
In this paper we examine the likely consequences for the sustainability of fiscal policy of pursuing
goals that rely on restrictive ceilings on deficits and debt. We provide a formal theoretical framework for
analyzing the sustainability of fiscal policy based on the government intertemporal budget constraint and
derive conditions that determine whether a given fiscal stance is sustainable. This framework generalizes the
existing literature in several important respects. We allow for time-varying interest rates, for the primary
deficit to be endogenous, for a finite planning horizon suitable for medium-term policy making, for possible
future policy shifts, we show how published forecasts can be used and we provide a measure of fiscal
pressure. We then apply this analysis to the fiscal positions of the United States and the European Union
countries since 1970 and to their planned positions over the next decade.
We find that many countries do not have a sustainable policy. The evidence in favor of sustainability
is strengthened for most countries when the data are extended to incorporate future fiscal consolidation plans,
reflecting the general shift toward fiscal austerity in recent years. In contrast. with a finite horizon we show
that the recent policy shift made the paths of future policies sustainable. However, imposing ceilings on debt
or deficit-to-GDP ratios throws most economies onto an unsustainable path unless governments undertake a
major tax or expenditure adjustment. High-debt countries can satisfy the 60 percent debt rule by 1999 only
by raising (reducing) the average tax (spending) rate substantially for five years. A 3 percent deficit plan by
1999, or a zero-deficit plan by the year 2002 puts an increasingly high pressure on most economies, including
the United States, requiring a gradual rise (decline) in the tax (spending) rate.
Key Words: Fiscal policy, sustainability, fiscal criteria, stochastic discount rate, forecasts.
JEL Classification: E6, H87, H6
The sustainability of fiscal policies is yet again in the headlines, both in the United States
and Europe. In the United States Congress and the Administration have proposed budgets
designed to reduce the deficit to zero by the year 2002, while European countries are trying to
meet the challenging fiscal constraints of the Maastricht treaty by 1997. The treaty requires
governments to run a deficit of no more than 3 percent of GDP with a maximwn debt-to-GDP
ratio of 60 percent. In most countries this implies a shift to austerity that is likely to entail a cost
of foregone output and higher unemployment. The authorities often justify their restrictive stand
by pointing to the necessity of pursuing "sustainable" fiscal policies that do not cause an excessive
build up of debt and therefore put pressure on monetary policy.
In this paper we examine the likely consequences for the sustainability of fiscal policy of
pursuing goals that rely on restrictive ceilings on deficits and debt. We provide a formal
theoretical framework for analyzing the sustainability of fiscal policy based on the government
intertemporal budget constraint and derive conditions that determine whether a given fiscal stance
is sustainable. This framework generalizes the existing literature in several important respects.
We allow for time-varying interest rates, for the primary deficit to be endogenous, for a finite
planning horizon suitable for medium-term policy making for possible future policy shifts. We
show how published forecasts can be used and we provide a measure of fiscal pressure. We then
apply this analysis to the fiscal positions of the United States and the European Union countries
since 1970 and to their planned positions over the next decade.
There is a large literature on the intertemporal budget constraint. The general conclusion
to emerge from this literature is that fiscal policy is sustainable if the government budget
constraint holds in present value terms. More precisely, the current debt should be offset by the
sum of expected future discounted primary budget surpluses (exclusive of interest payments).
Two approaches to analyzing the sustainability of fiscal policy have emerged in the
literature. The first consists of testing the stationarity of the debt and/or deficit. Results vary with
the specification of the budget constraint. Hamilton and Flavin (1986) reject nonstationarity of
constant-dollar undiscounted U.S. debt under the assumption of constant real interest rates.
Smith and Zin ( 1988) obtain the same result with a similar specification for the Canadian data.
I
Wilcox (1989) allows for stochastic interest rates and finds that discounted U.S. debt is
nonstationary. Other studies look for a cointegrating relationship linking the primary deficit, the
stock of outstanding debt and interest payments for the United States. The results of Trehan and
Walsh (1988), among others, find support for the sustainability of U.S. fiscal policy by showing
that the deficit inclusive of interest payments is stationary. In contrast, Kremers (1989), and
Hakkio and Rush (1991) show that in recent years fiscal policy violates the intertemporal budget
constraint. More recently, using historical data that goes back to 1700s Ahmed and Rogers
(1995) find strong evidence favoring sustainability of the U.S. fiscal policy and some support for
sustainability of the U.K. fiscal policy.
All of these tests of sustainability are based on the assumption that the processes
generating deficits and debt will continue into the future. In practice, however, altering fiscal
policy to achieve sustainability may be necessary. This suggests the need to analyze sustainability
allowing for expected future changes in fiscal policy. Using a sustainability indicator Blanchard et
al. (1990) find that most OECD countries are following sustainable policies in the medium-term.
This indicator consists of the difference between the current tax rate, and a sustainable tax rate
constant over a finite horizon that leads to a stable debt-to-GDP ratio. Using a similar approach,
Buiter, Corsetti and Roubini (1993), and Roubini (1995) find.that many OECD countries follow
unsustainable policies.
Although this approach is forward-looking, existing studies do not use forecasted values
for the fundamental variables; instead, they assume them to be constant. However, as the large
fluctuations in interest rates in 1980s show, any assumption of constancy is unrealistic in the
medium-term. Moreover, these studies calculate the current fiscal pressure arising from the need
for solvency. Yet, in reality, governments plan their fiscal policy over a medium-term horizon,
and smooth out the changes they plan to implement.
In this paper we examine the sustainability of fiscal policy in the U.S. and in the EU by
combining these two approaches. The contribution of this paper to the literature is twofold. First,
we extend the results of Wilcox ( 1989) to the case where the discount rate is stochastic and timevarying, and the discounted primary deficit can be either exogenous or endogenous. We show
that a necessary and sufficient condition for sustainability is that the discounted debt-GDP ratio
2
should be stationary. Second, we use forecasted values for the fundamental variables. In addition
to extending our sample, forecasted values allow investigation of the future fiscal stances of
governments, and calculation of the fiscal pressure for each year over the next five years. Thus,
we are able to capture the spirit of a fiscal plan by providing a profile of the evolving fiscal
pressure, while quantifying the impact of ceilings on each country's fiscal profile.
Our empirical results indicate that with an infinite-horizon, there is some evidence that the
fiscal stance in Denmark, the Netherlands and Ireland is sustainable, while fiscal policy in the
United States, Spain, Italy, Belgium, and Portugal is not Results improve for all countries when
we include future fiscal consolidation programmes by extending the sample into the year 2000.
Our simulation analysis shows that under current policies, the paths of future policies are
sustainable for all countries. However, imposing Oor 3 percent deficit limits, and 60 percent debt
limits within the next 3 to 6 years throws most governments' budgets onto an unsustainable path
unless they generate major primary surpluses.
The paper is organized as follows. In the first section, we derive the government's
intertemporal budget constraint for a stochastic discount rate adjusted for inflation and real GDP
growth. We discuss the implication of this constraint for the sustainability of fiscal policy under
both infinite and finite horizons. In the second section, we present the empirical results on the
sustainability of EU and U.S. fiscal policies. In the third section, we examine the effect of
imposing deficit and debt ceilings on government budgets. We present our conclusions in the
fourth section.
I. The government intertemporal budget constraint
The government intertemporal budget constraint can be written in nominal terms as
G,-T,+i,B,_ 1=t::..B, +t::..M,= -S,
where G=government expenditure, T=tax revenue, B=government debt at the end of period t,
3
(1)
M=monetary base, S=total budget surplus, i=interest rate on government debt. 1 The correct
implementation of the budget constraint requires the use of the net market value of debt. Net
debt is defined as gross debt minus financial assets.
Expressing (I) in terms of ratios to nominal GDP we get:
(2)
where the lower-case letters g, 't, b, m, ands denote the ratio of the corresponding upper-case
variables-to-nominal GDP, 1t, = (P,-P,. 1)/P,. 1 and TJ,=(Y,-Y,. 1)N,. 1, with P and Y standing for the
price level and real GDP. Equation (2) says that the interest-inclusive government deficit is
financed by new bond issues, base-money creation and seignorage. Equation (2) can be rewritten
as:
(3)
where d,=g,-'t,-tun,-( 1t,+11,)m,. 1 is the primary government deficit expressed as a proportion of
nominal GDP and p,=i,-1t,-T], is the real ex post interest rate adjusted for real output growth.
Equation (3) is a difference equation that can be solved forward to enable the current debt-GDP
ratio to be expressed in terms of expected discounted future primary deficits over the next n
periods:
n
b,=E, o,,.h, •• -E,
E
o,/f,.;
i=1
(4)
where o"", the time-varying real discount factor n periods ahead, adjusted for real GDP growth
rate, is defined as 2
1
Strictly speaking, the left-hand-side of equation (I) also includes a term reflecting changes in general
government net debt due to factors such as revaluations of financial assets, exchange-rate changes, and privatizations.
The debt series used in the analysis are consistent with this definition.
2
Most studies of sustainability assume a constant discount factor (see Hamilton and Flavin 1986, Trehan and
Walsh, 1988, Kremers, 1989, Blanchard et al., 1990). An exception is Wilcox (1989). For n periods, this assumption
would amount to a constant discount factor ll=(l+p)"".
4
n
o,,. = rr o+p,,,r 1
s=l
Equation (4) is often expressed by discounting each of the variables back to period 1. 3
Defining {a,; s=l ,2, ... ,t+n) to be the sequence of discount factors from period t back to period 1,
an arbitrary earlier point in time, {a.,.; s=l,2, ... ,t+n} to be the sequence of discount factors from
period s+n back to period 1, with
ct 0=0,
and o,..=a,.ja,, enables equation (4) to be written as
n
a/;,=E,a,+nbl<t-n
-E,L a,,t,,d, . .;
(5)
i= l
Let X, be the discounted value of the debt-GDP ratio, and
Z. be the discounted value of the
primary deficit-GDP ratio, both variables discounted back to period I. Equation (5) can then be
written more compactly as:
n
x,=E,X, •• -E,L z,.;
(6)
j,.J
Similarly, by multiplying equation (3) through by a,, we can write the government budget
constraint in discounted terms as:
Z,=llX,
(7)
1. Sustainability of fiscal deficits with an infinite horizon
A necessary and sufficient condition for sustainability is that as n goes to infinity the
discounted value of the expected debt-GDP ratio converges to zero. This is also known as the
transversality condition, and implies that no Ponzi games are allowed, meaning no new debt is
issued to meet interest payments. This condition can be expressed as4 :
3Toe choice of base period for calculating Ihe discount rale is unimportanl as ii does not affect Ihe test for
stationarity. It can be chosen on Ihe grounds of convenience.
"For band p constant, p must be positive. Ball et al. (1995) note !hat p, is negative for IO to 12 percent of Ihe
time wilh Ihe return on government debt less Ihan Ihe growlh rate of GDP. However, negative values of p, can still be
compatible wilh Ihe transversality condition as long asp, is predominantly positive.
5
(8)
It then follows that the current debt-GDP ratio is offset by the sum of current and expected future
discounted surpluses expressed as a proportion of GDP, implying that the governmen t budget
constraint holds in present value terms with
n
b,= -lim•• ~ E,L, l'>,.,d,.,
(9)
i=l
When the debt-GDP ratio is discounted back to period I, the transversality condition and the
government intertemporal budget constraint are:
(10)
X,= -lim •• ~
n
E,L, z,.,
i:: I
(11)
It may be noted that the transversality conditions in (8) and (10) do not require that the debt-GDP
ratio goes to zero, only that it does not grow faster than the growth-adjusted real discount rate.
In principle, current debt can be sustainable by any sequence of primary deficits or surpluses,
provided they satisfy equation (9) (or (11)), meaning that they offset the current level of debt.
The transversality condition has been tested in a number of ways, each depending on the
processes postulated for di and p,. Hamilton and Flavin (1986), and Trehan and Walsh (1988)
examine the case where di is strictly exogenous and p, is constant Wickens and Uctum (1993)
conduct the analysis with endogenous di and a constant p,. Wilcox considers the case with
exogenous
di but variable p,.
In this paper we consider a more general case where p, is stochastic and di (or z.J is
allowed to be either exogenous, or endogenous. We show that in both cases, the necessary and
sufficient condition for the intertemporal budget constraint to hold is that the discounted debt·
6
GDP ratio should be stationary. Before proceeding further, we define the concepts of exogeneity
and endogeneity employed in this paper.
z. is said to be exogenous with respect to X, if it is generated by a process
Zt =µ +e t =µ +0(L)E t,
00 =l
(12)
where e, is iid(O,o 2 ) and 0(L) is a polynomial in the lag operator L either with one root on .the
unit circle and the rest outside (when
Z. is 1(1)), or with all roots outside the unit circle (when Z1 is
1(0)).
Z. is said to be endogenous with respect to X, if there is feedback from discounted debt to
the discounted primary deficit, i.e., if it can be expressed by a functional form such as:
Z,=µ-aX,_ 1 +e,=µ-aX,_ 1 +0(L)e,
(13)
Equation (13) can be viewed as having been derived from a complete econometric model withe,
summarizing the effect of all of the other variables in the model, including any dynamics.'
PROPOSmON. Whether
Z. is exogenous or endogenous, a necessary and sufficient
condition for the transversality condition given by (10) to be satisfied is that X, is a stationary
process.
PROOF (see Technical Appendix).
An assumption underlying this conclusion is that the process generating
Z. is structurally
stable and will remain so in the future. In practice, if fiscal policy is currently unsustainable, it will
require to change in the future. Another assumption is that the time horizon is infinite. However,
both the EU and the United States governments are facing self-imposed deadlines that are less
than 10 years away. Thus, for practical purposes, we need to consider a finite-horizon version of
5One justification
for this negative feedback arises if government debt generates higher tax revenues which
reduce !he primary deficit. If government debt is regarded as net wealth, the additional spending increases both indirect
and, through higher income, direct tax revenues.
7
the sustainability problem, which pennits a change in future policy. We turn to this issue in the
next section.
2. Sustainability of fiscal policy with a finite horizon
In the medium tenn current fiscal policy can be said to be sustainable if it is able to achieve
a given target level of the debt-GDP ratio. Unlike the infinite horizon case, this level may be nonzero. Denoting the desired level of the debt-GDP ratio at the end of the planning period by b*,
from equation (4) the government intertemporal budget constraint becomes
n
b, -E/\,.b,:n= -E,L
a,,,d,.;
; .. 1
(14)
Equation (14) can be interpreted as follows. The left-hand-side (LHS) is the difference between
the current debt-GDP ratio and the expected value of the desired discounted debt-GDP ratio n
periods ahead. In other words, it is the desired change in the discounted debt-GDP ratio. The
right-hand-side (RHS) is the flow of discounted future primary deficit/surplus-GDP ratios,
expected at time t.
Equation (14) detennines the flow of future primary balances required to achieve the debtGDP objective, on the basis of the current fiscal stance. If equation (14) is satisfied then the
current fiscal stance can be said to be sustainable. If it is not satisfied, then either the debt
objective or the fiscal stance must be altered. If, for example, a government wishes to reduce its
debt burden then the LHS will be positive. To be sustainable, fiscal policy is then required to
produce a corresponding sequence of discounted surpluses. An example of this case is the U .K.
experience of the early 1980s.
A positive LHS may also be caused by a low discount ratio (or a high discount rate) due
to high real interest rates or low growth. This makes the present value of the future debt-GDP
ratio lower than the current value. In this case, even though there may be no change in the actual
level of debt (or in the debt-GDP ratio), pursuing a tight fiscal policy will still be necessary. This
example illustrates a need to adopt an active policy in the face of cyclical changes.
A change in the fiscal stance can be accomplished by changing taxes or government
8
expenditures or by money financing.• Blanchard et al. (1990) have considered the average tax
rate t* required to satisfy the government intertemporal budget constraint, and used it to derive a
measure of fiscal pressure under the assumption of a constant discount ratio. This measure is
defined as the discrepancy between the present tax rate and the sustainable tax rate. Their
assumption of a constant discount rate is restrictive and can be relaxed, for it is possible to
calculate the actual tax rate at every period in time and compare it with the sustainable tax rate at
every period in time.
Thus, if the discounted surplus is insufficient to permit the desired reduction in discounted
debt, then an increase in the average tax rate of
n
t'
n
1
-t=[E,L 11,,;1- [b,-E,6,,,b,:. +E,L 6,,,d,.;l
i=l
(15)
i=l
is required to satisfy the intertemporal budget constraint through to then-period horizon. The
change in the average tax rate will vary over time as new information becomes available on future
primary deficits/surpluses and the future discount rate. The size of the tax change required is a
measure of fiscal pressure. An alternative to a tax change would, of course, be to change
government expenditures.
The intuitive appeal of this fiscal indicator is that it is a simple measure that can be easily
implemented because it is based on obseivable variables or forecasts available at time t. Since the
terminal condition in the medium-term will not usually require the complete annihilation of
government debt, the indicator provides a useful medium-term policy tool connecting the desired
evolution of debt with expected future deficits.
II. Sustainability of U.S. and EU fiscal policies: solvency tes1s
In this section, after describing the data used in the analysis, we examine the trends in the
debt-GDP ratio defined in various ways. We then conduct a stability test of the discounted value
6
Strictly speaking, the primary deficit does not have to be generated by a structurally stable process since the
expectation operator allows an anticipated future change of fiscal policy to ensure that the government intenemporal
budget constraint is satisfied.
9
of government debt.
1. Data
There are two major issues that must be addressed when using government debt data:
whether to measure debt at market value or at face value (at par), and how to measure the
discount rate. The public debate on fiscal policy is based on official budget measures expressed in
nominal terms. However, as several authors pointed out (Eisner and Pieper, 1986, and Hamilton
and Flavin, 1986) this measure misstates the true fiscal stance of the government. From a
conventional accounting perspective, we need to consider not only the inflation rate but also the
market value of the debt and the deficit. The correct implementation of the government
intertemporal budget constraint requires the use of the discounted net market value of debt.
However, the available debt figures are usually expressed at par, since the.market value of debt
for the whole sample is available for just a few countries. 7 We obtain an estimate of the market
value of debt b, by multiplying the face value by the implied market price 1/( 1+pJ, where p, is the
yield on government debt. Thus, we calculate the market value of the debt-GDP ratio as b, =
(1 +p/ 1 bN,, where bN, is the government debt-GDP ratio at par.
The measurement of the yield on government debt is complicated by its heterogenous
composition. Government debt is commonly sold at a discount and redeemed at par, i.e., at face
value. At any point in time, the new debt issued will be redeemed at different dates in the future,
(i.e., will contain different maturities) and have different coupons. The appropriate measure of the
rate of return on government debt in any period is a weighted average of the one period holdingreturns for each of the maturities in existence, the weights being the share of any issue in the total
market value of outstanding debt. Since these weights are in general unknown, it is necessary to
estimate the average rate of return in a different way. In this paper, we adopt the most common
way that consists of dividing total net debt interest payments by the end of last period's (or the
7
Eisner and Pieper (1984) calculate the market value of the U.S. government debt by quantifying the market
value of tangible assets owned by the governmem.
10
beginning of this period's) face value of net debt 8
Our sample covers the period 1965 to 1994. 9 To capture any structural change that might
occur following the recent shift toward fiscal austerity, we also extend the sample period to the
year 2000 and include the forecast values of discounted and undiscounted debt
All data are annual and are taken from the OECD Economic Outlook (1994, 1995). The
same publication also gives forecasts of the GDP growth rate, the inflation rate and government
debt until 1997 and forecast of these variables (except net debt) for the year 2000, based on a
medium tenn base scenario. 10 Forecasts for the intervening years 1998 and 1999 are constructed
using a linear interpolation. Forecasts between the year 2000 and 2005 are kept constant.
Implicitly, this amounts to assuming that the dynamic path is converging toward a steady state at
the start of EMU. The forecasts for the primary deficit and net interest payments are from
unpublished OECD sources. The series for the high-powered-money until 1995 are obtained from
the International Financial Statistics 0ine 14). Between 1995 and the year 2000 we assume that
the base money grows at the same rate as nominal GDP.
Data for net debt after 1997 are not available, therefore we generate a net debt series until
2000 by assuming that net debt grows at the same rate as gross debt. This assumption implies
that the ratio of government financial assets-GDP is constant after 1997, or that nominal financial
assets owned by government grow at the same rate as nominal GDP. Hence, for t>l997, we use
the fonnula ~bN,=~b0 ,, where b0 is gross debt.
8
A problem in implementing this approach, which is usually overlooked, is that the published data on net
interest payments will be an underestimate of the figure required because it is based only on coupon payments. Thus, if
all debt were zero coupon (e.g., Treasury Bills), the published value of interest payments would be zero. In a
sustainability analysis, what is required is the implied interest payment. For zero coupon bonds this would be the
holding period return multiplied by the previous period's market value of that issue.
9nue to unavailability of government debt series, the starting dates are 1970 for France, and Italy, 1976 for
Spain, 1977 for Ireland, and 1980 for Austria.
1
°The figures for government debt refer to the debt of the general government sector, defined as the aggregate
of the central and local government sectors and the social security sector. We should note that due to data unavailability,
we conducted the analysis for Ireland and Portugal with gross debt.
11
2. Stylized facts
The rise in government debt has been a major concern for policy makers for the last
decade. The overall picture, however, reflects a general tendency for fiscal austerity in the
coming years. Inspection of the data reveals that in several countries the slippage of the early
1990s, which was preceded by major consolidation efforts, is expected to give way to fiscal
austerity toward the year 2000 (Chart I).
During the latter half of the 1980s fiscal austerity in most countries lead to a decline or
stabilization of the net (undiscounted) debt-GDP ratio. By contrast, in the early 1990s, following
the German unification and the European recession, debt-GDP ratios increased rapidly. The U.S.
debt-GDP figure has behaved similarly; in the second half of the 1980s, fiscal austerity and rapid
growth lead to a decline in this ratio, and the subsequent slowdown in the economy caused it to
increase.
As noted above, the transversality condition requires discounted debt to converge to zero.
A short-run indicator is that discounted debt should be falling over the forecast period.. Extending
the series by including forecasted values to the year 2000, the data indicate that several countries
have a declining discounted debt-GDP ratio, such as France, Denmark, the Netherlands, Belgium
and Ireland. 11 More formal tests based on unit root tests are reported in the next section.
3. Stationarity tests
In this section we conduct ADF and Phillips-Perron tests for the undiscounted debt (Table
1, first four columns) and for the discounted debt at par (Table 1, last four columns). We also
conducted the tests for the discounted debt at face value, but we do not report the results since
they are very similar. The tests were performed on actual data until 1994, and on a data set
extended to include the forecasts up to 2000.
The general conclusion to emerge from these tests is that the market value of the debtGDP is mean reverting in some countries, and there is a general improvement in fiscal stances
11
Chart I reveals two other points of interest. Firs~ in all countries the market value of debt is less than the
face value. This is consistent with the finding of Eisner and Pieper ( 1989) who show that by 1980, the market value of
the U.S. government debt was $66 billion less than its par value. Second, the discounted market value of debt lies
below the undiscounted value, except in high nominal growth countries.
12
toward the end of the century. The null hypothesis of a unit root cannot be rejected for the
undiscounted debt-GDP ratio but can be rejected for several countries when debt is discounted.
For the Netherlands and Denmark, the null hypothesis is rejected at the 10 percent significance
level with both tests. In the case of Ireland, the rejection occurs at IO percent with the PhillipsPerron test. The effect of extending the data period by including the OECD forecasts is slightly to
reduce the test statistics, reflecting the recent shift toward fiscal austerity. The United Kingdom,
and to a lesser extent Italy and Belgium now get closer to having a mean reverting debt process.
m. Sustainability of U.S. and EU fiscal policies: a simulation analysis
We now consider the sustainability of the current fiscal stance of the United States and the
EU countries over a 5-year horizon to the year 2000. Our aim is to determine whether the fiscal
deficit over the next several years is compatible with the evolution of government debt. We show
that most countries' fiscal stance is sustainable, but imposing Oor 3 percent deficit limits, or a 60
percent debt limit would throw most governments' budgets onto an unsustainable path unless a
policy change that generates primary surpluses is implemented.
For each period up to 1989 we calculate the ex post fiscal pressure resulting from the ex
post reduction in debt over the next five years (b,-61.11b1+0 ) and the ex post cumulative discounted
surplus -[l'>ub,._1• After 1990, we incorporate into the computations the forecast values for
growth, inflation, the interest rate, debt and the deficit. For example, the 1991 measure of the
fiscal stance is based on the actual values of 1992, 1993 and 1994, and the forecasts for 1995 and
1996. From 1994 onwards, the calculations use only forecasted values.
First, we consider the current situation and evaluate the fiscal pressure entailed in fulfilling
the sustainability criterion. This involves using equation (15) to calculate at each point in time the
change in the average tax rate required over the next 5 years (i.e., through the 5-period horizon)
to satisfy the intertemporal budget constraint. Second, we repeat this calculation imposing the
condition that countries have to satisfy the 60 percent Maastricht debt criterion. Third, we
examine the consequences of replacing this condition by a deficit limit of Oor 3 percent
13
1. Where do countries stand with respect to the sustainability criterion?
When the government debt and deficit are discounted with implicit rates, all countries
appear to satisfy the intertemporal budget constraint. Chart 2 displays our measure of fiscal
pressure. This is based on using the market value of debt and a seignorage-adjusted primary
deficit The horizontal axis denotes years (from 1980 to 2000) and the vertical axis shows the
fiscal pressure as a percentage of GDP. In each panel, the area above 0 represents positive fiscal
pressure, and indicates the need for a tightening of fiscal policy (by, for example), a higher tax
rate to satisfy the intertemporal budget constraint. In other words, positive fiscal pressure
indicates that to achieve the desired change in government debt, larger future surpluses are
required. Conversely, the area below 0 represents a negative fiscal pressure, showing that the
intertemporal budget constraint is not binding and that the government can even follow an
expansionary policy.
Chart 2 indicates that by the year 2000, almost no country needs to raise the average tax
rate because the fiscal pressure is non positive. In fact, under current policies, several countries
even have the option of fiscal expansion ranging from 0.1 to 4 percent of GDP without violating
the intertemporal budget constraint. The United States and Spain are the only exceptions. They
have been pursuing unsustainable policies that have severely reduced their room for maneuver
during the last decade, however, the required fiscal adjustment is modest. Both countries need to
raise their tax rate by 0.2 to 0.3 percent by the year 2000.
More insight into these results can be obtained by examining the behavior of the two
components of equation (15) depicted in Chart 2. These components are the desired change in
the debt-GDP ratio (the left-hand-side of equation (14)), and the cumulative discounted surpluses
(the right-hand-side of equation (14)). These are shown in the chart in appendix A. The left
panel is the desired change in the debt-GDP ratio. A negative slope violates the sustainability
condition because it indicates that the discounted value of debt five years ahead is larger than the
current outstanding stock of debt. The right panel represents the cumulated discounted primary
surpluses. The area under the curve is the cumulated future primary surplus (deficit) if it is
positive (negative).
For most countries the late 1980s is a period of rising desired debt accumulation (left
14
panel). The desired discounted debt is higher than the current stock of debt and the desired debt
accumulates at an increasing rate. In most cases this trend is reversed by the early 1990s when
the rate at which desired debt accumulation slows. By 1995, all countries are back to a
sustainable path of government finances with a lower desired discounted debt-GDP ratio than the
current one. The United States reached the sustainable path in 1993 and has been on a stable path
since then. The trend reversal is more dramatic in high-debt countries such as Italy, Belgium and
Ireland due to the maintenance of a high rate of desired debt reduction throughout the 1990s.
The right panel of the chart in appendix A shows how these countries finance their desired
debt accumulation. The second-half of the 1990s reflects a general tendency for fiscal
consolidation with cumulated primary surpluses in all countries. The most virtuous countries are,
again, the most indebted ones. Ireland, and Belgium have been running discounted cumulated
primary surpluses since the mid-1980s, and Italy since 1989. The United States had periodic
small discounted cumulated primary deficits until 1992, after which it achieved a balance. Thus,
fiscal consolidation in most countries and a reduction in the desired debt accumulation account for
the lack of fiscal pressure displayed in Chart 2.
To analyze the effect of inflation on deficit financing we also perform simulations with the
primary deficit unadjusted for seignorage. We find that although the inflation-tax turns out to be
insignificant for several countries, it still constitutes a nonnegligible source of finance for others.
Not surprisingly, seignorage revenues are highest in countries with an expected inflation rate of
greater than 3 percent at the end of the century. Between 1996 and 2000, Spain and Portugal are
expected to finance respectively 2 and 5 percent of their deficits through seignorage revenues,
while the United States and Ireland will use around 1½ percent of inflation financing.
2. Imposing the 60 percent debt ceiling of the Maastricht treaty
The Maastricht debt limit is imposed on gross government debt at par in 1999. 12 We need
to translate this restriction into a limit on the net market value of debL From the assumptions that
. 12S1rictly speaking, the debt limit imposed on gross public debt defined under the Maaslricht Treaty differs
somewhat from the corresponding OECD figures (for more details see OECD, I 994, p.17). However, since these series
start only in 1990 and they are not complete for all countries, we used the OECD series for the empirical analysis. For
consistency, we also used the same OECD series for the simulation analysis.
15
b0 ,- b0 ,.,= bN,- bN1_1 and b, = (I +pJ·' bN,, we can express the net market value of debt in terms of
gross par value of debt 13 b,=(l+pJ· 1c(l+p1_1)b1•1+b 0 ,-b 0 1•1]. Thus, for t=l999, and b0 1999=0.6
percent, the net debt-GDP ratio at par becomes b1999=(l+p 1999)° 1((l+p 1998 )b 1998+0.6-b0 , 998].
Imposing this debt requirement for the year 1999 would make several countries' fiscal
stance unsustainable and would require fiscal tightening. Not surprisingly, the 60 percent debt
requirement does not affect countries where the constraint is non binding (Germany, France, the
United Kingdom, Denmark, Austria) since this assumption is incorporated in the forecasts of the
OECD (Chart 3). It increases fiscal pressure moderately in countries where the constraint would
be binding because their debt-GDP ratio is slightly above this ceiling (the United States -if it had
to satisfy the criterion--, Spain, Portugal, Ireland and the Netherlands). However, in countries
with a debt-GDP ratio above 100 percent, the debt criterion is politically infeasible because. it
requires sizeable sacrifices (Italy and Belgium).
Fiscal pressure is, in general, mean reverting even in high-debt countries. By 1999, the tax
rate required for sustainability falls below the current level, indicating that the constraint becomes
nonbinding. The two exceptions are again the United States and Spain. In both countries fiscal
pressure remains small but positive and appears to continue beyond the year 2000.
To satisfy both the intertemporal budget and debt constraints over the course of 4 to 5
years, the medium-debt countries need to generate cumulated primary surpluses ranging from 6
percent to 12 percent of GDP. The Netherlands, Spain, and Portugal are required to raise their
tax revenues or reduce expenditures by a total of 5 to 6 percent of their respective GDPs within 7
years (starting from 1994 until the year 2000), amounting to an average of roughly I percent a
year. Clearly, the later the starting date, the larger is the required annual rise in the tax ratio.
Ireland, a medium-debt country since the end of the 1980s, needs a more radical fiscal
retrenchment, despite a cumulated discounted surplus of 70 percent between 1994 and 2000
(right panel, chart in appendix A). This will represent an increase in debt pressure of about 2
percent.
13
This expression is obtained as follows: From the first assumption we get b",=b 0 ,b0 0 ,+b"., . Replacing b0 , in
the second assumption we gel b,=(l+p,. 1)° 1(b0 ,b\1+b",.,). The second assumption also implies that b,. 1=(1+p,_ 1)" 1b",.,.
Solving for b",., in terms of b,., and replacing it in the previous equation, we get an expression for the net market value of
debt as a function of par value of gross debt.
16
If the United States had to comply with the Maastricht debt criterion, it would fall into the
category of medium-debt countries, and would be required to increase tax revenues (or reduce
expenditures) by a cumulated 1 percent of GDP over 7 years. Although small, this figure
represents an increase of 16 percent over what it would have had to generate if its aim were
simply to satisfy its intertemporal budget constraint over the same period.
Italy and Belgium constitute a third category where fiscal pressure on otherwise solvent
economies becomes very severe. Between 1994 and 2000, both countries have been and are
planning to generate primary surpluses of about 25 percent per year. However, to satisfy the debt
criterion and be solvent by 1999, Belgium must further raise its primary surplus by a cumulated 55
percent of GDP, or an average of 8 percent of GDP per year, bringing the annual primary surplusGDP ratio to 32 percent. The additional sacrifice is slightly lower for Italy, 35 percent (5 percent
per year), but still politically infeasible since it raises the total annual primary surplus-GDP figure
to 30 percent.
This analysis shows the economic and political infeasibility of the strict application of the
Maastricht treaty's debt criterion for governments with initial high indebtedness. In contrast, the
fiscal pressure entailed in satisfying the intertemporal budget constraint without imposing the debt
criterion is generally viable and still imposes a fiscal discipline.
3. Imposing deficit ceilings
In this section, we show that, like debt ceilings, deficit ceilings make the fiscal stance of
countries unsustainable unless tax ratios are raised significantly. However, unlike the debt ceiling,
the deficit ceiling creates a positive and persistent fiscal pressure on all countries, which lasts
beyond the date these limits are imposed.
The deficit limit in the Maastricht Treaty is defined inclusive of interest payments on debt
To impose a deficit ceiling, we calculate a debt-GDP ratio consistent with the constraint. From
equation (2) we can express the outstanding stock of current debt in terms of the deficit inclusive
of interest payments bi as
b, =h, +[ I -( TJ, +1t,) Jb,_ 1
17
(16)
A ceiling on total deficits can be expressed by imposing a limiting value on h.. We
consider the effects on the intertemporal budget constraint of two types of deficit limits: a 3
percent limit by 1999, as required by the Maastricht Treaty, and a Opercent limit as planned by
the U.S. government by 2002. In both cases a deficit limit imposed by the end of the century
forces all governments to adopt even more austere fiscal programmes even though they follow
otherwise sustainable fiscal policies.
To satisfy the balanced budget condition by the year 2002 and simultaneously be solvent,
between 1996 and 2000, the United States would have to maintain a positive growth rate of the
tax-GDP ratio of about 7 percent per year. This amounts to a cumulated surplus of 18 percent of
current GDP over the course of 4 years, which is 17 percent more than what is required to satisfy
solvency. Moreover, the rate of increase continues to be positive beyond the year 2000.
Although the fiscal retrenchment in Germany, and France is less severe, both countries
would need to increase the growth rate of their tax-GDP ratio by about 5 percent per year for 3
years until 1998, and keep the inflow of revenues at an annual constant rate of 2 percent of GDP,
thereafter. While in the second half of the 1990s the U.K. budget was not under any fiscal
pressure, it would now need to generate a primary surplus of an average of 1 percent of GDP.per
year for 5 years until 1998, and keep it constant at 1/2 percent of GDP afterwards.
Austria and the Netherlands were also previously among the countries that did not have
any fiscal pressure. Now, between 1996-1998, they would both be required to raise their tax
revenue respectively by I and 1/2 percent of GDP on average. Ireland would have to generate an
additional 3 percent annual primary surplus between 1994-98. All three countries would need to
keep their tax revenue at a constant I to 3 percent of GDP after 1998.
A third group of European countries would be required to make more sizeable sacrifices
to simultaneously satisfy the deficit constraint and the solvency condition. Until 1998, Belgium
and Portugal have to raise the annual growth of tax revenues to about 6 percent for two and four
years, respectively. For both Italy and Spain the increase would need to be at least lOpercent for
three years. After 1998, all four countries would need to keep their higher tax ratios at a constant
1.5 to 4 percent.
On average, between 1994 and 2000, the deficit ceilings create an excess fiscal pressure of
18
11 percent, pointing to the need for further fiscal consolidation. It is interesting to note that this
rise in the average primary surplus is required even though the OECD forecasts incorporate the
assumption that, by the year 2000, almost all of the countries satisfy the 3 percent deficit
constraint. This analysis shows the importance of pursuing fiscal retrenchment consistent with the
cyclical economic activity, instead of imposing arbitrary deficit limits. Taken in isolation, these
limits give no information about the true fiscal profile of the economy.
IV. Conclusion
In this paper we have derived conditions suitable for determining whether or not fiscal
policies are sustainable in the long run, in the medium term, and in the presence of debt and deficit
ceilings. We have applied these conditions to the fiscal stances of the United States and the EU
countries since 1970 and their planned positions over the next decade. On the basis of infinite
horizon-tests the broad conclusion is that many countries do not have a sustainable policy.
However, there is some evidence that the government discounted net debt is mean-reverting for a
few countries, implying that their fiscal policies are sustainable. The evidence in favor of
sustainability is strengthened for most countries when data are extended to incorporate future
fiscal consolidation plans. This reflects the general shift towards fiscal austerity in recent years.
In the medium-term analysis we have shown that in the absence of ceilings most countries
have sustainable fiscal policies. However, imposing Oor 3 percent deficit limits or 60 percent
debt limit in the next 3 to 5 years throws most government budgets onto an unsustainable path.
The recovery in Europe is currently being adversely affected by government attempts to
conform to arbitrary fiscal criteria such as rigid fiscal ceilings which do not take into account
cyclical factors. Our results suggest that satisfying the intertemporal budget constraint provides a
sufficient fiscal discipline for governments. Introducing arbitrary additional constraints could lead
to difficult political decisions.
19
Table 1: Integrability Tests for the market value of net debtt
Undiscounted debt-GDP ratio
Discounted debt-GDP ratio
x,
b,
p.p
ADF
p.p
ADF
:
1994
2000
1994
2000
1994
2000
1994
2000
The U.S.
-1.3
-1.0
-1.1
-0.9
0.1
-0.8
1.3
-0.2
Germany
0.1
-0.5
0.5
-0.3
-1.3
-1.8
-1.5
-2.1
France
1.7
-0.2
2.8
0.5
-2.9
-3.4
-4.3
-4.8
TheU.K.
-1.8
-2.0
-1.6
-1.8
-1.7
-2.5
-1.0
-1.6
Austria
-0.6
-1.2
-0.6
-1.2
-1.5
-1.8
-1.6
-1.9
Denmark
-2.1
-2.8
-0.6
-1.2
-3.0
-3.5
-0.9
-1.3
Netherl.
-0.8
-1.5
0.7
-0.2
-3.0
-3.2
-2.2
-2.3
Ireland*
-1.9
-1.5
-1.8
-1.2
-2.3
-2.7
-2.3
-2.9
Spain
-0.1
-1.1
0.2
-0.7
-1.0
-1.9
-0.3
-1.4
Italy
1.3
-1.2
1.2
-1.3
-1.2
-2.1
0.2
-1.7
Belgium
-0.9
-1.6
-0.2
-1.2
-2.0
-2.1
-1.6
-1.9
Portugal*
-0.3
-1.1
-0.2
-1.1
1.1
-0.7
1.8
-0.3
t ADF is the augmented Dickey-Fuller test with the regression equation:
A.x, =µ+(a-l)x 1• 1+I:P; A.x,.;+u,.
The null hypothesis is H0 : µ=O and a=l.
Critical values for the 5% and 10% confidence interval are -2.6 and -2.2 for the first subsample and
-2.5 and -2.2 for the second subsample.
*Gross debt (net debt figures are not available)
References
Ahmed, S. And J. H. Rogers, 1995, "Government budget deficits and trade deficits: are present value
constraints satisfied in long-term data?", Journal of Monetary Economics, 36, 351-374.
Ball, L., D.W. Elmendorf, N.G. Mankiw, 1995, ''The deficit gamble", NBER WP #5015.
Blanchard, 0., J.-C. Chouraqui, R.P. Hagemann, and N. Sartor, 1990, "The sustainability of fiscal policy:
new answers to an old question", OECD Economic Studies, 15, 7-36.
Buiter, W., G. Corsetti, N. Roubini, 1993 "Excessive deficits: sense and nonsense in the Treaty of
Mastricht", Economic Policy, Spring.
Eisner, R. And P.J. Pieper, 1984, "A new view of the federal debt and budget deficits", American Economic
Review, 74, 11-29.
Hakkio, C.S. and M. Rush, 1991, "Is the budget deficit too large?", Economic lnquiry,29, 429-445
Hamilton, J.D. and M.A. Flavin, 1986, "On the limitations of government borrowing: a framework
for empirical testing",American Economic Review, 76, 808-819.
Hansen, L.P. and T.J. Sargent, 1980, "Formulating and estimating dynamic rational expectations models",
Journal of Economics Dynamics and Control, 2, 7-46.
Kremers, J.J.M., 1989, "U.S. federal indebtedness and the conduct of fiscal policy'', Journal of Monetary
Economics, 23, 219-238.
OECD Economic Outlook, 1994 and 1995, 55, 56, 57 and 58.
Roubini, N., 1995, "The economics of fiscal bondage: the balanced budget amendment and other binding
fiscal rules", Yale University.
Smith G. W., and S. Zin, 1991, "Persistent deficits and the market value of government debt", Journal of
Applied Econometrics, 6, 31-44.
Trehan, B., and C.E. Walsh, 1988, "Common trends, the government budget constraint, and revenue
smoothing", Journal of Economics Dynamics and Control, 12, 425-444.
Wickens, M.R. and M. Uctum, 1993, "The sustainability of current account deficits: a test of the U.S.
intertemporal budget constraint", Journal of Economic Dynamics and Control, 17, 423-441.
Wilcox, D.W., 1989, "The sustainability of government deficits: implications of the present-value
constraint", Journal of Money Credit and Banking, 21, 291-306.
20
Chart 1: Government Net Debt as a % GDP
Face Value
--- Market Value .... Discounted Market Value
France
~-- United States
United Kingdom
olGDP
80
80
60
60
I·
80
80
60
60
--------
40
40
~ - " ~ , : ~ ' - .!.. , .... -----::-.. ~.
20t
0
1980
peic ...,
19!!4
1988
1992
1996
2000
rLa..,.u"'s.,1..,n.i,aa________...,.____
••
,· _____ :,.·.
•
40
~--=-~--.-.. . . . ·
"
0
1980
1984
1988
1992
, .,
20 •·
1996
2000
80
80
BO•-
60
60
60•·
/ ___ :,,."_J.
... - --
40
1992
1996
2000
•;;, B,~;"m
!•
0
1980
250
I I
1
150 I
I
1001
V
£~--~-=
~--·
,,,.-----
I
-
- - ---::-,..
··-•• ~.,. -
-
1984
1988
1992
1996
'r.rn-;o:; rlf'lhl (rlfll ilohl ft,11,.,_.., '"" no! m1,11111hlAl
0
1980
1984
80
. --1988
1992
1996
.,
1984
1988
/
40
/
100
2000
1992
20
1996
2000
0
1980
~I
1984
1988
1992
.. -
------ .... ..
150
-.
100
--
1988
~-,/
2000
0
1980
200
I I
150
I
100
1984
1988
1992
1996
2000
jPortugal*
.· · t
r
, ,•
-····-·
_
~---~
1984
1996
Ireland*
200
0
1980
-------
20
/
·····----------....
200
150
,... . . ··1
---------·· i
60
---
2000
t. .· · ·
I-
-
lt::ilv
50
50 I
0
1980
2000
-,,, _:,_... ----·······
.,_re..,,
200
1996
1992
,\··-,
' ..;:-~., "
/
f/',,,r•·':':_,,~,.,
·················-·
1988
1984
/
20
1984
;····--··
40•-
·----······
0
1980
~::.:- - - -......
1988
.....
20
>'
0
1980
40
I
olGOP I
·~
,,, i
./>i ____ _
1992
1996
2000
I
--1=1 ~1~=-
WI~
0
1980
I
1984
1988
1992
1996
2000
1980
1984
HlRA
1992
1996
2000
Chart 2: Fiscal Pressure With No Constraints*
proportion
of GDP
,--------------------------~
0.04
1-
0.02
~
United Kingdom
....
,, ,. "·
~
,,
·~
,_
-,
_,
h
...
United States
o,
-0.021----- ·············
-0.04 ._
France
-0.06 ._
-0.08 - ;
-0.1
.
1980
.
'
1982
'
'
.
'
1984
1986
'
'
'
1988
1990
1992
'
'
1994
'
'
1996
'
'
1998
'
'
2000
0.04
Portugal
0.02
Spain
o
·-···········
-0.02
.....
-0.04
Oervnarl<
/
-0.06
-0.08
-0.1
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
1994
1996
1998
2000
0.04
0.02
Ireland
-.
0
Netherlands
-0.02
-0.04
-0.06
....
-0.08
-0.1
1980
1982
1984
1986
1988
1990
1992
• The char! shows at each point In time the required change In the average tax (or spending) rate over the next five years to satisfy the lntertel11)0ral budget constramt (equation , 5).
A positive fiscal pressure (1he area above 0) Indicates that the intertent)oral budget constraint Is not satisfied, and future primary surpluses should be generated.
Chart 3: Fiscal Pressure With 60% Debt Limit in 1999*
proportion
of GDP
0.04
0.02
United States
United Kingdom
-0.04
France
Germany
-0.06
-0.08
-0.1
'
1982
1980
'
1984
1986
1988
1990
1992
1994
1996
1998
2000
0.04
Portugal
0.02
Spain
·,
0
-0.02
Austria
-··········-····-
. '·
-0.04
'••·····«.,::
- .• r -
,
·:.: ~: - . - .
-
.
Denmark
-0.06
-•
-0.08
-0.1
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
0.15
Belgium
····················•·
..... •···~
\
0.1
Ital
0.05
Ireland
- - ·-·.
... .. .
0
-0.05
"
..
___ ,,,-::.··
-
......... •"
Netherlands
•'
,;·
·0.1 w,.._.. :,_..... ..c~--- '---....: ..----'- --'----' --'----- '-----~ ------'-"----- -'
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
• See note in Chart 2 for further details.
Chart 4: Fiscal Pressure With Deficit Limits of
3% in 1999 for the EU and 0% in 2002 for the US*
proportion
of GDP
0.04
United States
0.02
United Kingdom
0
.
...... _.. ,..... ,. . ·:.·, .... .-
-0.02
-0.04
. .. - ..
France
Germany
-0.06
-0.08
-0.1
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
~
0.04
0.02
0
-0.02
Austria
...........
....
..............
~
Denmark
-. -... -
-0.04
-0.06
-0.08
-0.1
1980
1982
1984
1986
1988
1990
1992
1994
0.04
1996
1998
2000
1998
2000
.... -. -...
0.02
Ireland
_,,_.. ,.;•·:~•·;:·· --
~. -. -..
0
-,.:·r·
Netherlands
-0.02
-0.04
Belgium
-0.06
Italy
-0.08
-0.1
1980
1982
1984
• See note in Chart 2 for further details.
1986
1988
1990
1992
1994
1996
Technical Appendix
Proof of Proposition
I. Z, is exogenous
It is assumed that Z, is determined by Z,=µ+e,=µ+6(L)e., with 60=1 (equation 12). Two cases can be
distinguished: Z, is 1(0) or 1(1 ).
Case]: Z,isl/0)
From equation (6) we get:
limn_ E,X,+n=X,+limn . .oo
n
n
i•t
i•l
E,L z,.;=X,+limn_ nµ+limn_, E,L e,1-;
(la)
Using the methodology by Hansen and Sargent (1980), when at the limit n tends to infinity, the last term on
the right-hand-side can be expanded as follows:
n
E,L e,.;=([8(L)-8JL · +[8(L)-(8 +8 L)]L · +[8(L)-(8 +8 L+8 L )]L · +....) e,
1
i•l
2
0
·
1
2
0
1
3
2
=(8(L)L -l[I +L · 1 +L · 2 +... ]-L . 1(80 +81 +82 +... ]-L -2 [80 +8 1 +8 2 +... ]-...) e,
=(8(L)L - 1(1 +L · 1 +L · 2 +... ]-80 [L -l +L •2 +... ]-8 1[L • 1 +L · 2 +... ]-...) e,
[8(1 )-8(L) l e,
=
(1-L)
Substituting this back into equation (la), we obtain
E,X =X +lirn
lirn
n_,
t+n
t
n-.oo
n + [8(1)-8(L)] e,
µ
(1 -L)
(2a)
Equation (2a) tells us that the discounted value of debt converges to zero (i.e. limit when n-~ of E,X,..=O) if
two conditions are satisfied:
(i) µ=0, which means that there should be no deterministic drift in the Data Generating
Process of the
discounted primary deficit series.
(ii) AX, =Z, -6(1 )e,. But from (7) we know that AX, =Z, . Hence, the second condition is satisfied if Z,=O
(the discounted deficit is exactly balanced at each period in time), or if X, =6*(L)e,, where 6*(L)=6(L)/(l-L).
The latter means that for the transversality condition to be satisfied the process generating the discounted
debt must be stationary.
Case 2: Z, is I/ 1J
Suppose, for example that
21
It follows that
lim,_
n
n
i-I
i•I
i•l
j•O
E;x,., =X, +lim,-~ E,L z,.;=X, +lim,-L (iµ +Z, +E,L e,.;)
(4a)
Thus, the transversality condition will not be satisfied if Z, is I( I).
2. Z, is endogenous
z,
It is assumed that is determined by Z,=µ-aXt-l+e, =µ-aX,. 1+6(L)e,. (equation 13). Replacing X, from (7)
into (14) and taking first differences, we get:
AZ,= -aZ,. 1 +!:;.e,
(Sa)
We can rewrite equation (5a) as ·
Z,=!:;.e, [1-(1-a)L]" 1
=(I -L)6(L)[I -(I -a)L]" 1 =<i>(L)e,
z. is stationary if the roots of <j>(L) lie on or outside the unit circle. This will happen if e, is stationary.
(6a)
Since
t:;.X, =<l>(L)e,, we then obtain X,={ 6(L)/[ 1-(1-a(L))L)e,, implying once more that X, is stationary.
An exception, when <j>(L)•(I-L)6(L)/[1-(1-a(L))L), is where a(L)=(l-L)a with Ia I> I, for then 1(1-a(L))L=(I-L)(l+aL). In this case <j>(L)=6(L)/(l+aL), and so it will be necessary for 6(1)=0, i.e., for
6(L)=(l-L)6*(L). We then obtain X,=[6*(L)/(l+aL))e,, implying still that X, is stationary.
To summarize, whether Z, is exogenous or endogenous, we require that both X, and Z, be stationary
for the transversality condition to be satisfied.
22
Appendix A*
Desired Debt Accumulation
Cumulated Future Primary Balances
pr0p0rti0o of GDP
proportion cA GOP
0.6
0.6,-------------------
0.4
,
0.4
United States
United Kingdom
0.2
0
--~·-····
-0.2
/
0.2
.,.,.------.
,---······ .:..........
.>
~~:•:.· :............. -..:.·:·:..•:-:::-,.~~,-~••!'.:·.~:.:·; ::-•·--· ..
France
--0.4
-0.6 ' - - - - - - - ~ - - - ~ - ' - - - - - - - - ' - - - - ' - - ~ - - - - - - - - - - ' - '1982
1984
1986
1988
1990
1992
1994
1996
1998
'
2000
-,-·
-0.4
France
-0.6
0.6
I
L....-----'-'---~-----'-------- ----------'
1982
1984
1986
1988
1990
1996
1980
1992
0.6
0.4
__ .,. _ -·~n~~ing dom
-0.2
United States
Germany
1980
I=- ,_...... --_:··"·:·:·:·:·:·:·::·c,c:·-··,· --~".rmany
0
,,
1994
1998
2000
Portugal
0.4
. -·
/
0.2
Denmark
.
I
o
0.2
Portugal
~c"-~
-.·. ,
.... ,...-· -··:..~·~-.•:,,.",= - . •-1
...?'""
--0.2
•'
.
0
#.
......
--
~~':~~~___
..:;:.:., .......: ...___""'""•.:.,;:,_
~--..--=-··
_.,•
.. ·
Spain
-02
--0.4
······-·---•·
Austria
-0.4
Austria
-0.6 '-----'-- '---'--'-- ---'---'- '--'---'-- ''-----'-- '-'--'---' --'-"-'
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
0.6
-0.6
~-~---- '--'--'-. ..L.---' -------- '--~---- '---'-~- ----'
1980
1982
1984
1986
1992
1994
1996
1998
2000
0.4
Ireland
Belgium
02
02
Belgium
0
...
_• ...... __ ::
-···
0
~,c··
Italy
-0.4
1990
0.6 , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,
0.4
--0.2
1988
Netherlands
.. -·
-0.6 ' - - ~ - ' - - - ~ - - - - - - ' - - - - - - - - - - - - - - - ' - - ~ - - - - ~ - - - - -........
19ao
1952
1984
t986
,sae
1990
1992
1994
1996
1999
2000
I
_:.-··
_:;:.>
=
11
I
\.
-0.2
-0.4
Ireland
Netherlands
-0.6 .....________ _~ ~ - ~ ~ - ~ - - - - - - - ' - - - - - - - - - - - ----'
1980
1982
1984
1986
1986
1990
1994
1992
1998
1996
2000
• The chert shows the left-hand-side (left panel) end the right-hand-side (rigt,t pa11el) of equation (1-4).
Llfl.Jmaol.: e negative slope violates tho sustainability condition because it indicates Iha! the discounted value of debt five years ah,.arl
larger than the current oylstandlng slQCk of deb1. Bight onnel: the area under the curve is !he cumula!ed
fu1ure primarv !l11,rih1!'! /f"Hltfiri!l if It i.:: nnc;ltivp fn,ttr,1=11iv,,.1
"'
,.
@FedFRASER