Are taxes too low?

ELSEVIER Journal of Economic Dynamics and Control

20 (1996) 126331288

Are taxes too low?

Paolo Manasse

I. G. I. E. R., Opera, I-200Y0 Milan, Iialy

Dipartimento di Economia. Universitir Statale di Milano, Milano, Italy

(Received June 1994; final version received September 1995)

Abstract

Rising debt/GDP ratios have been a common feature of many developed countries since the 70s. In many cases, budget deficits have been the result of increased government spending coupled with relatively lower tax rates. In an economy where the government expenditure/GDP ratio follows a regulated Brownian motion, distortion smoothing is shown to imply a simple closed form relationship between the optimal tax rate, the process drift, and the optimal path of government debt. The paper shows that apparently ‘low’ tax rates are optima1 when a future stabilization of government spending is expected, so that the debt/GDP ratio rises on the optimal path as stabilization is approached. The expectation of future stabilization gives rise to a nonlinear relationship between tax rates and expenditures, similar to that observed in the data.

Ke_v words: Optimal taxation; Stochastic regime switch; Debt dynamics JEL classification: E62; F31; H63

1. Introduction

‘What is the optimal timing of taxation and debt issues?’ If taxes are non- distortionary, an operative bequest motive is at work, and markets are perfect, then the answer to this question is simple: the timing of taxation is irrelevant. However, as soon as we depart from the world of ‘Ricardian equivalence’, debt and the distribution of taxes over time have real effects.

I wish to thank Giuseppe Bertola, Allan Drazen, Carlo Favero, Alessandro Missale for their comments

and suggestions, as well as two anonymous referees. I am also grateful to Debbie Bloch, Bob

Hagemann, and Giuseppe Nicoletti of the OECD who assisted in providing the data for this paper.

0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved

SSDI 0165-1889(95)00899-7

1264 P. Manasse J Journal of Economic Dynamics and Conirol20 (I 996) 1263-1288

During the early 70s almost all OECD countries experienced an upward drift in the ratio of public spending to GDP. The spending bulge persisted well into the second half of the 80s. Unsustainable fiscal policies seemed the rule rather than the exception. In the presence of long-lasting shocks to spending, tax rates were adjusted upwards, but not as much as the tax smoothing model would

predict. According to this hypothesis (e.g., Barro, 1979) permanent shocks to spending should have been fully accommodated by changes in tax rates.’ This

‘undershooting’ of tax rates was all the more relevant the higher the levels of spending attained. Towards the mid 80s spending was eventually stabilized, while tax rates were generally kept steady or reduced. This paper asks the following question: can we explain high deficits and increasing debt/output ratios within this standard representative-consumer approach? In other words, were taxes set ‘too low’ from an optimal taxation point of view?

To answer this question, a simple model is set up. Following Bertola and Drazen (1993) government spending is modelled as a regulated Brownian motion, i.e., a random walk up to the point when it is eventually cut back. Through the application of the techniques popularized by the ‘target zone’ literature (e.g., Krugman, 1991), the optimal tax policy and debt path are characterized, and their profile is shown to depend on the stochastic process driving expenditures, the real interest rate, and the economy’s rate of growth. The main conclusion of the paper is that apparently ‘low’ tax rate can be optimal when a future regime switch in government spending is expected. In other words, when public expenditures cannot be continuously adjusted, the debt/GDP ratio rises on the optimal path as stabilization is approached. The

paper also shows that the expectation of a future stabilization gives rise to a nonlinear relationship between optimal tax rates and government expenditures, similar to that appearing in the data.

The paper is organized as follows. Section 2 takes a first look at the data and sets out a number of important stylized facts about taxes, expenditures, and debt in the major OECD countries since the 60s. In Section 3, a simple equilibrium model of stochastic optimal taxation in continuous time is presented. Section 4 provides an econometric test of the model’s main implications. A summary of the results concludes the paper.

2. A first look at the data

After the 60s virtually all industrialized countries experienced a sharp increase in the ratio of government spending in GDP as well as an increase

‘Political theories of fiscal policy have explained high budget deficits by either stressing the strategic

role of debt (e.g., Alesina and Tabellini, 1990; Persson and Svensson, 1989) or emphasizing the distributional conflict inherent to taxation (e.g., Alesina and Drazen, 1991; Drazen and Grilli, 1993).

P. Manasse /Journal qf’Economic Dynamics and Control 20 (IYY6) 1263-12&V 1265

Table I General government revenues (tax), current expenditures (cup), and net debt (debt) in % of GDP: Averages over corresponding periods

196&1970 1971-1980 1981-1993

Country ta9 rup debt tc1.y exp drht ra.x rup

Belgium 0.304

Canada 0.269

Denmark 0.316

France 0.368h

Germany na

Greece 0.224

Italy 0.279

Japan 0.188

Norway 0.370

Sweden 0.372

UK 0.317b

USA 0.282b

0.271

0.234

0.264

0.326”

;a197

0.269

0.131

0.300’

0.301

0.269h

0.273b

na 0.393 0.365 0.541 0.436 0.401

0.202 0.313 0.305 0.09 0.343 0.353

na 0.455 0.414 - 0.056 0.514 0.492

na 0.393 0.369 0.107 0.453 0.446

na 0.42 I 0.387 0.022 0.435 0.412

0.150 0.261 0.246 0.236 0.329 0.357

0.320” 0.291 0.318 0.512 0.388 0.378

- 0.058 0.226 0.185 0.023 0.290 0.223

na 0.476 0.40 1 - 0.016 0.486 0.449

na 0.486 0.457 - 0.263 0.534 0.549

na 0.338 0.327 0.573 0.363 0.359

0.343 0.301 0.299 0.23 I 0.305 0.315

I.132

0.36 I 0.2X6

0.241

0.225

0.65 I 0.872

0.177

- 0.154

0.058

0.398

0.294

Source: OECD (see Data Appendix for definitions).

‘.L Sample 1964 70, 1963--70, 1962-70, respectively.

in fiscal pressure and a build up of public debt. Table 1 shows the figures for 12 OECD countries. Between the period 1981-93 and the decade 196&70, the share of government consumption and the share of fiscal revenue in GDP rose on average by roughly 15 and 10 percentage points, respectively.’ In many

countries, the net debt/output ratios more than doubled in just over a decade. Typically, the turning point for the boost of spending and taxes took place in the early 7Os, and persisted well into the 80s. In this decade, fiscal imbalances were exacerbated by the overall increase in real interest rates that followed the ‘Volcker disinflation’ and the concurrent slowdown in growth rates.3

A study by the OECD (Blanchard et al., 1990) has computed indicators of the short- and medium-run sustainability of fiscal policies: the so-called ‘fiscal gaps’. These measures are calculated as the difference between the actual average tax rate and the rate that would be required to stabilize the debt/GDP ratio at its current level, given projections of primary deficits, real interest rates, and

‘Greece and Sweden exhibited the largest increases (government spending increased by 25 points of

GDP. and revenues by over 10 and 16 points in the two countries respectively). The US experienced

the lowest increase (15 and 8 points of GDP for expenditures and taxes respectively).

‘In the US the real interest rate net of the economy growth rate (measured as the difference between

the nominal yield on IO-year Treasury bonds and the growth rate of nominal GDP) averaged

- 3.15% in the decade 1968-78, in the years between 1979 and 1985 it climbed to + 2.31%, and

has remained at this level in the period 1986-93, despite the recent reduction of nominal rates.

1266 P. Manasse /Journal of Economic Dynamics and Control 20 (1996) 1263-1288

growth rates over the next three (short-run) or five (medium-run) years. These indicators (to which the reader is referred) document the following remark:

Observation 1 (&sustainability): Fiscal policies implemented by most OECD countries up to the$rst half of the 80s were unsustainable.

However, widespread improvement in budget imbalances occurred in the

second half of the 80s most notably in Canada, Denmark, Belgium, Ireland, and Sweden.4 Until stabilization, ‘shocks’ to government spending and tax rates proved highly persistent. This persistence is reported in the results of a battery of unit root tests (Adjusted Dickey - Fuller, ADF) that were run on annual data for

the share of government current noninterest spending in GDP (exp), the average tax rate (tax), and the ratio of general government net debt to GDP (debt); see Table 2.5.6 Standard unit root tests have been the object of a number of critiques: for example, if the underlying data generation process is stationary around a deterministic trend, but the latter shows a break, such tests would not reject the null hypothesis of unit roots even asymptotically (e.g., Perron, 1989; Hamilton, 1989). In general, these tests have low power in small samples.7 In our case, the spending and revenue/GDP ratios are bounded by definition. Hence, failure to reject the hypothesis of unit root must either be due to the low power of the tests or the short sample period (or both). Ideally, one should test for nonstationarity on series that stop before the stabilization occurs, but this would further shorten the sample and reduce the power of the tests even more. According to Table 2, the ADF tests fail to reject the presence of unit roots for government spending at 95% confidence for all the 12 OECD countries considered. The same happens for the tax rate and net debt ratio, respectively, with the exceptions of Norway, where the test is inconclusive. Rather than taking these tests at face value, these results are interpreted as suggestive of strong persistence:

Observation 2 (Persistence): Until stabilization, shocks in the share of government spending, in tax rates, and in the net debt ratios seemed highly persistent.

4The stabilization experiences differed from country to country, and a detailed description goes

beyond the scope of the present work. Those of Denmark and Ireland are well documented in Giavazzi and Pagan0 (1990).

‘See the Data Appendix for the definitions of the variables

6The tests were run on a model with (up to) four lags, both including and excluding an intercept

term. All yield the same results except for the two cases mentioned. Table 2 shows the results for the

model with one lag and no intercept.

‘A unit roots process with a mean-reverting error component is almost observationally equivalent

to a trend-stationary process (e.g., Cochrane, 1991; Blough, 1988) in finite samples,

P Manasse / Journal of’ Economic Dynamics and Control 20 (1996) 1263 12x8 1267

Table 2

Unit root tests: Adjusted Dickey-Fuller

Country 1UX cup debt

B&yium

(sample)

Canada

(sample)

Denmark

(sample)

France

(sample)

Germany

(sample)

Greece

(sample)

Italy

(sample)

Japan

(sample)

Norway

(sample)

Sweden

(sample)

UK

(sample)

USA

(sample)

- 2.010

( 1960-93)

- 1.103

(1960-93)

- 2.253

(1960-93)

- 0.795

(1963 -93)

- 1.208

(1960-93)

~ 0.286

(1960-93)

1.222

(1960-93)

- 0.319

(1960-93)

- 3.110*”

(1962293)

- 2.214

(1960-93)

- 2.664

(1963-93)

~ 1.860

(1963393)

- 1.641

(196O~m93)

- 0.554

(1960 -93)

- 1.074

(196Om 93)

- 0.318

(1963~m93)

~ 1.767

(1968 93)

~ 1.119

(1960 93)

~ 1.047

( 1960-93)

- 1.023

(1960-93)

- 0.920

(1960 93)

~ 0.703

(1960-93)

~ 1.017

(1963~93)

- 1.577

(I 963m~93)

0.317

(1970-93)

2.080

(1961 93)

-- 1.676”

(1970 93)

2.163

(1969-93)

- 0.230

(1960-93)

1.853

(I 960-93)

1.067

(1964493)

- 2.327

(1964-93)

-- 1.893

(1970 93)

- 0.955

(1969-93)

-- 1.885

(1966 93)

- 2.101

(1963 93)

Sourer: OECD (see Data Appendix for definitions).

The table reports the t-ratios associated with the coefficient of the lagged level of the dependent

variable, in the regression of its first difference on one lagged difference and on the lagged level. An

asterisk (*) denotes a t-ratio that is statistically different from zero at 95% level of confidence (the

critical value is - 2.9591). Hence the asterisk implies a rejection of the null hypothesis of unit root.

“In the model with linear trend the ADF for Denmark is - 3.878*. while for Norway is 0.99X.

Table 1 also shows that in all the OECD countries considered, tax rates (although rising consistently over the past two decades) did not keep pace with

the rise of government spending. a For 12 countries, the combinations of tax rates (on the y-axis) and expenditure ratios (on the x-axis), at various points in

“The paper concentrates on the analysis of tax and spending decisions, and largely neglects the issue

of real interest rates. With high capital mobility, the assumption of exogenous interest rates,

although inadequate for large countries, is not unacceptable for modelling fiscal policy in most other

countries.

1268 P. Manawe J Journal of’ Economic Dynamics and Control 20 (I 996) 1263- 1288

time, are depicted in Fig. 1 (with a 45” line). Proceeding from the 60s to the 90s one moves to the North-Eastern part of each graph. Typically, the slope of the curves is less than unity, and the curves flatten out as one moves up towards the upper right corner. Towards the beginning of the 80s increasing spending/ GDP ratios was often associated with declining tax rates, so that the curves often become concave. This nonlinearity appears quite clearly in Norway (1962289) and some episodes in Canada (1974-80), Denmark (1975&2), and Sweden (1973382). The early 80s in the US, the years of the Reagan administra- tion, witnessed tax cuts at times of high and increasing expenditures. France, Germany, and the UK present similar pictures. The (quasi) concavity of the relationship between tax and spending rates may simply be due to the function- ing of ‘automatic stabilizers’, such as a progressive tax system and unemploy- ment benefits, over the business cycle. In recession, the average tax rates should decrease when government outlays increase. For symmetry, however, we should see convexity of the curve during expansion, a feature seldom observed in the data, with the possible exception of Greece and Italy. In summary:

Observation 3 (Asymmetry and nonlinearity): Since the 70s the GDP share ofjiscal revenues moved by less than 1:1 with respect to the share of spending. In particular

BELGIUN CANAOA Sample 1960-93 Wple 1960-93

MNNW Smple 1960-93

SMEMN Samle 1960-93

WTE9 KIMJIM Smm 1963-93

UNITEO STAlES Samle 1960-93

Fig. 1. 7’ - G curves.

P. Manasse /Journal of Economic Dynamics and Control 20 (1996) 1263-1288 1269

tax rates have matched movements in expenditures at ‘low’ levels of spending, hut not at ‘high’ levels.

In the next section we present a simple model that captures the prolonged unsustainability of fiscal policy, the persistence of shocks to spending and tax rates, and the failure of the latter to fully adjust to spending rates. We ask the following question: Do such stylized facts necessarily point to suboptimality, i.e.. were taxes kept ‘too low’?

3. The model

The paper extends the model of Bertola and Drazen in two important aspects. First, taxes are assumed to be distortionary, so that Ricardian equivalence does not hold.’ Second, and more crucially, in addition to optimizing consumers, a new agent is introduced: a benevolent government that chooses taxes optimally, subject to consumers’ optimal plans and to its own budget constraint. This extension allows the time path of taxes, debt, and deficits to be determined, in addition to path of consumption and that of the current account.

3. I. Consumers ’ and government ‘s budget constraints

Consider a small open nonmonetary economy where an infinitely lived, risk-neutral consumer chooses the optimal consumption path and an infinitely lived, benevolent government chooses the sequence of tax rates that maximizes welfare, subject to its budget constraint and to a given stream of random public expenditures. The small open economy hypothesis allows us to consider the real interest rate, r, as given. Assume that taxation involves distortionary costs, X, that are increasing in the tax revenue, T, and decreasing in the tax base, Y: X = X(Y, T). The assumption is made that such costs are homogeneous in Y and T, X(Y,T) = x(z)Y, and that x’, x” > 0.” These costs are actually paid by consumers and add to their tax bill. It is easy to see (e.g., Bohn, 1990) that for this economy the objective of expected utility maximization is equivalent to minimization of the present value of deadweight losses. Let the consumers’ discount rate equal the real interest rate: l1 their objectives and constraints are

‘Bertola and Drazen briefly discuss distortionary taxation on pp. 24-25.

“C Y G T r = T/Y, A, B > 7 > 1 r n represent consumption, output, government purchases, tax revenue, , 3 the tax rate, real private asset, real government debt, the real interest rate, and the rate of growth.

Lower-case letters, 9, 7, etc., represent ratios of corresponding upper-case letters to output.

“This assumption is made in order to obtain the usual tax smoothing result. If the consumer discount

rate differed from the interest rate, consumption and hence the optimal path of taxes over time would not

be flat, and problems of time consistency in fiscal policy would arise; see, for example, Manasse (1991).

1270 P. Manasse 1 Journal of Economic Dynamics and Control 20 (1996) 1263-1288

given by

cc

W, = E, e-‘(s-l)Cs ds,

subject to

dA,/dt = rA, + Y,(l - z, - x(7,)) - C,. (1)

The government budget constraint is written, both in levels and ratio of GDP, as

G, + rB, - z, Y, = dB,/dt, gt + (r - n)bt - zt = db,/dt. (2)

Simply combining the expressions in (1) and (2), and defining net foreign assets as F = A - B, the standard national income identity is obtained, stating that the current account equals the excess of income (inclusive of interest payments) over absorption:

dF,/dt = rF, + Y,(l - x(r,)) - (C, + G,). (3)

By integrating (3) forward it is easy to see that, for a given path of government expenditures and output, welfare maximization coincides with minimization of the present value of deadweight losses:

s a,

W, = E, Y,(f - x(rJ)e- *(‘-‘)ds + F, - E, * I

p G, e-““-“ds,

lim Fse-‘(S-t) = 0. S-+03

(4)

This expression makes clear that, were it not for the presence of distortionary taxation, the choice of tax versus debt financing would not affect the consumer’s opportunity set and welfare.

3.2. Optimal taxation and consumption

A benevolent government chooses the sequence of tax rates z that maximizes (4) subject to (2). From the first-order conditions, the expected marginal distortion must be constant over time, E,(x’(r,)) = x’(r,) for s 2 t. Assuming the standard quadratic form for x(r) = 2’/2, the martingale property of optimal tax

P. Manasse J Journal of Economic Dynamics and Control 20 (1996) 1263 1288 1271

rates follows immediately:

E,(G) = tt for all s 2 t. (5)

The current tax rate must be the best guess of future tax rates. Simply by taking expectations of the government budget constraint in ratio of GDP and substituting the last equation, one obtains

my3e-“-“““-“ds 1 = (r - n)[bt + h,(y)],

(6) E, lim b,e-C’-“)C”p’) = 0. S-‘f-Z The optimal tax rate is the annuity value of ‘permanent expenditures’. This is defined as the sum of the growth-corrected interest payments on the existing stock of debt, plus the interest payments on the (present value of) current and

future anticipated expenditures, h(g). Consumers’ behavior is particularly easy to derive. First express the represen-

tative consumer’s variables in output ratios. From Eqs. (1) and (4)

s a

w, = E, e - (I - n)(s - 1) c, ds t

= E, %‘(I - x(rz))e-“-“““-“ds +ft - h,(g), (7)

where h(g) represents, as before, the present value of government purchases. From the assumption that the real interest rate equals the subjective discount rate, it follows that the ex ante consumption rate is constant, E,(c,) = c, for s 2 t. Substituting this into the first expression in (7), one obtains the standard result

CT = (r - n)w, = (r - n) E, (S

=(l - x(~$))e-“~“““-“ds +J - h,(g) (8) f

The current rate of consumption is a constant fraction of net wealth. This is given by the discounted stream of incomes, net of the share appropriated by the

government and net of tax collection costs, plus foreign asset. From (8) and (6) we see that consumers optimally spend a constant fraction of their ‘permanent income’, and similarly the government raises in each period a constant fraction of its ‘permanent expenditures’.

1212 P. Manasse / Journal of Economic Dynamics and Control 20 (I 996) 1263- 1288

3.3. The dynamics of government expenditures

Following Bertola and Drazen, y is conjectured to follow a Brownian motion with drift, 0, as long as no intervention takes place:12

dy = Odt + adw, 0 2 0. (9)

One motivation for the assumption in (9) is that a large share of public spending, such as pensions and health expenditures, results from some form of government commitment to welfare schemes. Many expenditures are driven by exogenous demographic factors (longer life expectation and lower birth rates), by labor market characteristics (unionized public sectors, with seniority and indexation as main reasons for wage increases), and by institutional features (soft budget constraints at lower levels of government). Similarly, many expenditures (subsidies, ‘local’ public goods) result from ‘lobbying’ activities by pressure groups. Once the government has committed to a welfare scheme or has given in to an interest group, it becomes politically costly to recede, and the new spending shares become ‘institutionalized’. The assumption that g follows a controlled Brownian motion is meant to capture these phenomena in a simple way. This formulation is also consistent with the evidence previously presented (cf. Obser- vation 2).

Pensions, the health service, and public employment cannot be reformed every year, if there are sunk costs associated to reforms. However, since government spending has to be financed by distortionary taxation, high spending/output ratios are costly as they imply high deadweight losses and low levels of welfare; see Eq. (4). Eventually, these distortions will exceed the costs associated with expenditure cuts. It is assumed that these cuts will be implemented as soon as y reaches the trigger point g”: at this point consensus is forced as the spending/output ratio is widely reckoned to be ‘unsustainable’.’ 3 Different triggers g” can be thought of as representing different ‘types’ of government in power. l4 Alternatively > by interpreting the ‘representative consumer’ as the median voter, one may think of this parameter as derived from the median voter preferences over private versus public goods.

In the ‘real world’, fiscal authorities are probably not immune from some of the demographic and political factors that shape spending, contrary to what is typically assumed in the optimal taxation literature. However, the simplification

“dw represents the increment of a standard Wiener process, with mean zero and variance u2 per

unit of time.

13Drazen and Grilli (1993) make a similar argument.

14A ‘democratic’ type, for example, may be characterized by a relatively ‘high’ trigger (i.e., it waits

longer before cutting expenditures).

P. Mannsse / Journal of Economic Dynamics and Control 20 (I 996) 1263 128X I273

adopted here is reasonable provided one is ready to assume that the costs associated with changing tax rates are of a lower order of magnitude than those incurred for stabilizing expenditures, up to the point where expenditure cuts are implemented.

When y is a driftless random walk (0 = 0), the optimal tax policy is immediately derived. Since both the optimal tax rate T* and g are martingales, the only way in which the budget constraint can be satisfied is if the following tax policy is adopted:

T* = (r - n)h + g. (10)

Thus, the optimal policy is to fully adjust the tax rate to government spending shocks and to keep the debt output ratio steady. With output growing at rate n, this requires running a budget deficit equal to nB (this is the standard result, e.g.. Barro, 1979). When public spending has a positive drift, the optimal tax policy and implied debt dynamics turn out to be (see the Appendix for derivation)

t: = Et@:) = (r - n)bt + gt + A,

(11)

Since G/Y is now expected to grow over time, the optimal tax-smoothing policy requires running a primary surplus, so that the fall in the debt/GDP ratio exactly offsets the expected increase in g (see Fig. 2). Since all shocks to the G/Y ratio are permanent, given our random walk formulation, tax policy works as a ‘shock absorber’ by reflecting deviations in g from its stochastic trend. In a sense, the optimal tax policy is to let debt ‘take care’ of the trend in the spending ratio, and to adjust the tax rate to the shocks in spending. It is for this reason that the tax rate is stochastic (a driftless random walk) while debt is not.’ ’

To close the model, the value of consumption can be recovered as follows. Since z* is a martingale under the optimal tax policy, the process driving distortionary costs x(z) can be found by applying Ito’s lemma. Substituting the expression for E,(x(T,*)) into Eq. (8) gives (see the Appendix), after some mani- pulations,

c: = 1 - x(7:) + (r - n)(ft - h,(g)) - 7 g2 (“,,“)$ 1). (12)

‘5Clearly, when 4 follows a Brownian motion with no controls, eventual violation of the budget

constraint is assured, so this result must be taken with care. We introduce barriers for (I in the next section.

1274 P. Manasse / Journal of Economic Dynamics and Control 20 (I 996) I263- 1288

b,

b

tl S

Fig. 2. Tax policy as a ‘shock-absorber’.

Both the first and the second moment of the spending process affect consumption decisions. When the expected present value of g rises, c* falls, as consumers’ permanent income is reduced by the required higher taxes and by the associated distortions [third and second term in the r.h.s. of (12)]. Since the cost of distortions is convex in the tax rate, a more volatile process for spending raises the expected future burden of taxation and lowers consumption [last term in the r.h.s. of (12)l.l 6

The (stochastic) equilibrium of the economy is now fully determined. Given the initial conditions for b(O),f(O), and g(O), and given the equations of motion forf, b, and g [Eqs. (3), (1 l), and (9)], Eqs. (6) and (12) determine the best policy function r* and the optimal consumption function c* that satisfy the house- hold’s and government’s budget constraints.

The tax smoothing model has two important implications. First, whenever shocks to spending are permanent, tax rates should either aim at keeping a steady debt/output ratio, or at reducing it, depending on whether the drift is null or positive. ’ 7 Second, movements in tax rates should match changes in the spending rate by one to one.

16This direct negative effect of expenditure volatility on consumption is a testable implication of

distortionary taxation, and is therefore absent in Bertola and Drazen.

“Provided that the stock of outstanding debt is positive and the long-run real interest rate exceeds the growth rate, this implies r > 9.

P. Manasse 1 Journal of Economic Dynamics and Control 20 (1996) 1263-1288 1215

Both of these predictions do not seem to square with the empirical evidence presented in the previous section. Hence the questions: were tax rates kept ‘too low’ and did they ‘under react’ to changes in expenditures?

3.4. The policy switch

We have to explore the implications of the expectation of a future stabilization on the optimal tax rate policy and on the dynamics of debt. In fact, absent stabilization, the government spending ratio will exceed unity with probability one in finite time. This is not feasible even in an open economy. For the reasons discussed earlier, it is assumed that spending cuts are resisted until g is pushed as high as g” Only at that point is spending stabilized. Two stabilization mecha- nisms are analyzed: ‘discrete intervention’, that is when government spending is instantaneously cut to gL whenever it reaches the upper trigger point gU and, conversely, ‘marginal intervention’, that is when g is prevented from rising above g”.‘8

3.4.1. Discrete intervention

We are looking for a closed form solution for the h(g) function in Eq. (6) under the present stabilization mechanism. The solution procedure for this problem is given in Bertola and Drazen (1993) and is reported in the Appendix. It yields

T: = (r - n)(b, + h(d)

(13)

From this and the flow budget constraint, the optimal debt dynamics is found to be19

db: -= dt

(14)

‘*We assume that trigger points are perfectly known in both cases. Bertola and Drazen (1993)

analyze the case when the trigger point is not perfectly known to consumers. The latter case will be

briefly discussed in the following section.

“From these expressions and from (3) and (9), the equilibrium path for consumption and the current

account are immediately derived. A.+ is a positive parameter.


A few observations are immediate. First, when for political (or demographic) reasons, the government spending ratio exhibits a nonnegative drift and stabilization is delayed until a trigger point is reached, tax rates should no longer aim at stabilizing (or reducing) the debt/GDP ratio. By comparing (13) and (14) with the corresponding expressions in (1 l), one can see that the effect of expectations of the future policy switch is to lower the optimal tax rate and to raise the accumulation of the debt ratio, until the trigger is reached. The additional terms in (13) and (14) reflect the expectation that ‘sooner or later’ the spending has to be cut if the government is to remain solvent. Provided the initial stock of debt and the spending drift are not too high, from Eq. (13) we can see that it may be optimal to run a primary deficit when the adjustment is imminent (gt is close to g”), and the forthcoming cuts are relevant (g” - gL is large).

Second, as long as stabilization is anticipated, shocks to G/Y become ‘tempor- ary’, and hence are no longer fully accommodated by tax changes, but are partly met by debt issues; cf. Observation 3. Moreover, by noting that g1 now appears also in the debt equation (14), it follows that the debt/GDP ratio becomes stochastic.

Third, for any given level of b, the optimal tax function is nonlinear in g (cf. Observation 3). This relationship is plotted in Fig. 3. At low values of g, random positive shocks in expenditures are almost perfectly matched by tax increases. However, as g is pushed up randomly, the tax rate responds less and less to movements in g. close to g”,

Eventually, when g is sufficiently random increases in the spending ratio may be associated with lower

t* - (r-n)b*

t

Fig. 3. The optimal tax function with discrete spending cuts.


db*/dt

Fig. 4. Debt dynamics with discrete spending cuts

tax rates.” In fact, as the trigger is approached, the present value of spending h(g) falls, since the spending cuts become closer.

Fourth, the assumed stabilization mechanism has an interesting implication for the behavior of r, and b, at times of stabilization. Note that the present value function h(g) does not change when g is cut (this follows from the ‘value matching’ condition, see the Appendix), and remember that the stock of debt is pre-determined, since default is ruled out by assumption. It then follows from the first line in (13) that the tax rate r* is constant at times of intervention, i.e., r*(g”) = r*(gL).

As a consequence, discrete cuts in the government spending are matched by 1: 1 movements in the budget deficit. Fig. 4 plots the change in the debt/output ratio against g. Debt issues accelerate as stabilization is approached. When g reaches the trigger level, g” and is cut back instantly to gL, the accumulation rate of the debt falls from point B to point C. The geometry of the figure is such that the segment BC = BA = g” - gL = the size of the adjustment.

3.4.2. Marginal intervention The previous mechanism has the implication that a benevolent government

may wish to reduce tax rates close to spending stabilization. However, in

“‘From Eq. (13), the derivative dr*/dg = 1 - ((g” - gL)/(ei’gu - e”‘g’))e*‘gl+ is more likely to be

negative the closer g is to g” and the larger the cut.

1278 P. Manasse j Journal of Economic Dynamics and Control 20 (1996) 1263-1288

t* - (r-n)b

Fig. 5. The optimal tax function with marginal intervention.

a number of countries tax rates were often kept steady at times of expenditure cuts. To accommodate this fact we consider the alternative mechanism of ‘marginal intervention’. This is the case in which, at the upper bound g”, the ratio of government spending to output is prevented from rising further, and g is pushed back ‘infinitesimally’. This may be representative of a situation where a relatively ‘weak’ government may at most resist to ‘new’ pressures, although ‘cuts’ into already established spending patterns may not be politically feasible, as they would undermine the government’s support. The solution for the optimal tax policy is particularly easy to characterize in this case (see the Appendix). It yields

e”“9r-8”’

r: = (r - n)b, + g, + & - 1+ db: 8 ea+ (9, -s”) - = - - dt

+p r-n 1+ .

(15)

The difference with the case of discrete intervention is that the downward sloping part of the solution for r* now disappears, see Fig. 5. As the upper bound for g is approached, the tax rate response flattens out, so that the properties of nonlinearity (‘low tax rates’ and less than 1: 1 adjustment) are preserved, but the actual cut in tax rates and the acceleration of debt issues approaching stabilization do not hold any longer.

P. Manasse /Journal of Economic Dynamics and Control 20 (1996) 1263-1288 12-D

Finally, we need to spend a few words on episodes of fiscal stabilization where tax rates were actually raised ‘close’ to expenditure cuts. Clearly, the tax

smoothing approach is not particularly well suited to fit such episodes, unless somehow specific assumptions are made about the nature of the g process. Suppose that, as in Bertola and Drazen, there is uncertainty as to whether g will be cut once it reaches a lower threshold level. However, if it is not cut at that point, it will have to be cut, with probability one, once the higher threshold g” is reached. Under these assumptions it is easy to see that whenever the lower limit is overcome with no intervention, the expected present discounted value of spending will be revised upward, and the tax rate will optimally jump up. As a consequence, higher taxes without expenditure cuts would typically signal a failed (or delayed) stabilization.*’

4. Back to the data

A thorough empirical analysis of the tax-smoothing hypothesis goes beyond the scope of this paper. The aim of this final section is: a) to check in a simple way whether the nonlinearities implied by the tax smoothing model with stabilization can be given some statistical support, at low frequencies; and b) to see whether the simple tax smoothing model without stabilization can be rejected. The following strategy is pursued: first, cointegration** between spending, tax, and debt rates is tested. In principle, lack of cointegration may be attributed to two causes: either the policy pursued was unsustainable, leading to a default in the long run, or the cointegration surface was nonlinear. When linear cointegration is rejected, nonlinear cointegration, in the spirit of Granger (1993), is investigated. When linear cointegration cannot be rejected, one implication of the tax smoothing model without stabilization is tested: the tax rate

should fully adjust to spending shocks in the long run. Table 3 presents Johansen’s (1988) tests for cointegration between tax, spend-

ing, and debt ratios. These tests are based on the maximal eigenvalues of the stochastic matrix of the VAR. 23 The first column tests the null hypothesis of

“The first illustrative example that comes to mind is the Italian experience of the past thirty years,

1969989.

“Notwithstanding the considerations made in Section 2, we proceed on the assumption of non-

stationarity of the series for tax, spending, and debt rates, since assuming stationarity when the

converse is true would invalidate the inference.

23For each country, the smallest order of the VAR that produces white noise residuals has been

chosen, and the presence of a linear and quadratic trend was tested. The resulting VAR character-

istics are reported in the fourth column of the table.

1280 P. Manasse 1 Journal of Economic Dynamics and Control 20 (I 996) 1263- I288

Table 3

Cointegration tests: Johansen

Country

(sample)

Ho (r = 0)

“S

H,(r = 1)

Ho (r I 1)

Z, (r = 2)

Ho 0. I 2) “S

H,(r=3)

VAR order

trend Rank r

Belgium

(1960-93)

Canada

(1961-93)

Denmark

(1970-93)

France

(1969-93)

Germany

(1968-93)

Greece

(1960-93)

Italy

(196493)

Japan

(1964 - 93)

Norway

(1970-93)

Sweden

(1969993)

i&L93

USA

(1963-93)

73.991 14.541

(22.002) (15.672)

17.485 12.348

(22.002) (15.672)

56.346 28.593

(20.967) (14.069)

29.414 11.43

(22.002 (15.672)

18.250 13.147

(22.002) (15.672)

43.540 9.266

(22.002) (15.672)

22.948 5.525

(20.967) (14.069)

52.598 12.424

(22.002) (15.672)

27.370 10.814

(20.967) (14.069)

67.208 16.095

(22.002) (15.672)

24.722 11.640

(22.002) (15.672)

18.128 5.598

(22.002) (15.672)

5.919

(9.243)

7.398

(9.243)

4.015

(3.762)

4.263

(9.243)

6.833

(9.243)

6.919

(9.243)

1.660

(3.762)

9.735

(9.243)

0.1566

(3.762)

5.782

(9.243)

7.105

(9.243)

1.721

(9.243)

3

no trend

4

no trend

I quadr trend

2

no trend

2

no trend

2

no trend

1

quadr trend

1

no trend ^ L

quadr trend

1 no trend

L

no trend

3

no trend

r=l

r=O

(r= I)

r=3

r=l

r=O

(r = 2)

r=l

r=l

r=l

r=l

r=2

r=l

r=O

The table reports tests ofjoint cointegration for tax, exp, and debt. The tests presented are based on

the maximal eigenvalues of the stochastic matrix for the system (those based on the trace of the

matrix yield similar results, unless indicated). The rank r represents the number of cointegrating

vectors in the system. The first column tests the null of rank r = 0 (no cointegration) against the

alternative of one cointegrating vector (r = 1), and similarly for the second and third columns.

A matrix of full rank (r = 3) implies that all the three variables are stationary. In parentheses the

95% critical values are reported. The fourth column (VAR order) indicates the order of the estimated

VAR. while ‘trend’ shows the presence or absense of trend in the model (for example ‘quadr trend’

means that the system in levels includes a quadratic trend). The last column summarizes the result of

the procedures; in parentheses we report the outcome of the tests based on the trace of the stochastic

matrix when it differs from the reported one.

rank r = 0 (no cointegration) against the alternative of one cointegrating vector (r = 1). The second and third columns have similar interpretations. The overall results based on 95% confidence interval appear in the last column. The presence of a single cointegrating vector for Belgium, France, Greece, Italy,

P. Manasse J Journal of Economic Dynamics and Control 20 (1996) 1263-128X 1281

Table 4

Parameter choice for present value calculations

Country

(sample)

Canada 0.394 0.275 0.0134 0.0032 6.019 (1961 93)

Germany 0.433 0.317 0.0674 0.0073 I.799 (1968 93)

USA 0.329 0.253 0.0100 0.0014 5.303

(1963-93)

g” represents the maximum value of the ratio of government current spending over GDP attained in

the sample period, and gL the minimum. d is the standard deviation of g’s first differences. r - n is the

average value of real long-term interest rate (nominal rate minus GDP deflator) net of GDP growth

rate, over the sample period. 1’ is calculated from the corresponding expression in the Appendix.

setting the drift H = 0.

Japan, Norway, and the UK, and of two cointegrating vectors for Sweden cannot be rejected. Conversely, cointegration in the US, Canada, and Germany can be rejected.24

Granger (1993) and Granger and Hallman (1991) have argued that, whenever economic theory suggests a nonlinear long-run equilibrium relationship, the traditional cointegration approach has to be generalized to allow for nonlinear attractors. In their spirit, for Germany, Canada, and the US, a test is carried out to find whether tax, spending, and debt rates tend to revert in the long run to the nonlinear surface described by the first expression in (15), that is, the optimal tax

policy with marginal intervention. 25 We estimate the parameters characterizing the random walk behavior for g [see Eq. (9)] and use them to recover the

parameters of the optimal tax policy (1’ is defined in the Appendix). The rather liberal assumption that the upper barrier g” is equal to the maximum value attained by g over the sample period is made, r - n is taken to be the sample average of real long-term yield on T-bills net of GDP growth, and the drift 0 is

set to zero (from which point estimates do not differ significantly). These parameter values, reported in Table 4, are then plugged into Eq. (15). The Johansen procedure is then repeated to test cointegration between taxes, debt,

and the present value of spending, g - e i*cg-g”‘/l+. The results are shown in Table 5. In US and Canada, where linear tests failed to detect cointegration, tax

24For the latter two countries the tests based on the trace yield different results from those based on

the eigenvalues. Results are also ambiguous for Denmark, where the result of full rank indicates that

all variables are I(O), contradicting the ADF tests (at 95% although not at 90%) of the previous table.

“The procedure was repeated for the case of discrete intervention, with similar results


Table 5

Nonlinear cointegration tests: marginal intervention

Country

(sample)

Ho (r = 0)

VS

Hr(r = 1)

H, (r I 1) VS

HI (r=2)

Ho (r I 2) VS

Ho (r = 3) Rank r

Canada (1961-93)

Germany (1968-93)

USA (1963393)

42.315

(22.002)

17.7916 (22.002)

34.995 (22.002)

19.665

(15.672)

13.5570

(15.672)

19.060 (15.672)

1.145

(9.243)

5.0707

(9.243)

1.384

(9.243)

r=2

r=O

(r = 1)

r=2

The table shows the Johansen tests for cointegration (see note of Table 3) between tax, debt, and the

nonlinear transformation for 9 relative to the case of marginal intervention ( = 9 - e”tg-~““/I),

based on Eq. (15).

In parentheses we report the outcome of the test procedure based on the trace of the stochastic

matrix when it differs from the reported one, based on maximum eigenvalue.

rates and debt appear strongly cointegrated with the present value of spending. The result for Germany (compare Tables 5 and 3) does not differ from those of the linear testz6

As a final exercise, we consider the countries for which linear cointegration was accepted, and conduct the following test of the tax smoothing model without stabilization. Under the conditional assumption of a unit root in g (and linear cointegration between r, g, and b), the tax rate should adjust 1: 1 to movements in g. Table 6 shows the estimates of a cointegrating regression of the tax rate on a constant, the spending and the debt ratios [compare with the first equation in (1 l)]. It appears that the (‘superconsistent’) estimates of the coefficient of g are well below unity, typically ranging between 0.4 and 0.6, and even lower for Denmark and Sweden (Japan being the exception).” We regard these tests as giving preliminary support of the model’s main message: seemingly unsustainable policies may be consistent with optimizing behavior when future stabilization is accounted for (the case of US and Canada). Moreover, the simple

Z6This is probably due to the fact that the standard deviation of the 9 process in Germany is much

higher than in the other two countries; see Table 4. This implies a very small value for the parameter

1+, so that the present value of spending does not differ from current spending but for a term that is

approximately constant.

“Negative coefficients for debt are not surprising as these coefficients should represent the difference

between the average real interest rate net of the growth rate over the sample period, which has been

negative for many countries. Eq. (12) however would imply that the constant should have the same sign of the debt coefficient.


Table 6

Cointegration regressions: tax = const + u exp + fi debt + e

Country const exp (8) debt (B) RZ DW

Belgium 0.151 0.597 0.042 0.949 1.055

(9.469) (12.994) (7.171)

Denmark 0.399 0.148 0.143 0.713 0.67 1

(6.670) ( 1.047) (3.298)

France 0.143 0.674 0.358 0.946 1.076

(0.156) (9.333) (8.265)

Greece 0.138 0.390 0.082 0.938 0.975

(11.573) (6.330) (4.820)

Italy 0.129 0.322 0.152 0.912 0.337

(1.837) (1.101) (3.237)

Japan 0.037 1.110 - 0.029 0.812 0.247

(1.134) (5.859) ( - 0.494)

Norway 0.414 0.151 ~ 0.007 0.136 0.548

(10.144) (1.515) ( - 1.141)

Sweden 0.282 0.453 - 0.006 0.644 0.468

(5.151) (4.380) (0.135)

UK 0.261 0.289 ~ 0.0153 0.466 0.5888

(5.391) (2.619) ( - 0.564)

t-statistics are shown in parentheses.

tax smoothing model without stabilization is clearly rejected by the data (in all countries with the possible exception of Japan).

5. Conclusions

During the early 70s almost all OECD countries experienced an upward drift in the ratio of public spending to GDP. The spending bulge persisted well into the second half of the 80s. In the presence of long-lasting shocks to spending, tax rates were adjusted upwards, but not as much as the tax smoothing hypothesis would predict. This ‘undershooting’ was all the more relevant the higher the levels of spending attained. Only towards the mid-80s was spending eventually stabilized, while tax rates were often kept steady or reduced. Were taxes set ‘too low’ from an optimal taxation point of view? My main conclusion is: not necessarily. Apparently ‘low’ tax rates may have resulted from expectations of a future regime switch in government spending. The paper has shown that the

1284 P. Manasse / Journal of Economic Dynamics and Control 20 (I 996) I263- 1288

debt/GDP ratio rises on the optimal trajectory as stabilization is approached. Also, the expectation of future stabilization gives rise to a concave relationship between optimal tax rates and government expenditures that resembles the one observed in the data. Some empirical support for this nonlinearity has been found for Canada and the US, while the standard tax smoothing hypothesis without stabilization has been unambiguously rejected by the econometric tests.

Appendix

A. 1. Derivation of Eqs. (11)

As long as no spending cuts are implemented and g evolves according to (9) the expectation of g at any future date is given by E,g, = gt + e(s - t), s 2 t.

Calculating the present value function h in Eq. (6) gives

h,Jg) G E, s mgse-~~-“W+js = df!_ 6 ~ t r - n + (I - n)2’ (A.1)

The closed form solution for the optimal tax rate function in Eq. (11) can be calculated from (A.l) and (6):

z: z E,(r,*) = (r - n)b, + gr + A, s 2 t. (A.21

When 8 = 0, the government spending/output ratio is expected to be constant. Thus, its expected present value, h,, is also constant and equal to g/(r - n).

The optimal path of the debt/output ratio can be recovered from the previous expression. Insert (A.2) into the budget constraint (2). This gives

db: e .

dt= - -

r - n’ i.e., b,* = b: - s 2 t.

By substituting the second expression in (A.3) into the optimal policy function (A.2), it can be checked that r* is a martingale:

E,rf = (I - n) bf + E,g, + +_ = r:, s 2 t, (A.41

with

z: = z: + a s

dw. t

P. Manasse / Journal of Economic Dynamics and Control 20 (I 996) 1263- 1288 1285

A.2. Derivation of Eq. (12)

By applying Ito’s lemma to the previous expression for T*, the process for distortionary costs x(r*) can be computed:

dxv = ;dv + (r:o)dw = f dv +(r: o)dw + a*(o - t), v > t.

Integration yields

a2 Xs = xt + y [(s - t) + (s - t)*] + (T: a)

s dw, s 2 t.

f

By substituting E,(x,) from this expression into the consumption equation (8) and integrating by parts,

s a,

“)(s-t)& _ E, xse-(‘-“)(spt)& +ft _ h,(g)

t

=(r-n) z_ ! a2 a2 -~ 2(r _ 42 (r - 43 +fr - hJg)

1

is obtained, from which (12) immediately follows.

A.3. Derivation of Eqs. (13) and (15)

The h(g) function in Eq. (6) can be written in an arbitrage form. Differentiating

with respect to time:

(r - n)h(g) = g + E,$.

By Ito’s lemma the expected change in h(g) is given by

E dh(g) - ’ dr

h’(g)0 + ;Q).

This can be inserted back into the previous condition to yield a standard second-order differential equation:

(r - n)h(g) = g + h’(g)fl + $l”(g). (A.5)

1286 P. Manasse 1 Journal of Economic Dynamics and Cortrol20 (1996) 1263-1288

Solving for h(g) gives

h(g) = Ae”+g + BeAmg + F(g), (A.6)

where I?‘(g) is given in (A.l) and represents the present value of expected government purchases, if no stabilization were ever to occur. The As are the positive and negative roots of the quadratic equation associated with the homogeneous part of (AS). Their expression is given by

L f = ( - 8 f Je2 + 2(r - n)o2)/0’.

The first two terms in (A.6) give the ‘bias’ in the present value function h, due to expectations of future intervention. Clearly, as g randomly increases towards the trigger point, g”, stabilization get closer and we anticipate the present value function h(g) to bend away from hN(g). Values of g closer to the trigger will assign a larger weight to lower values of g, so the present value function will decline as we get closer to the upper threshold. Thus we expect A < 0. Also, if g were pushed very low by a shock, the difference between h and hN should be ‘small’ or, at least, be bounded. This requires B = 0. The present value function h that satisfies these conditions is therefore

h(g) = Ae”+g + --L 9 ~ I - n + (r - n)2’ (A.7)

Next, the constant A remains to be determined. We impose the standard requirement of continuity of the solution. Since there is no uncertainty on the level of the trigger point, expectations must not be allowed to jump when stabilization actually occurs. This ‘value matching’ condition requires

4s”) = 4gL).

By simply substituting (A.7) into the above expression, A is recovered:

(s” - gL) 4S”? gL) = - (r _ n)(e”+g” _ eA+gL) < 0. 64.8)

Eq. (13) in the text is obtained by inserting Eqs. (A.7) and (A.8) into the optimal tax rule (6).

Derivation of Eq. (IS) Take the limits of the integration parameter, A in Eq. (A.S), for gL -+ g”:

lim A(g”,g) = - (L+(r - n)eLtg” -’ - 9+9”

) = 4s”).

P. Manasse / Journal of Economic Dynamics and Control 20 11996) 1263- 1288 1287

The government spending present value function is therefore

h(g) = 9 0 ea+(s-sL) ~ -

r - n + (r - n)* 3,+(r - n) ’

where it can be checked that h(g) satisfies the ‘smooth pasting condition’ h’(g”) = 0. By inserting this expression into (6) the optimal tax policy function and debt dynamics in (15) are recovered.

Data appendix

The data are from OECD Economic Outlook, Historical Statistics. They refer to the general gooernment accounts. For all countries, with the exception of Norway and the US, the average tax rate r is defined as the ratio to GDP of Current Receipts (YRG) net of Property Income Received = Revenue from Direct Taxes + Indirect Taxes + Social Security Transfers to Government + Other Current Receipts. For the US and Norway Other Current Receipts are not included.

The public spending ratio g for all countries, except US and Norway, is

defined as the ratio to GDP of Current Expenditures net of Property Income Paid = Consumption Expenditures + Subsidies + Social Security Transfers from Government + Other Current Expenditures. It does not include Govern- ment Investment nor Other Capital Transfers. For Norway and the US, Other Current Expenditures are not included.

The debt/output ratio, b, is defined as the ratio to GDP of total net general

government debt. I have not included government investment among expenditures on the

presumption that investment is made according to market criteria and will be self-financed. Similarly, I have considered net debt on the presumption that public assets yield market return and therefore are not transfers to debtors.

The interest rates used in the calibration exercise are from the OECD Main

Economic Indicators: for Germany, bond yield on the secondary market public bonds (3 to 15 years maturity). For the Canada, long term interest rate on federal government bonds (10 years maturity). For the US, yield on Treasury bonds (10 years or more).

References

Alesina, A. and A. Drazen, 1991, Why are stabihzations delayed?, American Economic Review 81,

1170-1188.

Alesina, A. and G. Tabellini, 1990, A political theory of fiscal deficits and government debt in

a democracy, Review of Economic Studies 57,403-414.

1288 P. Manasse / Journal of Economic Dynamics and Control 20 (1996) 1263-1288

Barro, R.J., 1979, On the determination of public debt, Journal of Political Economy 87,940-971.

Bertola, G. and A. Drazen, 1993, Trigger points and budget cuts: Explaining the effects of fiscal

austerity, American Economic Review 83, 1 l-26.

Blanchard, O., 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, 7736.

Blough, S.R., 1988, On the impossibility of testing for unit roots and cointegration in finite samples,

Working paper 211 (Johns Hopkins University, Baltimore, MD).

Bohn, H., 1990, Tax smoothing with financial instruments, American Economic Review 80,

1217-1230.

Cochrane, J.H., 1991, A critique of the application of unit root tests, Journal of Economic Dynamics

and Control 15, 2755284.

Dickey, D.A. and W.A. Fuller, 1981, Likelihood ratio statistics for autoregressive time series with

a unit root, Econometrica 49, 1057~1072.

Drazen, A. and V. Grilli, 1993, The benefits of crises for economic reforms, American Economic

Review 83, 598-607.

Fuller, W.A., 1976, Introduction to statistical time series (Wiley, New York, NY).

Giavazzi, F. and M. Pagano, 1990, Can severe fiscal contractions be expansionary? Tales of two

small European countries, in: 0. Blanchard and S. Fisher, eds., NBER macroeconomics annual

1990, 29333 10.

Granger, W.J., 1993, Modelling non-linear relationships between long-memory variables, Mimeo.

(University of California, San Diego, CA).

Granger, W.J. and J. Hallman, 1991, Long memory series with attractors, Oxford Bulletin of

Economics and Statistics 53, 1 l-26.

Hamilton, J.D., 1989, A new approach to the analysis of nonstationary time series and business

cycles, Econometrica 57, 357-384.

Johansen, S., 1988, Statistical analysis of cointegration vectors, Journal of Economics Dynamics and

Control 12, 231-254.

Krugman, P., 1991, Target zones and exchange rate dynamics, Quarterly Journal of Economics 56,

669-682.

Manasse, P., 1991, Fiscal policy in Europe: the credibility implications of real wage rigidity, Oxford

Economic Papers 43, 321-339.

OECD Economic Outlook, Historical statistics (1994).

OECD Main Economic Indicators, various issues.

Perron, P., 1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica 57,

1361l1401.

Persson, T. and L. Svensson, 1989, Why a stubborn conservative would run a deficit: Policy with

time-inconsistent preferences, Quarterly Journal of Economics 104, 325-345.

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