summary.

The purpose of this paper is to analyze the concept of automatic stabilization in various model formulations. It is found that in static models a positive relationship exists between the size of the tax rates and the normally used measures of automatic stabilization. When it comes to dynamic formulations, however, the concept of automatic stabilization has several dimensions, and whether high marginal tax rates have a stabilizing on the development of income will depend on the lag structure of both the tax function and other economic relationships. Consequently the policy implications of the conditions hi various sections cannot be worked out, until we know more about lag structures of the Danish economy.

Introduction

It is frequently argued that the growth of the public sector in relation to the private sector and the increased reliance on income dependent taxes - as opposed to for instance property taxes - have increased built-in flexibility¹ and subsequently also the dampening effect, which the public sector may have on autonomous changes in income and activity.

Some economists2 have even argued that due to a considerable time lag3
discretionary fiscal policy cannot be relied upon for stabilization and may in
fact accentuate the cyclical fluctuations in income. Instead, fiscal policy

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1. For any tax this concept is defined as dT/dY; i.e. the automatic change in tax revenue, when income or activity changes and the tax structure remains unchanged. The concept is further discussed in section 111.

2. See Friedman (194,8).

3. For a discretionary fiscal tax change the following lags are usually encountered: (a) Recognition lag, (b) reaction lag, (c) revenue effect lag, (d) adjustment lag for private demand, (e) adjustment lag for private output. An automatic stabilizer depends upon (c), (d) and [e), but not of course on (a) and (6).

I am indebted to J. Gelting, S. Hylleberg, G. Thorlund Jepsen, J. Vibe-Pedersen, C. Vastrup E. Yndgaardfor comments on an earlier draft of this paper.

should be confined to those automatic changes in the budget that, follow from
changes in income, possibly strengthened by automatic changes in the tax rates4.

Other economists have contended that automatic fiscal stabilizers can only reduce the secondary effects of an autonomous change in income, but cannot by themselves bring the economy back to full employment. In addition, a high degree of built-in-flexibility is a "mixed blessing" in that not only does it reduce the contractive effects of a decrease in income, but it also makes it more difficult to get an expansion started. The latter aspect has in recent years been interpreted as "fiscal drag"⁵, which implies that in order to secure full employment growth it is necessary to expand government expenditures or reduce tax rates, unless expansion in private consumption or private investment be relied upon.

Finally, some economists have argued that a high degree of built-in flexibility
have a destabilizing influence if we consider other endogenous
variables than total real income or if we apply a dynamic model.

Before discussing the various measures of automatic stabilization we should like to point out that some arbitrariness is bound to occur when a distinction is made between automatic and discretionary changes in the budget. On the revenue side a distinction is relatively easy, given the tax laws and the interpretation the laws, and the same applies to transfer payments on the expenditure However, when it comes to other expenditures, a distinction is less clear. Thus, in many countries public expenditures increase automatically with the general price level, and even for real expenditures one might say that some of them tend to follow an increased demand for public goods automatically, leaving very little for discretionary policies6.

In the following we shall be somewhat traditional in that mainly automatic in net taxes (tax revenue - transfer payments) are taken into account, but in the section on growth we shall also allow for automatic changes in real expenditures.

As a starting point for discussing some of the problems mentioned above
we take the traditional Musgrave-Miller measure7 which we subsequently

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4. The socalled "formula flexibility". See Musgrave (1959, p. 512). For a specific formula see Tanzi (1966).

5. This term was first used in the Economic Report of the President, January, 1962, U.S. Government Printing Office, Washington 1962. It refers to the automatic tendency towards a budget surplus at full employment, when the economy grows. This effect was an important argument for the American change in 1964, and we shall discuss the concept further in section 111. B.

6. See Bent Hansen (1969, pp. 18-19).

7. See Musgrave and Miller (1948).

extend to more complicated static models and to dynamic models in which both short run cyclical movements and long run growth are considered. The scope of the following is somewhat limited in that monetary effects of changes in the budget are not taken into account, and neither do we consider effects of revenue changes on the supply of labour.

I. Measures of automatic stabilization in a small static model

A. Musgrave and Miller

Let us consider the following models for a closed economy8:

(1)

where: T= income, C— consumption, /= investments, G= government
expenditures, and 7= net direct taxes. All variables are measured
in real terms.

In (1) all variables except T and C are exogenous, whereas in (2) Tis also
an endogenous variable.

Substituting into the equation for T and differentiating totally we obtain:

and9

(1.1)

According to Musgrave and Miller automatic stabilization can now be
denned as the proportion of income changes which is prevented due to income
dependent taxes:

(1.2)

where eT,Y is the elasticity of tax revenue with respect to income.

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8. Throughout the paper we disregard the error terms and the effects that various tax structures might have on the error terms; cf. Bent Hansen (1969, p. 23). Initially we also disregard wage- and price changes.

9. f'(Y) is the partial derivative of f(Y,q) w.r.t.T and f (q) the partial derivative w.r.t. the shift parameter q. For the sake of convenience we let d T equal/' (q) dq.

Thus a is really a comparison of income multipliers, one based on an
economic structure with endogenous taxes and a hypothetical one, where taxes
are assumed exogenous10.

It may be argued that the above measure does not correspond to the definition, the structural coefficients - in the above model only b - may change with the change in taxes, and this effect has not been taken into accoun ¹¹, 12. Alternatively, one might therefore take as a basis of comparison the income multiplier for a system with proportional taxes (T = t • T), as this measure will be less affected by changes in the marginal propensity to consume. This is done in:

B. Lusher

Using the same notation as above, but substituting a system with proportional
for model (1), we obtain13:

(1.3)

In a system with proportional taxes, the tax elasticity eT,Y = i> and the rate elasticity e/(y),Y =o. According to the measure in (1.3), a tax system will only be stabilizing if the revenue elasticity is > 1 or alternatively, if the rate elasticity is > o. Lusher's (1956) measure has the advantage that the structural coefficients of the actual system are much closer to those of the hypothetical system than in the Musgrave-Miller measure.

Cassidy (1970) has also applied a proportional tax as a basis of comparison,
but from an entirely different point of view. Thus he argues that if in a linear

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10. Rewriting the definition of a it is esay to see that it equals the relative reduction of the income multiplier.

11. In empirical estimates of automatic stabilization the usual procedure has been to first derive estimates for all structural coefficients, assuming taxes endogenous, and subsequently to estimate the same coefficients, assuming taxes exogenous. This procedure will of course solve the problem mentioned above, but a bias is introduced into the estimation when taxes incorretcly are assumed exogenous.

12. A similar problem may arise from the fact that the marginal propensity to consume varies cyclically, giving rise to asymmetric stabilization. See Eilbott (1966).

13. The relation between a and 8 is: 8 = a — . l-b(l-e_T,Y'Vr)

system we have a constant rate of growth of full employment real income, government expenditures proportional to real income and no autonomous elements in tax and expenditure functions, then a tax elasticity of i will be neutral, as the equilibrium growth will not be disturbed by the presence of the public sector. This definition, however, is relevant only under the rather strict conditions stated above, since for any other development in income or specificationof an elasticity of i is not desirable. Consequently, if we choose as reference base a tax system which either leaves a desirable position unchanged or moves the system from an undesirable position towards a desirableone, shall have to change the basis, whenever there are changes in the economic conditions, the economic structure, or the targets. This procedure has the disadvantage that computation of automatic stabilization becomes rather laborious. On the other hand, it emphasizes the fact that automatic stabilization is a relative concept, which ought to be evaluated in relation to the prevailing economic situation.

C. Brown

Brown (1955) has used the Musgravc-Miller concept in a slightly different way, as he defines automatic stabilization as the difference between amounts "of autonomous shifts in demand necessary to secure a unit shift in output or income" (p. 430) under respectively endogenous and exogenous taxes. Letting dl denote autonomous changes in demand we then have:

(M)

As pointed out by Brown, (1.4) will often make calculations easier, and q
will obviously vary positively with band/ (T). The limits are 1 and o as for
the Musgrave-Miller concept.

D. Pearse

Pearse (1961-62) has defined a measure which takes into account that automatic stabilization can only affect the secondary effects of an autonomous change in income. This is done by subtracting the primary change from both the hypothetical and the actual income change, but otherwise relying on the Musgrave-Miller measure. Assuming that the autonomous change occurs in private investment, we thus obtain:

(1-5)

It is interesting to note that Å = the automatic change in tax revenue, but
this interpretation is only valid, when the marginal propensity to consume is
assumed constant.

E. Bent Hansen

Bent Hansen (1959) has considered automatic stabilization in connection
with defining a measure of the total budget effect during a certain period.
Using the model explained above we obtain by subtracting dV% from dT\:

(1.6)

as an expression for automatic stabilization14,15. This measure depends positively
f'(T), b, and dV%, and it is important to note that the measure is
independent of the source of the income change.

In (1969) Bent Hansen defines dY\and dV% exclusive of di and da, and
we then find the following expression for dY\ — dT2:

(1.7)

According to Bent Hansen (1969, p. 24) this measure depends positively on "how much extra-budgetary disturbance there has actually been to dampen". in view of the equality given in (1.7) this conclusion does not seem appropriate, and it furthermore seems more natural to define automatic stabilization independent of the source of the disturbance¹⁶.

Before analyzing the extended model below we consider automatic stabilization relation to a target level of income, T*. Thus if a certain Y* is given exogenously, and if for model (2) above we assume proportional taxes above an exemption level (T= c -f- tY), we get the following expressions for the

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14. We let dT =/'(?). dq cf. footnote 9.

15. The same expression can be derived from Lars Mathiesen's (1961) definition of total and discretionary effects.

16. The economic structure may make the measure dependent upon the source of autonomous change, but this is an entirely different problem. For further discussion see n b.

required levels of the instrument (7" and c)*7 in respectively model (i) and
model (2):

(1.8)

For a given change in private investments the deviations from Y* will be respectively dY\ — dlj(i —b) and dY₂ — dl\{i —b(i — t)), and if we consider the necessary changes in the instruments to bring Y back to Y*, it may seem that more is required in model (1) than in model (2), since some change in tax revenue has already occurred in (2). However, in both cases the necessary change is dljb. This result is merely a reflection of the fact, that automatic works both ways.

II. Extended static model

In general automatic stabilization depends upon three factors:

a. The endogenous variable being analyzed.

b. The exogenous variable, causing the changes in the endogenous variable.

c. The tax instrument considered.

Generalizing the Musgrave-Miller measure, we may write:

fi and/2 will of course depend upon the economic structure, and as an illustration
shall consider the following linear model:

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17. We have selected cas the instrument; alternatively we might have chosen t.

18. This implies that other taxes are exogenous.

19. All variables are measured in factor prices.

20. The formulation of this function implies that all production takes place in companies which are subject to the corporate tax.

A. Automatic stabilization ofincome (T)

The reduced form equation for T will now be

(H.I)

whereas if all taxes were exogenous we would get:

(n.2)

Applying the Musgrave-Miller measure we then obtain:

(n-3)

As appears, a is positive and depends upon all three taxes, but if we want to analyze which tax is the more stabilizing, we can first define built-in flexibility subsequently derive the effect of changes in built-in flexibility on a. Defining built-in flexibility as the increase in tax revenue due to an autonomous in income, we have:

(n.4)

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21. In formulating the consumption function, we have assumed no money illusion. If certain parts of consumption are exempt from the indirect tax, h is a weighted average, which depends upon the proportion being exempt.

Fi will of course be affected by both the tax rates and the coefficients of
the structure, but if we confine ourselves to the former, we can derive the
following relations between the degree of built-in flexibility and a:

(n.*]

Thus in terms of changes in the degree of built-in flexibility the indirect
tax has the strongest influence on a, whereas the relative importance of Tp
and Tc depends upon b(i + fe)"1 compared with n.

If we alternatively wish to analyze the effects on the degree of automatic
stabilization of changes in the tax rates directly, we obtain:

fn.6)

For plausible values of the coefficients changes in h are seen to have the
smallest effect whereas the relative importance of t\ and h depends upon
3(i -—fh) (i -f h)~x compared with nfh.

Speaking loosely the difference in ranking between (11.5) and (11.6) may be explained by the fact that it takes a substantial change in h to change 2^3 by "one unit"22, and since ais positively affected by increases in h, a "one unit" change in F3 will have a rather large effect on a.

It is, perhaps, more interesting to note that the results in (11.6) are the reverse of what is often found for discretionary fiscal policy. Thus, it is well known23 that in order to obtain a certain reduction in total activity, £1 and h must be changed by equal amounts, whereas the required tax revenue changes are larger for 7i than for T%. Hence, for discretionary policy, changes in h are

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22. This is esaily seen by partial differentiation of (n.4).

23. See for instance Peacock and Shaw (1971, pp. 89-90).

in a sense more effective than changes in t\, but as seen from above the reverse
holds for automatic stabilization.

For all taxes we find that automatic stabilization depends positively on built-in flexibility and marginal tax rates. In the above model we have assumed money illusion, but if we alternatively formulate the consumption function as:

it is easy to show that the stabilizing effect of the indirect tax will increase. On the other hand, if we have "positive" money illusion - i.e. C is a positive function fø - the indirect tax will be destabilizing. This, however, does not seem very likely, so we shall not consider this case any further.

B. Automatic stabilization of T for various exogenous variables

In the results derived above we have not specified which exogenous variable is changing, but it is easy to see that the results will hold for changes in a, c, G, Ex and k. This similarity, however, is due to the rather simple structure of the model. In large scale econometric models24 it has been found that automatic stabilization very much depends on where the change takes place, and this again implies that a cannot be measured uniquely. In that case it seems most relevant to define a in relation to the exogenous component that has caused most of the changes and to evaluate the tax structure accordingly.

C. Automatic stabilization of other endogenous variables

Endogenous variables, which depend on income, normally will be stabilized, income is stabilized. Thus, if imports are a positive function of income, the balance of payment will change less when autonomous demand for domestic products is shifting. If, on the other hand, the autonomous shifts occur in demand for foreign products or in exports, the changes in the balance of payment are accentuated, when taxes are endogenous, since income is now restrained in bringing the economy towards balance of payment equilibrium²⁵.

If prices are a positive function of nominal income, endogenous taxes will
also reduce price movements, but this effect may be counterbalanced by the
fact that private demand and government expenditures are influenced by

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24. See Balopoulos (1967), Goldberger (1970), and Chalmers and Fischel (1967).

25. For derivation of a similar result from a somewhat different approach, see N. Kaldor (i960, P. 48).

price movements. Let us consider this problem in terms of the following simple
model:

where P denotes the level of prices and all variables are stated in nominal terms. We have assumed that investors are subject to money illusion - whereas consumers are not -, that the level of prices is determined partly autonomously and partly by income, and that the level of government expenditures is adjusted price movements.

Solving for P we obtain

if G is adjusted for price changes, and

(n-9)

if G is not adjusted for price changes.

Computing the Musgrave-Miller measure for changes in ey we get re
spectively:

and

(n.n)

ai is clearly positive, whereas kz may be negative - and the public sector thus destabilizing - if the adjustment factor for G is large and the tax rate small. With respect to the latter, two influences seem to be present under inflation. Thus most transfer payments vary directly with the level of prices causing a reduction in t, whereas the income tax schedule when based on nominal income an increase in t, if the tax schedule is progressive₂₆,₂₇.

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26. For further evaluation of automatic stabilization under inflation see Gel ting (1966, pp. 126-28).

27. A higher marginal tax rate, on the other hand, may give rise to demand for higher wages and subsequently higher prices. The effect of this destabilizing influence of a higher / can be worked out by making the autonomous element in the price equation a function of t.

Brown (1955) and Waldorf (1967) have analyzed built-in flexibility and the effect on price movements, using a stricter criterion than the one applied above. Thus they require, that the level of real taxes must increase, when prices rise, and this requirement leads - not surprisingly - to the conclusion that for autonomous changes in prices, the tax must be progressive in order to stabilize price movements.

III. Dynamic Models

When time is introduced into the model either through an explicit time variable or through lagged relationships, a number of problems arise, and as will appear from below, the interpretation of automatic stabilization becomes some what ambiguous. We shall analyze these problems in both short term models and under long term growth2B.

A. Short Term Models

Some of the problems mentioned above can be shown in the following very
simple model for a closed economy, where for the sake of simplicity we only
consider a personal income tax:

By substitution we get a first order difference equation in T:

(111.1)

The particular solution will be:

(m.2)

and the homogenous solution is:

(ra-3)(ra-3)

yielding a general solution:

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28. The analysis is incomplete, as we mainly study lags from the point of view of automatic stabilization the revenue side. Thus we do not at all deal with lags connected with discretionary changes in expenditures and revenues.

In order to determine the coefficient A we need an initial condition, which
we may specify as Yt 0= ¥0, and the complete solution, will then be:

(111.4)

If, on the other hand, we assume taxes exogenous, we get the following
complete solution:

(ra.s)

The last term in the above equations is the long term equilibrium for T,
and if we apply the Musgrave-Miller measure to the long run situation, we
obtain:

(m 6)

This expression can be interpreted as the relative reduction in the long run multiplier, a is positive, if the denominator is positive, and this again depends upon the stability of the long run equilibrium29. However, as appears, the endogenous income tax will increase the likelihood of having the stability condition satisfied.

For a short run analysis, however, a study of the long run equilibrium is not particularly interesting, as it may never be reached, and we therefore turn to the first term in equation (111.4), which will determine the time path of the system if an equilibrium situation is disturbed. This term consists of two parts, of which the first is the initial deviation from equilibrium and can be interpretedas scale factor. The second part determines both the actual time path and the time required for reaching the new equilibrium. When comparing equations (111.4) and (111.5), we find that endogenous taxes have reduced both the initial disturbance3o - and thus the scale factor - and the adjustment time, the latter because r will reduce the numerator and increase the denominatorin exponential part. In addition, endogenous taxes will increase the

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29. In general the measures derived for automatic stabilization in the static model will correspond to long run automatic stabilization in a dynamic model. Cf. Balopoulos (1967, p. 243).

30. Thus the static Musgrave-Miller measure is applicable for evaluating the "scale effect", as it is determined by the multipliers. See also Thalberg (1971, p. 307).

likelihood of obtaining a stable solution, for which the condition is that the
base of the exponential expression is absolutely < i.

For other models, however, the situation is less clear. Thus, when it comes to higher order difference equations, it is not unlikely that we may have tö compare systems, of which one is characterized by initially rather violent fluctuations, but a rapid adjustment time, and the other by a smooth movement towards the new equilibrium, but a rather long adjustment time. A possible way out may be to integrate over time the absolute deviations from equilibrium, a priori it is difficult to judge, which system is the more stabilizing.

In the following we shall analyze a few examples where the emphasis will be on various tax structures, whereas the specification of the other relations is constrained by the fact that difference equations of order higher than two are rather difficult to evaluate without simulations or empirical measures³¹.

i. Let us initially consider a model consisting of the following equations:

Solving the system for T, we get the solution33:

(m.7)

As appears, a high tax rate34 will reduce the scale effect - and thus reduce
the amplitude of the oscillations - but it will increase the adjustment time and

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31. The method used is similar to that applied by Smyth (1963) and Gørtz (1971).

32. This formulation resembles a tax system, which Denmark had until 1967.

33. See appendix 11.

34. When tax revenue in period tisa function of income in period t^—i we have to denne built-in flexibility in terms of a certain time period, If t and t^—i refer to years, and the time period is i year, built-in flexibility in the above model will be o, whereas it is r for a time period of 2 years. If the time period is 3 years built-in flexibility is < r, since we now have to take 7*t_i into account, and in the long run the built-in flexibility will be r/(i +r), which is less than r. In the following we shall use built-in flexibility and marginal tax rate interchangeably.

the likelihood of getting antidamped oscillations, which occur when r(i — v)j
(l—b — v) > i35.

If, on the other hand, the tax function had been Tt = rYt_i the complete
solution of T would have been:

(m.B)

Thus a high tax rate will reduce the scale effect, but it will still have a destabilizing influence on the adjustment path³⁶, although the influence of r is somewhat reduced, as bisin most cases less than i_— v. These results, however, much depends on the lag structure specified for the other equations.

2. Next, let us consider the following model:

which yields the following equation for 1"

(ra.9)

As we now have a second order difference equation, we shall confine
ourselves to analyzing the roots, as analysis of the complete solution gets rather
complicated.

Solving the above equation we find complex roots and fluctuations if (c + v— br)² < 4(z? + cr) (i^—b), and the fluctuations are damped for (v -f- cr)l(i —b)<i. If the roots are real, a stable solution is obtained³⁷ for (c — r {be))/ +c))/ — b) < i. Hence we find, that a high degree of built-in flexibility will increase the likelihood of getting a fluctuating adjustment path and of getting antidamped fluctuations, implying that a new equilibrium may not be reached. On the other hand, when the roots are real and the adjustment path smooth, a high degree of built-in flexibility will make the system converge faster to a new equilibrium.

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35. It is interesting to note that Thalberg (1971, p. 309) comes to the same conclusion; viz. that there tends to be explosive fluctuations in Y, when the lag in the tax function is longer than the lag in the consumption function.

36. Thus a high r will increase the likelihood of getting antidamped oscillations.

37. We can assume both roots positive, since for most cases br^—c^—v < o, v +cr> o and I—b1^—b > o. For further explanation of the method used for deriving the above conditions see app. 1.

3. Let us now change the tax function to38

whereas the other functions from case 2. are retained.

Solving for T we obtain

(m.io)

which will yield complex roots if (c(i — r{) — br₂ -f v)² < 4(^2 + v) (1 — £ (1 — fi)). The roots will furthemore yield antidamped fluctuations for (^2 -f- »)/(i — (1 — **i)) >1• From these conditions we observe that r\ will have a stabilizing influence, whereas a high r2 will be destabilizing. Thus, lags in the tax structure will tend to accentuate autonomous disturbances, and this effect would be even stronger, if we also took monetary effects into account.

When the roots are real, and we have a smooth adjustment path, the stability is₃₉ I—c—b~>—{b -f- c) (ri + r2), and we find that a high degree of built-in flexibility will increase the probability of convergence, regardless of whether taxes are lagged or unlagged.

4. Let us finally consider a case where consumption is assumed a distributed
lag function of disposable income:

where o<X< i and V Xi = i. By a Koyck-transformation this function
.= i
can be reduced to:

In order to avoid third order equations we assume the following equation
for respectively investment and taxes:

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38. The argument behind this formulation is that the tax payments occuring in year t for institutional reasons are determined by both present and past income. Consequently we also have to assume that consumption is determined by tax payments and not by tax liabilities. Another implication of the formulation is that r\ and f2 do not refer to actual tax rates.

39. We can again assume both roots positive, cf. app. i.

40. An investment function basedon the capital stock adjustment principle can be dealt with as the distributed lag consumption function, if the model is supplemented with a function for the capital stock (K_t - It + Kt-i).

Solving for T we then get41:

T_t(\ — Xb{\~ r)) — r*_i(A +») +Tt^v?.= {a +G)(1—A) (III.n) and the roots are complex if (A + v)² <vX(i—Xb(i— r), but yield damped fluctuations for vXj(i —bX(i— r)) < i. Hence, a high tax rate will increase the likelihood of a fluctuating but damped adjustment path.

For real roots and a smooth adjustment path, we get convergence for
(i — X) (i — v) — bX(i — r) >o, and a high tax rate is seen to stabilize the
adjustment.

The above four cases should only be considered illustrations of the appropriate analysis of automatic stabilization in dynamic short term models, and consequently conclusions we have arrived at are very tentative. However, under the present Danish personal income tax system, a high degree of built-in flexibility seems to have a stabilizing influence on the adjustment path, especially there are no lags in tax payments. It should be kept in mind, however, that the models used are extremely simple, and that any realistic model would lead to difference equations of order higher than two, which are too complicated discuss when the structural coefficients are unknown.

B. Long term growth

A number of economists⁴² have discussed the effects of discretionary fiscal policy on the equilibrium rate of growth and on deviations from this equilibrium. discussion, however, is not the subject of the following sections, where we merely analyze the stabilizing or destabilizing effects of a given tax and expenditure structure.

These effects are often thought of in terms of "fiscal drag", which conceptuallycorresponds the automatic budget reaction. Thus, if government expenditures are constant, "fiscal drag" will correspond to the automatic increasein revenue, when full employment income increases, and it is thereforeequivalent Pearse's static measure of automatic stabilization⁴³, 44.

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41. See appendix n.

42. See for instance: Kurihara (1956), Gurley (1953) and Musgrave (1959, ch. 20).

43. "Fiscal drag" should be distinguised from "fiscal leverage" (cf. Musgrave (1964)), which refers to dG-bf'(q) dq — i.e. the impact of a discretionary change in the budget — and from "changes in full employment budget surplus", which most often refers to dG—f (q) dq; i.e. the "unweighted" budget change at full employment, when discretionary changes occur. f(T,q) denotes the tax function p. 3) and b the marginal propensity to spend.

44. Note that "fiscal drag" is independent of the source of the income change; cf. Bent Hansen (1969, pp. 23-24).

Despite this resemblance, it does not seem appropriate to interpret automatic stabilization in a long run growth model exclusively as "fiscal drag". Whether a "fiscal drag" occurs will depend on the tax function compared with the government expenditure function⁴⁵, and whether the postulated contractive effect occurs depends upon the development in private expenditures compared with the government budget. In addition, however, it is of interest to analyze to what extent the existence of endogenous taxes may dampen possible deviationsfrom equilibrium path, and this problem is entirely different from that of "fiscal drag". Both questions will be dealt with in terms of a modified Harrod-Domar model and a neoclassical model.

Harrod-Domar model. The former has also been used by Peacock and Shaw46
and is specified as follows:

Assuming that p % of G is invested, and that the capital output ratio is
ijq the annual increase in productive capacity is qdYt_i where d—c-\-pg,
and the equilibrium rate of growth is47:

(ra. 12)

For given values of b,c> g,p and q it is possible to find a tax rate that will secure full capacity growth. Thus for b = 0.9, c = 0.15, g = 0.1, p = 0.5, and q = 0.40, the equilibrium tax rate is 0.1, and if that rate is adopted, the development income will be as shown in the second column of table 1, where column 1 indicates growth in productive capacity. If r < 0.1, income will grow faster than productive capacity, and if r > 0.1 income will grow less than capacity, implying that we have a "fiscal drag".

As the concept of automatic stabilization deals with the response of the

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45. Thus for the Danish economy there does not seem to be any problem of insufficient increase in government expenditures!

46. See Peacock and Shaw (1971» pp. 125 ff).

47. We have Y_t - b(i—r)r_t.i + cT_t + g?t or T_t = é(i—r)r«_i/(i—^{c— g)} or <fryr_{t _i} = b{i-*)i(i—c—g) —i which in equilibrium must equal the growth in capacity

economy to autonomous changes, we shall next turn to the question of how the existence of a public sector will affect this response. Let us assume that in period 1 c increases to 0.2 the other parameters remaining unchanged. This change will increase the capacity rate of growth from 8 % to 10 %, and we find a development in capacity as shown in column 3. If the tax rate remains unchanged at 0.1, income will grow as shown in column 4, and we notice an increasing discrepancy between capacity and income.

An expression for automatic stabilization corresponding to the static
measure might be obtained by deriving the effect on the demand determined
rate of growth with and without the public sector. This will yield:

(in. 13)

where n, and »2 denote the demand determined rates of growth. If this expression negative, the public sector will have a destabilizing influence, and whether this occurs will clearly depend upon the tax rate in relation to g, whereas the proportion of G that is spent on public investments does not have any influence.

Alternatively - and in our opinion more satisfactory - we can make the tax function more realistic by introducing a progressive rate structure and compare results of this change to either a system with proportional taxes or a system with no public sector at all. Following Peacock and Shaw we shall introduce progressive structure by making the tax rate a function of the rate of growth of income:

These formulations do not correspond completely to the actual determination a progressive rate structure, but if we take lags into consideration, the above formulations are quite realistic if T refers to tax payments and not to tax liabilities, and they furthermore simplify the calculations⁴⁸.

Starting with r«_i = 0.1 and assuming s = 0.0003 tne two formulations
yield the results shown in respectively columns 5 and 6in table 1. Peacock and

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48. We might alternatively let tax revenue be a function of the rate of change in income. This is done by Musgrave (1959, p. 512) to illustrate "formula flexibility".

Shaw only consider the second formulation of the progressive rate structure and conclude that a progressive structure is destabilizing compared with a proportional structure as it will overshoot the new equilibrium growth rate, when changes in the economic structure occur.

This conclusion, however, does not seem warranted, since a proportional rate, which is an equilibrium rate before the change in c, overshoots to a much greater extent than the progressive rate. In other words, with a proportional tax rate a discretionary policy is required to bring the economy back to equilibrium whereas under a progressive structure, the deviations from equilibrium growth are damped, although the system will not automatically be brought to the new equilibrium growth.

The first formulation of the progressive rate structure yields a desirable dampening influence immediately after the change in c, but later on the dampening is too strong, the deviations of V%^m from Vi^c becoming absolutely larger than the deviations of Vt^m.

The two formulations of the progressive rate structure have the common feature that they sooner or later will produce a "fiscal drag", unless g also changes with the rate of growth of income. In that sense a progressive rate structure may be undesirable and destabilizing. If, however, we consider stabilizationin to the system's response to autonomous changes, a "lagged" progressive tax is more stabilizing than a proportional rate structure and consequently also more stabilizing than a system with no public sector. As appears from the table the progressive tax will damp the deviations from equilibrium growth and thus give the government more time for introducing the appropriate discretionary changes. An. "unlagged" progressive structure, on the other hand, is destabilizing except for the first year, and it does not leave

much time for the authorities to introduce changes. Therefore, from the point
of view of stabilizing economic growth, a with-holding principle combined
with a strongly progressive structure seems unfortunate.

Neoclassical growth model. As pointed out by several authors there are many possibilities for affecting growth rates in a Harrod-Domar model, but when it comes to neoclassical models, the influence of fiscal policy is rather limited, as the equilibrium growth rate is determined exogeneously. However, with a given rate of growth, it is still possible to affect the level of consumption through fiscal interventions, and furthermore discretionary fiscal policy may shorten the adjustment time⁴⁹.

Furthermore, fiscal policy may still have an important influence on the actual rate of growth, which probably is a much more relevant variable from a policy point of view. Consequently, the practical use of the following section may seem very limited, as we have only analyzed automatic stabilization within a neoclassical equilibrium framework. The model used is as follows:

L and K denote respectively labour and capital, and / denotes net private and public investment. The rest of the notation is explained in earlier sections. As appears, we assume disembodied technical changes and a "natural budget reaction", since total tax revenue is spent on either consumption or investmen ⁵⁰.

The rates of growth of income and capital are respectively

and

(m.14)

(m-i5)(m-i5)

where q — the output/capital ratio, and d — the rate of depreciation.

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49. See Peacock and Shaw (1971), and Cornwall (1963 and 1965).

50. Alternatively we might assume G constant or a positive function of T. It makes no difference for the results derived below, as long as the model has to satisfy the condition that total investments = total savings.

(m.i6)

Thus, the equilibrium rate of growth is given exogenously and cannot be influenced by fiscal policy. If, however, an equilibrium situation is disturbed, the fact that we have endogenous taxes may influence the time required for returning to the equilibrium movement.

If we assume that the private propensity to save increases from say s₀ to si⁵¹, the actual rate of growth will deviate positively from the equilibrium rate of growth, but after some time the system will return to the equilibrium, where the output/capital ratio is now lower and the per capita income higher.

As pointed out by R. Sato (1963), the adjustment time may be considerable, we shall follow his method in analyzing the factors influencing the adjustment time. It is natural to consider the adjustment path for q, since q is constant in equilibrium. Consequently one can derive a difference equation for q, and on the basis of this equation it is possible to compute the following expression₅₂ for the time required to obtain k%of the total adjustment:

(in. 17)

Clearly, endogenous taxes affect adjustment time through the second term
in the numerator, and differentiating this term with respect to r, we obtain:

(m.18)

Hence, we find that endogenous taxes will prolong the adjustment time,
and a high degree of built-in flexibility will in that sense be destabilizing. This
result may seem surprising, but can probably be explained by the fact that the
existence of endogenous taxes will increase total investments and consequently
the rate of growth of capital during the disequilibrium period. Since the latter
00 o _
is characterized by K\K > TjT, the relative increase in KjK implies that it
o o
takes longer before KjK returns to the same level as TjY.

Concluding this section we note, that in a neoclassical growth model the
problem of "fiscal drag" is excluded by definition, as total savings always

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51. We have also analyzed disturbances caused by changes in p, n and b, but in no case did the tax rate have any influence on the adjustment time.

52. Derivation of this result has been omitted due to lack of space, but is available upon request from the author. The derivation follows very closely the method applied by R. Sato.

equal total investments. But, of course, this observation is not very helpful in
a situation, where "fiscal drag" is a problem.

Conclusion

In this paper we have endeavoured to analyze how automatic stabilization is defined and determined in various models with particular emphasis on the influence of the tax structure. The static conclusion that automatic stabilization is a positive function of marginal tax rates - or built-in flexibility - holds for some of the dynamic formulation.sbut not for others, and it becomes particularly when we turn to long run growth in a neoclassical setting.

Since any realistic empirical model will contain lagged relationships, the static measure is inappropriate, except for long run equilibrium changes. On the other hand, the dynamic measures are not very useful either, as long as the lag structure and the parameters of the model are unknown. Similarly, it should be borne in mind that both the static and the dynamic measures are relative in the sense, that whether they are "high enough" depends upon how much there is to stabilize. This, as well as a more exact evaluation of the automatic in a dynamic model, is only possible on the basis of empirical measurements, which, however, are beyond the scope of this article.