The Multiplier-Effect of Income and Consumption Taxes By

Erik Gsrtz

Contents: I. S u m m a r y of t h e E a r l i e r D i s c u s s i o n . - - I I . T h e B a s i c M o d e l of the D i s c u s s i o n . -- I I I . T h e R e s u l t s B a s e d o n t h e S i m p l e Model. -- IV. A D i a g r a m m a t i c a l P r e s e n t a t i o n . - - V. T h e B a l a n c e d B u d g e t M u l t i p l i e r s . V I . E x t e n s i o n o f t h e M o d e l . - - V I I . T h e P r i c e a n d W a g e F o r m a t i o n of t h e Model. - - V I I I . T h e M u l t i p l i e r s o f t h e M o d e l . - - I X . T h e E f f e c t o f t h e A s s u m p t i o n t h a t all C o n s u m p t i o n T a x e s a r e n o t P a s s e d o n t o t h e C o n s u m e r s . -- X. Closing R e m a r k s .

he discussions on public budget multipliers which took place in Nationalekonomisk Tidsskrifl and Weltzairtschaflliches Archly about ten years ago left a few problems unsolved. These are the subject of the present article.

T

I. Summary of the Earlier Discussion The point of departure was the opinion that the conventional GeltingHaavdmo theorem 1 (according to which the balanced budget multiplier of a tax-financed increase in expenditures on goods and services equals one) applies only to national income at market prices. If the tax increase 1 The theorem owes its name to the rediscovery of the income-effect of an income-taxfinanced increase in public expenditures on goods and services mentioned by some of t h e Mercantilists as early as in the middle of the seventeenth century and later rather explicitly by David Ricardo, Principles o/ Political Economy and Taxation, Ed. With Introd. Essay, Notes and Appendices, by Sir E. C. K. Gonner, London, 1927 Chap. X X X I : "On Machinery". --See thus Jorgen Gelting, "Noglo Bemmrkninger om Finansieringen af offentlig Virksomhed", Nationalakonomisk Tidsskritt, Bind L X X l X , Kobenhavn, x94x , pp. 2 9 3 s q q . - Trygve Haavelmo, "Multiplier Effects of a Balanced Budget", Econometrica, VoL X I I I , Chicago, IlL, x945, pp. 3xxsqq. - - For an extensive bibliography see Paul A. Samuelson,"The Simple Mathematics of Income Determination", in: The Colleaed Scientific Papers o / P a u l A . Samuelson, Ed. by Joseph E. Stiglitz, Vol. II, Cambridge, Mass., and London, x966 , p. x2o4. - - The Mercantilist literature on public finance is thoroughly discussed in Jorgen Gelting, Finansprocessen i det okonomiske kredsleb, Kobenhavn, I948, esp. Chap. 2.

212

Erik Gortz

is levied on consumption, the multiplier effect measured a t / a c t o r prices is reduced to zero 1. If expenditures are financed partly b y consumption taxes and partly b y income taxes the multiplier at factor prices is between zero and one, the exact size being determined by the relative importance of the two forms of taxation. This opinion was questioned b y NOrregaard Rasmussen 2. He argues that as the effect o/the balanced budget is real in the sense that the activity increases, the result is surprising. N0rregaard Rasmussen instead concludes that "the multiplier of the balanced budget financed b y consumption taxes is 1 when measured at factor prices, whereas it is 2 when measured at market prices ''3. In N0rregaard Rasmussen's opinion the abovementioned implicit conclusion thus still holds that the difference between the consumption tax multiplier measured at market prices and at factor prices is equal to one. As the balanced budget in case of consumption-tax-financed expenditures normally has a positive effect on national income at factor prices N0rregaard Rasmussen's objection is warranted. This effect, however, is not as great as the effect when expenditures are financed by income taxes, unless consumers' behavior is characterized by a certain kind of money illusion. His conclusion, therefore, implies the assumption that consumption taxes have the same effect on real consumption as the same amount of income taxes 4. But "consumption taxes reduce consumption more than does the same amount of income taxes, as the former are paid out of consumption only, whereas the latter are paid out of saving too ''~. For the revenue effect to be the same of the two forms of financing a greater change in the consumption tax rate than in the income taxe rate is thus required. From the discussion it m a y be concluded t h a t the multiplier at/actor prices of the balanced budget is one when expenditures are financed by income taxes, whereas it is normally between zero and one when expenditures See for instance Erich Schneider, Einl~hrung in die Wirtscha]tstheorie, T. I I I : Geld, Kredit, Volkseiftkommen u~td Beschd]tigu~r ix., verb. u. erw. Aufl., Tfibingen, i969, pp. 268 sqq. 2 p. Norregaard Rasmussen, "Markedspriser ctr. faktorpriser ved det balmlcerede budget", in: Til Frederik Zeuthen, 9- september 1958, Udg. af Nationalokonomisk Forening, Nationalekonomisk Tidsskrift, Bind XCVI, i958, Tillmgsh., Kobenhavn, i958 , pp. 3Iosqq. * Ibid., p. 312 (author's translation). * That this assumption implies money illusion in cousumers' behavior is shown by John Vibe-Pedersen, "Multiplikatorvirkningen af indirekte skatter", Nationalekonomisk Tidsskvi/t, Bind XCVIII, 196o , pp. 22osq. 5 Ibid., p. 22i (author's translation). This article was published at the same time as Winfried Vogt, "Einige Unklarheiten in der Diskussiou fiber die Multiplikatorwirkung eines ausgeglichenen Budgets", Weltwirtscha/tliches Archiv, Bd. LXXXV, I96o ]I, pp. 55sqq. The two articles reach the same conclusions in many fields.

The Multiplier-Effect of Income and Consumption Taxes

213

are financed by consumption ta,es. T h e e x a c t s i z e o f c o u r s e d e p e n d s o n t h e p a r a m e t e r s of t h e m o d e l , f r o m w h i c h t h e m u l t i p l i e r is d e r i v e d 1. A s r e g a r d s t h e difference between the multipliers a t m a r k e t a n d a t f a c t o r p r i c e s a l l a u t h o r s a g r e e t h a t i t is zero when expenditures are financed by income taxes. When expenditures are financed by consumption taxes the difference equals o n e a c c o r d i n g t o m o s t a u t h o r s . A s w i l l b e s h o w n i n t h e f o l l o w i n g t h i s is t h e c a s e o n l y if i n c o m e t a x e s a r e i n d e p e n d e n t of i n c o m e i n t h e y e a r i n q u e s t i o n , i.e. o n l y i / a "pay-as-you-earn" income

tax system is not introduced. H. The Basic Model of the Discussion T h e f o l l o w i n g c o n v e n t i o n a l assumptions u n d e r l i e t h e a b o v e - m e n t i o n e d results: (I) T h e m a r g i n a l s o c i e t y a s a w h o l e 2.

propensity

to consume

is constant

(2) T h e r e is n o m o n e y i l l u s i o n i n c o n s u m e r s ' c o n s u m p t i o n f u n c t i o n is i n r e a l t e r m s 8. (3) I n v e s t m e n t

and applies to

behavior,

i.e.

the

d e m a n d is e x o g e n o u s 4.

(4) F a c t o r p r i c e s a r e c o n s t a n t , i.e. a l l c o n s u m p t i o n o n t o c o n s u m e r s 5.

taxes are passed

(5) T h e r e is n o f o r e i g n t r a d e 6. (6) T h e t a x s c h e d u l e is p r o p o r t i o n a t e 7. The balanced budget multipliers are only briefly dealt with in Vibe-Pedersen, op. c/L The results will be proved below, however. t See e.g. Samuelson, op. cir., p. x2o5. - - Henry C. Wallich, "Income-Generating Effects of a Balanced Budget", The Quarterly Journal o! EcOnomics, Vol. LIX, Cambridge, Mass., x944/45, P. 8o. - - William J. Baumol and Manrice H. Peston, "More on the Multiplier Effects of a Balanced Budget", The American Eco~*omic Review, Vol. XLV, Menasha, Wise., x955, p. x42. s This is not always explicitly mentioned; see, however, Vibe-Pedersen, op. cir., where a rather extensive discussion on the effects of alternative consumption functions is given. 4 See e.g. Samuelson, op. cir., p. x2o5. - - Baumol and Peston, op. cir., p. I42. s This is not always explicitly mentioned either; see, however, Vibe-Pedersen, op. cir., p. 236. e This is only sometimes assumed. For the impact of various assumptions as regards foreign trade see William H. White, "Measuring the Inflationary Significance of a Government Budget", International Monetary Fund, Staff Papers, Vol. I, Washington, D.C., x95o--5I, Pp. 355 sqq. 7 This, too, is only sometimes assumed; as will be shown in the following, the level of the tax schedule affects some of the results presented below. The assumption will therefore be abandoned later.

214

Erik Gortz

These a s s u m p t i o n s m a y be summarized in the/ollowing model which is essentially equal to t h a t of Vibe-Pedersen: (I) Y m = C m + I m (2) Y, = C ~

+I,

(3) Yf = Cr" Pc + I t . Pi Yd

(4) Cr

= a + b 9

(5) I=

---- I r "

(6) Cm

= Cr" Pc" (1 + t)

(7) Yd

= Yf - - T

(8) T

=

Pc" (1 + t)

Pi

d 9 Yf

Notation (subscripts m, f, and r for m a r k e t , factor and c o n s t a n t prices, respectively) : (a) Y

= N a t i o n a l income

(b) C

= Consumption

(c) I

= Investment

(d) Pc and Pi

=

(e) Yd

---- Disposable income

(f) T

---- I n c o m e t a x e s

(g) d and t

= I n c o m e and c o n s u m p t i o n t a x rates, respectively

(h) a

= Constant (point of intersection between c o n s u m p t i o n function a n d o r d i n a t e axis)

(i) b

= Marginal p r o p e n s i t y to consume

F a c t o r prices of c o n s u m e r and i n v e s t m e n t goods, respectively

Of these It, Pc and Pi are exogenous variables 1, a a n d b are c o n s t a n t s , d a n d t are policy p a r a m e t e r s , a n d the r e m a i n i n g variables are endogenous. x This makes I exogenous too, of. equation (5). This equation is included for the sake of comparison with the following.

The Multiplier-Effect of Income and Consumption Taxes

215

In the following the effects o] changes in ,aUs. rather t h a n revenue will be discussed. Introduction of a pay-as-you-earn income t a x system in most countries (in D e n m a r k on J a n u a r y Ist, 197o ) m a k e s this appropriate, also with respect to income taxes. The quasi-multipliers x showing the effects of changes in the a m o u n t s of income and consumption taxes as well as the multiplier-effect of the balanced budget will be only briefly dealt with. IlL The Results Based on the Simple Model From the above model the following multipliers ~ can be derived: dY r (9) dd = - -

1 b 9 Yf Pc l + t - - b . ( 1 - - d )

(IO)

dYf dYr dd -- Pc" dd

(II)

dYm

dd

(12)

dY, dt

[38]

dYr -- Pc" (1 + t ) . dd 1

[39]

b 9 (1 - - d ) 9 Yf

Pc (1 + t ) - i l + - t .... b -(1 - - d))

Cf. Vibe-Pedersen, op. c/L, p. 2x5 (author's translation), where a quasi-multiplier is defined as: "The relation between the change in a dependent variable and the change in another dependent variable, the latter being regarded as a quasi-parameter." This concept will be illustrated in the subsequent discussion on the balanced budget multipliers. The concept is the same as Samuelson's pseudo-multiplier, see Samuelson, op. cir., p. i~o8: " . . . it is always possible to relate the resulting change in income to the resulting change in any other variable of the system . . . . Examples of such a pseudo-multiplier..." Samuelson criticizes the concept as: "Such a ratio can be called a multiplier if one pleases, but really it is a m ~ i s m u t a ~ i s concept and had better be recognized as a chameleon creature whose numerical value call be changed at will by specifying different combinations of variation in the basic parameters [in Samuelson's version a shift parameter of the whole tax schedule, government expenditures on goods and services, and net investment] . . . " If a specification of the relation between the changes in the basic parameters (exogenous variables) is made, however, the concept is not entirely without interest. t In the formulas the endogenous variables Yf, Pc and Cr appear. By this of course is meant the level of the variables in the equilibrium solution. They can be replaced b y their solution values expressed in parameters and exogenous variables only, but not without making the expressions rather complicated. The numbers in [ ] refer to Vibe-Pedersen, op. cir., where it is implicitly assumed, however, that prices are equal to I. This does not make any fundamental difference. The formulas (9) and (x2) have no such references since they are not directly mentioned by Vibe-Pedersen. This is due to the fact that they are identical with (io) and (x3) provided that Pc = I.

216

Erik

Gortz

(13)

dYf dY r dt -- pC" d~-

[46]

(14)

dYm dY, dt -- Cr'pc + PC" (1 + t) 9 d~

[4711

The interpretation of these multipliers is relatively easy ~. Re-arrangement of (9) gives the following expression: --b. (9a) A Y r

Y~ 9 Ad Pc" (1 + t) 1--d 1--b.-l+t

=

If the income tax rate is increased by Ad, disposable income will be reduced by Ad. Y f (divided by pc. (1 + t) when measured at constant prices). Multiplication by b yields the (negative) primary effect on consumption at constant prices. In order to get the total effect on national income at constant prices this primary effect must be multiplied by an ordinary "secondary-eBect-multiplier ''a of: 1

1--b'--

1--d l+t

the marginal propensity to consume being adjusted for the effect of income and consumption taxes per unit. As Pc and t are constants, the multiplier at constant prices must be multiplied by Pc and Pc " (1 + t) in order to get the multiplier at factor and at market prices, respectively (cf. equation (3) and equations (i) and (6) of the model, respectively). I n V i b e - P e d e r s e n , op. cir., p. 234, t h e f o r m u l a is: dYm dt

b 2" (1 - - d ) ~ 9 YI =

a'pc

--

(1 + t ) . ( l

+ t --b

. (l - - d))

I t is e a s i l y s h o w n , h o w e v e r , t h a t t h e t w o e x p r e s s i o n s a r e i d e n t i c a l . s See ibid., pp. 232, 234. 9 Defined as " t h e r e l a t i o n b e t w e e n t h e t o t a l effect a n d t h e p r i m a r y effect of a c h a n g e in a p a r a m e t e r " , cf. ibid., p. 216. T h i s c o n c e p t w i l l b e i l l u s t r a t e d in t h e f o l l o w i n g g r a p h s .

The Multiplier-Effect of Income and Consumption Taxes

217

The analogous formula (12) for the effects of changes in consumption taxes are interpreted in the same way: --b. (i2a)

A Y r ~-

(l--d).Yf pc'(1 + t) 1--d 1--b.-l+t

At l+t

Disposable income at constant prices must be multiplied by the relative change in the consumption tax rate in order to get the effect on disposable real income. The (negative) primary consumption effect at constant prices is obtained by multiplying this quantity by the marginal propensity to consume. Again the total effect on national income at constant prices is obtained by multiplying by the same "secondaryeffect-multiplier" as mentioned above. The multipliers at factor and market prices are analogous to (IO) and (ii), except for the first term in (14), which is most easily interpreted in telms of Figures I and 2. IV. A Diagrammatical Presentation Figure I presents a picture of the part of the model formulated at constant and factor prices, i.e. equations (2), (3), (4), (7) and (8). In the first quadrant equations (7) and (8) are depicted 1 as Yd = ( 1 - d) 9Yr. The dotted line represents the relationship when income taxes are increased. In quadrant II the relationship between disposable income at current and disposable income at constant prices is given, the dotted line representing the effect of an increase in consumption taxes. The third quadrant portrays equation (4) (the consumption function). By means of these three equations the "consumption function" Cr(Y~) in case of given tax rates can be drawn (e.g. starting at M on the Yf-axis one arrives at A, B and C on the Cr-axis in the three cases, viz. before a change in taxes, after a change in consumption taxes and after a change in income taxes). In quadrant IV the "consumption functions" Cr (Yf) in the three cases are thus depicted as A', B' and C'. These and the Cr-axis of course intersect at the level a, and the slopes of the lines are equal to the product of the slopes of the corresponding lines in quadrants I, II and III. Adding the exogenous investment level Ir to the three lines gives the parallel lines, A", B" and C" representing the corresponding relationships between Yr and Yf, e.g. for A": 1 In Figures i and 2 angles are denoted by slopes of the lines in q u e s t i o n .

Erik G~rtz

218

Figure I

I!

I

I

,

u p ~ ( t + t)

M Y~

' "~:i%~' :"-('':~)'-'-'....~.:'~i':~"'

Ill J"

|

IV

q and Y,

(15) Y r = ( a + I ~ ) +

b" (1 d) . Y, Pc" (I + t) -

-

From equations (2) and (3) another relationship between Yf and Yr is obtained. Re-arrangement of these two equations gives: (x6) Y r : l r ' ( 1 - - P : )

+lpc- " Yf

and this relationship is depicted in quadrant IV as L. An increase in consumption and income taxes changes the equilibrium levels o/Y~ and Yr as indicated by the vectors from D to E and F, respectively. The movement from D to e.g. F may be decomposed in the two vectors

The Multiplier-Effect of Income and Consumption Taxes

219

DH and HF. The vector DH shows the primary effect on C~ and Yr of the increase in direct taxes and is equal to the level of Yf at which the change takes place (i.e. V/f) multiplied by the differential coefficient with respect to d of the slope of the A"-function (cf. (15))

IDHI=

--Yr.

b Pc" (1 + t)

The vector H F indicates how this primary effect is increased due to the operation of the secondary-effect multiplier. This secondary-effect multiplier is obtained by calculating the extent to which the effect of H F in the Y~-direction increases ! DH ], i.e. by calculating the relation between the slope of L and the difference between the slopes of L and A ' . The total result of the decomposition thus is: 1 dY~ (9) d d --

~[f.

b Pc" (1 + t)" 1 Pc

Pc 1 b 9 V/f b" ( l - - d ) ----p-~" 1 + t - - b ' ( 1 - - d ) Pc" (1 + t)

Similarly the effect on Yr of an increase in consumption taxes m a y be calculated as the primary effect I DG[ multiplied by the secondaryeffect multiplier, i.e. by the effect of GE in the Yr-direction: 1

(12) dYr = __ m/f. b" (1 - - d) 9Pc. dt (Pc'(l+t)) * 1 P~ 1 Pc

Pc = b" (l--d) Pc'(l+t)

b- ( l - - d ) 9 Yf (l+t).(l+t--b.(1--d))

From quadrant IV it is easily seen that the multiplier-effects at factor prices are equal to the multiplier-effects at constant prices multiplied by the reciprocal of the slope of L, i.e. by pc. The first quadrant in Figure 2 shows the same result, presented in consumption at factor and at constant prices, as the effects on consumption are equal to the effects on national income, Ir and Pi being exogenous. In quadrant II the effects on consumption (and thus on national income)

220

Erik Gortz

at market prices are shown. The market-price multipliers of changes in consumption and income taxes are seen to be equal to the factor-price multipliers multiplied by (1 + t); in case of an increase in consumption taxes, however, one must add the vestor I GE equal to the level Cf multiplied by the differential coefficient ( = 1) of the slope of the undotted line. Figure 2 c, p~

/

~f r x

-

_

~: ..........

~

"

14-t

C, p~ (1 + t)

V. The Balanced Budget Multipliers 1 The multiplier-effect of changes in public expenditures on goods and services 2 can

be

found

as

the

quasi-multiplier-effect

of c h a n g e s

1 These multipliers will be discussed in order to clarify some problems of the earlier discussion and to give an example of the quasi-multiplier concept. By this is not intended to " . . . suggest that there is something unique about the theory of the balanced budget multiplier." That this is not the case is one of the main themes of the valuable article by Baumol and Peston, op. cir., from which the quoted passage is taken (p. x4o), and of John G. Gurley, "Fiscal Policies for Full Employment, A Diagrammatic Analysis", The Journal o1 Political Economy, Vol. LX, Chicago, II1., I952, p. 525. In the latter paper the balanced budget multiplier is shown to be a special case of a general analysis, only. Baumol and Peston, op. cir., p. x4o, stress that the multiplier of the balanced budget (and all other multipliers) of course cannot be determined " . . . without the use of any empirical material." "...

2 Not transfer expenditures and not expenditures on capital account, since the former merely redistributes income, and adds no governmental effective demand for o u t p u t

The Multiplier-Effect of Income and Consumption Taxes

221

in Im1, a part or all of the investment thus being considered as public expenditures on goods and services: dYf --l+t dYf dIr Pi" 1 + t - - b 9 (1 - - d) 1 + t (17) Cilm ~ di m = = Pi 1 + t--b 9 (l--d) dL

From equation (3) the following expression for the effect on real consumption of changes in I m is obtained: (18) dYf

dCr

and thus after substitution: dCr 1 b " (1 - - d) (19) dI--~ = p-~" 1 + t - b " (1 - - d ) As the amounts of income and consumption taxes equal: (8)

T ~ d - Yf

(income taxes),

to that of the public," and since the effect of the latter is on liquidity only (Baumol and Peston, op. clt., pp. x4 I, x43, respectively). When the expansionary or contractionary effect of a given budget has to be determined in practice, simple exclusion of the two forms of expenditures is not a recommendable procedure, however, since they of course have some effect on effective demand. I n the former case the redistributive effect often changes the propensity to consume in an aggregate consumption function; and in the latter case much esp. investment activity is carried out only if public loans are granted. Following Ralph Turvey, "Some Notes on Multiplier Theory", The American Economic Review, Vol. X L I I I , i953, p. 285, the best procedure perhaps would be: " i . Calculate the change in each type of expenditure and each tYpe of revenue at the initial equilibrium position. 2. Multiply each such change in expenditure b y its (positive) income-creating coefficient and each such change in revenue by its (negative) income-destroying coefficient. 3. Obtain the algebraic sum. This is the multiplicand. 4. Multiply by the multiplier calculated with the new marginal tax and expenditure rates." Another possibility might be the construction of "inflative and defiative indices" of government receipts and disbursements both in case of full employment and in case of idle resources, as suggested by G. L. S. Shackle, "The Deflative or Inflative Tendency of Government Receipts mid Disbursements", Oxlord Economic Papers, Vol. V I I I , x947, PP. 46sqq. t This multiplier is easily obtained, see e.g. Vibe-Pedersen, op. cir., p. 230 , formula (3x).

Erik Gortz

222

and (20)

Tc

=

(consumption taxes),

C r - Pc" t

the effect on public tax revenue of changes in

Im

is obtained as:

9 dC~+ d .dYf--b't'(1--d) +d.(1 +t) di m di m 1 + t--b. (1--d)

(21) aTe dT d-~m+dfmm = p ~ ' t

In order to arrive at the factor income multipliers of the balanced budget one must calculate the extent to which the tax rates d or t must be changed /or this revenue effect plus the revenue effect o/ the changed rates to be equal to the change in public expenditures. As mentioned in the above discussion specification of the relation between the changes in the exogenous variables is a necessary condition of meaningful quasimultipliers. (a) In case of a change in the income tax rate equations (io) and (2) yield the following expressions: (IO) --dYf = dd

b . Yf

1 + t--b.

( 1 - - d)

and, as C r = Y r - - It: dCr dYr (22) d d = d--d-=

-

-

_

1 b 9 Yf Pc l + t - - b . ( 1 - - d ) ~

_

.

By means of the above-mentioned expressions for the amounts of income and consumption taxes the following revenue effect of a change in d is obtained: dTc dT dC~ dYf (23) ~d- + ~ = Pc" t . dd + Yf + d . dd = --

b 9t 9 Y f l+t--b-(1--d)

(l+t)

+ Yt--

b 9d 9 Yf l+t--b-(1--d)

9 (1--b).Y,

l+t--b.(1--d)

If the budget must be balanced the sum of the two revenue effects must be equal to the change in Ira, i.e. : (24) Aim = ARevenue = (1 + t) 9 ( l - - b ) l+t--b.(1--d)

9 Yf . Ad +

+d.(1 + b-t.(1--d) 1 +t--b.(1--d)

+t)

AI m

The Multiplier-Effect of Income and Consumption Taxes

223

or after re-arrangements:

(25) Ad = (1 - - d ) .

_1. Yf

Aim

This means t h a t the/actor income multiplier o] the balanced budget is equal to 1 when expenditures are "financed" b y an income t a x r a t e changO, since: (26) hYf =

1+ t 1 +t--b.(1--d) l+t 1 +t--b'(1--d)

9 Aim

" Aim--1

b " Yf 1 +t--b.(1--d)

9 Ad

b-(1 --d) +t--b'(1--d)

9 AI m

---- Aim, provided t h a t the income t a x r a t e is changed according to the abovementioned restriction. AYe = AI m means t h a t ACt = 0 and thus AT c = 0. As the a m o u n t of consumption taxes is not changed AYf = AYm, i.e. the difference

between the market price and the /actor price multiplier o] the balanced budget equals zero, when expenditures are "financed" b y income taxes. (b) In case of a change in the consumption tax rate the following factor income multiplier of the balanced budget is obtained b y similar, though rather complicated calculations: (27) A y f =

a 'Pc" (1 + t) * 9 Aim a.pe.(l+t)i+b. (l--d) 9 (Ya--Cra)

This expression shows t h a t the of this paper (viz. t h a t t h e / a c t o r equals zero if expenditures are correct only if a = 0, or Pc = 0,

opinion t a k e n as t h e point of departure

income multiplier o] the balanced budget "financed" b y consumption taxes) is or t ------ - 1, b u t sensible only i] a pro-

1 As mentioned b y Baumol and Peston, op.c/t., p. x42, foottxot~ 6: "[M. H.] Peston ["A Note on the Balanced Budget Multiplier", The American Economic Review, Vol. XLIV, x954 (PP. x29--3o)] has shown, however, (challenging Turvey otx this point) t h a t the balanced budget theorem is consistent with the existence of other leakages, and with indirect taxes in part~ular" (italics mine). As formula (26) shows it is independent of whether or not a "pay-as-you-earn" income t a x s y s t e m is introduced, too. Weltwlrt.~haftlie.heaAre~v Bd. CIV.

x5

224

Erik Gertz

portionate consumption [unction x is given. As appears this is independent el the size o/d, i.e. independent of w h e t h e r or not a " p a y - a s - y o u - e a r n " income t a x system is introduced. The multiplier is not equal to one either, since t h a t would require b = 0 , or d = l , or zero saving, i.e. Y d - - C m = 0 . The last case is the counterpart of the previous conclusion t h a t consumption taxes reduce consumption more t h a n does the same amount of income taxes, as the former are paid out of consumption only, whereas the l a t t e r are paid out of saving too. If saving and the marginal propensity to consume is positive and if 0 < d < 1, the factor income multiplier of the balanced budget is between zero and one when expenditures are "financed" b y consumption taxes. The corresponding market income multiplier appears to be: (28) AY m = a - Pc" (1 + t) 2. (2 - - d) + b . (1 - - d) 9 ( Y d - - Cm) . Aim, a . p c . (1 + t) 2 + b ( l - - d ) 9 ( Y a - - C m ) and the difference between the multipliers at market and/actor prices is then: (29) AYm - - AYf ----a - Pc" (1 + t ) ' + b - ( 1 - - d ) . ( Y d - - C m ) - - a . Pc" (1 + t) 2. d a . Pc" (1 + t) 2 + b . ( l - - d ) 9 ( Y c - - C m )

9 AIm

This difference equals one only if a = 0, or Pc = 0, or t = - - 1 or d = 0, i.e. meaningfully equal to one only i / a proportionate consumption

/unction is given or i / a "pay-as-you-earn" income tax system is not introduce#. Comments to equations (27)--(29) show t h a t the multiplier at m a r k e t prices equals one and the multiplier at factor prices equals zero only if a = 0. For the m a r k e t price multiplier to be equal to two and the x I n this case the consumption function coincides w i t h a proportionate consumption function in current m a r k e t prices. As shown b y Baumol and Peston, op. cir., p. I43, footnote io, (referring to the model in Turvey, op. cir.): " . . . such a s i t ua t i on yields a m u l t i p l i e r of zero because he assumes a 'money illusion' which leads people to keep t he i r mone y expenditures unaffected b y sales t axes." g In both cases the amount of income taxes is not changed, and as t he b u d g e t has to be balanced, then A T c = A Im. In the former case this is because A y f = 0, (cf. e q u a t i o n (27)) and in the l a t t e r case d = 0.

225

The Multiplier-Effect of Income and Consumption Taxes

factor price multiplier to be equal to one both zero saving and d = 0 is required 1. Once again, however, it must be mentioned, that these multipliers are derived on the assumption t h a t the consumption tax rate is changed to such an extent that the budget is balanced (and of course on the assumptions of the whole model). VI. Extension of the Model

So far the effects of changes in consumption and income taxes have been discussed exclusively as eBects coming via consumption disregarding a lot of further effects which are essential for the income formation process. This is a normal procedure in ordinary multiplier theory and is realistic in so far as the effects on consumption are the primary effects of changes in taxes (most investigations of the determinants of consumption show rather short time lags in consumers' reaction). The consumption effect of course produces a simultaneous e~ect on employment implicit in the above model. Shortly afterwards, however, this can be expected to bring about a change in the wage rates, which jointly with the effect of the change in employment on the marginal product and on the climate for price formation will soon change prices. L a t e r on all these effects change the demand for investment goods, but according to most findings a much longer time lag is involved here. The latter effect is therefore omitted from the following model, i.e. investment at constant prices is still exogenous. The total effect of changes in consumption and income taxes is thus being considered within a horizon of say 6----9 months 3. Accordingly in the following the assumption of constant factor prices is given up with the implication that the assumption of all consumption taxes being passed on to the consumers must also be abandoned. This requires a submodel describing the price [ormation of consumer and investment goods (equations (9~176 In equations (9~ and (IO ~ it is presumed that prices are equal to marginal costs multiplied b y a factor x Not zero saving and a = 0, since t h a t would mak e t he denominator of the mul t i pl i e rs equal to zero. The meaning of this of course is t h a t it is impossible to determine equilibrium in a demand oriented model, if a 45 ~ -consumption function is assumed. t According to the results of an unpublished Danish investigation b y K n u d Liittichau

the average time lag between changes in e m p l o y m e n t an d wage r a t e is less t h a n one q u a r t e r in the total postwar period. I n the investigation the percentage change i n prices, too, significantly effects the percentage change in the wage rate, but an average time l a g of more than six months is involved. a This of course is not completely correct, since not all links of the m u l t i p l i e r process

appear w i t h i n 5 - - 9 months. The first few links of the process, however, form b y far the greatest part of the t o t a l multiplier-effect. xS*

226

Erik Gartz

depending on employment in the two sectors 1 (equations (II ~ and (I2~ Equations (I3 ~ and (14~ are the production functions of the two sectors, and equation (15 ~ defines the total wage bill. Equation (i6 ~ has a rather strong resemblance to the Phillips-curve. As independent variable is used the number of employed rather than the unemployment rate. This is not a serious problem, however, as simple re-arrangements can easily give these variables common denomination. As dependent variable is used the wage rate rather than the percentage increase in the wage rate. This is a more serious problem, but as it is only intended to describe the effects of changes in taxes in the relatively short run (say 6 - - 9 months or something like that), the wage rate m a y be interpreted as the sum of a given size at the beginning of the period and the change in the wage rate depending on some labour market variable. Hereby the equation approximates the Phillips-curve. Finally equation (I7 ~ is an identity describing total labour demand. In accordance with these changes in the model, national income at /actor prices is divided into wage income and residual income*. It is assumed as usual that the two groups have different marginal propensities to consume. Instead of a constant income tax rate an income tax/unaion is introduced. This, however, does not raise serious problems as the marginal income tax rates are taken to be constant at the income level of the two groups. In order to calculate the effects of changes in income tax rates a shift parameter, d, is introduced in the income tax functions. In the reformula2ed model only equations (1~ ~) and (50)--(6 ~) are thus unchanged: (I ~ Y m = C m + I m

(2 ~) Y , - - - - C r + I r (3~) Yf = C r ' p c + I r . p i (4 ~) C, = a + b w " (5~)

Wd +bR" Rd Pc" (1 + t) Pc" (1 + t)

Im = I r . p i

t Hereby an element of monopoly is introduced. t The effect on the balanced budget multiplier of assuming two or more sectors in the economy is discussed in Baumol and Peston, op. cir. (and in their "Reply", The American Economic Review, Vol. XLVI, x956, pp. x6osqq). - - Alvin H. Hansen, "More on the Multiplier Effects of a Balanced Budget: Comment", ibid., pp. x57sqq. - - Richard A. Musgrave, The Theory o/ PuNic Finance, A Study in Public E~aomy, New York, Toronto and London, x959, pp. 4 3 8 s q q . - - T h e results of these works are generalized in Rolla Edward Park, "Redistributional Aspects of the Balanced Budget Multiplier: A Comment on the Musgrave, Baumol-Peston, and Hansen Contributions", The Review o/Economics and Staiaics,Vol.XLIX, Cambridge, Mass., x967, pp. x x 9 s q q . - Here, too, a most illustrative diagrammatic presentation is given. In these works, however, factor prices are assumed constant.

227

The Multiplier-Effect of Income and Consumption Taxes

(6 ~)

Cm = C r ' p o ' ( 1

(7~ (a)

Wd = W - - T w

(b)

Ro = Yf - - W - - T R

(8 ~) (a) 1 T w = T o w +

(kw+d)'W

(b)' T R = T o R + (9 ~) (io ~ (II ~

+t)

(kR+d)'(Yt--W)

dCr w - qc = Pc" - dNc dI, w 9 qi = Pi" - dNt q, = fl(N~)

(12~

qi = fi(Ni)

(13 ~

Cr = f,(N~)

(14 ~

It = fi(Nl)

(15 ~

W = w. N

(16 ~

w = fs(N)

(17 ~

N -~-

Nc

+ Ni

T h e reformulation has i n t r o d u c e d t h e following new variables: (j) bw and b R

=

Marginal p r o p e n s i t y to consume of t h e two groups, respectively.

(k) Wd and Ra

= Disposable income of the two groups, respectively.

(1) Tw and TR

= I n c o m e t a x e s of t h e two groups, respectively.

(m) W

= Wage bill.

(n) Tow and ToR = Constants (the [negative] levels at which t h e linear t a x schedules and the o r d i n a t e axis intersect). (o) (kw + d) = Marginal income t a x rates of t h e two groups, and (kR + d) respectively (d being a shift p a r a m e t e r ) . (p) w

= Wage rate.

(q) qc and qi

= Relation b e t w e e n prices and m a r g i n a l costs of the two sectors, respectively.

(r) N e, N i and N = E m p l o y m e n t of t h e two e m p l o y m e n t , respectively.

sectors

and

total

1 In the following formulas the marginal income tax rates are mentioned as d w {being equal to k w + d) and dR (being equal to k R + d).

228

Erik G~rtz

VII. The Price and Wage Formation of the Model

From the part of the model describing price and wage formation (equations (9~176 the effects of a change in fiscal policy on the relative changes in factor prices of investment goods, wage bill, and production of consumer goods at constant prices are found to be proportional to the percentage change in factor prices of consumer goods 1. The relative change in the four variables per unit of tax rate change is denoted by t3i, W, Cr, Pc. E.g. the latter is defined as (3o) f ) c ~ 1 .dpc Pc dx this part of the model being differentiated with respect to x (dx can be used to express a change in the consumption as well as the income tax rate, since neither d nor t enter). (3I)

13i = KI" ISc

(32)

~V = K , ' p ,

(33)

Cr = K 3"15c

In these expressions the factors are equal to: l"Ew, N (34) K1 =

1E

1 IEdCr 1

~l- "[ 1 +Ew,

]

~]

(35) K , -l

l

[ Eacr

]

a The reasons why the endogenous variables except the four above-mentioned can be eliminated when differentiating the submodel (equations (9~176 is that they do not enter into equations (i~ ~ and thus are not needed when differentiating that part of the model. The derivations are omitted for convenience, as the interpretation of the results is relatively easy.

The

Multiplier-Effect of Income and Consumption T a x e s

229

1 - - "

Nc

(36) K8 =

EC

,

r

Nc

i .Ew,~ 1 [E_dCr ] -~-~" [ dNc' No__ Eqc, Nc

N-

E denoting the elasticity of the function to which the subscript refers, e.g.

(37) Ew,

N dw ----

The interpretations of the three factors are rather similar and will therefore be carried out for K x only (the ratio between relative factor price changes). As seen K 1 is determined by the extent to which the wageinduced price change in the consumer good sector is augmented or reduced by relative changes in marginal productivity and "price climate" (the denominator). As Ir is exogenous the latter effects are absent in the investment good sector (the numerator). The total impact of changes in marginal productivity and price climate probably tends to make K1 smaller than one. There is no doubt that the elasticity of the relation between price and marginal costs of consumer goods is positive. As for the elasticity of the marginal product the sign of the effect is more doubtful. Neo-classical theory tells that this elasticity is negative (thus having the same effect as the elasticity of qc), but most recent investigations show the opposite result x. If the elasticity of the marginal product is positive, it is most probable, however, that it is not greater than the elasticity of qc. This means that the effect of a change in fiscal policy is relatively greater on factor prices of consumer goods than on factor prices of investment goods. VIII. The Multipliers of the Model (l~

Using the above-mentioned results and differentiating equations ~ with respect to d and t generates the following multipliers*:

1 See e.g. Jorgen H. Gelting, "Beskmftigelse og produktivitet", in: Udviklingslinicr i makrogkonomisk teori, Udg. af Niels Thygesen og P. Norregaard Rasmussen, Kobeahavn, 1969, pp. zo6sqq., and the studies mentioned in t h a t article, e.g. the survey article b y Marc Nerlove, "Notes on the Production and Derived D e m a n d Relations Included in Macro-Economic Models", International Economic Review, Vol. VIII, Osaka, x967, pp. 223 sqq. 2 The generalized balanced-budget multiplier in Park, op. c/t., p. r22, presents certain features that are similar to some of those of the following multipliers. From a theoretical point of view, the most important difference is t h a t Park, assuming constant factor prices, does not use marginal weights explicitly. Given Park's assumptions, however, marginal weights and average weights are equal.

230

Erik

{9~ a) i4Y, dd

~

____

1

Gortz

~

Pc [bw" ~w + bR " (1 - - ~w)]" Yf l+t--bw.(1--dw).~--b

R. (1--

dR) . ~ +

S Ka . C r . p c

dYf dY, dd ---- P c ' q f " dd

(I0 ~ a) - -

dYm ( I I ~ a) - dd -

--Pc" ( l + t ) ' q m "

dY, dt

dYr dd

1 p,

(I2 ~ a) - -

[bw" ( 1 - - d w ) . = w + bR" (1 - - dR) 9 (1 - - ~ w ) ] " Yf + S (l+t).

[l + t - - b w . ( 1 - - d w ) . , ~ v - - ,b

a'(1-dR)-~+

s

]

Ka. Cr. Pc

dYf dY r (I3 ~ a) -~- = Pc" qf" dt dYm

dYr dt

(14~

The n e w variables introduced in the above formulas are defined as: W ( 3 8 ) =W ~

yf

( K 2 - - 1) 9 W

(39)~ ~ ,

Ka" Cr'pc K 1 9 Im +

(40) ~R --=

(Ka+ Ka"

1) 9 C, 9 pr Cr

9

Pc

(4I) S = - - bw" Tow - - bR" TOR 1 (42) qf ~ 1 + ~

K 1 9I m + Ka . C r . p c

1 (43) qm ~ 1 + ~

+

K1 9 Ka~m

Yf

, --

0tW

The Multiplier-Effect of Income and Consumption Taxes

231

The interpretation of these results seems rather complicated, since as previously the multipliers are expressed in the parameters of the model and the levels of some of the endogenous variables in the equilibrium solution. The interpretation will be much easier, however, when the multipliers are expressed in relative changes using formulas (31)--(33). As the multipliers in the more restrictive model above have been rather comprehensively discussed, only the differences between the multipliers derived from the two formulations of the model will be commented on. As for the numerators of the multipliers at constant prices the marginal propensity to consume in (9) and the marginal propensity to consume multiplied by the marginal disposable income share in (12) are in (9 ~ a) and (12 ~ a) replaced by weighted averages of the same parameters of the two groups. These weights are the relative shares of national income at factor prices of the two groups. This result was to be expected and will not be commented on further. Similarly, the product of the marginal propensity to consume and the marginal disposable income share (i.e. ( 1 - d)) in the denominators is replaced by some kind o~ a "weighted average" of these products for the two groups. This averaging, however, has two interesting features. First it is marginal "weights," which enter into the formulas. As differentiation of equation (3) of the basic model results in: dYf dCr dpc dpi (44) a T = dx " P c + ~ " C r + ~ "Ir ----= [Cr + f)c]' Cr "Pc + Pi" I m

Yr. Yf

the following expressions appear, when K 1, K s and K s are substituted by their values expressed in relative changes according to formulas (31)--(33) (multiplying by 1)c in n u m e r a t o r and denominator): ,

(39 a) ~w - -

( K s - - 1) 9 W K3 9 Cr " Pc

(xTv - -

w

Yf" Y~ - - 15~" Cr" p~ - - t3i " Im

232

Erik Gertz

and ,

(40 a) ~R =

K 1 9 Im +

(K 8 +

1) 9 C r 9 Pc - - Y f

--

K s 9 Cr . Pc ~)i" I ~ +

(Cr + P*)" C ~. p ~ - -

15c 9 Y f

t

m Qtw

Cr " Cr " Pc

( Y , - tic)" Yf = ~f. Y f

- - P c " C r " Pc - - fii"

Im

-,4

The "weights ''1 are thus equal to the change in factor income of the groups minus the effect on real consumption transactions of the change in factor prices of consumer goods, relative to the real change in factor income 2. Second the sum o/the "weights" is less than 1 since: o

(44) ~

+

~R =

~rf . y f _ _ Pc " Yf ~ < 1, Yf " Yf - - Pc 9 Cr " Pc - - Pi "Im

if Pc > !5i3. This is so because real consumption transactions are d e t e r m i n e d b y (disposable) income at factor prices and b y prices of consumer goods (not of all goods). If the relative changes in the prices of consumer and investment goods are different, the effect on d e m a n d (the n u m e r a t o r ) and on income formation (the denominator) of the changes in prices will be different. Similar marginal weights are obtained in the denominators of the multipliers based on a sector-devided model in J o h n Vibe-Pedersen, National Income and Aggregate Income Distribution, Acta Jutlandica, X X X V I : z, Samfundsvidenskabelig Serie xx, Skrifter fra Aarhus Universitets Okonomiske Institut nr. z7, Aarhus and Kobenhavn, x964. I n that paper, however, the effects of fiscal policy are not treated. s Not relative to the change in real income. If the model were formulated as to permit changes in real investment also differentiation of (3 ~ with respect to ). would yield: Yf" Y f - - P c . C r ' P c - - P i " I m = Cr" Cr" Pc + I t " Im This expression equals the change in real income only if factor prices equal one, since differentiation of (2~ yields: Y r ' V r = Cr" Cr + i t " Ir In the former case the relative real changes are multiplied by the factor income components, whereas in the latter case they are multiplied by the real income components. I n this model Ir is exogenous (i.e. Ir = 0). The relation between the two quantities then equals Pc. Generally, however, the relation is equal to a factor price index. The weights of this index a re the changes in the real income components, s Cf. the discussion above.

The Multiplier-Effect of Income and Consumption Taxes

~33

A correction term S due to the assumption that the tax functions are not proportionate enters into the multipliers. S of course is positive as the point of intersection between the tax functions and the ordinate axis is negative 1. Further comments will be postponed till later. As above, the multipliers at factor and market prices are equal to the multipliers at constant prices multiplied by Pc and Pc" (1 + t) respectively, and for the consumption tax multiplier at market prices, consumption at factor prices is added. Further, however, the correction ]actors q, and qm enter into the formulas. Re-arrangements of these yield the following expressions:

(42a) qf

=

1 K1 " Im 1 + ~ + K3 . Cr" Pc

(Cr+pc)"

Cr "p~ + P i ' Im Cr 9C~ 9 Pc VZ~ 9 Vf

~r . y , _ _ p c

" Cr " P c - - Pi" Im

and (43 a)

1 qm = 1 4" -~3 +

K1 " I= K3" Cm

(C, +lOc) 9 Cm + ? i " I,.

Cr " era It is impossible to convert qra to an expression analogous to qf without specifying, which of the fiscal policy parameters is changed ~. This, however, does not preclude the simple interpretation presented below. 1 As the marginal tax rate is greater t h a n the average tax rate, i.e. the elasticity of the tax function is greater than 1. 9 This does not mean that the correction factor qm is different in case of changes in consumption and income taxes. I t only means that the relative change in national income at market prices cannot enter into the expression for q ~ without specifying, which of the fiscal policy parameters is changed. This is so because one of the parameters (the consumption tax rate) enters into the expression for national income at market prices, as Ym ~ Cr "Pc" 9 (1 + t) + Ir "Pi. From this is seen that the expression for Ym would be different in ease of a change in consumption and income taxes. The counterpart of this result is that consumption at factor prices enters into the market price multiplier of a change in consumption taxes and not into the same multiplier of a change in income taxes.

234

Erik Gortz

As appears the multipliers at constant prices must as before be multiplied by Pc but now also b y qf, i.e. the relation baween the nominal and the real change in national income at/actor prices, in order to arrive at the corresponding multipliers at factor prices. Thus calculations at factor prices differ from calculations at constant prices b y the effect of changes in factor prices of consumer and investment goods. The interpretation of qm is analogous to the interpretation of qf. IX. The Effect of the Assumption that all Consumption Taxes are not Passed on to the Consumers

The above-mentioned results are based among other things on the assumption that marginal costs and factor prices of consumer and investment goods are not constant, which implies that all consumption taxes are not passed on to the consumers. This, however, is not the only way in which the above-mentioned models are different. In order to isolate the effects o~ the assumption o~ variable/actor prices and marginal costs from the effects especially of different marginal propensities to consume and of different marginal income tax rates of the two groups, the model must be changed somewhat. Thus equations (9~ ~ are changed rather thoroughly, whereas (1~ ~) remain unchanged. In the following it is assumed that the wage rate, the relations between prices and marginal costs, marginal products and therefore factor prices are constant 1. This assumption eliminates equations (9~176 (14~ and (i6~ and instead of (I3~ (15 ~ and (17 ~ the following equations are introduced: (13 ~ ) (15 ~ )

cr = G + g - N c , W = w 9N

,

(17 ~ )

N =No

,

G, g, w, and

Ni

+ Ni

and

being constants (g = marginal product).

Differentiation of the model (I~176 (I3~176(I5~176and (17 ~176with respect to d and t gives the following multipliers at constant prices: dYr 1 (9~ b) . . . . . . . . dd Pc [bw " ~w + l+t--b

w.(1-dw).

.

l

.

Pc

bR " (1 - - *r .

w .

g

bR,1 ~

" Yf ~

(1

_

1 _

Pc

9

1 I t might be assumed that the effects of variable marginal products and variable relations between prices and marginal costs would counterbalance each other. This, however, would not change the results.

The Multiplier-Effect of Income and Consumption Taxes

(12 ~ b)

dYr dt

1

Pc

[b w" ( 1 - - d w )

(1 + t )

9

1 +t--b

9 (1--~w)]

235

9 ~w + b e 9 ( 1 - - d R )

w.(1-d

1

w ) . Pc

9 W

g

9 Y~+S

The numerators of the multipliers are the same after the changes in assumptions. This, of course, was to be expected, as the primary effects of changes in fiscal policy are independent of whether or not factor prices are constant. It is the secondary-effect-multiplier that is influenced by changes in factor prices. That the correction term S enters only into the numerator of the multiplier of a change in consumption taxes and not into the denominators of the two multipliers is due to the same fact. The correction term enters into the multipliers because disposable income o] the two groups is measured at constant prices (i.e. divided by Pc and (1 + t)) in the consumption ]unction. When one or both of these deflators are changed this influences the real value of disposable income, and as disposable income is influenced among other factors b y the level of the tax schedule a change in the deflators also changes the level-effect of the tax schedule. When the consumption tax rate is changed this changes the level-effect of the tax schedule, and as this effect is a part of the primary effect of the tax rate change the correction term enters into the numerator of the multiplier of a change in consumption taxes. Only when factor prices are assumed to be variable a corresponding effect enters into the secondary-effect-multipliers, i.e. into the denominators of the multipliers. The weights in the denominators of the multipliers are also changed as a result of the change in assumptions. In the first place the sum o] the two weights is equal to 1. The effect on the weights of different changes in factor prices of course is eliminated, when factor prices are assumed to be constant. In the second place t h e y are differently expressed, but they are still marginal weights. This is seen from differentiation of equations (3), (13 ~176and (15~176 (45)

dYf dC, -- P c ' - dx dx

(46)

dCr dNc dx -- g ' - ~ -

236

Erik Gortz

(47)

dW dNc -- w . - dx dx

Elimination of dC~ dx and dNc ~ gives the following expressionX:

(48)

dW dx

1 w . . . . Pc g

~r. W

dYf -- ~rf. y f dX

Changing the assumption from variable to constant factor prices has an analogous effect on both multipliers. The impact of this change in assumptions is not limited to the problem of passing on consumption taxes to the consumers. The multipliers at factor and m a r k e t prices are analogous when factor prices are presumed to be variable and when t h e y are presumed to be constant. In the latter case, however, both qf and qm are equal to 1. X. Closing Remarks In the above article some /urther elements of the multiplier process than the effect on consumption have been t r e a t e d in a static model. As shorter or longer time lags of course are involved in all the elements and in all further effects a dynamization of the model would be desirable. This is the main theme in some o/the articles following Haavelmo's original treatment in Econometrica 2. F u r t h e r some aspects of the dynamic multiplier x I t can easily be shown that the marginal weights implied by this expression are equal to the average weights (i.e. equal to ~W and (1 - - t , W ) ) used in the numerators only if in both sectors marginal products equal average products, marginal products are the same, and factor prices are the same. This means that: Cr = g" Nc Ir = g 9 Ni and thus Yr=g-N Using the common price level the marginal weights are equal to average weights, as: 1

w

w

N

g = p'Vr

W

= v,-

s See R. M. Goodwin, "Multiplier Effects of a Balanced Budget: The Implication of a Lag for Mr. Haavelmo's Analysis"; G. Haberler, "Multiplier Effects of a Balanced Budget: Some Monetary Implications of Mr. I-Iaavelmo's P a p e r " ; Everett E. Hagen, "Multiplier Effects of a Balanced Budget: Further Analysis"; and Trygve Haavelmo, "Multiplier Effects of a Balanced Budget: Reply", Econometrica, Vol. XIV, x946, pp. x5osq., pp. x48sq., pp. x52sqq., and pp. x56sqq., respectively.

The Multiplier-Effect of Income and Consumption Taxes

237

are discussed in the article of Turvey 1. The point of departure is here the analysis o/the Stockholm School with its distinction between perspective (ex ante) and retrospective (ex post) magnitudes. The most advanced procedure would be to treat the effects of changes in fiscal policy within the/rameworks o / a comprehensive econometric model. The comparative static multiplier concept in that case would be replaced by short-term and long-term multipliers or description o/the whole time-path of the variables called forth by fiscal policy changes. In the author's opinion, however, some of the above-mentioned results would be valid in these more advanced procedures also. This probably holds as regards the effects of dividing the community into sectors, of assuming variable factor prices and different marginal propensity to consume of the various sectors, as well as of using non-proportionate taxes. $

$

$

Z u s a m m e n f a s s u n g : Der Multiplikatoreffekt der Einkommen- und Verbrauchsteuern. - - Die Abhandlung stellt die Wirkungen auf gew6hnliche finanzpolitische Multiplikatoren bei Aufgliederung der Volkswirtschaft in Sektoren, bei A n n a h m e variabler Faktorpreise und unterschiedlicher marginaler Konsumneigungen yon Lohnbeziehern und Empf/ingern yon Residualeinkommen sowie bei E r h e b u n g nichtproportionaler Steuern dar. Es wird gezeigt, dab die Elemente der regelm~Bigen Multiplikatoren bei Variierung dieser A n n a h m e n durch gewogene Durchschnitte der gleichen Elemente ersetzt werden. Diese Werte sind manchmal marginale und manchmal Durchschnittmgewichte. Die Summe dieser Gewichte ist nicht notwendigerweise gleich eins. Ferner implizieren die /~nderungen in den A n n a h m e n die Verwendung gewisser Berichtigungsglieder in den Formeln. Es werden die gew6hnlichen Multiplikatoren bei .~nderungen der Einkommenund Verbrauchsteucrs/itze in Schaubilderu dargestellt und einige Probleme der Multiplikatoren eines ausgeglichenen Budgets diskutiert. Zum SchluB werden einige Aspekte einer dynamischen Darstellung errrtert.

R r s u m r : L'effet multiplicateur des i m p r t s sur le revenu et la consommation. - L'article drcrit les effetm sur les multiplicateurs usuels de politique financi~re, q u a n d on prend l'~conomie par secteurs, q u a n d on suppose des prix de facteurs variables et des penchants marginaux de consommation diff~rents pour les salarirs et pour ceux qui ont un revenu restant, et pour le cas d ' i m p r t s non-proportionnels. I1 est drmontr6 qu'en variant ces suppositions, les 616ments des multiplicateurs r~guliers sont remplac~s par des moyennes pond~rres des m~mes 616ments. Ces valeurs sont ou des poids marginaux, ou bien des poids moyens. Le t o t a l de ces poids n'rgale pas n~cessairement L Si l'on change les suppositions, il faut introduire dans les formules certains ajustements. Sont dorm, s des diagrammes des multiplicateurs usuels sous une fluctuation des t a u x d ' i m p r t sur le revenu et la consommation, et sont 6tudirs certains probl~mes a Turvey, op. cir., pp. 287sqq.

238

Erik Gortz The Multiplier-Effect of Income and Consumption Taxes

des multiplicateurs d ' u n budget 6quilibr6. Finalement, quelques aspects presentation dynamique sont discut6s.

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R e s u m e n : E1 efecto multiplicador de los impuestos sobre la r e n t a y sobre el gasto de consumo. - - E1 presente estudio ofreee una exposici6n de los efectos de los multiplicadores fiscales corrientes, teniendo en cuenta la divisi6n de la economia en sectores y suponiendo que los precios de los factores de produeci6n son variables, que los asalariados tienen otra propensi6n marginal al consumo que los receptores de ingresos residuales, y que el estado recauda impuestos no-proporcionales. Se demuestra pot un lado, que los elementos de los multiplicadores corrientes quedan substitufdos por promedios ponderados de los mismos elementos, si se alteran los supuestos. Estos valores a veces son ponderaeiones marginales, a veees ponderaciones por t~rmino medio. La suma de estas ponderaciones no es necesariamente igual a la unidad. Adem~s se demuestra, que al variar los supuestos hay que variar tambien las f6rmulas en una manera determinada. El trabajo contiene varios diagramas que expresan la cuantia de los multiplicadores corrientes al variar las tasas de los impuestos sobre la renta y sobre el consumo. Finalmente, se discuten varios problemas del multiplicador del presupuesto equilibrado, ast como algunos aspectos de una presentaci6n dinAmica.

R i a s s u n t o : L'effetto del moltiplicatore delle imposte sul reddito e di quelle di consumo, I1 saggio espone gli effetti su usuali moltiplicatori politico finanziari nella suddivisione delFeconomia in settori, nell'accettazione di prezzi variabili dei fattori e di differenti tendenze marginali dei consumi di salariati e beneficiari di reddito residuale come anche nella riscossione di imposte non proporzionali. Viene mostrato che gli elementi dei moltiplicatori regolari sono sostituiti, nella variazione di queste accettazioni, da medie ponderate degli stessi elementi. Questi valori a volte sono marginali e a volte pesi medi. La somma di questi pesi non necessariamente uguale ad uno. Inoltre i m u t a m e n t i nelle accettazioni implicano l'impiego di termini di rettifica nelle formule. Gli usuali moltiplicatori sono rappresentati in grafici nei m u t a m e n t i dei tassi dellqmposta sul reddito e dell'imposta di consumo e sono discussi alcuni problemi dei moltiplicatori di un bilancio preventivo in pareggio. Alla fine sono dibattuti alcuni aspetti di una rappresentazione dinamica. -

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