Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

6

Deficit

iuivalence theorem, and to g model of consumption

financing rule, and d be measured.

I

he name suggests, by the i), who immediately dispaper, however, the new that the Ricardian equivyields important policy

mounts to the following: qd used to finance these mption, investment, and res are financed by means ans of the private sector

(es are equivalent. delayed taxation: if the today, private agents will ,, re through higher taxaorem is valid, the Blinder er 2) is seriously flawed. t wealth, which includes

Chapter 6: The Government Budget Deficit

government debt! Under Ricardian equivalence, government debt in the hands of the public should not be counted as net wealth since it is exactly matched by the offsetting liability in the form of future taxation.

6.1.1 A simple model

Suppose that historical time from now into the indefinite future is split into two segments. The first segment (called period 1) is the present, and the second segment (called period 2) is the future (obviously, by construction, there is no period 3). There is perfect foresight on the part of both households and the government. We look at the behaviour of the representative household first. It lives as long as the government does, and achieves utility by consuming goods in both periods. Labour supply is exogenous and household income consists of exogenous "manna from heaven". Lifetime utility V is given by:

V = U(C1) +( 11+ p) U(C2)'	(6.1)

where Ct is consumption in period t (= 1,2), U(.) is the instantaneous utility function, p is the pure rate of time preference, representing the effects of "impatience". The higher p, the heavier future utility is discounted, and the more impatient is the household. At the end of period 0 (i.e. the "past"), the household has financial assets amounting in real terms to Ao over which it also receives interest payments at the beginning of period 1 equal to rAo, where r is the real rate of interest, which is assumed fixed for convenience. The exogenous non-interest income payments are denoted by Y1 and Y2, respectively, so that the budget restrictions in the two periods are:

Al = (1+ OA° + (1— ti)Yi — (6.2)

A2 = (1 + r)Ai + (1 — t2)Y2 — C2 = (6.3)

where t1 and t2 are the proportional tax rates on income in the two periods, and A2 = 0 because it makes no sense for the household to die with a positive amount of financial assets (A2 < 0), and it is also assumed that it is impossible for the household to die in debt (A2 > 0). (Below, we modify the model and show that households with children may wish to leave an inheritance.) Note that (6.2)—(6.3) incorporate the assumption that interest income is untaxed.

If the household can freely borrow or lend at the going interest rate r, A l can have either sign and equations (6.2)—(6.3) can be consolidated into a single lifetime budget restriction. Technically, this is done by substituting out A l from (6.2)—(6.3):

Al = C2 - (1 - 1-2)/72		= (1 ± OA° + (1 — 1-1)171 —C1
	1 + r
C1 + C2	= (1 + r)Ao + H,		(6.4)
1+r
			135

The Foundation of Modern Macroeconomics

where the right-hand side of (6.4) represents total wealth, which is the sum of initial financial wealth inclusive of interest received, (1 + r)Ao , and human wealth, H:

	H (1 - ti)Yi +	(1 - t2)Y2	(6.5)
		1 + r

Equation (6.4) says that the present value of consumption expenditure during life must equal total wealth.

In order to demonstrate the Ricardian equivalence theorem, we need to introduce the government and its budget restriction. We start as simple as possible by assuming that the government buys goods for its own consumption (G1 and G2), and finances its expenditure by taxes and/or debt. There is no money in the model, so money financing is impossible. The government, like the household, exists for two periods, and can borrow or lend at the interest rate r. In parallel with (6.1)-(6.3), the government's budget identities are:

(D1 =) rBo + Gi - =B1 - Bo, (6.6)

(D2 ) rBi + G2 — t2 Y2 = B2 B1 = (6.7)

where Di and Bi denote, respectively, the deficit and government debt in period i (i = 1, 2), respectively, and B2 = 0 because the government, like the household, cannot default on its debt and is assumed to remain solvent (no banana republic!). Using the same trick as before, equations (6.6)-(6.7) can be consolidated into a single government budget restriction:

(1+ r)B0 + Gi - t1 Y1 =			t2 Y2 — G2
			1 + r
(1 ± r)B0 + +	G2	= t1Y1 + t2 Y2		(6.8)
	1+r		1+r'

where the left-hand side of (6.8) represents the present value of the net liabilities of the government, and the right-hand side is the present value of net income of the government (i.e. the tax revenue).

Since government bonds are the only financial asset in the toy economy, household borrowing (lending) can only take the form of negative (positive) holdings of government bonds. Hence, equilibrium in the financial capital market implies that:

Ai = Bi, (6.9)

for i = 0, 1, 2.

The first demonstration of the Ricardian equivalence theorem is obtained by solving the government budget restriction for (1 + r)Bo, and substituting the result into

Ube household bu

1.1

C1 C2 =(1+ r)

= tl Y1

I

=

_A express, n

, Jdgfa restriction altog spending

of Cl and C Nit way in which the g

,

.1 consumption pia ,ver is, of c,

the subseq,.., , Ja 1,4 I: one that yield

L, 'CO = log Ct .

tithe •ehold choc.

..e

ming the Lagrange n

lug C1 +

- =	1

•( 1 + p)k,2

to taird conthzio

Dining (6.13)-(6.141

• = • =	1+r
Cl	(1

16.15) s-ys • isehoid wishes to er

:able in vi. aou.w.ciold has a lui

- 7tion.

136

'h, which is the sum of initial

,,and human wealth, H:

(6.5)

ption expenditure during life

heorem, we need to introduce s simple as possible by assumumption (G1 and G2), and is no money in the model, so the household, exists for two r. In parallel with (6.1)-(6.3),

(6.6)

(6.7)

7overnment debt in period i !rnment, like the household, olvent (no banana republic!). ) can be consolidated into a

(6.8)

t value of the net liabilities of It value of net income of the

I

t in the toy economy, house- ,2ative (positive) holdings of capital market implies that:

(6.9)

111

theorem is obtained by solvKI substituting the result into

Chapter 6: The Government Budget Deficit

the household budget restriction (6.4) taking (6.9) into account:

+	C2			(1 - t2)Y2 ]
	1 r = (1 + r)B0 +[(1 ti)Yi +			1 + r
		t2 Y2	G2		1 - t2) Y2
	-t1 Y1 ++ (1 ti)Y1				1 + r
		1 r	1 r
	= Y1 -	Y2-G2 = 52.			(6.10)
			1 + r

The final expression shows that the tax parameters drop out of the household's udget restriction altogether. Only the present value of (exogenously given) government spending affects the level of net wealth of the household. Consequently, the choice of C1 and C2 do not depend on the tax parameters t1 and t2 either. The way in which the government finances its expenditure has no real effects on

consumption.

So if consumption plans are unaffected by the timing of taxation, then what is? The answer is, of course, household saving. In order to demonstrate this, and to facilitate the subsequent discussion, we use a specific form for the utility function U(.); one that yields very simple expressions for the optimal consumption and saving plans:

U(Ct ) = log Ct . (6.11)

The household chooses C1 and C2 such that (6.1) is maximized subject to (6.10) and given the utility function (6.11). Again the optimality conditions can be obtained by using the Lagrange multiplier method. The Lagrangean is:

log C1	+ 1 ) log C2 + ), [S2 U1		C2 1	(6.12)
	1 p	1 + r
so that the first-order conditions are:
a.c 1				(6.13)
act
ac _	1	=0,		(6.14)
a C2 (1 ± p)C2 1 + r

and the third condition, aLiax = 0, yields the budget restriction (6.10). By combining (6.13)-(6.14), the so-called consumption Euler equation is obtained:

x =	1	=	1 + r	C2 1 + r	(6.15)
	C1		+ p)C2	C1 1 + p

In words, (6.15) says that, for example, if r > p, C2 ICi > 1 or C2 > C1. The household wishes to enjoy relatively high consumption in the second period. This is understandable in view of the fact that a low value of p (relative to r) implies that the household has a lot of patience, and hence a strong willingness to postpone consumption.

137

(dV =

(d51 >

The Foundation of Modern Macroeconomics

Equation (6.15) determines the optimal time profile of consumption, i.e. it shows consumption in the future relative to consumption now. The level of consumption is obtained by substituting (6.15) into the household budget restriction (6.10):

	= (1 -Fp\ 2,		=(1+/- \	•	(6.16)
	2-Fp)	z	2-Fp)

The expression for household saving (S1) is determined by the identity S1 A l -A0 = B1 - Bo, or:

	Si = rBo + (1 - ti)Yi	(1 + p 0	(6.17)
		2+p)-'

from which we see immediately that the tax rate t1 does not vanish from the expression for household saving in the first period.

Now consider the following Ricardian experiment. The government reduces the

tax rate in the first period (dt1 < 0) but keeps its goods consumption (G1 and G2) constant. Then equation (6.17) implies that

dSi dti > 0, (6.18)

(as dS2 = drBo = 0) but the government budget restriction (6.8) implies that taxes in the second period must satisfy:

Yi dti ( Y2 dt2 = 0	dt2 =	( (1 + r)Yi) dti > 0,	(6.19)
--FT)		Y2

as the present value of government liabilities are unchanged by assumption. Hence, the reaction of the household to this Ricardian experiment is to increase its saving in the first period 0) in order to be able to use the extra amount saved plus interest in the second period to pay the additional taxes. In Figure 6.1, the experiment has been illustrated graphically.

The initial income endowment point is EK, . It represents the point at which the household makes no use of debt in the first period (i.e. B1 = 0) and simply consumes according to (6.2)-(6.3). Since the household can freely lend/borrow at the going rate of interest r, however, it can choose any (C1, C2) combination along the budget line AB. Suppose that the optimal consumption point is at Ec, where there is a tangency between an indifference curve 0) and the budget line. The optimal consumption levels are given by CI and q, respectively. As a result of the Ricardian experiment, income rises in the first period and falls in the second period, but the net wealth of the household (Q) is unchanged. Hence, the income endowment point shifts along the given budget line in a south-easterly direction to ET. The optimal consumption point does not change, however, since nothing of importance has changed for the household. Hence, the only thing that happens is that the household increases its saving in the first period and it does so by purchasing more bonds from the government.

C2

Figure 6.1. kg

are many theort. z iirnre theorem. In the ix

:tA.vnt symposium on

:rsin thesz:: .._

12 Distorting ta ,

• point we ha - . to imagine ui,:z

- inc - me depends 0 :

J write Y1

-146e 'I income endow=

,equen: F ciLii simpler exam

--_•odel introduced ab

_.:e is a ihianuons (6.2)-(6.3) are

=Bo + (1 — t1 ) [

=B1 + (1 - t2)[1 . _

.e .‘- e already become

C2

CI- 1 + r( 1 - t2)

138

111-

mption, i.e. it shows level of consumption

• triction (6.10):

(6.16)

' ntityS1 = Al - Ao =

(6.17)

I

t vanish from the

-nment reduces the mption (G1 and G2)

(6.18)

3) implies that taxes

(6.19)

assumption. Hence, o increase its saving !xtra amount saved ;. In Figure 6.1, the

point at which the rid simply consumes )orrow at the going cm along the budget , where there is a line. The optimal suit of the Ricardian nd period, but the ncome endowment irection to Er. The ing of importance happens is that the purchasing more

Chapter 6: The Government Budget Deficit

C2

A

cr	Y1 (1-	t]) Y1
	+(1 +6B0 +(1	+ r)B0

Figure 6.1. Ricardian equivalence experiment

There are many theoretical objections that can be levelled at the Ricardian equivalence theorem. In the next subsections we discuss the most important theoretical reasons causing Ricardian equivalence to fail. The interested reader is referred to the recent symposium on the budget deficit for further details (see Barro (1989) and other papers in the same issue of the Journal of Economic Perspectives).

6.1.2 Distorting taxes

Up to this point we have assumed that income in the two periods is exogenous. It is easy to imagine that, for example due to an endogenous labour supply decision, income depends on the tax rate on labour income (see Chapter 1). If that is the case, we should write Y1 (t1 , t2) and Y2(ti, t2), and the path of taxes may directly influence the income endowment point, and potentially also the level of net household wealth. Consequently, Ricardian equivalence should be expected to fail.

An even simpler example of a distorting tax can be provided with the aid of the model introduced above. Assume that non-interest income is exogenous but that there is a comprehensive income tax, and that interest income is also taxable. Equations (6.2)-(6.3) are modified to:

B1 = Bo + (1 - [Yi + rBo] -	(6.20)
B2 = + (1 t2) [Y2 + - C2 = 0,	(6.21)

where we have already incorporated (6.9). The consolidated budget restriction for the household becomes:

	C2	ti)Yi +	(1 - t2)Y2	(6.22)
	+ 1 + r(1 — t2) = [1 + r 1 ti)1130 +[		1 ± r(1 - t2)
				139

future to the present.

The Foundation of Modern Macroeconomics

The budget restrictions for the government are also suitably altered:

(D1	) rBo + —ti [Y]. + rBo] =B1 — Bo,	(6.23)
(D2	) rBi + G2 — [Y2 ± rBil = —Bi,	(6.24)
so that the consolidated government budget=restrictiony rt2(ilys2:_ t2)
	G2	(6.25)
[1 + r(1 — 1-1)] Bo + G1 + 1 + r(1 — t2)

Failure of the Ricardian equivalence theorem is demonstrated by solving the government budget restriction for (1 + r(1 — ti))Bo, and substituting the result into the

household budget restriction:

. "11111k

Cl	C2	= Y1 G +	Y2 G2	= S2(t2)-	(6.26)
	1 + r(1 — t2 )		1 + r(1 — t2)

This expression shows that the income tax in the second period does not drop out of the household budget constraint. Consequently, optimal consumption plans are affected by the timing of taxation. Obviously, t1 does not appear in (6.26) because it operates like a lump-sum tax. Households are taxed on their interest income in the first period and can do nothing to avoid having to pay that tax (since Bo is predetermined and is hence a "sitting duck" for the tax man). The tax in the second period changes the intertemporal price of consumption now versus later, and as a result distorts the saving decision. 1

Intermezzo

The two-period consumption model. Because the two-period consumption model has played such an important role in the macroeconomic literature it pays to understand its basic properties well. Assume that the representative household's lifetime utility function is given in general terms by:

V = V(Ci, C2 ),

(a)

where C, is consumption in period i, and we assume positive but diminishing marginal utility of consumption in both periods, i.e. Vi Es. aviaci and Vu

a2 viac < O. Note that (6.1) is a special case of (a) incorporating a zero cross

I Indeed, optimal C1 and C2 are modified from (6.16) to:

Ci = 1	P Q)	M/ C2 = fl+r(1 — t2)) S.2(t2)
(2 + p		2+p

from which we conclude that aci / at2 > 0 and ac2 /at2 = —(r/(2 + p))(Yi — G1) < 0. So the tax leads to a shift of consumption from the

140

		Chapter 6: The Government Budget Deficit
suitably altered:		C)2 v ocia C2. In the general case considered here, no such
derivative1712
(6.23)	restriction is placed on V12.
(6.24)	Abstracting from taxes, the household's periodic budget identities are given
	by Ai + Ci = (1 + ro)Ao + Y1 and C2 = (1 + ri)Ai + Y2 which can be consolidated

ion is:	to yield the lifetime budget constraint:

	t, Y2	(6.25)
	4- r( 1 -
		t2) •

• 'nstrated by solving the govsubstituting the result into the

1

(6.26)

, nd period does not drop out ►ptimal consumption plans are 5 not appear in (6.26) because ed on their interest income in 7 to pay that tax (since Bo is ix man). The tax in the second ion now versus later, and as a

+	C2	=-- (1 + ro)A0 + [Yi +	Y2	=	(b)
	1 ri		1 +

where Yi is exogenous non-interest income in period i, Ao is initial financial wealth, 52 is initial total wealth (i.e. the sum of financial and human wealth), and r1 is the interest rate in period i. The household chooses C5 and C2 in order to maximize lifetime utility (a) subject to the lifetime budget constraint (b). The first-order conditions are given by (b) and the Euler equation:

	Vi Wi t C2) =	(c)
	V2 (C1, C2)
		I

where we indicate explicitly that Vi in general depends on both C1 and C2 (because 1712 0 0 is not excluded a priori).

Equations (b)-(c) define implicit functions relating consumption in the two periods to the interest rate and total wealth which can be written in general terms as Ci = Ci(E2, ri) for i = 1, 2. To find the partial derivatives of these implicit functions we employ our usual trick and totally differentiate (b)-(c) to obtain the following matrix expression:

two-period consumption acroeconomic literature it that the representative

.-ral terms by:

(a)

positive but diminishing e. avoci and Vii

incorporating a zero cross

1 )(Yi — G1) < 0. So the tax leads to

	[ dCi		C2	(d)
		+	1+r1)2 dr1,
	dC2
		0	V2

where the matrix A on the left-hand side of (d) is defined as:

		1	1
	A		1 +	(e)

VIA - (1 + ri)V12 V12 - (1 + ri)V22

where we have already incorporated Young's theorem according to which Vi2 = V21 (Chiang, 1984, p. 313). The second-order conditions for utility maximization ensure that the determinant of A is strictly positive (see Chiang (1984, pp. 400-408) for details), i.e. I A I > 0. This means that the implicit function theorem can be used (Chiang, 1984, p. 210).

141

C2 - Y2

The Foundation of Modern Macroeconomics

Let us first consider the effects of a marginal change in wealth. We obtain from (d):

a ci V12	(1 +ri.) V22	0
-		<	(f)
	164

	ace = (1+ ri)vi2 - >0	(g)
	< •

Several observations can be made reading these expressions. First, the effect of wealth changes on consumption in both periods is ambiguous in general. Second, if lifetime utility satisfies V12 > 0 then aCi/aS2 > 0 for i = 1, 2, and present and future consumption are both normal goods. Third, if V12 < 0 then either present consumption or future consumption may be an inferior good (aci l as2 < 0). It follows from (b), however, that at, most one good can be inferior, i.e.:

	aci	1 ac, =1.	(h)
	ac	1 +rl ) a s

Next we consider the effects of a marginal change in the interest rate rl . It follows from the budget restriction (b) that a change in r 1 not only changes the relative price of future consumption (on the left-hand side of (b)) but also affects the value of human wealth (and thus total wealth) given in square brackets on the right-hand side of (b). Indeed, in view of the definition of C2, we find as 21 ari —172 1(1 +1.1 )2 < 0, i.e. an increase in the interest rate reduces the value of human capital because future wage income is discounted more heavily. By taking this (human) wealth effect into account we obtain the following partial derivatives from (d):

	ac1	= (171 - (1 -+-ri)V22)	111	/ 1	1 +		(i)
	ar	16,1	1+ r1	IL\1) ( V2		)
	ac2	+ ri)112 - Vl l	( A 1 )	(I)	0,		(1)
	ar	Ipi		lAi
	Fs!

where we have used the second period budget identity, (1 + ri)A 1 =

to simplify these expressions. Again several observations can be made regarding the expressions in (i)-(j). First, without further restrictions on V12 and A l the effects are ambiguous. By differentiating the lifetime budget equation (b) we find:

	aci (	1 \ ac2		A 1	(k)
	ar,	1 +	art	1 +

from which we deduce that for an agent who chooses to save (Ai > 0) either present or future consumption (or both) rise if the interest rate rises. Second, if

-ims as ( and

-

ane

142

income effect,

human wealth effect,

substitution effect

pummunr

!nge in wealth. We obtain

(f)

(g)

- nressions. First, the effect is ambiguous in general. C > 0 for i = 1, 2, and roods. Third, if V12 < 0 option may be an inferior it at most one good can be

(h)

in the interest rate r1 . It ri not only changes the cl side of (b)) but also affects qz;ven in square brackets

le definition of Q, we find erest rate reduces the value scounted more heavily. By Main the following partial

172	<
1 +		(i)

or

-itity, (1 + r1)A1 - C2 Y2 cms can be made regard-

._ strictions on V12 and Ai time budget equation (b)

(k)

uses to save (A 1 > 0) either crest rate rises. Second, if

Chapter 6: The Government Budget Deficit

Al > 0 and V12 > 0 then aCi/ar 0 and aC i > O. Third, if the agent's utility maximum happens to coincide with its endowment point (so that A i = 0) then it neither saves nor dissaves and it follows that aci lar < 0 and actor > 0.

In the literature it is often assumed that the utility function is homothetic. A homothetic utility function can be written as V(C1, C2) = G [H(C1, C2)] where G[.] is a strictly increasing function and H(C i , C2) is homogeneous of degree one in C1 and C2 (see e.g. Sydsxter and Hammond, 1995, p. 573). We recall the following properties of such functions from the intermezzo in Chapter 4: (P1) H 1 C1 + H2C2 = H, (P2) H1 and H2 are homogeneous of degree

zero in C1 and C2, (P3) H12 —(Ci /C2)Hii = —(C2/Ci)H22 and thus Hil =- (C2/C1)2H22, and (P4) a12 —d log (C1 /C2)/d log (H1 /H2 ) 1/1 1/2 /(H/42) 0. Since H11 < 0 it follows from (P3) that H12 > 0 and from (f) to (g) that present and future consumption are both normal goods. To study the effect of a change in the interest rate we note that the first-order condition (c) becomes Hi /H2 = 1 + Since the Hi are homogeneous of degree zero, this Euler equation pins down a unique Ci /C2 ratio as a function of 1 + r1. By loglinearizing the Euler equation and the budget restriction (b) (holding (1+ro)A0, Y1, and Y2 constant) we obtain the following expression:

	-	dC1
col	— col	ct	(A1/ 0)	dri	(1)
—1	1	tiC2	0-12
	-	C2	[

where Ni C1 Q and 1 — 0)1 C2/((1 ri ) S2) are the budget shares of, respectively, firstand second-period consumption. Solving (1) we obtain the comparative static effects:

aci =	+	[(1	col)	Y2	(1 — 04)042 ,	(m)
art				1+ rig2
aC2	C2— [0		(0_,	Y2	coicri2	(n)
art	1 ± r1			(1 r1)Q

where we have also used (1 + r1)A1 = C2 — V2. The three terms appearing in square brackets on the right-hand sides of (m) and (n) represent, respectively, the the and the (see also Obstfeld and Rogoff (1996, p. 30) for this terminology). We illustrate these effects in Figure 6.2.

The ultimate effect of an increase in the interest rate r1 is given by the move from E0 to E 1 . This total effect can be decomposed into the usual Hicksian fashion. In doing so we exploit the fact that for homothetic utility functions the slope of the indifference curves is the same along a straight ray from the origin. Two such rays are drawn in Figure 6.2, one for the old and one for the new interest rate. The move from Eo to E' is the substitution effect (SE) and the move from E' to E" is the income effect (IE). If the household were to have

143

The Foundation of Modern Macroeconomics

Ak E"

Figure 6.2. I ncome, substitution, and human wealth effects

no non-interest income in the second period (Y2 = 0) this would be all as the human wealth effect would be absent. If Y2 is positive, however, the increase in the interest rate reduces the value of human capital and shifts the budget restriction inward. Hence, the human wealth effect (HWE) is represented by the move from E" to E1 . Students should check their understanding of homothetic utility functions by drawing the case for which the substitution effect is zero. Further results on the two-period model are presented by Obstfeld and Rogoff (1996, ch. 1).

6.1.3 Borrowing restrictions

In the basic case we have assumed that households can borrow/lend at the same rate of interest as the government. In practice this is unlikely to be the case, as is evidenced by the prevalence of credit rationing of young agents with high earning potential but no tangible appropriable collateral (slavery is not allowed, so future labour income typically cannot serve as collateral). Furthermore, households are more risky to lend to than (stable) governments, suggesting that the former may pay a larger risk premium than the latter. It turns out that borrowing restrictions can invalidate the Ricardian equivalence proposition.

For simplicity we assume that a household is unable to borrow altogether but can lend money at the going interest rate r. In the case discussed so far, this would be no problem because the household chose to be a net lender in the first period. Let us now augment the scenario by assuming that income is low in the first period and high in

1111.M.1.41414

- lei cast

144