Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

u,

c, wealth effects

this would be all as the however, the increase l and shifts the budget

[WE) is represented by the -standing of homothetic )stitution effect is zero. by Obstfeld and Rogoff

borrow/lend at the same inlikely to be the case, as is ■,.! agents with high earning is not allowed, so future irthermore, households are Ming that the former may that borrowing restrictions

D borrow altogether but can ssed so far, this would be no *he first period. Let us now the first period and high in

Chapter 6: The Government Budget Deficit

C2

A

0

C°

c 1

B

Figure 6.3. Liquidity restrictions and the Ricardian experiment

the second period. This case has been drawn in Figure 6.3. The income endowment point is q, and the optimal consumption point in the absence of borrowing restrictions is q, . This point is not attainable, however, since it involves borrowing in the first period, which is by assumption not possible for the household. The effective choice set is consequently only AqC70 and the optimal consumption point (C7, OP is at the kink in the budget line (in point E,1", ).

If we now conduct the Ricardian experiment of a tax cut in the first period matched by a tax increase in the second, the income endowment point shifts along the unrestricted budget line AB, say to point El. . As a result, the severity of the borrowing constraint is relaxed and the consumption point (C1, CI) moves to point Er. The effective choice set has expanded to AEr C10 and real consumption plans (and household utility) have changed for the better.

Obviously, a similar story holds in the less extreme case where the borrowing rate is not infinite (as in the case discussed here) but higher than the rate the government faces. In that case the budget line to the right of the income endowment point is not vertical but downward sloping, and steeper than the unrestricted budget line AB (see the dashed line segments). As a result, the Ricardian experiment still leads to an expansion of the household's choice set and real effects on the optimal consumption plans.

6.1.4 Finite lives

Everybody knows that there are only two certainties in life: death and taxes. Hence, one should feel ill at ease if Ricardian equivalence only holds if households live

145

The Foundation of Modern Macroeconomics

end of the world

	•	young
• old	•
government

3 time

Figure 6.4. Overlapping generations in a three-period economy

forever. In the example discussed so far, households, the government, and the entire economy last for two periods, which effectively amounts to saying that the household has an infinite life. Suppose that we change the model slightly by introducing two households, that each live for only two periods, and that the government and the economy last for three periods. The old household lives in periods 1 and 2. whilst its offspring, the young household, lives in periods 2 and 3. The structure of the overlapping generations is drawn in Figure 6.4.

We describe the old generation first. They are assumed to possess the following lifetime utility function:

V° = log C(i) +	1	log C2° + aVY , a > 0,	(6.27)
	1 + p

where the superscript "0" designates the old generation, and "Y" the young generation. Equation (6.27) says that if a > 0, the old generation loves its offspring, in the sense that a higher level of welfare of the young also gives rise to a higher welfare of the old. The old can influence the welfare of the young by leaving an inheritance. Assume that this inheritance, if it exists, is given to the young just before the end of period 2 (see Figure 6.4). The inheritance is the amount of bonds left over at the end of the old generation's life, i.e. B. Clearly, it is impossible to leave a negative inheritance, so that the only restriction is that 13° > 0.

The consolidated budget restriction of the old generation is derived in the usual fashion. The periodic budget restrictions are:

13? = (1+ r)B0 + (1 — ti)Y? — C°I ,	(6.28)
13° = (1+ r)B? ± (1— t2)11 —	(6.29)

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41111E

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,,N41041111

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tor

111.1.1 -31

146

end of the world

3 time

economy

!rnment, and the entire saving that the houseslightly by introducing at the government and 'es in periods 1 and 2, and 3. The structure of

possess the following

(6.27)

" Y" the young generaits offspring, in the e to a higher welfare of

eaving an inheritance.

.; just before the end F bonds left over at the ble to leave a negative

is derived in the usual

(6.28)

(6.29)

Chapter 6: The Government Budget Deficit

from which 13° can be eliminated to yield:

co Bo			(1 — 1-
ro 2 '	2	= (i r)B0 + (1 ti)11) +		)	(6.30)
			2)	11
l+r			1 + r

where S2° is total wealth of the old generation, and the term in square brackets is human wealth of the old generation denoted by H°. Equation (6.30) says that the present value of consumption expenditure (including the bequest to the young) during life must equal total wealth. In order to determine the appropriate size of the bequest, the link between the size of the inheritance and lifetime utility of the young generation must be determined, i.e. we must find in' = (1)(B?).

By assumption the young generation has no offspring (presumably because "the end of the world is nigh"), does not love the old generation, and hence has the

standard utility function which only depends on own consumption levels:
VY = log Cr +	log CI'.	(6.31)

Its consolidated budget restriction is derived in the usual fashion. The periodic budget restrictions are:

13217 = (I — t2 )117 —	,						(6.32)
= (1 + r)[B° + 13217] + (1 — t3 )YI —				= 0,			(6.33)
from which BY can be eliminated to yield:
CY	(1		(1 — t3)Y1			Y ,	(6.34)
CY + 113?= +		t2)Y2	1	+ r	3
1 + r

where QY is total wealth of the young generation, and the term in square brackets is the human wealth of this generation denoted by HY.

The optimal plan for the young generation is to choose CI' and CI' such that (6.31) is maximized subject to (6.34). The solutions are similar to those given in (6.16):

CY = ( 1+ /9 )C = 1 r Q Y				•	(6.35)
2	2 p	3	2+p

By substituting these optimal plans into the utility function (6.31), we obtain the expression relating optimal welfare of the young generation as a function of the exogenous variables, including the inheritance B°:

	1 p			1 r		(2 + p )
VY CB?) = log ( 2 + p		1( p1	) log (	2 + p	+	1 + p	log S2- Y
= +	(2 + p log [B° + HY ] .						(6.36)
	1 + p

The old generation is aware of the relationship given in (6.36), and uses it in the decision regarding its own optimal plan. Hence, the old generation chooses ci),

147

The Foundation of Modern Macroeconomics

cis), and B° such that (6.27) is maximized subject to (6.30), (6.36), and the inequality restriction BY > 0. The first-order conditions are obtained by postulating the Lagrangian:

	log C° +			1 log C° + ci(l)(B2°) +			co	CS) + B°
							C2° 1	2	(6.37)
			(1					1 +r
				± p)	2
so that the first-order conditions are:
aL _ 1									(6.38)
ac?		=0'
aL	=		1 a.		=0'				(6.39)
a c°)ci+ p , 1 + r
aL		[ (dvY)			0, B° 0,	B°	ar)	=0.	(6.40)
aBS)	= a dB°			1 + r			aB°
							2

(The fourth condition, min. = 0, yields the budget restriction (6.30).) Equation (6.40) is the Kuhn-Tucker condition for the optimal inheritance BS) that must be used because of the inequality restriction (see e.g. Chiang (1984, ch. 21) and the Mathematical Appendix). The mathematical details need not worry us at this point because the economic interpretation is straightforward. If a = 0 (unloved offspring), then equation (6.40) implies that aLlaB° = —A/(1 + r) < 0 (a strict inequality, because (6.38) shows that A. = 1/Ci) > 0) so that BS)(aLlaBS)= 0 implies also HY = 0. In words, no inheritance is given to offspring that is unloved. More generally, if a is so low that a.ciaB° < 0, giving an inheritance would detract from the old generation's lifetime utility, which means that the inheritance is set at the lowest possible value of B° = 0.

Hence, a positive inheritance implies that the first expression in (6.40) holds with

equality. Using (6.36) it can then be written as:
> 0	ac	= 0 <#.	a(2 + p)	1	(6.41)
	aB-		(1+ p) [B° HY] 1 r	1+ p)CSP

where we have also used (6.39) in the final step. Furthermore, (6.38)-(6.39) can be combined to yield the familiar Euler equation for consumption.

( 1 + r o
1 + p Cl.	(6.42)

4.111111 • TNIL,

- C

148

I

5.30), (6.36), and the inequal- , obtained by postulating the

Co	'
co -2		(6.37)
I -1 + r

I

(6.38)

(6.39)

(6.40)

t'et restriction (6.30).) Equa31 inheritance 13° that must ;. Chiang (1984, ch. 21) and Is need not worry us at this itforward. If a = 0 (unloved

- + r) < 0 (a strict

-

) that Biy(ar/aB(2) = 0 implies spring that is unloved. More heritance would detract from it the inheritance is set at the

I ,ression in (6.40) holds with

1 - = (6.41)

+r (1 + p)C°

I

hermore, (6.38)-(6.39) can be Gumption.

(6.42)

Chapter 6: The Government Budget Deficit

By using (6.30), and (6.41)-(6.42), the solutions for optimal consumption and the (positive) inheritance can be solved:

c(1)	= (1 + p) [Q ° + HY 1(1 + r)]	(6.43)
	(2 + p)(1 + a)
	(1 + r)S2° + HY	(6.44)
	(2 + p)(1 + a)
132,
	= a(1 + r)C2° - HY	(6.45)
	(1 + a)

These results are intuitive. First, if a is very large (unbounded love for the offspring) the old generation consumes next to nothing, and the inheritance approaches its maximum value of (1 + r)Q°. Second, if there is a lot of growth in the economy, HY is high and the young have high human wealth. This means that the marginal utility of bequests falls, so that the inheritance is reduced (agpaHY < 0). Since the offspring is wealthier, the old generation consumes more in both periods of life

(acs /aHY > 0 and acpaHY > 0).

It can now be demonstrated that, provided the optimal bequest stays positive, Ricardian equivalence holds in this economy despite the fact that households have shorter lives than the government. The government budget restriction is now:

	G2 G3	0	_L t-2(y° +	+	6171		(6.46)
( 1 + r)B0 + + =					+ 0	2.
1	+ r (1 + r) 2 1 + r (1

Consider the following Ricardian experiment: the government reduces the tax rate in period 1 (dti < 0) and raises it in period 3 (dt3 > 0), such that (6.46) holds for an unchanged path of government consumption, i.e.:

0 = rfidti +	YY	(balanced-budget).	(6.47)
	(1 +3 r)2 dt3

What do (6.43)-(6.45) predict will be the result of this Ricardian experiment? Clearly, from (6.43) we have that:

n (1 + p)[dS2° + (1/(1 + r))dH Y ]			(6.48)
de{ =	(2 + p)(1 + a)
But (6.30) predicts that
dS2° = - Y?dti > 0,			(6.49)
and (6.34) says that
cill Y =	YY	= (1 ±	(6.50)
	3 dt3
	1 + r
			149

The Foundation of Modern Macroeconomics

(where we have used (6.47) to relate and (6.48) is reduced to

dC? 0 dti

and, of course, also (by (6.44))

dC2° = n. dti

dt3 to dti ) so that dQ° (1/(1 + 0)dllY = 0,

(6.51)

(6.52)

The Ricardian experiment does not affect the consumption plans of the old generation at all. What is the intuition behind this result? The answer is found in (6.45).

dB° = all + 0dA° - dHY	dti
1 + a
(-a(1 + 031) - (1 + 011		(6.53)
1 + a	dti = -(1 + r)17Pti > 0.

The entire tax cut is simply added to the inheritance. In period 1 the old generation buys government bonds (that have just been emitted by the government to finance its deficit, hence no upward pressure on the bond price!) on which it receives interest. The additional bonds plus interest are added to the inheritance so that the young generation is able to meet its higher tax bill. Equations (6.34)- (6.35) and (6.53) therefore predict that the consumption of the young generation is unchanged as well.

	CMY = C14 —	YY
		3 ) dt3
		1 + r
	= -(1 + 011)dti - ( 11		11)(1 + 02 = 0,	(6.54)
			K
		1 + r
	which implies that
	dC1' = dCK =0.			(6.55)

In conclusion, the fact that individual lives are finite does not mean that Ricardian equivalence automatically fails. Provided future generations are linked to the current generation through a whole chain of operative bequests, the unbroken chain of connected generations ensures that Ricardian equivalence holds. Of course, once a single link of the chain snaps (zero bequests, childless couples), generations are no longer linked and Ricardian equivalence does not hold in general. Leaving no inheritance is the optimal strategy if the degree of "altruism" a is low, or if future

150

dS2° + (1/(1 + r))dHY = 0,

-, tion plans of the old genmull? The answer is found

I	(6.53)
) dti > 0

e. In period 1 the old genemitted by the government le bond price!) on which it re added to the inheritance 'r tax bill. Equations (6.34)-

n of the young generation

I	(6.54)
D,

(6.55)

•-s not mean that Ricardian Mons are linked to the curquests, the unbroken chain :Ice holds. Of course, once ss couples), generations are , ld in general. Leaving no ruism" a is low, or if future

Chapter 6: The Government Budget Deficit

income growth is high.2 Students should test their understanding of this material by showing that Ricardian equivalence also fails, even if there are positive inheritances, if there is an inheritance tax that is varied in the experiment.

6.1.5 Some further reasons for Ricardian non-equivalence

A further reason why Ricardian equivalence may fail is the occurrence of net population growth. Intuitively, the burden of future taxation is borne by more shoulders, so that the burden per capita is lower for future generations than for current generations. Hence, one expects real effects from a Ricardian experiment that shifts taxation to the future. (We demonstrate this with a formal model in Chapter 14 below.)

A fifth reason why Ricardian equivalence may fail has to do with issues such as irrationality, myopic behaviour, and lack of information. Households may not be as farsighted and rational as we have assumed so far, and may fail to fully understand the implications of the government budget restriction. Furthermore, they may simply not have the cognitive power to calculate an optimal dynamic consumption plan, and simply stick to static "rule of thumb" behaviour like "spend a constant fraction of current income on consumption goods".

A sixth reason why Ricardian equivalence may fail has to do with the "bird in the hand" issue. A temporary tax cut, accompanied by a rise in government debt, acts as an insurance policy and thus leads to less precautionary saving and a rise in private consumption (Barsky et al., 1986). The main idea is that the future rise in the tax rate reduces the variance of future after-tax income, so that risk-averse households have to engage in less precautionary saving. A temporary tax cut thus has real effects, because it is better to have one bird in the hand than ten in the air. This critique of Ricardian debt equivalence relies on the absence of complete private insurance markets. A related reason for failure of debt equivalence is that people are uncertain of what their future income and thus also what their future bequests will be (Feldstein, 1988). People may thus value differently, on the one hand, spending a sum now, and, on the other hand, saving the sum of money and then bequeathing.

Finally, a frequently stated but incorrect "reason". A popular argument is that government debt matters in as far as it has been sold to foreigners. The idea is that in the future our children face a burden, because they have to pay higher taxes in order for the government to be able to pay interest on and redeem government debt to the children of foreigners. A rise in government debt is thus thought to constitute a transfer of wealth abroad. However, the original sale of government debt to foreigners leads to an inflow of foreign assets whose value equals the present value of the future amount of taxes levied on home households which is then

2 Barring transfers in the opposite direction, i.e. from child to parent.

151

The Foundation of Modern Macroeconomics

paid as interest and principal to foreigners. Hence, this critique of Ricardian debt equivalence turns out to be a red herring.

6.1.6 Empirical evidence

The Ricardian equivalence theorem has been the subject of many tests ever since its inception by Barro (1974). The existing literature is ably surveyed in a recent paper by Seater (1993). There is a substantial part of the empirical literature that finds it hard to reject the Ricardian equivalence theorem. Nevertheless, the jury is still out as solid tests with microeconomic data still have to be performed. Even though Seater (1993) concludes that debt equivalence is a good approximation, Bernheim (1987) in his survey comes to the conclusion that debt equivalence is at variance with the facts. Even though debt equivalence is from a theoretical point of view invalid and according to most macroeconomists empirically invalid as well, one might give the supporters of Ricardian debt equivalence, for the time being, the benefit of the doubt when they argue that the Ricardian proposition is from an empirical point of view not too bad. Hence, in the following section we see what role there is for government debt if Ricardian equivalence is assumed to hold.

6.2 The Theory of Government Debt Creation

Is there any role for government debt if it barely affects real economic outcomes such as investment and consumption? According to the neoclassical view of public finance, there is still a role for government debt in smoothing intratemporal distortions arising from government policy. In particular, government debt may be used to smooth tax and inflation rates and therefore private consumption over time. Such neoclassical views on public finance give prescriptions for government budget deficits and government debt that are more or less observationally equivalent to more Keynesian views on the desirability of countercyclical policy. After a simple discussion of the intertemporal aspects of the public sector accounts, we review the principle of tax smoothing. In the light of this discussion we are able to comment on the golden rule of public finance.

6.2.1 A simple model of tax smoothing

Assume that the policy maker can only raise revenue by means of a distorting tax system (e.g. labour taxes). Assume furthermore, that there are costs associated with enforcing the tax system, so-called "collection costs", and suppose that we can measure the welfare loss of taxation (LG) as a quadratic function of the tax rates

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152

rG = r,

golden rule of government finance:

Git ,

critique of Ricardian debt

A of many tests ever since its y surveyed in a recent paper npirical literature that finds ievertheless, the jury is still be performed. Even though d approximation, Bernheim it equivalence is at variance a theoretical point of view irically invalid as well, one

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As real economic outcomes neoclassical view of public hing intratemporal distorvemment debt may be used - consumption over time. pis for government budget Pservationally equivalent to clical policy. After a simple tor accounts, we review the we are able to comment

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y means of a distorting tax Te are costs associated with and suppose that we can function of the tax rates

Chapter 6: The Government Budget Deficit

(t1 and t2), and a linear function of income levels in the two periods (Yi and Y2).

t2 Y2

LG t? yi + 2 (6.56) 1+ PG

where AG is the (policy maker's) political pure rate of time preference. We continue to assume that household income is exogenous. The government budget restriction is augmented somewhat by distinguishing between consumption and investment expenditure by the government, denoted by GF and respectively (t = 1, 2). Instead of equations (6.6)-(6.7) we have:

(131 )	rBo +	+	tiYi = 13 1 -13o,	(6.57)
(D2	rBi +		t2 Y2 = B2 - B1 = -13 1	(6.58)

where le2 is the gross return on public investment obtained in period 2, so that the rate of return rG can be written as:

R12 = (1 + rG)Gli .

(6.59)

Obviously it makes no sense for the government to invest in period 2 since the world ends at the end of that period (hence GI2 = 0). Note furthermore that (6.57)-(6.58) also imply the following relationship between the deficits in the two periods and the initial debt level:

D1 + D2 + 130 = 0.

(6.60)

To the extent that there is an initial debt (Bo > 0), the sum of the deficits in the two periods must be negative (i.e. amount to a surplus). The consolidated government budget restriction can be obtained in the usual fashion:

(1+ r)B0 + + -		= t2 Y2 ± (1 ± rG)Gli -		B1]
	r)Bo + r +	GC	1+r	t2 Y2
			(r- GI
[ :="] (1 +		2	1+ r =	+ 1+r '	(6.61)
		1 + r

where 2 i is the present value of the net liabilities of the government. We immediately see the as long as government investment expenditure can be debudgeted from the government budget constraint. In words, public investments that attain the market rate of return give rise to no net liability of the government and hence do not lead to present or future taxation. They can be financed by means of debt without any problem.

The growth rate of income in this economy is defined as y _--_-Y21171-1, so that we can write 172 = ( 1 + y) Yi, and everything can be written in terms of Y1. Specifically, the right-hand side of (6.61) can be rewritten as:

	6	1 = tl	( 1 + y	(6.62)
			+ r ) t21

where i is net government liabilities expressed as a share of income in the first period.

153

The Foundation of Modern Macroeconomics

The policy maker is assumed to minimize the welfare loss due to distortionary taxation, subject to the revenue requirement restriction (6.62). The Lagrangean is:

t	v	1t2	x	±± yr	t2	(6.63)
2-12	- 1	2 2

so that the first-order conditions are:

a				(6.64)
at1	= ti Yi =
a		( 1 + y )	= 0,	(6.65)
=t1+ Y
ate	21 + pG Y1	X1 + r

and the third condition, ariaa. = 0, yields the revenue requirement restriction (6.62). By combining (6.64)-(6.65), the "Euler equation" for the government's optimal taxation problem is obtained:

=	=	± r t2171.	t =	++Prc	t2•	(6.66)
		1 + pc)

This expression is intuitive: a short-sighted government (pG greater than r) would choose a low tax rate in the current period and a high one in the future. In doing so, the "pain" of taxation is postponed to the future. The opposite holds for a very patient policy maker.

Equations (6.62) and (6.66) can be combined to solve for the levels of the two tax rates:

	tl =	(1 + r)2	(1 + 026		(6.67)
			+ (1 + y)(1	+pGY
	t2 =	(1 + pG)(1 + r)1			(6.68)
		(1 + r)2	+ (1 + y)(1 + PG)

where the optimal path for government debt is also implicitly determined by equations (6.67)-(6.68). We observe that the existing debt exerts an influence on the optimal tax rates only via In that sense it is only of historical significance. The debt was created in the past and hence leads to taxation now and in the future. The optimal taxation problem is illustrated in Figure 6.5. The straight line through the origin is the Euler equation (6.66), and the downward sloping line is the revenue requirement line (6.62). The concave curves are iso-welfare loss curves (i.e. combinations of t1 and t2 for which LG is constant, or dLG = 0). The closer to the origin, the smaller the welfare costs of taxation. The given revenue is raised with the smallest welfare loss in a point of tangency between the revenue requirement line and an iso-welfare loss curve. This happens at point E.

A special case of the tax-smoothing theory is obtained by assuming that r = pG. In that case, (6.67)-(6.68) predict that the two tax rates are equal in the two periods:

1 + r

tl = t2 = ( 2 + r y (6.69)

Debt is used to keep the tax rates constant, hence the name "tax smoothing".

	ti
,(1 +	1

F . ;ure 6.5.

ft-hand • inctAue.

=

in period 1 C

D1 rP-

- — - Y1 -

6.6. The 01 - AlIti the aid of ti art -.1ment'- setts earning a

Cr' ion spc- - --as i ta_1/4.. e the weiii

tr

'14 not raise tht' = -(1 +

e kit.

tti..8)

Mai taxes remain u

154