Discussion Paper No. 791

PUBLIC DEBT PLACES NO BURDENON FUTURE GENERATIONSUNDER DEMAND SHORTAGE
Takayuki Ogawa
andYoshiyasu Ono

September 2010

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Public Debt Places No Burden on Future
Generations under Demand Shortage∗
Takayuki Ogawa† and Yoshiyasu Ono‡

Abstract
In a Diamond-type overlapping-generations setting public debt issuance placesno burden on future generations including those who repay the debt if prices andwages are ﬁxed and unemployment occurs in the periods in which public bonds areissued and repaid. Whether the collected fund is spent on government purchases
or transfers to the present generation, public bond issuance stimulates aggregate
demand and creates additional employment of future generations, which yields additional income that is large enough to cover their tax burden. This property istrue whether the debt is repaid by children or grandchildren.
Keywords: Demand shortage, Overlapping generations, Public debt, Unemployment, Welfare eﬀects
JEL Classiﬁcation Numbers: E24, E62, H63∗

The authors would like to thank Koichi Futagami, Shinsuke Ikeda, and seminar participants at

Kobe University and Osaka University for their helpful comments and suggestions. Ogawa is ﬁnancially
supported by the Grant-in-Aid for Young Scientists (Start-up) 18830085, the Japan Society for the
Promotion of Science (JSPS), and the Grant-in-Aid from the Zengin Foundation for Studies on Economics
and Finance. Financial support to Ono’s research by the GCOE programs, the Ministry of Education,
Culture, Sports, Science and Technology is also acknowledged.
†

Faculty of Economics, Osaka University of Economics, 2-2-8 Osumi, Higashiyodogawa-ku, Osaka

533-8533, Japan. Tel: +81-6-6328-2431; Fax: +81-6-6327-2390; E-mail: tkogawa@osaka-ue.ac.jp.
‡

Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-

0047, Japan. Tel: +81-6-6879-8578; Fax: +81-6-6878-2766; E-mail: ono@iser.osaka-u.ac.jp.

11Introduction

Public debt burden is a long-discussed issue in the literature. For example, Tobin (1980,
p. 49) states:
Do government deﬁcits absorb private saving? Does public debt diminish
private demand for stocks of productive capital assets? Can the burden of
current government expenditure be shifted to future generations? These are
old questions. Today they are once more in the forefront of economic controversy. Few issues of economic theory and fact evoke such polar disagreement.
The contesting views carry radically divergent implications for public ﬁscal
and ﬁnancial policy.
This issue has become more relevant especially after the ﬁnancial crisis of 2008. Many
countries face serious economic turndown and are forced to expand ﬁscal spending by
issuing public bonds, which provokes serious worries about debt burdens on future generations. Particularly in Japan serious stagnation continues for twenty years and a hugeamount of public debt has accumulated. We reconsider this issue in the presence of demand shortage and involuntary unemployment and show that there is no public debt
burden on future generations.
Public debt is widely believed to place burdens on future generations in the literature.
Using an overlapping-generations model with ﬁnitely-lived agents, for example, Diamond(1965) ﬁnds that internal debt reduces capital accumulation (if the saddle-point stability isvalid) and deteriorates steady-state welfare (if the economy is dynamically eﬃcient). Blanchard (1985) and Weil (1989) obtain similar results in alternative overlapping-generations
models with inﬁnitely-lived agents.1 They ﬁnd that public debt makes future generations
worse oﬀ on the dynamically eﬃcient path since it creates persistent income transfers

1

Blanchard (1985) assumes that each agent anytime faces a common and constant instantaneous

probability of death and thus all generations have the same average rest of life. Weil (1989) does nottake account of the death probability. However, since the role of ﬁnite lifetime is negligible in the present
context, as Buiter (1988) discusses, these two analyses have basically the same implication.

2

from future generations to current generations.2 In these models public debt burden isdue to intergenerational redistribution.3 While they consider the case of full employment, we use an overlapping-generations model with involuntary unemployment since a
government more likely issues public bonds under stagnation so as to ﬁnance extra ﬁscal
spending and create new demand. In this setting we show that there is no public debt
burden on future generations although the economy is dynamically eﬃcient.4
There are not many studies that explore the implication of public debt in the presenceof demand shortage and involuntary unemployment. Among those, Tobin (1980) suggeststhat a helicopter drop of government bonds raises demand for current consumption andreduces unemployment whereas Modigliani (1961) argues that government bonds placeburdens on future generations even in the presence of unemployment. However, both ofthem use ad hoc models without rational agent behavior. Rankin (1986) uses a Diamondtype overlapping-generations model with nominal price rigidity and analyze public debt
2

Lopez-Garcia (2008) obtains basically the same result as Diamond (1965) in a model with endogenous

labor supply. By developing an endogenous growth model with overlapping generations and production
externalities, Saint-Paul (1992) shows that public debt reduces the growth rate and the welfare of somefuture generations. The implication of Diamond (1965), Blanchard (1985), and Weil (1989) has been
extensively investigated in various directions. See e.g., Drazen (1978) for human capital accumulation,
Persson (1985) for an open economy, Lapan and Enders (1990) for endogenous fertility, and Sen (2002)
for monopolistic competition in the commodity market.
3

There are other sources that make Ricardian equivalence invalid. They are e.g. capital market

imperfections and distortionary taxation. See Bernheim (1987) and Romer (2005, ch. 11) for the conditionof Ricardian equivalence to be invalid.
4

Since neither population growth nor physical capital depreciation is assumed, the equilibrium pathis dynamically eﬃcient in our model. With full employment public debt issuance makes future generations worse oﬀ on the dynamically eﬃcient path, as mentioned above. On the dynamically ineﬃcientpath, however, public debt issuance may make future generations better oﬀ because it remedies capital
overaccumulation (see Diamond 1965 and Tirole 1985). Chalk (2000) and Rankin and Roﬃa (2003) examine whether there is a path toward the dynamically ineﬃcient steady state where the government rolls
over the debt forever, implying that future generations are not worse oﬀ on that path. In an uncertain
economy setting, Bohn (1995), Ball et al. (1998), and Blanchard and Weil (2001) distinguish the rate of
return on public bonds from that on physical capital to ﬁnd a positive probability of deﬁcit sustainabilityon the dynamically eﬃcient path.

3

burden. He shows that in the presence of excess supply in the commodity and labormarkets, viz. “Keynesian unemployment” in Malinvaud’s (1977) terminology, a permanent increase in public debt raises steady-state output and welfare under some reasonableconditions, including saddle-point stability.
The model of this paper has a similar structure as that of Rankin (1986) but diﬀers
from it in the following two aspects. First, we take into account employment of not only
the young but also the old. Second, while Rankin focuses on the steady state only, we
analyze the eﬀect of a temporary increase in bonds in the transitional state. This extension
enables us to distinguish the following three policies: (i) public debt is redeemed only bytaxing on the current young, (ii) additional taxes are levied on not only the current young
but also the generations that are not yet born, and (iii) deﬁcit-covering bonds are issued
to redeem the current bonds.5 We show that in any of these three cases there is no debt
burden on future generations. Even if the old are employed due to ﬁscal spending andthe young are taxed, employment of the young eventually increases enough to cover thetax payment, yielding no public debt burden on the young.
When analyzing public debt burdens on future generations in the presence of unemployment, Ogawa (2005) employs a model á la Blanchard (1985) and Weil (1989). He
shows that whether or not public debt issuance makes future generations worse oﬀ depends
on the amount of outstanding bonds. In that model all generations, including the current
generation, coexist even in the future and hence a government can anytime redistributedebt burdens through appropriate taxation. In the Diamond model, in contrast, such
redistribution is infeasible because the current generation does not exist when imposing
a tax. Such a diﬀerence in the demographic structure between the two types of models
leads to a diﬀerent conclusion. This paper ﬁnds that public debt issuance never makes
future generations worse oﬀ in the Diamond model.
Steigum (2001), Kaas and von Thadden (2004), and Josten (2006) take account ofreal wage rigidity due to the bargaining power of trade unions. Among them, Steigum(2001) shows that debt burdens on future generations that arise from postponement of
5

We explicitly introduce equities whereas Rankin (1986) assumes that ﬁrm revenues are always trans-ferred to the old. This diﬀerence does not alter the main implication.

4

the labor income tax is larger with union power than without it. Kass and von Thadden (2004) compares unemployment dynamics under balanced and unbalanced budgets.
Josten (2006) uses an endogenous growth model with production externalities and ﬁnds
that an increase in the debt-to-capital ratio forces the government to raise the labor income tax, which worsens unemployment and decreases economic growth. Whereas they
consider structural unemployment, we treat involuntary unemployment due to demand
shortage. Using a New Open Economy Macroeconomics model with short-run nominal
price rigidity, Ganelli (2005) considers a permanent tax cut ﬁnanced by debt issuancethat is maintained even after nominal prices fully adjust. He concludes that it leads tocurrent account deﬁcits and decreases long-run consumption in the home country. We incontrast examine the eﬀect of debt issuance only in the period where nominal prices are
ﬁxed and ﬁnd that consumption of future generations does not decline.
The rest of the paper is organized as follows. Section 2 presents the basic structureof the model and formulates the dynamics. Section 3 discusses the main implication of
public debt burdens in the presence of Keynesian unemployment. Section 4 extends theanalysis to the case of deﬁcit-covering bond issuance. Section 5 concludes the paper.

2

The Model

Individuals live for two periods and two generations coexist at any point in time. The
population size of each generation is normalized to unity. Nominal price Pt and nominal wage Wt are ﬁxed and Keynesian unemployment occurs at least in the period inwhich bonds are issued and the period in which they are repaid. Note that the following
argument holds whether prices and wages are rigid or ﬂexible in the subsequent periods.6
Utility of an individual born in period t depends on consumption when young and old,

6

Rankin (1986) assumes nominal prices and wages to be ﬁxed in all periods since he treats the eﬀectof bond issuance in steady state. In contrast, we analyze the eﬀect on economic variables in the period
in which the bonds are issued and the period in which bonds are repaid in the presence of unemployment. Hence price/wage rigidities are required only in the relevant periods. See Ono (2001) for a recent
discussion about public debt burdens under a liquidity trap and price/wage sluggishness.

5

cyt and cot+1 respectively, and real money holdings mt (= Mt /Pt ) at the end of her youth–i.e.,
Ut = u(cyt ) + v(mt ) + θu(cot+1 ),(1)

where u0 > 0, u00 < 0, v 0 > 0, v 00 < 0, θ > 0.
oGiven real wage wt (or wt+1 ), each individual earns wage income wt lty (or wt+1 lt+1
) byoo
supplying labor lty (or lt+1
) and pays lump-sum tax-cum-subsidy zty (or zt+1
) when young(or old). She accumulates three kinds of interest-bearing assets, viz. public bond bt+1 ,physical capital kt+1 , and equity νt+1 , when young, and receives returns (1 + rt+1 )(bt+1 +kt+1 + νt+1 ) when old, where rt+1 is the real interest rate. The budget equations when
young and old are respectively
cyt + mt + (bt+1 + kt+1 + νt+1 ) = wt lty − zty ,cot+1 =

1oomt + (1 + rt+1 )(bt+1 + kt+1 + νt+1 ) + wt+1 lt+1
− zt+1,
1 + πt+1(2)(3)

where πt+1 represents the inﬂation rate. These two equations reduce to the lifetime budget
equation:
cyt +

Rt+1
1
mt +
co = Ωt ,1 + Rt+1
1 + rt+1 t+1(4)

where Rt+1 and Ωt are respectively the nominal interest rate and the capitalized value of
lifetime disposable income:Rt+1 ≡ (1 + rt+1 )(1 + πt+1 ) − 1,
)
(
1
o
o
− zt+1.
wt+1 lt+1Ωt ≡ wt lty − zty +1 + rt+1(5)(6)

The individual chooses (cyt , mt , cot+1 ) to maximize (1) subject to (4) and hence theoptimal conditions areu0 (cyt ) = (1 + rt+1 )θu0 (cot+1 ),(7)v 0 (mt )
Rt+1
.
y =
0
u (ct )
1 + Rt+1(8)6

Equations (4), (7) and (8) yield the following demand functions:cyt = cy (Ωt , rt+1 , Rt+1 ),(9)

mt = m(Ωt , rt+1 , Rt+1 ),(10)

cot+1 = co (Ωt , rt+1 , Rt+1 ),(11)

where 0 < cyΩ < 1, mΩ > 0, mR < 0.
See Appendix A for the abovementioned properties of the demand functions.
The ﬁrm sector has the following production function:yt = F (kt , lt ),satisfying Fk > 0, Fkk < 0, Fl > 0, Fll < 0, Fkl ≥ 0,

(12)

where kt and lt are inputs of capital and labor respectively. Physical capital depreciation
is assumed away, for simplicity. Since Pt and Wt are ﬁxed in the relevant periods, whichleads to shortages of commodity demand and labor demand, the ﬁrm sector minimizes
the total cost of producing output yt and thenFk (kt , lt )Fl (kt , lt )
=
> 1.
rt
wt
This equation gives kt as a function of lt , rt and wt :kt = k(lt , rt , wt ),where kl > 0, kr < 0, kw > 0.

(13)

Because of the shortage of commodity demand the ﬁrm increases neither kt nor lt although
the marginal productivity of each input exceeds each price, and hence the ﬁrm value νt
is strictly positive. Since k is given by (13), the dynamics of ν is
νt+1 = (1 + rt )νt − F (k(lt , rt , wt ), lt ) + rt k(lt , rt , wt ) + wt lt .

(14)

In the presence of labor demand shortage we have to assume a rule of job allocation
between the young (lty ) and the old (lto ). It is
lty = ly (lt ),lto = lt − ly (lt ),where 0 < lly ≤ 1.

(15)

Substituting (5), (9), (13) and (15) into (2) leads towt ly (lt ) − zty = cy (Ωt , rt+1 , (1 + rt+1 )(1 + πt+1 ) − 1)
+ mt + bt+1 + k(lt+1 , rt+1 , wt+1 ) + νt+1 .
7(16)

From (5), (6), (10), and (15) we ﬁnd rt+1 and Ωt to satisfy
mt = m (Ωt , rt+1 , (1 + rt+1 )(1 + πt+1 ) − 1) ,{}1oΩt = wt ly (lt ) − zty +wt+1 [lt+1 − ly (lt+1 )] − zt+1.
1 + rt+1(17)

These two equations solve rt+1 and Ωt as functions of lt and lt+1 for given price parametersoPs and Ws for s = t, t + 1, and policy parameters Mt , zty and zt+1
. Applying them into(16) gives lt+1 as a function of lt and νt+1 . Shifting (14) by one period and applying rt+1
obtained from (17) into the result gives νt+2 as a function of lt , νt+1 and lt+1 , where lt+1
has already been given as a function of lt and νt+1 . Therefore, lt+1 and νt+2 are represented
as functions of lt and νt+1 –i.e., l and ν formulate an autonomous dynamics. Note that we
consider the case where nominal prices and wages are ﬁxed and so are (wt , wt+1 , πt+1 , mt )in the relevant periods.
Totally diﬀerentiating the dynamics of l and ν gives eﬀects of bond issuance bt+1 andotax burdens zty and zt+1
on the dynamic paths of l and ν:


At 


dlt+1








dbt+1




 + Ct  dzty  ,


dνt+1
o
dzt+1
dlt

 = Bt 

dνt+2
where

At ≡

αt ζt −βt ²tζtBt ≡ 

γt ζt −δt ²t
ζt− δζtt,−(1 − cyΩt )wt llyt

ζt −(1+rt+1 )βtζtmΩt wt llyt)δt
− (1+rζt+1t
Ct ≡ 

− βζtt10cyΩ

1 − cyΩt

−

−mΩt

m t
− 1+rΩt+1

8

t
1+rt+1



.


,

(18)

αt ≡ −

cyΩt wt+1 (1 − llyt+1 )− klt+1 (< 0),
1 + rt+1oocyΩt (wt+1 lt+1
− zt+1)βt ≡ −+ cyrt+1 + (1 + πt+1 )cyRt+1 + krt+1 ,2(1 + rt+1 )mΩt wt+1 (1 − llyt+1 )γt ≡ −(< 0),
1 + rt+1oomΩ (wt+1 lt+1
− zt+1)δt ≡ − t+ mrt+1 + (1 + πt+1 )mRt+1 ,2(1 + rt+1 )²t ≡ (Fkt+1 − rt+1 )klt+1 + Flt+1 − wt+1 (> 0),ζt ≡ νt+1 + kt+1 − (Fkt+1 − rt+1 )krt+1 (> 0).

3

No Burden on Future Generations

Given that initially there is no public debt, this section considers the case where thegovernment issues bonds to ﬁnance ﬁscal spending in period T and repays them by taxing
on the old and/or the young in period T +1 so that the original path is recovered in period
T + 2 and after. Section 4 treats the case where a tax is imposed also on the generation
born in period T + 2 but not on the generations born afterwards.
The government unanticipatedly issues bonds dbT +1 (> 0) in period T in order toﬁnance government purchases dgT (≥ 0) and subsidy −dzTo (≥ 0) to the old, and in period
T + 1 taxes dzTo +1 (≥ 0) on the old and dzTy +1 (≥ 0) on the young so as to redeem thebonds. That is,{dbt }∞
t=0 = (0, ..., 0, dbT +1 , 0, ..., 0),(19)

dbT +1 = dgT − dzTo ,(20)

dzTy +1 + dzTo +1 = (1 + rT +1 )dbT +1 .

(21)

Furthermore, in order to focus on the pure eﬀect of bond issuance on the generations
born in periods T and T + 1, we consider the bond issuance and the subsequent taxationthat do not aﬀect the utility levels of the generations born in period t ∈ [T + 2, ∞].
Formally, in period T + 1 the tax-cum-subsidy on the young dzTy +1 is adjusted lest the
dynamic path should change from the original one in the subsequent periods t ∈ [T +2, ∞].
Therefore, we substitute dlT +2 = 0 and dνT +3 = 0 into the dynamics (18) for t = T + 1
9and thereby ﬁnd such dzTy +1 to satisfy−(1 − cyΩT +1 ) dzTy +1 = BT +1 mΩT +1

dlT +1,dνT +2

which reduces to
dzTy +1 = wT +1 llyT +1 dlT +1 ,dνT +2 = 0.

(22)

The implication is straightforward. If in period T + 1 the government levies a 100%
tax on the additional labor income of the young that is created by ﬁscal expansion (i.e.,dzTy +1 = wT +1 llyT +1 dlT +1 ), the budget of the young is unaﬀected. Thus, in the subsequent
periods employment and the ﬁrm value remain unchanged (i.e., dlt = 0 and dνt+1 = 0 fort ∈ [T + 2, ∞]).

3.1

No Burden on Children

Public debt is widely believed to place a burden on children by generating an income
transfer from them to their parents. Theoretical studies, such as Diamond (1965), Blanchard (1985) and Weil (1989), support this view in the case of full employment. We incontrast treat the case of Keynesian unemployment and ﬁnd that such burdens do notarise.
Because (19)–(22) imply no welfare eﬀect on the generations born in period t ∈ [T +2, ∞], we can obtain the pure welfare eﬀect on the children (viz., the generation born inperiod T ) of bond issuance by assuming the following policy:
Policy 1. Public debt is fully redeemed by taxing on the old in period T +1, –i.e., dzTo +1 =(1 + rT +1 )dbT +1 (> 0) and dzTy +1 = 0.
Under (19)–(22) and Policy 1 a unique consistent equilibrium obtains as follows:


 

  
1
y
dlT
dl
0
t
wT l


=
 =   for ∀t ∈ [T + 1, ∞].
(23)
 lT  dbT +1 , dνT +1dνt+1
00
The period-T value of the additional tax imposed on the children when they are old is
1
dzTo +1
1+rT +1= dbT +1 . If full employment holds, it makes the children worse oﬀ since10there is no room for employment expansion. However, in the presence of involuntary
unemployment ﬁscal expansion ﬁnanced by bond issuance stimulates aggregate demandand increases employment since from (12) and (23) where kT is predetermined we derive
dyT = FlT1
dbT +1 ,wT llyTdlT =1
dbT +1 .
wT llyT

Accordingly, the children earn additional labor income that exactly oﬀsets the additionaltax since (15) and the second equation above givewT dlTy = wT llyT dlT = dbT +1 .
This result implies that the children’s welfare is not aﬀected by the bond issuance.
The formal proof is as follows. Since from (23) dlT +1 = 0, dνT +2 = 0 and dνT +1 = 0 andprices are invariant in the relevant periods, (14) in which t = T + 1 and (17) in whicht = T respectively yield²T
1
1 + rT +1
dlT +1 + dνT +2 −dνT +1 = 0,
ζT
ζT
ζT
mr
+ (1 + πT +1 )mRT +1
drT +1 = 0,dΩT = − T +1
mΩT

drT +1 =dπT +1 =0.
Therefore, from (1), (5), and (9)–(11) we derivedcyT = 0, dcoT +1 = 0, dUT = 0.
In sum, the parents (viz., the generation born in period T − 1) receive income througheither transfers or government purchases and use it up before they die (i.e., within period
T ).7 Hence, in the presence of Keynesian unemployment their children deﬁnitely enjoy anincrease in labor income that exactly oﬀsets the tax burden due to public bonds. Fromthe second property of (23) with Policy 1, there is obviously no welfare eﬀect on the
grandchildren (viz., the generation born in period T + 1). We summarize this result in
the following proposition:
7

As analyzed in Barro (1974), if the current generation chooses to bequeath a legacy to future gen-erations due to altruistic motives, the Ricardian equivalence holds and an issue of public debt has noeﬀect.

11

Proposition 1. Under (19)–(22) and Policy 1 public debt places no burden on future
generations.
Remark 1. If government purchases are used to build facilities that yield utility for future
generations, they are better oﬀ by the magnitude of the utility.
Note that Policy 1 improves the welfare of the parents. To show it we combine (3)and (14) and obtaincoT =1
mT −1 + (1 + rT ) bT + kT + νT +1 + F (kT , lT ) − wT lT + wT [lT − ly (lT )] − zTo ,1 + πT

where kT is predetermined. Totally diﬀerentiating this equation and applying (1), (20)and (23) into the result yields[)](
dcoT = (FlT − wT ) + wT 1 − llyT dlT − dzTo

(24)

FlT − wT llyT
Fl T
dgT −
dz o > 0,=ywT llTwT llyT T
dUT −1 = θu0 (coT )dcoT > 0.
An increase in aggregate employment makes the parents better oﬀ since it raises not only
the value of equities held by them but also their wage earnings. Those two eﬀects are
represented by the ﬁrst and the second term of the coeﬃcient of dlT in (24), respectively.
Direct transfer −dzTo (> 0) obviously beneﬁts them. Thus, Policy 1 makes them betteroﬀ. The two expressions of dcoT in (24) imply that the welfare of the parents increases as
the marginal expansion of their employment (1 − llyT ) is larger, and that they receive alarger gain from direct transfer −dzTo than from government purchases dgT .

3.2

Tax on Grandchildren

We have so far considered the case where all the tax burden due to public debt restson children. This section examines the case that a part of it is placed on grandchildren,which is broadly believed to make them worse oﬀ. To address this issue we consider thefollowing policy in place of Policy 1 in the context of (19)–(22):
Policy 2. Public debt is redeemed by taxing on both the old and the young in period T + 1:
dzTy +1 + dzTo +1 = (1 + rT +1 )dbT +1 (> 0), dzTo +1 ≥ 0, and dzTy +1 > 0.
12

Note that this policy can be interpreted as a combination of Policy 1 and an income
transfer from grandchildren to children in period T + 1.
Under (19)–(22) and Policy 2 we have a unique consistent equilibrium that satisﬁes



 
  
1
y
dlT +1dl
0t
wT +1 ll y
= =   for t ∈ [T + 2, ∞]. (25)T +1 dzT +1 , dνT +2dνt+1
00With full employment the additional tax makes the grandchildren worse oﬀ by raising
the children’s consumption. In the presence of unemployment, however, the rise in thechildren’s consumption increases the grandchildren’s employment in period T + 1. From(15) and (25), the additional labor income of the grandchildren is
wT +1 dlTy +1 = wT +1 llyT +1 dlT +1 = dzTy +1 ,which exactly oﬀsets their tax burden. Consequently, the grandchildren’s welfare is unaﬀected whereas the children’s welfare increases: The same mechanism as Proposition 1
works in period T + 1. While the formal proof is given in Appendix B, this property issummarized as follows:
Proposition 2. Under (19)–(22) and Policy 2 public debt issuance makes the generation
born in period T better oﬀ and the generations born in period T + 1 and after unchanged.
Thus, there is no burden on future generations.
Remark 2. The larger the marginal expansion of children’s employment (1 − llyT +1 ) is, thegreater their welfare is.
As for the parents’ welfare, in Appendix C we obtain dcoT and dUT −1 :dcoT =FlT − wT llyT
ΦT
Fl T
o
dgT −
dz y
R 0,
y
y dzT +
wT llTwT llT(1 + rT +1 )wT llyT ΛT T +1

dUT −1 = θu0 (coT )dcoT R 0,
where ΛT is positive but ΦT can be positive or negative. Thus, public debt may or may
not make the parents worse oﬀ.8
8

Rankin (1986, pp. 497–498) imposes saddle-point stability and some other assumptions to show thatpublic debt issuance raises output and welfare in steady state. Diamond (1965) also assumes saddlepoint stability to show that public debt burden arises under full employment. Similarly, we need someassumptions to show that public debt issuance makes the current generation better oﬀ.

13

4

Deﬁcit-Covering Bonds

This section examines the implication of deﬁcit-covering bonds issued to redeem bonds.
The amount of public bonds is zero for any point in time on the original path and isunanticipatedly increased in period T by dbT +1 (> 0) to ﬁnance government purchases
and transfers to the old. Additional bonds are issued in period T + 1 by dbT +2 (> 0) tocover ﬁscal deﬁcits and are redeemed in period T + 2 by collecting taxes from the old andthe young. That is,{dbt }∞
t=0 = (0, ..., 0, dbT +1 , dbT +2 , 0, ..., 0),
dbT +1 = dgT − dzTo ,

(26)
(27)

dbT +2 + dzTy +1 + dzTo +1 = (1 + rT +1 )dbT +1 ,(28)

dzTy +2 + dzTo +2 = (1 + rT +2 )dbT +2 .

(29)

To focus on the pure eﬀect of the additional bond issuance on the future generations,this section assumes that the tax-cum-subsidy on the young dzTy +2 is adjusted in period
T + 2 lest the dynamic system for subsequent period t ∈ [T + 3, ∞] should change fromthe original one. Such dzTy +2 is obtained by substituting dlT +3 = 0 and dνT +4 = 0 into
the dynamics (18) for t = T + 2:
dzTy +2 = wT +2 llyT +2 dlT +2 ,dνT +3 = 0.

(30)

In addition, this section imposes the following policy to obtain the pure welfare eﬀectof the additional bond issuance.
Policy 3. Public debt is redeemed by taxing on only the old in period T + 2: dzTo +2 =(1 + rT +2 )dbT +2 (> 0) and dzTy +2 = 0.
Under (26)–(30) and Policy 3, additional bond issuance yields a tax burden on the
grandchildren when they are old whereas there is no welfare eﬀect on the generations born14

in period t ∈ [T + 2, ∞]. The consistent equilibrium must satisfy


 
1
dl
)
wT +1 lly(y
 T +1  = T +1
 dbT +2 + dzT +1 ,dνT +2
0

  
0
dlt
 =   for t ∈ [T + 2, ∞].

dνt+1
0

(31)

By comparing (25) and (31) we ﬁnd that the implication of Proposition 2 is valid in
Proposition 3 and that deﬁcit-covering bonds play exactly the same role as the tax on the
grandchildren when young (dzTy +1 ). Issuing additional bonds in period T + 1 generates
an income transfer from the grandchildren to the children in the form of tax reduction
and the children increase consumption. Since the children use up all the gains before theydie (i.e., within period T + 1), the grandchildren deﬁnitely receive the additional incomethat exactly covers the additional tax burden. Thus, the welfare of the grandchildren is
unaﬀected by the additional bond issuance whereas the children are better oﬀ.
We summarize the above result in the following:
Proposition 3. Under (26)–(30) and Policy 3, issuance of deﬁcit-covering bonds enhances the welfare of the generation born in period T while leaving unchanged the welfareof the generations born in period T + 1 and after. Public debt thus places no burden on
future generations.
The welfare eﬀect on the parents is ambiguous for the same reason as discussed below
Proposition 2.9 If public bonds are permanently rolled over, Rankin’s (1986) result on
the steady-state welfare is duplicated.10 Rankin imposes some conditions to show that apermanent increase in public debt raises the steady-state welfare. In contrast, we showthat bond issuance always places no burden on future generations while it may or may
not make the current generation better oﬀ. This result is consistent with Rankin’s since
the eﬀect on the steady-state welfare is a mixture of the two.
9

See Appendix C for the formal proof.

10

As already mentioned in Footnote 3, we explicitly introduce equities whereas Rankin (1986) assumes

that the ownership of ﬁrms is always in the hands of the old. Thus, there is a slight diﬀerence between
the results of the two.

15

5

Conclusion

Government purchases and transfers to the present generation ﬁnanced by public debt
issuance require additional taxes on future generations and thereby make them worse
oﬀ as long as full employment always holds, as shown in the literature. If there aredemand shortages and involuntary unemployment at least in the periods in which bonds
are issued and redeemed, however, the future generations who are taxed to redeem thebonds earn a large enough wage income to cover the additional tax burden while theeconomic situations of the subsequent generations are unaﬀected. Thus, there is no debt
burden on future generations. This result still holds even if deﬁcit-covering bonds areissued and the additional taxes are imposed on yet unborn generations.

16Appendices
Appendix A

The Property of Demand Functions

This appendix derives the properties of demand functions (9)–(11). Totally diﬀerentiating(4), (7) and (8) yields
1
> 0,
Γt
[
( 00 o)]u0 (cot+1 )u (ct+1 )cot+1
y
crt+1 =1− −,Γt (1 + rt+1 )2 u00 (cot+1 )u0 (cot+1 )
[
( 00
)]
v (mt )mtv 0 (mt )ycRt+1 = −1− − 0,Γt (1 + Rt+1 )2 v 00 (mt )v (mt )
cyΩt =

mΩt = ∆t cyΩt > 0,
mrt+1 = ∆t cyrt+1 ,[mRt+1 = ∆t cyRt+1 +]v 0 (mt )2
Rt+1
u00 (cyt )[]mΩt v 0 (mt )mt(1 + Rt+1 )2(1 + Rt+1 )2=− 0+ 2 00 y +< 0,2(1 + Rt+1 )2v (mt )
Rt+1 u (ct )(1 + rt+1 )2 Rt+1
θu00 (cot+1 )

where
2
Rt+1
u00 (cyt )u00 (cyt )Γt ≡ 1 ++> 1,(1 + Rt+1 )2 v 00 (mt ) (1 + rt+1 )2 θu00 (cot+1 )Rt+1 u00 (cyt )
∆t ≡
> 0.
(1 + Rt+1 )v 00 (mt )Hence, we have
1 − cyΩt =Γt − 1
> 0,
Γt

[
]
[
]
(1 + πt+1 )mΩt v 0 (mt )> 0.
− cyΩt mrt+1 + (1 + πt+1 )mRt+1 + mΩt cyrt+1 + (1 + πt+1 )cyRt+1 = −2
Rt+1
u00 (cyt )(A.1)

Appendix B

The Proof of Proposition 2

Because dzTy = 0 and dzTo +1 = (1 + rT +1 )dbT +1 − dzTy +1 from (20) and (21), the dynamics
given by (18) for t = T is

 


dlT
1 − cyΩT
dlT +1+ = BT AT dνT +1dνT +2
−mΩT
17

cyΩ

T

1+rT +1mΩT
1+rT +1


dbT +1dzTy +1.

Under (19)–(22) and Policy 2 we have a unique consistent equilibrium


 
1
dlT
1
wT l y =DT dzTy +1 ,
 lT  dbT +1 +ywlT +1 lT +1
dνT +1
0


 
1
dl
wT +1 lly y
 T +1  = T +1
 dzT +1 ,dνT +2
0

  
dlt
0

 =   for t ∈ [T + 2, ∞],
dνt+1
0


where

Dt ≡ Bt−1 −−cyΩ wt+1 llytt+1

1+rt+1mΩt wt+1 llyt ²t+ αt ζtζ−βtt+1

1+rt+1t ²t+ γt ζtζ−δ
t

(A.2)


.

Since the utility levels of the generations born in period t ∈ [T + 1, ∞] are not aﬀectedby bond issuance under (19)–(22) and Policy 2, it is suﬃcient to examine only the welfare
eﬀect on the children. Linearizing (14) in which t = T + 1 and applying (A.2) to theresult yields
1
1 + rT +1²T
dlT +1 + dνT +2 −dνT +1
ζT
ζT
ζT
FlT +1 + (1 + FkT +1 )klT +1[ (1+R] dzTy +1 > 0,
=−
0 (co
)u
)
T
+1
wT +1 llyT +1 RT +1 (1+rT +1 )u00T(c+1o ) + (1 + FkT +1 )krT +1

drT +1 =

(A.3)

T +1

where ²T and ζT are given below (18). From (17) we obtaindΩT = −

mrT +1 + (1 + πT +1 )mRT +1
drT +1 .
mΩTEquations (5), (9) and the above equation lead to[]yy
crT +1 + (1 + πT +1 )cRT +1 drT +1[]][−cyΩt mrt+1 + (1 + πt+1 )mRt+1 + mΩt cyrt+1 + (1 + πt+1 )cyRt+1

dcyT = cyΩT dΩT +
=

mΩT

(A.4)
drT +1 > 0,

where the sign of the numerator is given by (A.1). Since u0 > 0 and u00 < 0, from (7),(A.3) and (A.4) we obtaindcoT +1 =u0 (coT +1 )u00 (cyT )ydc−
drT +1 > 0.
(1 + rT +1 )θu00 (coT +1 ) T (1 + rT +1 )u00 (coT +1 )
18

(A.5)

Accordingly, (1), (A.4) and (A.5) givedUT = u0 (cyT )dcyT + θu0 (coT +1 )dcoT +1 > 0.
Since a decrease in llyT +1 raises drT +1 given by (A.3), it increases dcyT in (A.4) and dcoT +1
in (A.5) and hence increases dUT , implying Remark 2.

Appendix C The Welfare Eﬀect of Public Debt on Parents under(19)–(22) and Policy 2This appendix shows that the welfare eﬀect of public debt on parents depends on thesigns of dνT +1 and dlT , which are ambiguous. Totally diﬀerentiating (14) where t = T + 1
and substituting (25) into the result gives²T
1
ζT
dlT +1 +dνT +2 −drT +1
1 + rT +1
1 + rT +1
1 + rT +1
ζT²T
dzTy +1 −=drT +1 ,y(1 + rT +1 )wT +1 llT +1
1 + rT +1

dνT +1 =(A.6)

where ²T (> 0) and ζT (> 0) are given below (18). An income transfer from the grandchildren to the children, dzTy +1 > 0, directly increases labor employment dlT +1 and ﬁrm valuedνT +1 . It also raises the interest rate (as shown in (A.3)), which in turn reduces dνT +1 .
Accordingly, the total eﬀect on dνT +1 is ambiguous. In fact, substituting (A.3) into (A.6)
yieldsdνT +1 =

ΘT
dz y
R 0,
(1 + rT +1 )ΛT T +1

(A.7)

where
[

]
(
)
(FkT +1 − rT +1 )klT +1 + (FlT +1 − wT +1 ) (1 + RT +1 )coT +1
u0 (coT +1 )ΘT ≡− 00 oRT +1 (1 + rT +1 )u (cT +1 )coT +1][− (FlT +1 − wT +1 )krT +1 − (νT +1 + kT +1 ) (1 + FkT +1 )klT +1 + FlT +1 R 0,[]
(1 + RT +1 )u0 (coT +1 )yΛT ≡ −wT +1 llT +1+ (1 + FkT +1 )krT +1 > 0.
RT +1 (1 + rT +1 )u00 (coT +1 )Note that this is the same as dνT +1 given in (A.2). The denominator of (A.7) is positive
and thus the sign of ΘT determines the sign of dνT +1 .

19

Totally diﬀerentiating (16) where t = T and substituting (20), (25), and (A.6) into
the result generates−αT
1βT
drT +1yy dlT +1 +yy dνT +1 +y
(1 − cΩT )wT llT(1 − cΩT )wT llT(1 − cΩT )wT llyTcyΩT
1+dbT +1 +dz y
wT llyT(1 − cyΩT )(1 + rT +1 )wT llyT T +1
wT +1 llyT +1 cyΩT − (1 + rT +1 )αT + ²T y
1o=(dgT − dzT ) +dz
wT llyT(1 − cyΩT )(1 + rT +1 )wT llyT wT +1 llyT +1 T +1

dlT =

+

(1 + rT +1 )βT − ζT
drT +1 ,(1 − cyΩT )(1 + rT +1 )wT llyT

where αT (< 0), βT , ²T (> 0), and ζT (> 0) are given below (18). Thus, while the income
transfer from the grandchildren to the children directly increases labor employment dlT ,it also raises the interest rate (see (A.3)), which makes the eﬀect on dlT ambiguous. In
fact, substituting (A.3) into the above equation gives
dlT =

ΞT
1
o
dzTy +1 R 0,y (dgT − dzT ) +ywT llT(1 + rT +1 )wT llT ΛT

(A.8)

where
[
] (
)
(1 + FkT +1 )klT +1 + FlT +1 cyT
u0 (cyT )ΞT ≡− 00 y y
RT +1
u (cT )cT[]()
(1 + FkT +1 )klT +1 + (FlT +1 − wT +1 ) (1 + RT +1 )coT +1
u0 (coT +1 )+− 00 oRT +1 (1 + rT +1 )u (cT +1 )coT +1[]+ wT +1 (1 + FlT +1 )krT +1 − (νT +1 + kT +1 ) (1 + FkT +1 )klT +1 + FlT +1 R 0.
Note that this is the same as dlT given in (A.2). Under (19)–(22) and Policy 2, dzTy +1 isin the following range:
0 < dzTy +1 ≤ (1 + rT +1 )dbT +1 = (1 + rT +1 )(dgT − dzTo ).
If ΞT is positive, public debt issuance unambiguously increases labor employment dlT . If
ΞT is negative, in the case of dzTy +1 = (1 + rT +1 )(dgT − dzTo ), dlT takes its lower bound(dlˆT ) and may become negative. It is given bydlˆT =

ΥT(dgT − dzTo ),wT llyT ΛT
20

where[] ()(1 + FkT +1 )klT +1 + FlT +1 cyT
u0 (cyT )ΥT ≡− 00 y y
RT +1
u (cT )cT
{
[
]}
y
)
(1 + FkT +1 )klT +1 + FlT +1 − wT +1 (1 − llT +1 ) (1 + RT +1 )coT +1 (u0 (coT +1 )+− 00 oRT +1 (1 + rT +1 )u (cT +1 )coT +1[]+ wT +1 (1 − llyT +1 )(1 + FlT +1 )krT +1 − (νT +1 + kT +1 ) (1 + FkT +1 )klT +1 + FlT +1 R 0.
If ΥT is positive, the total eﬀect on dlT is always positive. If ΥT is negative, dlˆT is negative
and thus the total eﬀect on dlT is ambiguous depending on the magnitude of dzTy +1 .
From (1), (3), (14), (A.7), and (A.8), we obtain[()]
dcoT = dνT +1 + (FlT − wT ) + wT 1 − llyT dlT − dzTo
FlT − wT llyT
Fl T
ΦT
o
=
dgT −
dzTy +1yy dzT +
ywT llTwT llT(1 + rT +1 )wT llT ΛT

(A.9)
R 0,

dUT −1 = θu0 (coT )dcoT R 0,
where
[
] (
)
)
(
(FlT − wT llyT ) (1 + FkT +1 )klT +1 + FlT +1 cyT
u0 (coT +1 )u0 (cyT )ΦT ≡− 00 y y + φT − 00 o
RT +1
u (cT )cTu (cT +1 )coT +1][+ FlT wT +1 (1 + FkT +1 ) − wT llyT FlT +1 (1 + rT +1 ) krT +1[]− (νT +1 + kT +1 )FlT (1 + FkT +1 )klT +1 + FlT +1 R 0,][
FlT (1 + FkT +1 ) − wT llyT (1 + rT +1 ) klT +1 + FlT (FlT +1 − wT +1 )
φT ≡
> 0.
[
]−1
RT +1 (1 + rT +1 ) (1 + RT +1 )coT +1
If ΦT is positive, public debt issuance is unambiguously beneﬁcial to parents. If ΦT isnegative, the case of dzTy +1 = (1 + rT +1 )(dgT − dzTo ) gives the lower bound of dcoT (dcˆoT ),which may become negative. It is given bydcˆoT =

ΨgT
ΨzT
dgT −
dzTo ,yywT llT ΛT
wT llT ΛT

where
[
] (
)
(
)
(1 + FkT +1 )klT +1 + FlT +1 cyT
u0 (coT +1 )u0 (cyT )gc− 00 y y + ψT − 00 o
RT +1
u (cT )cTu (cT +1 )coT +1[]− ψTgk krT +1 − (νT +1 + kT +1 )FlT (1 + FkT +1 )klT +1 + FlT +1 R 0,

(FlT − wT llyT )
g
ΨT ≡

21

[
]
FlT (1 + FkT +1 ) − wT llyT (1 + rT +1 ) klT +1 + FlT (FlT +1 − wT +1 ) + wT +1 llyT +1 (FlT − wT llyT )
> 0,
[
]−1
RT +1 (1 + rT +1 ) (1 + RT +1 )coT +1
[
]
gk
y
y
y
ψT ≡ wT llT (FlT +1 − wT +1 llT +1 ) + wT +1 FkT +1 (1 − llT +1 ) − FlT wT +1 (1 − llyT +1 )(1 + FkT +1 ) R 0,
ψTgc ≡

and
[
] y (
(
)
)
y
(F
−
w
l
)
(1
+
F
)k
+
F
cT
u0 (coT +1 )u0 (cyT )
l
T
k
l
l
T
T
+1
T
+1
T
+1
lT
z
zc
ΨT ≡
− 00 y y + ψT − 00 o
RT +1
u (cT )cTu (cT +1 )coT +1[]− ψTzk krT +1 − (νT +1 + kT +1 )FlT (1 + FkT +1 )klT +1 + FlT +1 R 0,
[
]
y
F
(1
+
F
)
−
w
l
(1
+
r
)
klT +1 + FlT [FlT +1 − wT +1 (1 − llyT +1 )]
l
k
T
T
+1
T
T
+1
l
T
zc
ψT ≡
> 0,
[
]−1
RT +1 (1 + rT +1 ) (1 + RT +1 )coT +1[]
ψTzk ≡ wT llyT FlT +1 (1 + rT +1 ) − FlT wT +1 (1 − llyT +1 ) + FkT +1 R 0.
If ΨgT is positive, government purchases ﬁnanced by public bonds always make parents
better oﬀ. If ΨgT is negative, the total welfare eﬀect of government purchases on parents
is ambiguous depending on the magnitude of dzTy +1 . Similarly, the subsidy to parents
ﬁnanced by public bonds may or may not improve the welfare of parents depending on
the sign of ΨzT and the magnitude of dzTy +1 . Consequently, we conclude that the welfare
eﬀect of public debt on parents is ambiguous under (19)–(22) and Policy 2. By replacing
dzTy +1 with dbT +2 + dzTy +1 in (A.9), we ﬁnd the welfare eﬀect of public debt on parents tobe ambiguous under (26)–(30) and Policy 3.

22

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25