DBJ Discussion Paper Series, No.1502

Public Debt as a Burden on the Future Generation:
A Keynesian Approach

Masayuki Otaki(Institute of Social Sciences, University of Tokyo)

October

2015

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in orderto protect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Public Debt as a Burden on the Future Generation: A Keynesian Approach
Masayuki Otaki
The Institute of Social Sciences, The University of Tokyo
ohtaki@iss.u-tokyo.ac.jp

Abstract
By using an overlapping generations (OLG) model in the context of the production
economy, I show that public debt lowers the future generation’s well-being even in thesituation of imperfect equilibrium. According to the dynamically extended multiplier
theory, which has a rigorous microeconomic foundation, the effect of the redemption of
public debt is equivalent to that of the balanced-budget multiplier because theredemption increases autonomous spending by old generation while the same amount oftax is levied on the young generation. Thus, the aggregate disposable income remains
before the issuance since the value of the balanced-budget multiplier is unity. However,
it is evident that real GDP before tax reduction increases; it costs more resources toearn the same disposable income. Therefore, it is unavoidable that the issuance of
public debt impairs the welfare of future generation even in a Keynesian framework.

Keywords: Redemption of Public Debt; Burdenon

the Future Generation;

Balanced-Budget Multiplier; Keynesian Economics; Fiscal Discipline
JEL Classifications: E21; E24; E62; H62; H63

1. Introduction
Almost all advanced countries are concerned by the huge amounts of public debts
carried over from the past. It seems to be unredeemable within a generation, and thusintergenerational conflicts in this regard are likely to become increasing. However,besides the seminal work of Diamond (1965), who shows that the segregated sequence ofintergenerational economic decisions ends in market failure (i.e. dynamic inefficiency)
and that the issuance of public debt prevents the sufficient capital accumulation, otherfamous studies are rather optimistic about the redemption of public debt despite theaforementioned grievous fact.
Ricardo (1821) is an exception. He argues that since the issuance of public debt implies
the same amount of future tax liability, it only results in income transfer from thetaxpayers to those who are exonerated from taxation (most of whom are debt holders).
He warns that this is, therefore, prone to impair the cohesion of the society. In addition,
since the levied tax calls for the same amount of savings, government expenditure
financed by such tax is entirely cancelled out by the savings, and hence income does notincrease by the issuance of public debt.1 One must note that Ricardo is acutely aware of
the devastating nature of the conflict between tax payers and recipients, even thoughsuch an income transfer is implicitly assumed to belong to the same generation.
Lerner (1944) radically alleges that there is no future burden accrued from theissuance because the redemption is only a kind of transfer within a country. Ricardo andLerner offer the same argument in the sense that tax associated with the redemption is

According to Ricardo (1821: Ch.17), `But independently of his consideration, it is by nomeans certain, that political utility would gain any thing by the sacrifice of political
integrity; it does by no means follow, that the party exonerated from the payment of theinterest of the national debt would employ it more productively than those to whom
indisputably it is due. By cancelling the national debt, one man’s income might beraised from ￡1,000 to ￡1,500. But another man’s would be lowered from￡1,500 to￡1,000. These two men’s incomes now amount to￡2,500. They would amount to no
more than then. If it be the object of Government to raise tax, these would be precisely
the same taxable capital and income in one case, as in the other. It is not, then, by thepayment of the interest rate on the national debt, that a country is distressed, nor is itby the exoneration from payment that it can be relieved. It is only by saving fromincome, and retrenching in expenditure, that the expenditure diminished by theannihilation of the national debt. It is by the profuse expenditure of Government, and of
individuals, and by loans, that the country is impoverished; every measure, therefore,
which is calculated to promote public and private economy, will relieve the publicdistress: but it is error and delusion to suppose, that a real national economy difficulty
can be removed, by shifting it from the shoulders of one class of the community, whojustly ought to bear it, to shoulders of another class, who, upon principle equity, oughtbear no more than their share.’1

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a transfer within the same nation as far as public debt is not external but internal
although Lerner seems too optimistic about levying tax2. This comes from his emphasison the role of “functional finance,’’ which is nowadays called the aggregate demand
management policy. One of important aims of this study is to examine his assertion
critically; that is, whether public debt becomes a burden to future generation when an
economy is entrapped by imperfect employment equilibrium.
As a case for the redemption and taxation across generations, Barro (1974) suggeststhat if each generation is altruistic to only their subsequent generation, it maximizes
the utility by integrating its own budget constraint and its descendant. Accordingly, heasserts that the monetary gain from the issuance of public debt is entirely canceled out
by the corresponding future tax liability. Thus, the Ricardian equivalence theorem, inthe narrow sense that public debt does not comprise the net wealth in an economy as awhole, can be extended by adopting the local altruism.
Nonetheless, if such altruism were prevalent, there would be no serious
intergenerational conflict as we observe presently. In this paper, I examine how theissuance and redemption of public debt affect the intergenerational resource allocationand income distribution in a monetary economy using a dynamically extended
multiplier theory developed by Otaki (2007, 2009, and 2015).
This theory is characterized by the property that the equilibrium price level sequence
is unrelated to the nominal money supply. This property is vital for negating thequantity theory of money as in Lucas (1972).
Money can function as a store of value because everyone is confident that money is the
only transaction medium and vice versa, and hence both functions are inseparable as
Otaki (2015 Ch.14) exactly proves. This fact acutely suggests that current value (theinverse of the price level) of money crucially depends on its future value without
conferring the condition of nominal supply. Accordingly, as far as the rational
expectation on the future value of money is stable, current price level becomesendogenously fixed and independent of the level of aggregate demand. This property ofthe theory immediately implies that, without unprecise and arbitrary assumption ofmenu cost (or staggered pricing rule) associated with the utility function which contains
the real cash balance, one can describe the state of imperfect employment as an
Pareto-inferior equilibrium. This theory also derives Keynes’s investment multiplier

According to Lerner (1944: p.303), "the national debt is not a burden on posterity
because if posterity pays the debt it will be paying it to the same posterity that will bealive at the time when the payment is made."22when the lifetime utility function from consumption is homothetic and additively
separable from the disutility of labor.
Based on this theory, I analyze how the issuance and redemption affect the related
generational well-being. Whenever the government expenditure, which is financed bypublic debt, increases, the disposable income of the current generation also increases(although the value of the multipliers differ depending on whether or not theexpenditure is directly transferred to individuals), and heightens its economic welfare
under some weak condition.
Nevertheless, the true problem is not located there. The effect of the redemption ofsuch public debts to the subsequent generation is problematic. This generation is leviedby the same amount as the sum of principal and interest of the issued public debt. In
another aspect, however, the old generation, which enjoys much consumption during theprevious period, holds much money because of the redemption. This stimulates theaggregate consumption. Consequently, one will find that the macroeconomic effect of theredemption is represented by the balanced-budget multiplier, the value of which is unity.
Hence, the disposable income of the subsequent generation is kept at the same level asthat of the preceding generation in the issuance of public debt.
It must be noted, however, that keeping the disposable income unchanged never meansthat economic welfare is not affected by the redemption. This is because more resources
are demanded for the redemption despite the invariant disposable income. Thus, it isevident that redemption aggravates the welfare of the subsequent generation. In thissense, the issuance of public debt becomes a future burden.
This paper is organized as follows. Section 2 constructs the model and shows theresults of the comparative statics concerning the issuance and redemption of public debt.
I consider why fiscal discipline is prone to be dysfunctional in Section 3, based on themodel in Section 2. Section 4 provides the brief concluding remarks.

2. The Model
I consider a two-period overlapping generations (OLG) model with an infinite time
horizon where output markets are under monopolistic competition. This model isdeveloped by Otaki (2007, 2009) and generalized by Otaki (2015). There are
non-atomistic firms that monopolistically produce differentiated goods in the density ofpoints located within the interval  0,1 . Let the character of a firm be denoted  . The
only production resource is labor, and for simplicity, unit labor produces unit

3

differentiated goods. Each firm  can employ its attached individuals, whose density isalso located within  0,1 . That is, individuals totally exist in the density of the interior
points of the square  0,1   0,1 .
Each individual possesses the same utility function, U ,
1

1
 1
 1 1,   1,
Ut  u  c1t , c2t 1    t   , ckl    ckl   
d 
 0

1

(1)

where u is a concave and homothetic function.  denotes the disutility of labor.  t is adefinition function, which takes value unity when employed and zero when unemployed.
There are two kinds of store of value in this economy. One is money. The other is public
debt. The public debt accrues units of money at gross interest rate  , which is set at the
government discretion. Accordingly every individual wishes to save his or her income interms of the public debt because there is no uncertainty. For clarifying the role of the
public debt, I dare to assume that the supply of the public debt is too scarce to satisfy
the whole demand for saving. One must note that this analysis never loses generality
even though the interest is attached to all money as Lucas (1972) assumes. In addition,
it is assumed that tax is levied only on firms.
The maximization of lifetime consumption leads us to the following indirect utility
function concerning consumption and the corresponding price index:

 yd 
ht  f  t  ,   t  

1

1
P
 1
 1
Pt     pt    d  ,  t  t 1 (2)
0
Pt



d

where yt is the real disposable income in terms of current goods. f is a monotonously
increasing and invertible function. In accordance with Otaki (2007), it is assumed thatthe labor markets are in interior equilibrium (i.e. there is imperfect employment
equilibrium) and the equilibrium nominal wage stacks at the nominal reservation wage.
Equations (1) and (2) give the nominal equilibrium wage, Wt , asWt  Pt  t   f 1   .

(3)Here I assume that the concept of goods markets is a symmetric Nash equilibrium, andhence obtains the following optimal pricing rule:4

P  t   f  
Wt
 t
1
1 
1   1
1Pt 

1   1    t   1
   *  .
f  

(4)

Thus, the equilibrium inflation rate becomes constant over time. In addition, byapplying Roy’s identity to Equation (2), I obtain the aggregate consumption function,

Ct , asCt  c  *  ytd ,(5)where ytd is the aggregate real disposable income evaluated at the current price level.3
For closing the model, one needs to specify the money-supply rule, or equivalently, thebudget constraint of the government. To purify the effect of issuance and redemption of
public debt, I must eliminate the effect of changes in the real cash balance owing to
inflation. Accordingly, I obtain the following budget constraint of the government:

M t  M t 1
MM1 g  1  *  m  g , m  t  t 1 , (6)
Pt
Pt
Pt 1
  
where g denotes the real government expenditure towards some wasteful objects. m is
the real cash which is assumed to be constant over time.
I specify the time schedule of public debt management to elucidate the effects to theeconomy and its welfare implications. At period t  1 , there is no asset other than money.
This period becomes the reference point of the comparison. During period t , public debts
are newly issued the nominal sum, which amounts to D . The corresponding revenue isentirely transferred to the individuals who are born during the previous period t  1 .
Such debts expire at the beginning of period t  1 , and I assume that no rollover ispossible. The necessary nominal tax, D , is young generation’s due.
As such, the equilibrium condition of the aggregate goods market and equilibrium real
GDP is classified into the following three cases:

Note that the opportunity cost for current consumption is not the real rate of interest
but the inflation rate because I assume that the equilibrium effective demand issufficiently high so that the aggregate saving exceeds the issued public debt.
3

5Case 1: Period t  1

yt 1  c  *  yt 1  g 
 yt 1  ytd 

M t 1 c  *  yt 1  m
Pt

(7)

m
,
1  c  * 

where the third term on the right-hand side at the beginning of Equation (7) denotes theconsumption of the old generation. When I take the government’s budget constraint,
Equation (6), into consideration, the final form of Equation (7) is obtained.
Case 2: Period tDDyt  c  *   yt    1    m 
Pt 
Pt

1  c  *     1  D  m Pt yt 
1  c  * 

(8)

D
m
Pt
d
 yt 
.
1  c  * The terms inside of the square bracket on the right-hand side on the top of Equation (8)are the real aggregate disposable income in terms of current goods.

  1 D
Pt

corresponds to the income accrued from the interest on the public debt. The third term
represents the transfer to the old generation financed by the public debt.
Case 3: Period t  1yt 1  c  *   yt 1 
 D  m D
Pt 1 Pt 1m
 yt 1 D*
1  c   Pt 1
 ytd1 (9)m.
1  c  * 

The term of the inside of the square bracket at the beginning of Equation (9) expresses
the real disposable income of the young generation since tax is levied on the young
generation to redeem the public debt issued in the previous period. The third term on

6

the right-hand side of this equation refers to the real expenditure of the old generation,
who were born at the beginning of period t , from the redemption.
The obtained result is summarized in Figure 1. There are two important facts. First, as
the figure indicates, the real disposable income is unchanged before and after theissuance of the public debt, and the real disposable income increases only during theperiod (period t ) in which the issuance occurs. Second, the issuance of the public debt isbeneficial for both for the old and young generations who live during the corresponding
period. This fact implies that every period offers the incentive for enriching the economy
via fiscal deficit. As discussed in the next section, this is quite a precarious propensity
when one assigns much importance to sustainability (or the well-being of the residentsof the economy over time).
Finally, let us analyze implications of welfare economics on the issuance of public debt,
that is, whether public debt becomes a burden to future generation. To clarify the
analysis one needs define what constitutes a “burden on the future generation.’’ That is,
Definition 1

I posit that the issuance of public debt becomes a burden on the future generation
whenever such issuance lowers the welfare of the subsequent generation compared withthat of the generation living in the advent of the issuance.
As par the above definition, I should compare the economic welfare of the generation
born at the beginning of period t  1 (the t  1 th generation) with that of the t  1 th
generation. One find that both generations achieve the same level of the real disposable
income (i.e.,m). This is because the economic environment surrounding the1  c  * t  1 th generation coincides with the implementation of the additional balanced-budget
expansionary fiscal policy at period t  1 . That is, the increase in the consumption of theold generation associated with the redemption of the public debt incurs the sameamount of taxation from the young generation at period t  1 .
Nonetheless, this does not imply that the economic welfare, which each generation
enjoys, remains unchanged by the issuance of public debt at period t . This is because
higher GDP is required for the t  1 th generation to achieve the same disposable income.
The fact that higher GDP is necessary means that the t  1 -th generation uses more
input to obtain the same well-being as the t  1 th generation. Accordingly, from7

Definition 1, it is clear that the issuance of public debt is surely a burden on the future
generation. More rigorously, one can prove the following theorem:
Theorem 1

The issuance of the public debt at period t becomes a burden on the future generation in
the sense of Definition 1.
Proof
Since the equilibrium nominal wage stacks to the nominal reservation wage, a worker’s
utility never changes in conjunction with the employment level. Accordingly, from theproperty of the lifetime indirect utility function (2), one can focus on the change of firm’s
profit to prove the theorem. From the optimal pricing rule (4) and the symmetry of
goods markets equilibrium, the net profit after the deduction of tax for the redemption,

SURt 1 , which is equal to the total economic surplus of generation t  1 , is
SURt 1   1 yt 1  T .

(10)

As is clear from Equations (7) and (9), the economic circumstance of period t  1 isequivalent to that in period t  1 plus the balanced-budget fiscal expansion whichamounts to


Pt 1

 D . Since the value of the balanced-budget multiplier is unity, bydifferentiating Equation (10), I obtaindSURt 1   1  1 dT  0.

(11)

Equation (11) completes the proof.

3. The Importance of Fiscal Discipline
The discussion in the previous section reveals following two precarious properties of theissuance of public debt.
(i) Despite the unavoidableness of redemption, every two succeeding generations have apropensity to issue public debt because such a deed enriches them both.
(ii) As Theorem 1 asserts, since the redemption of public debt necessarily aggravates thewell-being of the young generation at that time, it is likely to be postponed as far as
possible.

8

As such, once people experience the ephemeral pleasure obtained by fiscal deficit, there
is an appalling tendency that the issuance of public debt ceaselessly continues until thegovernment declares the (at least partial) default of its debt. The enforcement of fiscal
discipline is, therefore, urgently required.
One might argue that autonomous expansion of outstanding public debt associatedwith increases in interest payment will automatically lead to full-employment
equilibrium. However, it must to be noted that the stock of public debt exploded withsuch an irresponsible debt-management policy, which sets no limit. Accordingly, such an
economy is forced to declare the default sooner or later, and thus the interest must be
paid to the debt holders by levying the tax to preserve the stock of public debt (i.e.,maintain it at a constant level) over time and make the economy sustainable .
Let the sustainable real public debt be denoted as d ( d 

Dt  jPt  j 1). d satisfies  1 
1  c  *  1  *  d   m  *  d




m
1
(12)
1  c  *  *
d 
 .
c  * 1  c  * 
From Equation (10), it is apparent that whenever the total social surplus, SUR , isnegative, and the taxation for collecting the interest payment exceeds the benefit, whichstems from achieving full employment, full employment is unattainable. Accordingly, if 1
 d   1
*


(13)

holds, an irresponsible pubic-debt management policy, which satisfies properties (i) and(ii), triggers the default sooner or later. It is evident that an extremely low interest
policy (including a zero-interest policy), which sets  close to unity, is inevitable foravoiding the default as the experiences of the world economy show. The default of public
debt denotes income redistribution at the government’s discretion between theunfortunate generations, the occurrence of which Ricardo (1821) acutely warns us
against.
Nevertheless, one must note that this paper neglects alternative investment
opportunities such as real capital the rate of return of which is thought to besignificantly positive. In such a case, the nominal rate of interest  is not dependent on

9the government’s discretion, and thus, a zero-interest policy is no longer feasible at leastin the long run. Consequently, one might experience much difficulty in mitigating acrisis caused by default of public debt. That is, fiscal discipline is the bedrock ofsustaining a sound monetary economy in reality.

4. Concluding Remarks
This study considers whether public debt becomes a burden on the future generation
mainly under imperfect employment equilibrium. The results indicate the followings.
First, the issuance of public debt becomes a burden on the future generation because
more taxes are necessary to regain the same disposable income earned at the advent of
the issuance. Second, unlike Barro’s (1974) altruistic assumption, if the concern of anindividual is confined to his or her own well-being, every two succeeding generations arelikely to prefer issuing public debt. This fact owes much to the propensity, that is, theold generation in question can enjoy affluent consumption financed by the public debt,
and at the same time, the young generation that coexists is also enriched because theaggregate demand is stimulated by the increased expenditure of the old generation. One
must note that this is quite a precarious tendency because such ephemeral prosperity
postpones the redemption of public debt as far as possible, and, far worse, accelerates
the accumulation of debt.
Finally, one might assert that the state of full employment hinders such a worrisome
propensity. However, as the equations in Section 3 reveal, unless propensity to save or
the degree of monopoly (the inverse of the elasticity of demand to its relative price),
which is the very origin of the social surplus of the young generation in this model,4 issufficiently high, full employment is hardly achievable as far as the zero-interest policy
is not sustainable at least in the long run. This assertion implies that a partial defaultof public debt, which is equivalent to an income readjustment by the monetary
authority between the old and young generations living during the period in question, isunavoidable.
I must mention that the model neglects the effect of capital accumulation, which
enriches the aggregate saving. Taking such a factor into consideration would require amore precise analysis of the public-debt crisis.

See Otaki (2015: Ch.2) for a discussion on why the monopolistic factor heightens
economic welfare against naïve insights.

410

References
Barro RJ (1974). Are government bonds net wealth? Journal of Political Economy 82:
pp. 1095-1117.
Diamond PA (1965). National debt in a neoclassical growth model, American Economic
Review 60: pp. 1126-50.
Lerner AP (1944). The economics of control: principles of welfare economics, Macmillan,
London.
Lucas RE, Jr. (1972). Expectations and the neutrality of money, Journal of Economic
Theory 4: pp.103-124.
Otaki M (2007). The dynamically extended Keynesian cross and the welfare-improving
fiscal policy, Economics Letters 96: 23-29.
Otaki M (2009). A welfare economics foundation for the full-employment policy,
Economics Letters 102: 1-3.
Otaki M (2015). Keynesian economics and price theory: re-orientation of a theory ofmonetary economy, Springer-Japan, Tokyo.
Ricard D (1821). On the principles of political economy and taxation, 3rd edition.
http://www.econlib.org/library/Ricardo/ricP4.html#17.4
Accessed on 13th September 2015.

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Disposable Income

θ

D
+m
Pt
1−c

m
1−c

t−1

t

t+1
Time

Figure 1: Public Debt and Disposable Income

12