Discussion Paper No. 642

WELFARE ANALYSIS OF DEBT POLICYDURING RECESSIONS

Takayuki Ogawa
The Seventh ISER-Moriguchi Prize (2005)
Awarded Paper

September 2005

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanWelfare Analysis of Debt Policy during Recessions
Takayuki Ogawa1
Kyoto University, Japan

Abstract
This paper develops an overlapping-generations model with nominal wage rigidities andexamines the welfare effects of debt policy when unemployment exists. Issues of public debt
stimulate aggregate consumption demand and create employment. Future generations thenface both increased wage incomes and higher taxes. If the amount of outstanding bonds
is already large, debt policy deteriorates the welfare of future generations by levying heavy
taxes. By contrast, if the outstanding bond issue is relatively small, debt policy can be
Pareto improving by creating more employment. Therefore, the welfare implications of debt
policy during recessions can be discriminated from those during booms.

JEL Classification: E12, E24, E62, H63
Keywords: Debt policy, Overlapping generations, Welfare effects
Correspondence: Institute of Economic Research, Kyoto University, Yoshida-Honmachi,
Sakyo-ku, Kyoto 606-8501, JAPAN. E-mail address: ogawa@kier.kyoto-u.ac.jp

1

This paper received the Osaka University Institute of Social and Economic Research Moriguchi Prize inFebruary 2005. I thank Koichi Futagami, Shinsuke Ikeda, Hiroshi Nakaota, Yoshiyasu Ono, Tadashi Shigoka, andseminar participants at the 2004 Spring Meeting of the Japanese Economic Association and Osaka University for
their helpful comments. This research is financially supported by the Grant-in-Aid for JSPS Fellows, the Ministryof Education, Culture, Sports, Science and Technology.

I.

Introduction

Tobin (1980) asserts that in conditions of under-employment further issues of public debt
stimulate aggregate demand and increase output and employment. Inspired by Tobin, Rankin(1986) demonstrates that when there are demand shortages and unemployment, such Keynesiandebt policy brings higher steady-state social welfare. And more recently, Ono (2001) contends
that, in the context of liquidity traps, current debt issue never places a burden on future generations. By way of contrast with the Keynesian scenario, Modigliani (1961) argues that debt
issue puts a strain on future generations, even with under-employment situations. Accordingly,
whether debt policy during recessions is beneficial to future generations remains a controversial
issue. By introducing nominal wage rigidities into a monetary overlapping-generations model àla Weil (1987), this paper clarifies the welfare implications of debt policy during recessions.
In considering the problem of a burden on future generations of public debt, the definitionof future generations is crucial. According to Lerner (1948), future generations are defined as
people who exist at future dates irrespective of birth dates. On the other hand, Bowen et al.
(1960) regard people who were born at the same date as the same generation, and define future
generations as people who will be born after the present date. The analyses that approach the
problem by using overlapping-generations models, including the current paper, follow the latter
definition.
Another important point involved in analyzing this problem is whether the analysis uses
an overlapping-generations model à la Diamond (1965) or one à la Blanchard (1985) and Weil(1987, 1989). In the Diamond model, people undoubtedly die and hence the current generationthat enjoys benefit from debt issue never exists at far future dates. Therefore, the focus of theanalysis is mainly drawn to the future level of output. Assuming full employment, Diamond(1965) shows that debt issue interferes with capital accumulation and lowers the steady-statelevel of output and welfare (if the economy is dynamically efficient). By contrast, developing1the Diamond model with fixed prices and wages, Rankin (1986) finds that debt issue raises thesteady-state level of output and welfare. Consequently, in the Diamond situation, debt policy isharmful to future generations during booms, whereas it is beneficial during recessions (as longas the economy is dynamically efficient).2
On the other hand, the Blanchard-Weil model examines the situation where the current
generation exists even at far future dates.3 Then, the literature on the lines of Blanchard
and Weil focuses mainly on the income redistribution effect between different generations ofgovernment policy. As shown in Blanchard (1985) and Weil (1989), debt finance means an
income transfer from future generations to existing generations. It is because part of taxes
associated with debt issue is levied on future generations, which are not connected with existing
generations. Based on this intergenerational income redistribution effect, debt policy during
booms is beneficial to existing generations and harmful to future generations (unless the economy
is dynamically inefficient).4
The purpose of this paper is to examine the welfare implications of debt policy during recessions in the context of Blanchard and Weil. Taking into account both the intergenerational
income redistribution effect and the existence of unemployment, the result differs from that of Diamond (1965), Rankin (1986), and Blanchard (1985)-Weil (1989). A further issue of public debt
2

Sen (2002) points out that, in the Diamond model with monopolistic competition, debt finance can be Pareto

improving depending on production technology.
3

Blanchard assumes that people face a constant instantaneous probability of death throughout their lifetimes

and thus the average rest of lives at some date is identical for all generations. Buiter (1988) concludes that, inthe Blanchard model, the role of finite lifetimes is negligible and the appearance of new generations is rather
more important, like the Weil model. Weil assumes that infinitely-lived agents appear at a constant rate. One
justification of such a situation is primogeniture through inheritance of bequest. See Weil (1989) for otherjustifications.
4Saint-Paul (1992) also demonstrates that in an endogenous growth model with overlapping generations andproduction externalities, debt policy reduces the growth rate and welfare of some future generations.

2stimulates aggregate consumption demand and decreases unemployment, thereby supporting
the Keynesian scenario. However, future generations face both this beneficial effect of increased
wage incomes and the harmful effect of higher taxes. If the amount of outstanding bonds is
relatively small and there is high population growth, debt policy enhances the welfare of future
generations through creating more employment and is Pareto improving (even if the economy is
dynamically efficient). Conversely, debt policy diminishes the welfare of future generations bylevying heavy taxes if the outstanding issue is already large.
Section II presents the structure of the model. By using a simplified model without capital
accumulation and wage adjustment, Section III clarifies the essence of debt policy during recessions. Section IV considers two extensions. The first incorporates capital accumulation. Itshows that, as Modigliani (1961) is concerned, debt issue discourages investment and can leadto low levels of steady-state output if the prevailing interest rate lies at relatively low levels.
This possibility barely changes the prime welfare implication obtained in Section III, because
the economy is in dynamic-inefficiency at low levels of the interest rate. The second studies therole of wage adjustment. It shows that while a big fall in the nominal wage rate leads to full
employment, the welfare implication of debt policy is basically the same as that of Section III if
wage adjustment is sufficiently sluggish to preserve unemployment. Section V summarizes andconcludes.

II.

The model

II.A. Consumers
The basic structure of the model is the same as that of Weil (1987), i.e., a hybrid of amoney-in-utility model and an overlapping-generations model, except for nominal wage rigidities.
Economy starts at time 0. The size of initial population is normalized to unity. At each point
in time, new infinitely-lived consumers appear at the rate n(> 0). The total population at time3

t is hence obtained as N (t) = ent .
∞Given the time paths of the real interest rate {r(t)}∞t=s , the nominal interest rate {R(t)}t=s ,∞the real wage rate {w(t)}∞t=s , and the lump-sum tax {τ (s, t)}t=s , a consumer representative of

the cohort born at time s(≥ 0) chooses {c(s, t), k(s, t), b(s, t), m(s, t)}∞t=s to maximize the lifetime
utility,Z ∞(1)

U (s, s) ≡[(1 − α) ln c(s, t) + α ln m(s, t)]e−ρ(t−s) dt,

ssubject to(2)da(s, t)= r(t)a(s, t) + w(t)x(s, t) − c(s, t) − R(t)m(s, t) − τ (s, t),dt(3)a(s, t) ≡ k(s, t) + b(s, t) + m(s, t),(4)c(s, t) ≥ 0,m(s, t) ≥ 0,(5)a(0, 0) > 0,a(s, s) = 0 for ∀s > 0,(6)

lim a(s, t)e− s r(v)dv ≥ 0,

Rt

t→∞

where the constant parameter α satisfies α ∈ (0, 1), ρ(> 0) is the subjective discount rate,and c(s, t) is consumption at time t(≥ s) of a consumer born at time s.5 For simplicity, laborendowment of each consumer is normalized to unity. To consider unemployment, however, therealized labor supply is denoted by x(s, t). Equation (3) implies that total non-human wealtha(s, t) consists of capital k(s, t), public bonds b(s, t), and real money balances m(s, t). Equation(5) indicates that the initial generation is endowed with positive non-human wealth; whereas, forlack of bequest motives of pre-existing generations, none of new generations receives non-humanwealth at birth.

5

To characterize the instantaneous utility function by a more general CES form does not add further insights.

4

The first-order conditions aredc(s, t)
dt

(7)
(8)

= [r(t) − ρ]c(s, t),

R(t) =

α c(s, t),
1 − α m(s, t)

and the above constraints. Equations (2) and (6)–(8) yield the individual consumption function(9)c(s, t) = ρ(1 − α)[a(s, t) + h(s, t)],where h(s, t) is the net human wealth defined as
Z ∞
(10)

h(s, t) =Rz[w(z)x(s, z) − τ (s, z)]e− t r(v)dv dz.

tDefine the aggregate consumption per capita c(t) as c(t) =Rc(0,t)+ 0t c(s,t)dN (s). Same definitions
N (t)hold for other variables: a(t), k(t), b(t), m(t), x(t), τ (t), and h(t). By applying this aggregation
rule to (8) and (9), I have
α c(t),
1 − α m(t)(11)

R(t) =(12)c(t) = ρ(1 − α)[a(t) + h(t)].

II.B. Production
Using physical capital and labor, competitive firms produce a commodity used for consumption and investment. Firms have identical production technology. For analytical tractability,
the production function is specified by the Cobb-Douglas form: y(t) = θk(t)β l(t)1−β , which isexpressed in aggregate per capita. y(t) is the aggregate output per capita and l(t) the aggregate
labor demand per capita. θ(> 0) and β respectively measure productivity and capital share in£¢
output. In particular, this paper assumes β ∈ 0, 21 of which analytical meaning is presented in5Section IV.A.6 The optimization by firms provides·¸
k(t) β−1r(t) = βθ
,
l(t)
·
¸
k(t) βw(t) = (1 − β)θ
.
l(t)

(13)
(14)

II.C. Government
The government budget constraint isdb(t)= [r(t) − n]b(t) − µ(t)m(t) − τ (t),
dt

(15)

which is expressed in aggregate per capita. I omit the role of fiscal spending and distortional
taxes for purpose of comparison with the literature. µ(t) is the expansion rate of nominal money
supply. Letting π(t) be the inflation rate, this paper assumes that µ(t) = π(t) + n for ∀t, so thatm(t) remains at m̄(> 0). Unless µ(t) = π(t) + n holds for ∀t, either full employment or zero
employment is eventually attained through the Pigou effect.7
Following the existing studies, I characterize debt policy as follows. At time 0, government
increases b(0) through a helicopter drop; and after time 0, aims to keeping it constant at thatvalue, b̄(> 0).

III.

Debt policy during recessions

III.A. Simple setting
To elicit the essence of debt policy during recessions, I establish the following simple setting.
First, in this section, the nominal wage rate is permanently fixed at a given level, W̄ .8 Hence,
unemployment may arise.
6

This range regarding β includes its plausible values used in the standard RBC literature. For example, Kingand Rebelo (1999) set β equal to one-thirds.
7

See e.g., Ono (1994, 2001).

8

Rankin (1986) also follows this exogenously-fixed-prices approach à la Barro and Grossman (1971) and Ma-

linvaud (1977). Blanchard and Kiyotaki (1987) show nominal price/wage rigidity to arise as a consequence of6Second, throughout the paper, I assume the distribution rule that labor demand and lump¡¢sum taxes are identical across agents: x(s, t) = min 1, l(t) and τ (s, t) = τ (t).9 Then, h(s, t) is
an age-independent variable: h(s, t) = h(t).
Third, this section imposes β = 0, so that the role of capital accumulation is set aside andlabor is a unique production input. Given the nominal wage rigidity, the commodity market isin equilibrium if and only if PW̄(t) = θ, where P (t) denotes the price level. P (t) is instantaneously
adjusted to satisfy this value, which is henceforth labelled P̄ . As the inflation rate is always
zero, the nominal value of m(t) remains at M̄ (= m̄P̄ > 0) and that of b(t) at B̄(= b̄P̄ > 0).
The real interest rate equals the nominal interest rate: r(t) = R(t).

III.B. Steady state and welfare
Keeping the distribution rule in mind, (2), (5), (8), (10), and (12) together derive the dynamic
equation for c(t),
(16)

dc(t)
= [r(t) − ρ]c(t) − ρn(1 − α)a(t).
dtThis equation implies that the amount of assets affects aggregate consumption streams for lackof bequest motives.10 From (11), (16), and a(t) = B̄+ M̄, the aggregate dynamic system isP̄P̄characterized bydr(t)B̄ + M̄= r(t)2 − ρr(t) − ρnα.
dtM̄As seen in Figure 1, there is a unique steady state, r∗ , with r(t) > 0, which is determined by(17)r∗ 2 − ρr∗ − ρnαB̄ + M̄= 0.
M̄

monopolistic agents’ behavior in the presence of menu costs. See Mankiw and Romer (1991) for microfoundationsof price/wage rigidities.
9

I can employ the age-dependent income schedule used in Blanchard (1985), but it does not alter the main

result of this paper.
10

Refer to Blanchard (1985) and Weil (1989) for this point.

7

dr(t)
dt

1
2ρ

ρ
r∗

0

Θ

r(t)

M̄
−ρnα B̄+M̄

Figure 1: Aggregate Dynamics.

The steady state associated with r∗ is unstable. If r(t), or equivalently c(t), evolves along thedivergent paths, either the intertemporal budget constraint of the consumer or the non-negativity
constraint on consumption is violated. Therefore, r∗ must be chosen at the initial point in time.
α θ P̄In Figure 1, Θ ≡ 1−αrepresents the interest rate associated with full employment. To takeM̄

account of unemployment, I set Θ above r∗ .
As in Blanchard (1985) and Weil (1987, 1989), the non-negativity on individual consumption
requires r∗ ≤ ρ + n.11 Substituting B̄ = 0 into (17) gives the lower bound of r∗ , which is labelledr∗ and satisfies r∗ > ρ. As a result, r∗ must be in the following range:¯¯Condition 1. r∗ < r∗ ≤ ρ + n.
¯

11

From (5) and (9), the non-negativity of individual consumption is satisfied if and only if h(s, s) is non-negative

for all s > 0. Equation (12) implies that h(s, s) ≥ 0 is equivalent to c(t) ≥ ρ(1 − α)( B̄+ M̄), which, togetherP̄P̄
with (17), generates this condition.

8

From (15), the steady-state lump-sum tax τ ∗ is
Ã

B̄ M̄
τ =r
−n
+
P̄
P̄
P̄
∗

(18)

∗ B̄

!
.

Next, I derive welfare. In the absence of the transition process, substituting (8) into (1)yieldsZ ∞U (s, s) =ln c(s, t)e−ρ(t−s) dt −

s

α
α
α
ln R∗ + ln
,
ρ
ρ 1−α

where R∗ (= r∗ ) denotes the steady-state nominal interest rate. The first term in the right-handside of this equation represents the gross lifetime utility derived from both consumption andholding money. The second term implies the opportunity costs of holding money. By use of (7)and (9), welfare is reduced to12
(19)

U (s, s) =α11ln r∗ [a(s, s) + h(s, s)] − ln R∗ + ln αα (1 − α)1−α .
ρ
ρ
ρ
∗

∗

Letting c∗ be the steady-state consumption, h(s, s) is obtained as h(s, s) = c r−τfrom (10) and∗the equilibrium condition in the commodity market. Equation (19) is eventually rewritten as(20)

U (s, s) =1α1ln[r∗ a(s, s) + c∗ − τ ∗ ] − ln R∗ + ln αα (1 − α)1−α ,ρρρwhich says that the gross lifetime utility is increasing with respect to the net income at birth,that is, r∗ a(s, s) + c∗ − τ ∗ .

III.C. Comparative statics
This subsection examines the effects of an increase in B̄. First, I show its implication as theKeynesian demand-management policy, which is emphasized by Tobin (1980).
Lemma 1. Debt policy raises the interest rate.
∗

Proof. Differentiating (17) totally and using Condition 1, I get dr= (2r∗ρnα
> 0.
dB̄
−ρ)M̄
12

¥

See e.g., Futagami and Shibata (1999, 2003) and Mino and Shibata (2000) for details of the derivation. In

deriving this formula, I used the approximate expression of ln(1 + ²) = ².

9

Proposition 1. Debt policy stimulates aggregate consumption demand.
∗ρn(1−α)Proof. Use of (11) and Lemma 1 gives dc= (2r∗ −ρ)P̄ > 0.
dB̄¥Consideration of the intergenerational income redistribution effect of debt finance is sufficient
to explain this proposition. That is, unless wage income changes, debt policy increases total
wealth of existing generations and decreases that of future generations through higher taxes.
Therefore, existing generations consume more and future generations less. The total effect on
aggregate consumption is definitely positive because part of future generations that consume lessdoes not yet appear in the economy. Consequently, the larger amount of assets induces moreaggregate consumption demand for lack of a consumption-smoothing motive across generations.
Since the resultant rise in wage income does not disturb this explanation, increasing non-humanwealth through debt policy becomes an effective demand-management policy.
As implied by (20), the level of the net income at birth is crucial for welfare. I next examine
the effect on r∗ a(s, s) + c∗ − τ ∗ .

Lemma 2. Debt policy raises the net income of the initial generation at birth.
∗∗]
Proof. Taking notice of a(0, 0) = B̄+ M̄and using (18) and Lemma 1, I get d[r a(0,0)−τ
=
P̄
P̄
dB̄
∗
∗ −τ ∗ ]M̄ dr∗+ P̄n > 0. Together with Proposition 1, d[r a(0,0)+c> 0.
P̄ dB̄dB̄

¥The interpretation of Proposition 1 is also applied to this lemma. Because part of taxes
associated with debt issue is levied on future generations, which are not connected with existing
generations, public bonds become net wealth for existing generations. That is, debt policy raisesr∗ a(0, 0) − τ ∗ . Combined with the demand-stimulating effect (Proposition 1), the net incomesof existing generations undoubtedly rise.
Meanwhile, I have the following:
Lemma 3. Debt policy raises the net incomes of future generations at birth if r∗ < r∗ < 23 (ρ+n)
¯
¢
¡ ∗ 2
and reduces those if max r , 3 (ρ + n) < r∗ ≤ ρ + n.
¯
10Proof. Notice a(s, s) = 0. Differentiating (18) totally and combining it with Proposition 1 yield£¤−3r∗ r∗ − 23 (ρ+n)
d(c∗ −τ ∗ )=in which B̄ is eliminated by use of (17). Condition 1 ensures thedB̄
(2r∗ −ρ)P̄denominator to be positive. The numerator is positive if 0 < r∗ < 23 (ρ + n) and negative if
2∗3 (ρ + n) < r . Taking into account Condition 1, the proof is completed.
2 ∗

¥
∗

dr
The intuitive interpretation of this lemma is as follows. Since ddB̄c2 = −2ρn(1−α)< 0 is(2r∗ −ρ)2 P̄ dB̄

satisfied, the marginal effect on aggregate demand decreases. The source of this result is thatan expansion in consumption raises the interest rate simultaneously and consumers postpone
consumption demand through intertemporal choices. On the other hand, the relation betweenτ ∗ and B̄ satisfiesdτ ∗dB̄
d2 τ ∗
dB̄ 2

(21)
(22)

=
=

B̄ dr∗ r∗ − n+,P̄ dB̄
P̄
2[3r∗ (r∗ − ρ) + ρ(ρ + nα)] dr∗> 0.
(2r∗ − ρ)2 P̄
dB̄

The first term in the right-hand side of (21) represents increased tax by a rise in the interest
rate. The second term in the right-hand side of (21) is negative when r∗ < n but positive whenr∗ > n. As illustrated in Figure 2(a) and 2(b), there are two possible cases concerning theM̄schedule of τ ∗ . In the figures, B̄ P ≡ ρα[n − ρ(1 + α)] is the B̄ corresponding to r∗ = n, andM̄
B̄ U ≡ ρα[n+ρ(1−α)] represents the upper bound of B̄ that gives r∗ = ρ+n. Figure 2(a) depicts

the case with a Ponzi scheme (i.e., B̄ P > 0), in the words of Weil (1989). At B̄ ∈ (0, B̄ P ), r∗ < n
is satisfied and lump-sum transfers exceed seigniorages (τ ∗ < −n M̄
). At B̄ ∈ (B̄ P , B̄ U ], I haveP̄r∗ > n and τ ∗ > −n M̄, so that τ ∗ is increasing with respect to B̄. Figure 2(b) describes the caseP̄
without a Ponzi scheme (i.e., B̄ P ≤ 0). In this case, debt issue always involves higher taxes. As
shown in (22), the marginal effect on taxes increases because debt issue raises the interest rate
simultaneously.
For these reasons, the schedule of c∗ − τ ∗ falls into two patterns, Figure 3(a) and 3(b). In
M̄
(−2ρ2 + 4n2 + 2ρn − 9ρnα) is the B̄ that attains r∗ = 32 (ρ + n), andthe figures, B̄ M ≡ 9ρnαM22

2 M̄ (ρ +2n )satisfies that dB̄
> 0. If population growth is relatively low, I obtain Figure
dn = 9
ρn2 α

11

τ∗
(1−α) M̄(ρ + n)
α
P̄

B̄ P
B̄ U

0

−n M̄
P̄

Figure 2(a): B̄ P > 0.

τ∗(1−α) M̄(ρ + n)
α
P̄

B̄ P
0

B̄ U

−n M̄
P̄

Figure 2(b): B̄ P ≤ 0.

12

B̄

B̄

c∗ − τ ∗

B̄ U
B̄ M

0

B̄

Figure 3(a): B̄ M ≤ 0.

c∗ − τ ∗

B̄ U
0

B̄ M

Figure 3(b): B̄ M > 0.

13

B̄

3(a) with B̄ M ≤ 0, where debt policy always reduces the net incomes of future generations bylevying heavy taxes. On the contrary, if there is enough high population growth, I get Figure3(b) with B̄ M > 0. Because high population growth brings the large demand-stimulating effect,
debt policy raises the net incomes of future generations at B̄ ∈ (0, B̄ M ). However, the demandstimulating effect decreases with issuing of bonds. When B̄ reaches B̄ M , debt policy reducesc∗ − τ ∗ like Figure 3(a).
Remark 1. It is easy to prove that B̄ P > 0 is not necessary condition but sufficient condition
for B̄ M > 0. At B̄ ∈ (B̄ P , B̄ M ), debt policy raises the net incomes of all generations despite a
non-Ponzi scheme.
Lemma 2 and 3 directly derive the welfare implication:2−α(ρ + n), debt policy is Pareto improving.
Theorem 1. If r∗ < r∗ < 3−α¯³´
If max r∗ , 2−α(ρ+n)< r∗ ≤ ρ + n, debt policy improves welfare of the initial generation but¯ 3−αdeteriorates that of future generations.
Proof. See Appendix A.
2Remark 2. 2−α3−α (ρ + n) < 3 (ρ + n), which reflects the fact that an increase in B̄ brings thehigher interest rate and the higher cost of holding money.
Differently from models with full employment (e.g., Blanchard 1985 and Weil 1989), when there
is high population growth and the amount of outstanding bonds is relatively small, debt policy
during recessions is Pareto improving by creating more employment (even if the economy is
dynamically efficient). However, as public debt accumulates, the demand-stimulating effectdeclines whereas the cost through taxes swells. Consequently, when the outstanding issue islarge, future generations are made worse off by heavy taxes. The result is also different from
Rankin (1986) where debt policy during recessions is always beneficial to future generations.

14

IV.

Extensions

IV.A. Capital accumulation
Modigliani’s (1961) original concern is the crowding-out effect of public debt: capital accumulation is discouraged by raising the interest rate and welfare of future generations falls.
The contribution of Rankin (1986) challenges this negative scenario by showing that debt issue
increases the steady-state output and welfare. For purpose of comparison, this section pays
attention also to the effect on capital accumulation.
The optimal conditions (13) and (14) generate"W̄r(t) = βθ(1 − β)θP (t)

(23)

# β−1
β

.

Since the equilibrium condition of asset markets ensures a(t) = k(t) + PB̄(t) + PM̄(t) , the dynamic
equation for c(t) becomes
(24)

"
#
dc(t)
B̄
M̄
= [r(t) − ρ]c(t) − ρn(1 − α) k(t) ++.
dt
P (t) P (t)Keeping y(t) = r(t)k(t)in mind, the equilibrium condition of the commodity market is
β
(25)

dk(t)
r(t)k(t)=− c(t) − nk(t).
dtβThe Fisher equation and (11) yield
(26)

"
#
dP (t)α c(t)P (t)=− r(t) P (t).
dt
1 − α M̄

Equations (23)–(26) constitute the autonomous dynamic system.
Appendix B proves that the steady state is uniquely determined if it exists, and is dynamically stable. The analysis assumes the existence of the steady state. See Appendix B for themathematical proof of the following argument.
To explain the effect on the steady-state values (c∗ , k ∗ , P ∗ ), I propose three channels displayedin Figure 4. By the same logic as Proposition 1, Channel 1 signifies that taking (r∗ , k ∗ , P ∗ ) as
15

B̄ ↑

c∗ ↑ (Channel 1)
r∗ ↑
P∗ ↑

c∗ ↓
B̄
M̄
P∗ + P∗ ↓(Channel 2)

r∗ ↑

k∗ ↓
k∗ ↓

c∗ ↓ (Channel 3)k∗ ↑Figure 4: Channels.

given in (24), debt policy directly raises c∗ . Starting this point, there are two secondary effects
denoted by Channel 2 and 3. From (23) and (26), I find that an increase in c∗ raises P ∗ .13 In
turn, r∗ rises from (23) and PB̄∗ + PM̄∗ falls. Equation (24) implies that these secondary effects
put downward pressure on c∗ (Channel 2). Equation (25) shows that there are conflicting effects
on k ∗ : the demand-stimulating effect of Channel 1 directly raises k ∗ but indirectly reduces it
through raising r∗ . I can prove that k ∗ totally falls. Equation (24) implies that this secondary
effect puts downward pressure on c∗ (Channel 3). Dynamic stability (Det< 0) ensures that c∗
totally rises through Channel 1–3. I conclude that debt policy increases c∗ and P ∗ and decreasesk∗ .
Responding to this result, the total effect on the steady-state output, y ∗ = c∗ + nk ∗ , isambiguous. When the prevailing interest rate is relatively low (i.e., r∗ < nβ(3β−2)
2β−1 ), the marginal
13

This result depends on β ∈ 0, 12 . If β exceeds 12 , an increase in c∗ reduces P ∗ . Then, the total effects arenegative on c∗ , ambiguous on k∗ , and positive on P ∗ .

16crowding-out effect is large and debt policy is harmful to aggregate output. Modigliani’s (1961)
scenario holds true in this case: debt policy reduces future aggregate output through crowding
out investment. However, the total effect including a change in taxes is much more important.
Assuming that the economy is always in steady state, I obtain the welfare implication.14
Proposition 2. If r∗ < r∗ < 2−3β−α(1−β)
3−4β−α(1−β) (ρ + n), debt policy is beneficial to all generations.
¯³´
If max r∗ , 2−3β−α(1−β)(ρ+n)< r∗ ≤ ρ + n, debt policy improves welfare of the initial gener¯ 3−4β−α(1−β)
ation and deteriorates that of future generations.
Proof. See Appendix B.
2−3β−α(1−β)
Remark 3. 3−4β−α(1−β)(ρ + n) < 2−α3−α (ρ + n), which reflects the fact that an increase in B̄accompanies the negative effect through crowding out investment.
Basically, the implication of Theorem 1 is inherited even if capital accumulation is incorporated.
Although there is the possibility that aggregate output drops, the economy is in dynamicinefficiency at low levels of the interest rate. Therefore, if there is high population growth
and the amount of outstanding bonds is small, debt finance improves welfare of all generations
whether aggregate output drops or not.

IV.B. Wage adjustment
In the real world, deflation or a reduction in the nominal wage rate is often observed during
recessions. This subsection examines the effects of wage adjustment on debt policy. From thetheoretical viewpoint, they are divided into (i) the rate-of-change effect and (ii) the level effect.
The effect of a rate of change of prices is first investigated. Assume that the nominal wage˙W (t)rate is adjusted by the rate φ ( W(t) = φ). The price level is also adjusted by the rate φ

14

Under the same assumption, Futagami and Shibata (2003) examine the welfare effects of government budget

deficits in an endogenous growth model with overlapping generations.

17

˙

(t)
(t)
(π(t) = PP (t)= φ) to satisfy W
P (t) = θ. The aggregate dynamic system is then rewritten as

dr(t)b̄ + m̄= [r(t) − ρ][r(t) + φ] − ρnα.
dtm̄To obtain a unique unstable steady state with r(t) > 0, the analysis assumes nα + φ > 0. The
steady state, which must be chosen at the initial point in time, is characterized by (20),(r∗ − ρ)(r∗ + φ) − ρnαr ∗ = R∗ − φ =b̄ + m̄
= 0,
m̄

α c∗
− φ,
1 − α m̄τ ∗ = (r∗ − n)b̄ − (φ + n)m̄.
¡¢
Proposition 3. If r∗ < r∗ < r∗ M , debt policy is beneficial to all generations. If max r∗ , r∗ M <¯¯r∗ ≤ ρ + n, debt policy improves welfare of the initial generation and deteriorates that of future√[(2−α)(ρ+n)−2φ]+ [(2−α)(ρ+n)−2φ]2 +4φ(3−α)(ρ+n)
MM∗∗
generations. r
is given by r≡.
2(3−α)
Proof. See Appendix C.
Remark 4. r∗ M > 2−α3−α (ρ + n) for φ < 0.
Deflation raises r∗ for given R∗ and reduces seigniorages. Hence, aggregate consumption demand
∗

∗

−τ )
diminishes and taxes increase ( d(c dφ> 0). According to the mechanism explained in Section

III, this means that the marginal welfare effect of debt issue increases (Remark 4). Obviously,
this rate-of-change effect barely changes the prime implication of Theorem 1.
The effect of a level of prices is established in the following proposition.

Proposition 4. A fall in the nominal wage rate W̄ stimulates aggregate consumption demandthrough the Pigou effect, but it does not alter the welfare implication of Theorem 1.
∗

∗

∗

c
< 0.
Proof. From PW̄(t) = θ and (17), I obtain ddrW̄ = 0. Combined with (11), this gives ddcW̄ = − W̄∗2∗

U (s,s)τ
From (18), I have ddτW̄ = − W̄
. Keeping Theorem 1 in mind, these results bring ddB̄d= 0 forW̄∀s ≥ 0.

¥18

Remark 5. If full employment is achieved by a big fall in W̄ , the welfare implication alters
significantly, that is, the Blanchard-Weil result is applied.
This proposition indicates that the duration of recessions is crucial in assessing a burden on
future generations of public debt. However, it should be noted that if policy makers can pay backthe debt as soon as reaching full employment, Theorem 1 is directly applied during recessions
without modification.
The two propositions demonstrate that the welfare implication of Theorem 1 is basicallyinherited as long as wage adjustment is sufficiently sluggish to preserve unemployment.

V.

Conclusion

The present paper develops an overlapping-generations model with nominal wage rigidities
and examines the welfare effects of debt policy during recessions. Further issues of public debt
stimulate aggregate consumption demand and create employment. In so doing, future generations face both this beneficial employment effect and higher taxes. As public debt accumulates,
the beneficial effect gradually declines whereas the harmful effect steadily swells. If the outstanding issue is already large, heavy taxes lower the welfare of future generations. In contrast,
if the outstanding issue is relatively small and there is high population growth, the beneficial
effect dominates the harmful effect and debt policy is Pareto improving (even if the economy is
dynamically efficient). These welfare implications are different from the results obtained in boththe Blanchard-Weil model with full employment and the Diamond model with unemployment.
There are some directions for future research. For instance, the following is useful in evaluating a burden of future generations accurately: (i) to consider the transitional dynamics ofcapital accumulation; to employ more realistic (ii) distribution rules across agents; (iii) fiscal
and monetary policy; (iv) wage adjustment processes.

19Appendix A. Proof of Theorem 1
Differentiating (20) totally yields
dU (0, 0)dB̄
dU (s, s)
dB̄

=
=

1
d[r∗ a(0, 0) + c∗ − τ ∗ ]
α dr∗
−
,
ρ[r∗ a(0, 0) + c∗ − τ ∗ ]ρr∗ dB̄dB̄
1d(c∗ − τ ∗ )α dr∗−for ∀s > 0.
ρ(c∗ − τ ∗ )ρr∗ dB̄dB̄

Rewrite r∗ a(s, s) + c∗ − τ ∗ as a function of r∗ and B̄ by using (11) and (18). Eliminate B̄ fromthe result and Lemma 2 by use of (17). Utilizing these results, Lemma 1, and 3, the welfare
effect of a change in B̄ is calculated asdU (0, 0)dB̄

=dU (s, s)dB̄=n(2 − α)> 0,ρ[a(0, 0) + h(0, 0)](2r∗ − ρ)P̄£¤−(3 − α) r∗ − 2−α3−α (ρ + n)
for ∀s > 0.
ρh(s, s)(2r∗ − ρ)P̄

2−α
The former equation is positive. The latter equation is positive if 0 < r∗ < 3−α(ρ + n) and
2−αnegative if 3−α(ρ + n) < r∗ . Taking account of Condition 1 completes the proof.

¥

Appendix B.
Steady state and dynamic system
From (23)–(26), the steady state is, after tedious manipulations, characterized byΦ(r∗ ) ≡ r∗ 3 − (ρ + nβ)r∗ 2 − ρnαB̄ + M̄B̄ + (1 − β)M̄ ∗r + ρn2 αβ
= 0.
M̄
M̄

r∗ must satisfy the following conditions: (i) r∗ ≥ ρ, which is required for c∗ ≥ 0 from (24); (ii)r∗ ≥ nβ, which is required for k ∗ ≥ 0 from (25); (iii) r∗ ≤ ρ + n, which is the non-negativity
constraint of individual consumption; (iv) r∗ > r∗ , which defines the lower bound of r∗ . I¯
M̄M̄> 0, Φ(ρ) = −ρ2 n(1 − α) − ρnα B̄+(ρ − nβ) < 0 for ρ ≥ nβ, andhave Φ(0) = ρn2 αβ B̄+M̄M̄

Φ(nβ) = −ρn2 (1−α)β 2 < 0, so that r∗ is uniquely determined if it exists. Since r∗ > max(ρ, nβ),¯
the range of r∗ is summarized as
Condition 2. r∗ < r∗ ≤ ρ + n.
¯20The dynamic system linearized around the steady state is

 
h
dc(t)
 dt

 

 

 
 dk(t)  =  dt  

 

 
dP (t)dt

1−α M̄
α P ∗21−β ∗ 2M̄+ ρnα B̄+β r
M̄r∗ − ρ

−ρn(1 − α)−1r∗ −nββ1−α M̄ 1−β r∗2
α P ∗ 2 β r∗ −nβ

α P ∗21−α M̄

02β−1 ∗β ri∗  c(t) − c 




  k(t) − k ∗  .






P (t) − P ∗For this coefficient matrix,
Trace = 3r∗ − (ρ + n),
Det = −

1 − β r∗ψ(r∗ ) < 0
β 2 r∗ − nβin which B̄ is eliminated by use of the above steady-state condition. The function ψ(r∗ ) is givenbyψ(r∗ ) ≡ 2r∗ 3 − (ρ + 4nβ)r∗ 2 + 2nβ(ρ + nβ)r∗ − ρn2 αβ 2 .
∗)∗∗Det< 0 is proved as follows. The first derivative of ψ(r∗ ) is dψ(rdr∗ = 2(r − nβ)[3r − (ρ + nβ)].

First, consider the case where ρ+nβ
< nβ. Because I have ψ(nβ) = ρn2 (1 − α)β 2 > 0, ψ(r∗ ) > 03
is satisfied within Condition 2. Next, consider the case where ρ+nβ
≥ nβ. Since ρ+nβ
< ρ is3
3
satisfied, it is sufficient to show ψ(r∗ ) > 0 for r∗ ≥ ρ. I have ψ(ρ) = ρ(ρ−nβ)2 +ρn2 β 2 (1−α) > 0.
Resultantly, ψ(r∗ ) is always positive within Condition 2 and hence I get Det< 0.
As k(0) is predetermined and c(0) and P (0) are not, the steady state is dynamically stable:
the dynamic path is either saddle-point stable or indeterminacy. In the case of saddle-point
stability, divergent paths are inconsistent with the optimality of individual consumers.

Comparative statics
Differentiating the steady-state conditions totally, the steady-state effect of debt policy is
obtained as
dc∗
dB̄
dk ∗
dB̄
dP ∗
dB̄

ρn(1 − α)(2β − 1)r∗ (r∗ − nβ)> 0,
β 2 P ∗ Det
β 2 [r∗ − n(2β − 1)] dc∗=< 0,(2β − 1)(r∗ − nβ)2 dB̄βα P ∗ 2 dc∗= −> 0.
(2β − 1)r∗ 1 − α M̄ dB̄
=21

As explained in the text, Det< 0 and the restriction on β produce the above signs. From theseresults, I get the effect on the steady-state output, y ∗ = c∗ + nk ∗ :
"
#
dy ∗
r∗
nβ(3β − 2) dc∗nβ(3β − 2)
∗
= ∗
r −
≥ 0 if r∗ ≥.
(r − nβ)22β − 12β − 1dB̄
dB̄
By use of w∗ x∗ = (1 − β)y ∗ , the restriction on β, and (15), the effect on the steady-state net
wage income w∗ x∗ − τ ∗ is derived as
¤
£
(ρ
+
n)
(3 − 4β)r∗ r∗ − 2−3βdc∗d(w∗ x∗ − τ ∗ )2 − 3β3−4β=≥ 0 if r∗ ≤(ρ + n).
ρn(1 − α)(2β − 1)
3 − 4βdB̄
dB̄
Assume that the economy is always in steady state. Following the similar procedure to
Theorem 1, I obtain the welfare effect:dU (0, 0)dB̄

=dU (s, s)dB̄=1−[2 − 3β − α(1 − β)] dc∗> 0,ρ[a(0, 0) + h(0, 0)] ρ(1 − α)(2β − 1) dB̄
hi∗ − 2−3β−α(1−β) (ρ + n)
[3
−
4β
−
α(1
−
β)]
r
3−4β−α(1−β)
dc∗1for ∀s > 0.
ρh(s, s)ρn(1 − α)(2β − 1)dB̄Notice that 3 − 4β > 2 − 3β > α(1 − β).

dU (s,s)dB̄

2−3β−α(1−β)
is positive if r∗ < 3−4β−α(1−β)(ρ + n) andnegative otherwise. Taking into account Condition 2, the proof of Proposition 2 is completed.

Appendix C. Proof of Proposition 4
Following the same procedure to Theorem 1, I get the welfare effect:dU (0, 0)db̄
dU (s, s)db̄==n[(1 − α)r∗ + (r∗ + φ)]ρr∗ [a(0, 0) + h(0, 0)][2r∗ − (ρ − φ)]> 0,ζ(r∗ )for ∀s > 0,ρr∗ h(s, s)[2r∗ − (ρ − φ)]where the function ζ(r∗ ) is given byζ(r∗ ) ≡ −(3 − α)r∗ 2 + [(2 − α)(ρ + n) − 2φ]r∗ + φ(ρ + n).
r∗ must be in the following range:
Condition 3. r∗ < r∗ ≤ ρ + n.
¯22

where r∗ satisfies r∗ > max(ρ, −φ). This ensures dUd(0,0)
to be positive. Letting r∗ M be the largerb̄¯¯root of ζ(r∗ ) = 0 and r∗ M − be the smaller root, it is satisfied that r∗ M − < r∗ and r∗ M < ρ + n¯
if the steady state is unique (nα + φ > 0). Taking account of Condition 3, dUd(s,s)
is positive ifb̄¡¢r∗ < r∗ < r∗ M and negative if max r∗ , r∗ M < r∗ ≤ ρ + n.
¯¯¥References
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