Discussion Papers In Economics
And Business

Welfare Analysis of Debt Policy during Recessions

Takayuki Ogawa

Discussion Paper 04-14

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare Analysis of Debt Policy during Recessions

Takayuki Ogawa

Discussion Paper 04-14

July 2004
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare Analysis of Debt Policy during Recessions∗Takayuki Ogawa†
Osaka University, Japan

Abstract
This paper develops an overlapping-generations model with nominal wage rigidities and examines the welfare eﬀects of debt policy during recessions. Issues of public
debt stimulate aggregate consumption demand and create employment. Future generations then face both increased wage incomes and higher taxes. If the amountof outstanding bonds is already large, debt policy deteriorates the welfare of future
generations by levying heavy taxes. By contrast, if the outstanding bond issue isrelatively small, debt policy can be Pareto improving by creating more employment. Therefore, the welfare implications of debt policy during recessions can bediscriminated from those during booms.
Keywords: debt policy, overlapping generations, welfare eﬀects.
JEL Classification: E12, E24, H63.
∗

I thank Shinsuke Ikeda, Yoshiyasu Ono, and Tadashi Shigoka for their helpful comments. This re-search is financially supported by the Grant-in-Aid for JSPS Fellows, the Ministry of Education, Culture,
Sports, Science and Technology.
†

Mailing Address: Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,

Ibaraki, Osaka 567-0047, JAPAN. E-mail Address: tkogawa@iser.osaka-u.ac.jp

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 Keynesian debt 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, evenwith 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 à la 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, thedefinition of 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. 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.1
Another important point involved in analysing 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 the1

Refer to Iwamoto (2002).

1

current generation that enjoys benefit from debt issue never exists at far future dates.
Therefore, the focus of the analysis 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-state level of output and welfare (if the economy is
dynamically eﬃcient). By contrast, developing the Diamond model with fixed prices andwages, Rankin (1986) finds that debt issue raises the steady-state level of output andwelfare.2 Consequently, in the Diamond situation, debt policy is harmful to future generations during booms, whereas it is beneficial during recessions (as long as the economy
is dynamically eﬃcient).3
On the other hand, the Blanchard—Weil model examines the situation where the current generation exists even at far future dates.4 Then, the literature on the lines ofBlanchard and Weil focuses mainly on the income redistribution eﬀect between diﬀerent
generations of government policy. As shown in Blanchard (1985) and Weil (1989), debt
finance means an income transfer from future generations to existing generations. It is
2

Rankin (1986) additionally finds that debt issue reduces welfare despite raising output, if consumers

supply labour elastically and the economy is suﬃciently close to Walrasian equilibrium.
3

Ono (2001) stresses that debt policy during recessions never places a burden on all future generations

even if the transitional dynamics are considered in the Diamond situation. Sen (2002) points out that,in the Diamond model with monopolistic competition, debt finance can be Pareto improving depending
on production technology.
4

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, in the 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 ofbequest. See Weil (1989) for other justifications.

2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).5
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), Blanchard (1985), and Weil (1989).
A further issue of public debt 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. Conversely, debt policy diminishes the welfare of
future generations by levying 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, Section III clarifies the essence of debt policy during recessions.
Section IV considers capital accumulation. It shows that, as Modigliani (1961) is concerned, debt issue discourages investment and can lead to 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 dynamic5

Saint-Paul (1992) also demonstrates that in an endogenous growth model with overlapping gen-erations and production externalities, debt policy reduces the growth rate and welfare of some future
generations.

3

ineﬃciency at low levels of the interest rate. Section V summarizes and concludes.

II.

The Model

II.A. Consumers
The structure of the model is the same as that of Weil (1987), i.e., a hybrid of a moneyin-utility model and an overlapping-generations model. 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 time t is hence obtainedas 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,(1)

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

subject 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,

t→∞

Rt

where ρ(> 0) is the subjective discount rate and c(s, t) is consumption at time t(≥ s) of4

a consumer born at time s.6 Total non-human wealth a(s, t) consists of capital k(s, t),public bonds b(s, t), and real money balances m(s, t). The constant parameter α satisfiesα ∈ (0, 1). For simplicity, labour endowment of each consumer is normalized to unity.
To consider unemployment, however, the realized labour supply is denoted by x(s, t).
Equation (5) indicates that the initial generation is endowed with positive non-humanwealth; whereas, for lack of bequest motives of pre-existing generations, none of new
generations receives non-human wealth at birth.
The first-order conditions aredc(s, t)= [r(t) − ρ]c(s, t),dt
α c(s, t),
R(t) =
1 − α m(s, t)

(7)
(8)

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

Z ∞
t

Rz

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

Define the aggregate consumption per capita c(t) as c(t) =Rtc(0,t)+ 0 c(s,t)nens ds
.
N (t)

Same

definitions 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)].

6

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

insights.

5

II.B. Production
Using physical capital and labour, competitive firms produce a commodity used for
consumption and investment.

Firms have identical production technology.

For an-alytical tractability, the production function is specified by the Cobb-Douglas form:y(t) = θk(t)β l(t)1−β , which is expressed in aggregate per capita. y(t) is the aggregate
output per capita and l(t) the aggregate labour demand per capita. θ(> 0) and β respectively measure productivity and capital share in output. In particular, this paper assumes£ ¢β ∈ 0, 12 of which analytical meaning is presented in Section IV.7 The optimization byfirms provides
(13)
(14)

∙

k(t)
r(t) = βθ
l(t)

¸β−1
∙

,

k(t)
w(t) = (1 − β)θ
l(t)

¸β

.

II.C. Government
The government budget constraint is
(15)

db(t)
= [r(t) − n]b(t) − µm(t) − τ (t),dtwhich is expressed in aggregate per capita. µ is the expansion rate of nominal money
supply. This paper assumes µ = n to obtain the steady state of m(t), so that the nominal
value of m(t) remains at M̄ (> 0). I omit the role of fiscal spending and distortional taxes
for purpose of comparison with the literature.
Following the existing studies, I characterize debt policy as follows. At time 0, government increases the nominal value of b(t) through a helicopter drop; and after time 0, aims
7This range regarding β includes its plausible values used in the standard RBC literature. For example,

King and Rebelo (1999) set β equal to one-thirds.

6

to keeping it constant at that value, 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, throughout this paper, the nominal wage rate is permanently fixed at agiven level, W̄ .8 Hence, unemployment may arise.
Second, throughout the paper, I assume the distribution rule that labour demand and¡¢lump-sum taxes are identical across agents: x(s, t) = min 1, l(t) and τ (s, t) = τ (t).9Then, 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
and labour is a unique production input. Given the nominal wage rigidity, the commodity
market is in 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, m(t) remains at M̄
(> 0) and b(t) at B̄(> 0). The real interestP̄P̄
rate equals the nominal interest rate: r(t) = R(t).
8

For example, Blanchard and Kiyotaki (1987) show nominal price/wage rigidity to arise as a conse-

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

I can employ the age-dependent income schedule used in Blanchard (1985), but it does not alter themain result of this paper.

7III.B. Steady State and Welfare
Keeping the distribution rule in mind, (2), (5), (8), (10), and (12) together derive thedynamic 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 lack of bequest motives.10 From (11), (16), and a(t) = B̄+ M̄, the aggregate dynamicP̄P̄
system is characterized 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 determinedby(17)r∗ 2 − ρr∗ − ρnαB̄ + M̄= 0.
M̄

The steady state associated with r∗ is unstable. If r(t), or equivalently c(t), evolves along
the divergent paths, either the intertemporal budget constraint of the consumer or the
non-negativity constraint on consumption is violated. Therefore, r∗ must be chosen atα θ P̄
the initial point in time. In Figure 1, Θ ≡ 1−αrepresents the interest rate associatedM̄with full employment. To take 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 labelled r∗ and satisfies r∗ > ρ. As a result, r∗ must be in the following range:¯¯
10Refer to Blanchard (1985) and Weil (1989) for this point.

11

From (5) and (9), the non-negativity of individual consumption is satisfied if and only if h(s, s) is
´
³
M̄
,
+
non-negative for all s > 0. Equation (12) implies that h(s, s) ≥ 0 is equivalent to c(t) ≥ ρ(1−α) B̄P̄P̄which, together with (17), generates this condition.

8

Condition 1. r∗ < r∗ ≤ ρ + n.
¯
From (15), the steady-state lump-sum tax τ ∗ is
Ã
!
B̄
B̄
M̄
(18)
τ ∗ = r∗ − n
+
.
P̄
P̄
P̄
Next, I derive welfare. In the absence of the transition process, substituting (8) into
(1) yields
U (s, s) =Z ∞sln c(s, t)e−ρ(t−s) dt −αααln r∗ + ln
.
ρ
ρ 1−α

The first term in the right-hand side of this equation represents the gross lifetime utility derived from both consumption and holding money. The second term implies theopportunity costs of holding money. By use of (7) and (9), welfare is reduced to12
(19)

U (s, s) =1α1
ln 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 atbirth, 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 the Keynesian demand-management policy, which is emphasized by Tobin (1980).
12

See e.g., Mino and Shibata (2000) and Saint-Paul (1992) for details of the derivation. In deriving

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

9

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
Proposition 1. Debt policy stimulates aggregate consumption demand.

¥∗ρn(1−α)Proof. Use of (11) and Lemma 1 gives dc= (2r∗ −ρ)P̄ > 0.
dB̄Consideration 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 less does not yet appear in the economy. Consequently,
the larger amount of assets induces more aggregate consumption demand for lack of aconsumption-smoothing motive across generations. Since the resultant rise in wage income does not disturb this explanation, increasing non-human wealth 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
=
dB̄
P̄
P̄
d[r ∗ a(0,0)+c∗ −τ ∗ ]M̄ dr ∗
n+>0.
Togetherwith
Proposition
1,
> 0.
dB̄
P̄ dB̄
P̄

¥

The interpretation of Proposition 1 is also applied to this lemma. Because part oftaxes associated with debt issue is levied on future generations, which are not connectedwith existing generations, public bonds become net wealth for existing generations. That10

is, debt policy raises r∗ a(0, 0)−τ ∗ . Combined with the demand-stimulating eﬀect (Proposition 1), the net incomes of 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∗ <¯¡¢2
(ρ + n) and reduces those if max r∗ , 23 (ρ + n) < r∗ ≤ ρ + n.
3¯
Proof. Notice a(s, s) = 0. Diﬀerentiating (18) totally and combining it with Proposition£¤−3r ∗ r ∗ − 23 (ρ+n)
d(c∗ −τ ∗ )1 yield dB̄
=in which B̄ is eliminated by use of (17). Condition 1(2r ∗ −ρ)P̄ensures the denominator to be positive. The numerator is positive if 0 < r∗ < 23 (ρ + n)
and negative if 23 (ρ + n) < r∗ . Taking into account Condition 1, the proof is completed.¥
2 ∗

∗

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

is satisfied, I find that the marginal eﬀect on aggregate demand decreases.13 On the other
hand, the relation between τ ∗ and B̄ satisfies
(21)
(22)

dτ ∗
B̄ dr∗ r∗ − n
=
+
,
dB̄
P̄ dB̄
P̄
2[3r∗ (r∗ − ρ) + ρ(ρ + nα)] dr∗d2 τ ∗=> 0.
dB̄ 2(2r∗ − ρ)2 P̄
dB̄

The first term in the right-hand side of (21) represents increased tax by a rise in theinterest rate. The second term in the right-hand side of (21) is negative when r∗ < nbut positive when r∗ > n. As illustrated in Figure 2(a) and 2(b), there are two possibleM̄
cases concerning the schedule of τ ∗ . In the figures, B̄ P ≡ ρα[n − ρ(1 + α)] is the B̄
M̄
corresponding to r∗ = n, and 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),13

The source of this result is that an expansion in consumption raises the interest rate simultaneously.

As c∗ (or r ∗ ) is determined so as to clear (17), a small rise in c∗ (or r∗ ) is enough to oﬀset an increase inB̄ when c∗ (or r ∗ ) has a large value.

11

in the words of Weil (1989). At B̄ ∈ (0, B̄ P ), r∗ < n is satisfied and lump-sum transfers). At B̄ ∈ (B̄ P , B̄ U ], I have r∗ > n and τ ∗ > −n M̄, soexceed seigniorages (τ ∗ < −n M̄
P̄P̄that τ ∗ is increasing with respect to B̄. Figure 2(b) describes the case without a Ponzi
scheme (i.e., B̄ P ≤ 0). In this case, debt issue always involves higher taxes.14
For these reasons, the schedule of c∗ − τ ∗ falls into two patterns, Figure 3(a) and 3(b).
M̄In the figures, B̄ M ≡ 9ρnα(−2ρ2 + 4n2 + 2ρn − 9ρnα) is the B̄ that attains r∗ = 23 (ρ + n),
M

2

2

)
and satisfies that dB̄dn = 29 M̄ (ρρn+2n> 0. If population growth is relatively low, I obtain
2αFigure 3(a) with B̄ M ≤ 0, where debt policy always reduces the net incomes of future
generations by levying heavy taxes. On the contrary, if there is enough high population
growth, I get Figure 3(b) with B̄ M > 0. Because high population growth brings thelarge demand-stimulating eﬀect, debt policy raises the net incomes of future generationsat B̄ ∈ (0, B̄ M ). However, the demand-stimulating eﬀect decreases with issuing of bonds.
When B̄ reaches B̄ M , debt policy reduces c∗ − τ ∗ like Figure 3(a).
Remark 1. It is easy to prove that B̄ P > 0 is not necessary condition but suﬃcientcondition 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:
Theorem 1. If r∗ < r∗ < 2−α(ρ + n), debt policy is Pareto improving.
3−α
¯
¡
¢
If max r∗ , 2−α(ρ+n)< r∗ ≤ ρ + n, debt policy improves welfare of the initial generation¯ 3−αbut deteriorates that of future generations.
Proof. See Appendix I.
14

2 ∗An increase in B̄ raises the interest rate at the same time. This is the reason of ddB̄τ2 > 0.

12

Remark 2. 2−α(ρ + n) < 23 (ρ + n), which reflects the fact that an increase in B̄ brings
3−αthe higher interest rate and the higher cost of holding money.
Diﬀerently from models with full employment (e.g., Blanchard 1985 and Weil 1989), debt
policy during recessions is Pareto improving by creating more employment if there is highpopulation growth and the amount of outstanding bonds is relatively small. However, as
public debt accumulates, the demand-stimulating eﬀect declines whereas the cost throughtaxes swells. Consequently, when the outstanding issue is large, future generations are
made worse oﬀ by heavy taxes. The result is also diﬀerent from Rankin (1986) wheredebt policy during recessions is always beneficial to future generations.

IV.

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 showingthat debt issue interferes with investment but totally 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(23)r(t) = βθ"W̄(1 − β)θP (t)

# β−1
β

.

Since the equilibrium condition of asset markets ensures a(t) = k(t) + PB̄(t) + PM̄(t) , thedynamic 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)13Keeping y(t) = r(t)k(t)in mind, the equilibrium condition of the commodity market is
β(25)r(t)k(t)dk(t)=− c(t) − nk(t).
dtβThe Fisher equation and (11) yield"#dP (t)α c(t)P (t)=(26)− r(t) P (t).
dt
1 − α M̄
Equations (23)-(26) constitute the autonomous dynamic system.
Appendix II-A 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
II-B for the mathematical proof of the following argument.
To explain the eﬀect on the steady-state values (c∗ , k ∗ , P ∗ ), I propose three channels
displayed in Figure 4. By the same logic as Proposition 1, Channel 1 signifies that taking(r∗ , k ∗ , P ∗ ) as given in (24), debt policy directly raises c∗ . Starting this point, there are twosecondary eﬀects denoted by Channel 2 and 3. From (23) and (26), I find that an increasein c∗ raises P ∗ .15 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 decreases k ∗ .
Responding to this result, the total eﬀect on the steady-state output, y ∗ = c∗ + nk ∗ ,is ambiguous. When the prevailing interest rate is relatively low (i.e., r∗ < nβ(3β−2)
),
2β−1
15

¢
¡
This result depends on β ∈ 0, 12 . If β exceeds 12 , an increase in c∗ reduces P ∗ . Then, the total

eﬀects are non-positive on c∗ , ambiguous on k ∗ , and non-negative on P ∗ .

14the marginal 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 intaxes is much more important. Assuming that the economy is always in steady state, I
obtain the welfare implication.
Proposition 2. If max(ρ, nβ) ≤ r∗ < 2−3β−α(1−β)(ρ + n), debt policy is beneficial to all
3−4β−α(1−β)´³
generations. If max ρ, nβ, 2−3β−α(1−β)(ρ+n)≤ r∗ ≤ ρ+n, debt policy improves welfare
3−4β−α(1−β)
of the initial generation and deteriorates that of future generations.
Proof. See Appendix II-B.
Remark 3. 2−3β−α(1−β)(ρ + n) < 2−α(ρ + n), which reflects the fact that an increase in
3−4β−α(1−β)
3−αB̄ accompanies 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 dynamic-ineﬃciency at low levels of the interest rate. Therefore, if there is highpopulation growth and the amount of outstanding bonds is small, debt finance improves
welfare of all generations whether aggregate output drops or not.

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 issuesof public debt stimulate aggregate consumption demand and create employment. In sodoing, future generations face both this beneficial employment eﬀect and higher taxes. As

15

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 welfareof future generations. In contrast, if the outstanding issue is relatively small and there ishigh population growth, the beneficial eﬀect dominates the harmful eﬀect and debt policy
is Pareto improving. These welfare implications are diﬀerent from the results obtainedin both the Blanchard-Weil model with full employment and the Diamond model with
unemployment.

Appendix I: Proof of Theorem 1
Diﬀerentiating (20) totally yields
dU(0, 0)1
α dr∗d[r∗ a(0, 0) + c∗ − τ ∗ ]=−,ρ[r∗ a(0, 0) + c∗ − τ ∗ ]ρr∗ dB̄dB̄
dB̄
dU (s, s)1
α dr∗d(c∗ − τ ∗ )=−for ∀s > 0.
ρ(c∗ − τ ∗ )ρr∗ dB̄dB̄
dB̄
Rewrite r∗ a(s, s) + c∗ − τ ∗ as a function of r∗ and B̄ by using (11) and (18). EliminateB̄ from the 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)n(2 − α)=> 0,dB̄ρ[a(0, 0) + h(0, 0)](2r∗ − ρ)P̄£¤−(3 − α) r∗ − 2−α(ρ + n)dU (s, s)3−α=for ∀s > 0.
dB̄ρh(s, s)(2r∗ − ρ)P̄
The former equation is strictly positive. The latter equation is positive if 0 < r∗ <2−α(ρ + n) and negative if 2−α(ρ + n) < r∗ . Taking account of Condition 1 completes the
3−α
3−α

¥

proof.

16

Appendix II
A. 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 + (1 − β)M̄ ∗B̄ + M̄r + 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. These conditions are summarizedasCondition 2. max(ρ, nβ) ≤ r∗ ≤ ρ + n.
M̄
> 0 and Φ(nβ) = −ρn2 (1 − α)β 2 < 0, r∗ is uniquely determined
Since Φ(0) = ρn2 αβ B̄+M̄

if it exists.
The dynamic system linearized around the steady state is
 


h
i
dc(t)
1−β
2
1−α M̄
B̄+M̄
 dt   r∗ − ρ −ρn(1 − α) α P ∗ 2 β r∗ + ρnα M̄   c(t) − c∗ 

 



 



∗
 dk(t)  =  −1  k(t) − k ∗  .
r −nβ1−α M̄ 1−β r∗2 dt  βα P ∗ 2 β r ∗ −nβ

 



 



dP (t)
2β−1 ∗
α P ∗2∗0r
P (t) − Pdt
1−α M̄β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 given byψ(r∗ ) ≡ 2r∗ 3 − (ρ + 4nβ)r∗ 2 + 2nβ(ρ + nβ)r∗ − ρn2 αβ 2 .
17

∗

)
Det< 0 is proved as follows. The first derivative of ψ(r∗ ) is dψ(r= 2(r∗ − nβ)[3r∗ − (ρ +dr ∗< nβ. Because I have ψ(nβ) = ρn2 (1−α)β 2 > 0,nβ)]. First, consider the case where ρ+nβ3
ψ(r∗ ) > 0 is satisfied within Condition 2. Next, consider the case where ρ+nβ
≥ nβ. Since
3
ρ+nβ
3

< ρ is 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 dynamicallystable: 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.

B. Comparative Statics
Diﬀerentiating the steady-state conditions totally, the steady-state eﬀect of debt policy
is obtained asρn(1 − α)(2β − 1)r∗ (r∗ − nβ)
dc∗
=
≥ 0,
β 2 P ∗ DetdB̄dk ∗β 2 [r∗ − n(2β − 1)] dc∗=≤ 0,(2β − 1)(r∗ − nβ)2 dB̄dB̄βα P ∗ 2 dc∗
dP ∗
= −
≥ 0.
(2β − 1)r∗ 1 − α M̄ dB̄
dB̄
As explained in the text, Det< 0 and the restriction on β produce the above signs. Fromthese results, 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∗ ≥.
2(r − nβ)2β − 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£¤(3 − 4β)r∗ r∗ − 2−3β(ρ+n)
d(w∗ x∗ − τ ∗ )dc∗
2 − 3β3−4β=≥ 0 if r∗ ≤(ρ + n).
ρn(1 − α)(2β − 1)
3 − 4βdB̄
dB̄
18Assume that the economy is always in steady state. Following the similar procedure
to Theorem 1, I obtain the welfare eﬀect:dU(0, 0)−[2 − 3β − α(1 − β)] dc∗
1=> 0,ρ[a(0, 0) + h(0, 0)] ρ(1 − α)(2β − 1) dB̄dB̄hi
2−3β−α(1−β)
∗
[3
−
4β
−
α(1
−
β)]
r
−
(ρ
+
n)
3−4β−α(1−β)
dc∗
1dU (s, s)=for ∀s > 0.
ρh(s, s)ρn(1 − α)(2β − 1)dB̄dB̄Notice that 3 − 4β > 2 − 3β > α(1 − β).

dU (s,s)dB̄

is positive if r∗ ≤ 2−3β−α(1−β)(ρ + n)3−4β−α(1−β)

and negative otherwise. Taking into account Condition 2, the proof of Proposition 2 iscompleted.

19

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Mino, Kazuo and Akihisa Shibata, “Growth and Welfare Eﬀects of Monetary Expansion
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21

dr(t)
dt

1
ρ
2

ρ
r∗

0

r(t)
Θ

M̄
−ρnα B̄+M̄

Figure 1: Aggregate Dynamics

τ∗(1−α) M̄(ρ + n)
α
P̄

B̄ P
B̄ U

0

−n M̄
P̄

Figure 2(a): B̄ P > 0

22

B̄

τ∗
(1−α) M̄(ρ + n)
α
P̄

B̄ P
0

B̄ U

B̄

−n PM̄¯∗

Figure 2(b): B̄ P ≤ 0c∗ − τ ∗

B̄ U
B̄ M

0

Figure 3(a): B̄ M ≤ 0

23B̄c∗ − τ ∗

B̄ U
0

B̄

B̄ M

Figure 3(b): B̄ M > 0B̄ ↑c∗ ↑ (Channel 1)

P∗ ↑

r∗ ↑
B̄
+ PM̄∗ ↓P∗c∗ ↓(Channel 2)

r∗ ↑

k∗ ↓
k∗ ↑

Figure 4: Channels

24

k∗ ↓

c∗ ↓ (Channel 3)