Bubbles and unemployment in an endogenous
growth model
Ken-ichi Hashimoto
Ryonfun Im
December 2014
Discussion Paper No.1431

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPAN

10 December 2014

Bubbles and unemployment in an endogenous growth model *†‡Ken-ichi Hashimoto and Ryonfun Im
Abstract

We construct a continuous-time overlapping-generations model with labor market friction inorder to examine the relationship between bubbles, economic growth, and unemployment.
We show that the existence of bubbles is contingent upon the equilibrium unemployment rate.
Asset bubbles can (not) exist when unemployment is low (high), which leads to higher(lower) interest rates and economic growth through labor market efficiency. Hence, economic
growth under the bubble regime where bubbles can exist is higher than that under the
non-bubble regime where bubbles cannot exist. Furthermore, policy or parameter changes
that have a positive effect on the labor market shift the economy from a non-bubble regime toa bubble regime.
Keywords: overlapping

generations,endogenous

growth,labormarketfriction,unemployment
JEL Classification Number: J64, O41, O42* We are grateful for helpful comments from Takashi Kamihigashi, Tadashi Morita and Yoshiyasu Ono. We
acknowledge financial support from the Japan Society for Promotion of Science through a Grant-in-Aid forYoung Scientists (B) No. 24730209. All remaining errors are, of course, our own.
†

Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe 657-8501, JAPAN; E-mail:hashimoto@econ.kobe-u.ac.jp.
‡

Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe 657-8501, JAPAN; E-mail:tdrgsk@gmail.com.

1. Introduction
Bubbles have occurred many times in the past and have had major impacts on the macro
economy. In particular, bubbles have been frequently observed when economic activity isbooming and the growth rate of GDP is high (Martin and Ventura, 2012; Farmer andSchelnast, 2013, ch. 6). Also, empirical studies show that asset bubbles are accompanied by areduction in the unemployment rate (Phelps, 1999; Fitoussi et al., 2000). Since there are no
studies that address the theoretical interaction between bubbles, growth and unemployment,
the purpose of this paper is to examine the relationship analytically.
In his seminal study, Tirole (1985) examines the condition of the existence of bubbles on
intrinsically useless assets in an overlapping generations model. Grossman and Yanagawa(1993), King and Ferguson (1993), and Futagami and Shibata (2000) extend Tirole’s modelto an endogenous growth framework. They re-examine the conditions that are necessary forbubbles to exist and show the relationships between bubbles and economic growth. In theirstudies, when a bubble arises in the economy, it diverts savings from capital accumulation
and retards economic growth. For an alternative perspective on bubbles, Olivier (2000)
considers bubbles not as useless assets, but as assets tied to capital goods. As such, he is able toshow that bubbles can increase economic growth. However, these do not consider the
possibility of unemployment.
There are theories that state that equilibrium unemployment occurs as a result of frictionin the labor market. Diamond (1982), Mortensen (1982), and Pissarides (1985) develop
search and matching models of unemployment (which scholars have since applied to a wide
variety of fields 1 ). Eriksson (1997) introduces labor market frictions into the standard
dynamic optimizing (Ramsey) model with capital stock externalities through learning by
doing in order to ensure long-run economic growth. He then examines the effects of various
policies (capital taxes and unemployment benefits) on both economic growth andunemployment. 2 However, unlike in an overlapping-generations model, rational bubbles

1

See Pissarides (2000) for an introduction to search friction models.

2

See Bean and Pissarides (1993), Aghion and Howitt (1994), Caballero and Hammour (1994), andHaruyama and Leith (2010) for other models of the relationship between growth and unemployment that address
labor market frictions.

1cannot be generated in an infinitely lived, representative-agent model (Ramsey-type model)(Tirole, 1985; Santos and Woodford, 1997).3
To the best of our knowledge, no studies have used an endogenous growth model toanalyze the connection between bubbles and unemployment. In order to fill this void, wedevelop a theoretical framework with which to examine the conditions necessary for bubbles
to exist in an economy with endogenous unemployment, and we explore the relationships
between bubbles, economic growth, and unemployment. 4 In order to do so, we adopt
Eriksson’s (1997) endogenous growth and labor market friction approach, and merge theRamsey framework with Weil’s (1989) continuous-time overlapping-generations model.
In our model, where unemployment stems from labor market friction, labor market
efficiency is reflected in the interest rate because the marginal productivity of capital isinfluenced by it. Then, because asset returns are related to the interest rate, the existence of
bubbles depends on conditions in the labor market. As such, we find that the equilibrium
unemployment rate is a key factor in the existence of bubbles; when it is below a certain level
and interest rate is high, bubbles asset can exist. When the conditions are satisfied for bubbles
to exist, we say that the economy is in a “bubble regime”; conversely, when it is not possible
for them to exist, we say that the economy in a “non-bubble regime.” In a bubble regime,
there are multiple equilibria, such that a steady state can exist either with bubbles or without.
We show that bubbles divert savings away from physical capital and lower the output growthrate, which is a common finding in the literature. On the other hand, when we compare
bubble regimes to non-bubble regimes, we find that the output growth rate is always higher
under the former than the latter.
Additionally, with our model we can examine the effects of labor market policy orparameter changes on bubbles, economic growth, and unemployment. For example, we findthat, because unemployment benefits raise the value of unemployment, they have a negative
impact on employment (which is a standard conclusion among models with search friction).

3

As Santos and Woodford (1997) point out, it is hard to generate rational bubbles in an infinitely lived agent
model without market frictions. For an alternative approach that focuses on the friction in the financial market see
Hirano and Yanagawa (2010) and Martin and Ventura (2012); these authors examine the existence of bubbles
and show that they can be growth enhancing or growth impairing depending on the restrictiveness of thecollateral constraint.
4

Miao et al. (2012) and Kocherlakota (2011) present studies that are similar to our own. Miao et al. (2012)
investigate the relationship between unemployment and stock market bubbles in an economy with labor market
friction and financial market friction. Kocherlakota (2011) assumes that output is determined by household
demand, and as such, he does not consider the firm’s behavior in an economy with matching frictions andbubbles. However, these studies do not consider economic growth endogenously.

2Thus, if unemployment benefits are reduced, then the employment rate should increase andthe labor market should become more efficient, which would raise the interest rate andconsequently shift the economy from a non-bubble regime to a bubble regime. In this case,there would be a negative relationship between unemployment and economic growth. 5
The remainder of this paper is organized as follows: Section 2 outlines the features of themodel. Section 3 describes the steady-state equilibrium with and without bubbles, andcompares the effects of policy and parameter changes under the two regimes on bubbles,
economic growth, and unemployment. The final section summarizes our findings andconcludes the paper.

2. The Model
There are many infinitely lived dynasties in the economy. At each point in time, a new andidentical infinitely lived dynasty appears at rate n. The total population at time 0 is N0; thus,
the total population at time t is Nt = N0ent. Unless it is necessary, time notation is suppressed
for the remainder of the paper.
2.1. Matching
In the labor market, unemployed workers and firms with vacant positions strive to find eachother. We include mating frictions in the labor market in order to generate unemployment. By
denoting the number of successful of matches as f, this process can be described by thematching functionf(uN, uN),where u and u represent the unemployment rate and the vacancy rate, respectively. As such,uN is the number of unemployed workers and uN is the number of vacant jobs in the economy.
Following the standard assumptions, 6 the matching function is satisfied to be concave,homogeneous of degree one, and increasing in both of its arguments. If the tightness of thelabor market is defined as

5

In fact, many empirical studies show a negative relationship between unemployment and economic growth(Ball and Moffitt, 2001; Muscatelli and Tirelli, 2001; Staiger et al., 2001; Tripier, 2006; Pissarides and Vallanti,2007).
6

See Petrongolo and Pissarides (2001) for a discussion on matching functions.

3

θ≡

uN u
= ,
uN u

then the probability that a firm with vacancies will be matched to an unemployed worker canbe given by the following property:f (υN , υN )∂q (θ)
1 = f  ,1 ≡ q (θ) , where
<0.
∂θ
υN
θ 

2.2. Firms
There are N identical firms in the economy. The number of firms is assumed to equal thenumber of consumers. The production function of firm j is described by

y j = Ak αj (k l j )1−α , 0 < α < 1,
where A, kj, and lj represent productivity, capital stock, and the number of workers employedby firm j, respectively. k represents labor productivity, which is assumed to be driven by
spillovers (similar to those proposed by Romer (1986) that emanate from a firm’s
accumulated investments per worker). In order to ensure the existence of a long-run growthNpath, it is assumed that k takes the form k =  ∫ k j dj  N , which represents the average 0

capital stock.
In order to create matches, firm j must put out job vacancies. If we let νj be the numberof firm j’s vacancies, then q(θ)νj workers are hired by firm j in every moment.7 Furthermore,
firm j fires or loses workers at a rate of δlj, where δ represents the exogenous separation rate.
By summing these two flows, we find that the size of the labor force changes according to theequationlj = q (q)v j − δl j .

(1)

In accordance with Eriksson (1997), Mortensen and Pissarides (1999), and Pissarides andVallanti (2007), we assume that the cost to recruit workers for a vacancy is proportional to
the wage rate, γw (where w is the wage rate and γ is a cost parameter).8 In order to simplify
the model, we do not consider the adjustment cost of investment. Consequently, firm j’s
profit maximization problem can be written as
7

As θ is given for all firms, the probability q(θ) is the same for all firms.

8This assumption is required in order to ensure that there is a balanced growth path.

4

max ∫ ( y j − rk j − wl j − γwv j )e ∫t di , subject to (1).
i∞− rι dιt(2)This inter-temporal maximization problem can be solved using the current-value Hamiltonian
function H = (yj − rkj − wlj − γwvj) + χ(q(θ)vj − δlj), where χ is the current shadow value oflabor. Assuming that the market shares are small enough that each firm takes average capital( k ) and market tightness (θ) as constant, the first-order conditions are ∂H/∂kj = 0, ∂H/∂νj = 0,and ∂H/∂lj = rχ − χ . Combining these conditions yields:

 ∂y j =  αAk αj −1 (k l j )1−α = r , ∂k 
 j (3)

 ∂y j γw w q ′(θ)θ 
 r + δ − + .
=  (1 − α) Ak αj k 1−α l −j α = w +
 ∂l

θ
θ
(
)
(
)
q
w
q


 j 

(4)

All firms are considered identical because of the symmetry in production technology,
and as such, k = kj = k in equilibrium. Also, the vacancy rate is equal to the number of firm j’svacancies, u ≡  ∫0 v j dj  N = vj = v. Thus, the tightness of the labor market is determined byN

the ratio of the number of vacancies per firm to unemployment rate, θ (≡ uN/uN) = v/u.
Furthermore, from the definition of the employment rate, we get:N
1 − u ≡  ∫ l j dj  N = l .
 0


(5)

As a result, per capita output can be written as
y = Ak (1 − u )1−α .

(6)Following the example of other studies that include search friction (Pissarides, 2000), we
assume that wages are negotiated by the worker and the firm after they meet. When a matchis made, the firm can put the worker into production and save on the vacancy cost. Hence, theupper bound on the wage is the marginal benefits of labor, which is determined by themarginal product and the marginal value of the saved vacancy cost (∂y/∂l + θγw).9 The lower
bound, on the other hand, is a worker’s opportunity income, that is, the unemployment
benefit, lw (which is proportionate to the wage rate, l ∈ (0, 1)). It is assumed that a
The upper bound on the wage can be derived as follows: ∂(y − γwv)/∂l with v = θu = θ(1 − l). This isconsistent with the marginal benefit of labor with an internalized unemployment rate.
95

negotiation between a firm and a worker results in a wage that is somewhere between these
two extremes (Eriksson, 1997); such that w = (1 − β)lw + β(∂y/∂l + θγw), where β ∈ (0, 1)
denotes the worker’s bargaining power.10 Consequently, the wage rate can be expressed asfollows:β∂y⋅ .
1 − (1 − β)l − βγθ ∂l

w=

(7)

2.3. Households
We approach households in much the same way as Weil (1989). In our model, there are many
infinitely lived households in the economy. Households do not have any assets when theyfirst enter into the economy, though we assume that the first generation has a set of initialassets. We distinguish this first generation of households from those that are born at time 0(denoted 0+) by denoting them 0-. The lifetime utility of a representative household ingeneration s is∞

∫ ln c(s, i)e−ρ ( i −t )

t

di ,

where c(s, t) represents the consumption of generation s at time t, and ρ is the subjective
discount rate. Each household allocates its total assets (z) between physical capital (k) and the
bubble asset (m) (the bubble asset is an intrinsically useless paper asset, specifically money).
Because of the arbitrage condition, the price of the bubble asset in terms of goods (1/p) mustsatisfy the condition (1 / p) /(1/p) = − p / p = r, or rather, the return on one unit of the bubble
asset must be equal to r. By using this relation, we can obtain the following flow/stock budget
constraint in terms of real goods:11dz ( s , t )= rz(s, t) + w(1 − u) + lwu + vI − τ − c(s, t),dt(8)z(s, t) = k(s, t) + m(s, t),where z(s, t) represents the total asset holdings of generation s at time t, lw is theunemployment benefit, τ is a lump-sum tax, and vI is the income from vacancies (defined as

10This formulation of the wage equation is broadly used in the literature on unemployment that includes
search friction (Aghion and Howitt, 1994; Caballero and Hammour, 1996; Eriksson, 1997). See Pissarides(2000) and Hall (2005) for discussions on the determination of the wage equation.
11

See Appendix A for a detailed explanation of the derivation.

6

vI =  ∫0 γwv j dj  N = γwv). In order to eliminate any uncertainty regarding employment, weN

assume that each household has a large number of members. The first-order maximization
conditions are given bydc ( s , t )= (r − r)c( s, t ) ,
dt

(9)
t

and the transversality condition is given by limt →∞z ( s, t )e ∫s− ri di= 0 . Using Equations (8) and(9), we can obtain the following consumption function:c(s, t) = ρ[z(s, t) + ht],
where hτ ≡ ∫

∞

τ

(

(10)

)

i

wi (1 − u i ) + λwi u i + v − τ i e ∫τ di .
Ii− rι dι2.4. Government
The government supplies the useless paper asset, B, which is priced in terms of goods at 1/p.
We define the real value of the supply of this asset as M ≡ B/p. The government gives thisasset to the first generation (0-) at time 0 and continues to supply it to each household atconstant rate of µ (= B / B ). Since the government’s real revenue is the sum of the lump-sumtax (τN) and the bubble expansion (µM), and its expenditure is the unemployment benefit(lwuN), we can write its budget constraint as
τN + µM = lwuN.

(11)2.5. Market clearing conditions
We define the aggregate variable Xt as follows (Weil, 1989):t

X t ≡ x(0 −1 , t ) N 0 + ∫ x( s, t )ne ns N 0 ds .
0

By combining Equations (8), (9), and (10) with the above definition, we are able to formulate
the following economy-wide dynamic equations:12

12C = (r − r + n)C − nrZ ,(12)Z = rZ + ωN − C ,(13)Equation (12) is derived by using Equations (9) and (10), and z(0, 0) = 0. Equation (13) is derived byusing Equation (8). Equation (14) is derived by using M = B/p with B /B = µ and − p / p = r.

7

M = (µ + r ) M ,(14)

where Z = K + M and ω ≡ w(1 − u) + γwv + µM/N.
N

The aggregate output is defined as Y ≡ ∫ y j dj = yN. As such, by using Equation (6) we0

can getY = AK (1 − u )1−α .

(15)Also, using Equations (13) and (14), we obtain the goods market equilibrium condition13
Y = C + K .

(16)

2.6. The equilibrium conditions on a balanced growth path
In this subsection, we derive the conditions on a balanced growth path. Using Equations (1)and (5), in conjunction with the definition v = θu, we obtain u = − θq(θ)u + δ(1 − u).
Consequently, the following condition is satisfied in the steady state:u=δ,δ + θq(θ)(17)

where ∂u/∂δ > 0 and ∂u/∂θ < 0.14 Equation (17) represents the Beveridge curve; this implies
that the unemployment rate rises when the separation rate increases or the labor market
becomes less tight.
Also, based on Equation (3), and using the fact that k = kj = k, the interest rate can begiven byr = αA(1 − u )1−α .

(18)With this we can easily confirm that there is a negative relationship between theunemployment rate and the interest rate.
We define the growth rate of aggregate output (Y (= yN)) as g ≡ Y / Y . Based on
Equations (5), (6), and (7) and the fact that ∂y/∂l = (1 − α)Akl−α, we find that (in a steady state
where u and θ are constant) the wage growth rate equals the per capita output growth rate
such that13

See Appendix B for the derivation of (16)Based on the matching function property provided in subsection 2.1, we obtain ∂(θq(θ))/∂θ > 0, which
implies ∂u/∂θ < 0.
148w / w = g − n.

(19)

As such, substituting Equations (7) and (19) into Equation (4) yields: (1 − β)(1 − λ)− q q (q) = r + δ − g + n .

βg



(20)

We define the variables normalized by aggregate output as c ≡ C/Y and m ≡ M/Y. Using
the definitions (c, m) and the fact that Z = K + M, as well as Equations (12), (14), and (15)andrecalling thatthe steady statelevels

of

c

and

m

are

constantwhenC / C = K / K = Y / Y = g on a balanced growth path, we are able to derive the following
equilibrium conditions:
1+ m  = 0 ,(ρ − ρ + n − g )c − nρ
1− α A(1 − u )(21)(µ + r − g)m = 0.

(22)

By dividing Equation (16) by Y and using Equations (15) and (18), we obtain the following
expressionc = 1− g

α
.
r

(23)

Based on Equations (23) and (18), and the requirement that the consumption-output ratio be
positive (c > 0), the condition g < r/α = A(1 − u )1−α must be satisfied. In the steady state,
Equations (17), (18), (20), (21), (22), and (23) give the equilibrium values of c, m, g, r, u, andθ.
If we consider the case in which the population growth rate is zero (n = 0), as in a
representative-agent (Ramsey) model with one dynasty (Eriksson, 1997), then we can easily
confirm that g = r − ρ for c > 0 (based on Equation (21)). Consequently, (based on equation(22)) the value of asset bubbles must become zero (m = 0). As a result, in a Ramsey-type
representative-agent model, bubbles cannot exist in equilibrium.

93. The conditions for, and consequences of bubbles
3.1. The conditions for bubbles
In this section, we derive the conditions under which bubbles may exist in an economy. Undera bubble regime, there are multiple steady states, the positive bubble equilibrium (m > 0) andthe bubble-less equilibrium (m = 0). Using Equation (22), we are able to obtain the aggregate
output growth rate (g) with a positive bubble:
g = r + µ.

(24)

In the positive bubble equilibrium, the output growth rate depends positively on the interest
rate and the bubble’s expansion rate.
The equilibrium bubble value can be found by substituting Equations (18), (23) and (24)into Equation (21), and then rearranging them:µ= (n − ρ − µ)(1 − α) nρ 
α  ρ
α 


 .
(
)
g
n
n
ρ
−
−
ρ
−
µ
−
ρ
=
−
µ
+





nρρ  αnρρn − ρ − µ  1− α (25)

Given unemployment rate, u (which gives the interest rate from Equation (18)), both theaggregate output growth rate and the bubble satisfy Equations (24) and (25). Based on thefirst equality in Equation (25), a positive value of m must require n − ρ − µ > 0 under thecondition r/α > g for c > 0 from Equation (23). Also, from the second equality, we see that abubble regime requires:
ρ>ρ ≡

α 
nρ 
 µ +
.
1− α 
n − ρ − µ 

(26)Then, using r = αA(1 − u )1−α from (18), we obtain the condition of the unemployment rate fora bubble regime:1

 1

nρ  1−α µ +
 .
u < u ≡ 1− 
−
α
)
A
n
(
1
−
ρ
−
µ




(27)

Thus, we are able to attain the following proposition from Equation (27):

Proposition 1: If Equation (27) is satisfied in equilibrium, then bubbles can exist in theeconomy; if not, then bubbles cannot exist.

10If the equilibrium unemployment rate is below a certain threshold ( u ), and the interest rate is
greater than r , the bubbles asset can exist.
Under full employment (u = 0), since the interest rate is given by a production parameter: r= αA, the existence of bubbles depends on the condition αA > r . With unemployment,
however, the equilibrium level of unemployment plays a crucial role in the existence of
bubbles. The following subsection examines the equilibrium unemployment rate in greater
detail.
3.2. The properties of Non-Bubble regime
In this subsection, we consider the steady state equilibrium under a non-bubble regime. Using
Equations (18), (21), and (23), we denote the growth rate in a non-bubble regime as gNB, thendefine it asgNB = Γ(r) ⇔ρ( g − ρ − n + ρ) g −  − nρ = 0 ,
α


(28)

where Γ(0) < 0.15 In equilibrium g < r + n − ρ must be satisfied because g < r/α. Also, by
taking the total derivative of Equation (28), we obtain the following property of Γ(r):Γ ′(ρ ) =∂g (ρ / α − g ) + (ρ + n − ρ − g ) / α
=
> 1.
∂ρ
(ρ / α − g ) + (ρ + n − ρ − g )Furthermore, (using Equations (24) and (28)) we can confirm that at the threshold level r(given in Equation (26)) the growth rate in a non-bubble regime is equivalent to the growthrate in a bubble regime; g = r + µ = Γ( r ), as shown in Figure 1.
Next, we consider the equilibrium unemployment rate, which determines the interest
rate r = r(u). We know from Equation (17) that θ is given by a function of θ = θ(u; δ); ∂θ/∂u< 0 and ∂θ/∂δ > 0. Then, by substituting Equation (28) and r = r(u) into Equation (20), we
find that the equilibrium unemployment rate in a non-bubble regime (uNB) is determined byΦ L (u NB ; β, λ, γ, δ) = Φ R (u NB ; δ, n) , (1 − β)(1 − λ)where Φ L (u NB ; β, λ, γ, δ) ≡ − θ(u NB ; δ) q (θ(u NB ; δ)) ,βγΦ R (u NB ; δ, n) ≡ r (u NB ) + δ − g (r (u NB )) + n ,15

See Appendix C for the explicit expression for Γ(r).

11

(29)

Φ ′L (u ) > 0 , and Φ ′R (u ) > 0 .
By defining Ψ(u) ≡ ΦL(u) − ΦR(u), we can obtain the following conditions: Ψ(0) < 0 andΨ(1) > 0. Furthermore, if we assume that the equilibrium is unique, then Ψ′(u) > 0 (Φ′L(u) >Φ′R(u)) must be satisfied. Figure 2 (a) shows the equilibrium unemployment rate under anon-bubble regime at the point ENB. Additionally, we can confirm that the intersection ofΦR(u) and δ + n − µ gives the threshold level of u in Equation (27), since gNB = g = r + µ
at the threshold. From Equations (18) and (28), we obtain the equilibrium interest and growth
rates under the non-bubble regime, rNB = αA(1− uNB)1−α and gNB = Γ(rNB). This relationship is
depicted at the point ENB in Figure 3.
With Equation (29), we can examine the properties of the equilibrium level of uNB
depending on various parameters, including β, λ, γ, δ, and n. Consequently, we find that thepartial derivatives of uNB indicate∂u NB (1 − λ)q
= 2
> 0,
∂β
β γΨ ′∂u NB (1 − β)(1 − λ)q∂u NB (1 − β)q
> 0,
=
=
> 0,
∂γ
∂λ
βγΨ ′
βγ 2 Ψ ′(30)  ∂θ 
∂u NB
1
∂u NB
1   ∂q  (1 − β)(1 − λ)
,
=
> 0.
=
− θ   > 0
1 + q − ∂nΨ′∂δΨ ′  ∂θ βγ  ∂δ 

Using Equation (30) in conjunction with (18) and (28), we get
sign

∂u NB
∂g NB
∂r NB
= sign= sign −
,
∂i
∂i
∂i

(for i = β, λ, γ, δ, n ).

(31)

Thus, we obtain the following proposition from Equations (30) and (31).
Proposition 2: Under a non-bubble regime, the bargaining power of labor (β),
unemployment benefits (l), the vacancy cost (γ), the separation rate (δ), and the population
growth rate (n) all serve to increase the unemployment rate, so that the equilibrium interest
and output growth rates fall.

Intuitively, greater bargaining power among workers (β) results in higher wages, this implies
that the incentive for job creation would decrease and firms would reduce vacancies. Then,the tightness of the labor market (θ = v/u) would decrease and the steady state unemployment
rate would rise. The effects of the vacancy cost (γ) and unemployment benefits (l) are thesame as mentioned above. In addition to the above effect, a higher separation rate (δ) would
12also shift the Beveridge curve and raise unemployment. These findings regarding the effects
on the unemployment rate are in line with those of the standard search friction model(Pissarides, 2000).
Moreover, we can see from Equation (19) that the wage growth rate decreases as the
population growth rate (n) increases. A decrease in the growth rate of wages means that
future vacancy costs will be lower; hence, firms would postpone creating them. As such, thelabor market would become more restrictive and, in turn, the unemployment rate would rise.
As a result of this effect on the unemployment rate, the changes in the parameters(β, λ, γ, δ, n) that increase the unemployment rate would also have a negative effect on theinterest and output growth rates (see Equations (18) and (28)).
3.3. The properties of Bubble regime
In this subsection, we analyze the properties of bubble regime. Under this regime, there are
two equilibria: the positive bubble equilibrium (B) and the bubble-less equilibrium (N). We
denote the variables in this regime as u ιB , rιB , and g ιB where ι = (B, N). The variables inthe bubble-less equilibrium are given by u NB , rNB , and g NB and are equivalent to those in the
non-bubble regime described in the previous subsection. The bubble-less equilibrium
unemployment rate ( u NB ) is given by ΦL(u) = ΦR(u) and shown at the point E NB in Figure 2(b). Then, from Equations (18) and (28), we obtain the equilibrium interest and growth rates,
rNB = αA(1− u NB )1−α and g NB = Γ( u NB ). This relationship is depicted at the point E NB in Figure

3.
Next, we consider the positive bubble equilibrium unemployment rate. When there is apositive bubble, the growth rate can be given by g BB = r + µ. By substituting this expression
into Equation (20), we can obtain the equilibrium condition for the unemployment rate ( u BB ),Φ L (u BB ; β, λ, γ, δ) = δ + n − µ .

(32)

As shown at the point E BB in Figure 2 (b), the equilibrium unemployment rate with apositive bubble is larger than that without a bubble; that is, u BB > u NB , which implies rBB <rNB . The positive bubble equilibrium interest and growth rates are rBB = αA(1− u BB )1−α andg BB = rBB + µ. This relationship is depicted at the point E BB in Figure 3.

By taking the partial derivatives of u BB we find13

∂u BB (1 − λ)q
= 2
> 0,
∂β
β γΦ ′L∂u BB (1 − β)(1 − λ)q∂u BB (1 − β)q
=
> 0,
=
> 0,
∂γ
∂λ
βγΦ ′Lβγ 2 Φ ′L

(33)

∂u BB
  ∂θ ∂u BB∂u BB − 1∂q  (1 − β)(1 − λ)
1  
1
,
,
q
=
+
−
−
θ
1
>
0
0
>
<0.
=
=
 
  ∂δ ∂δΦ ′L  ∂θ βγ∂n Φ ′L∂µ Φ ′L  

Then, using Equation (33) in conjunction with (18) and (24), we also find
sign

∂r B
∂g BB
∂u B
= sign B = sign − B < 0 ,
∂i
∂i
∂i

sign

(for i = β, l, γ, δ, n),

(34)

∂u B
∂g BB ∂rBB
∂rBB
= sign − B > 0 ,=+1 > 0.
∂µ
∂µ
∂µ
∂µ

Consequently, we obtain the following proposition from Equations (33) and (34).
Proposition 3: In a positive bubble equilibrium, the bargaining power of labor (β),
unemployment benefits (l), vacancy cost (γ), the separation rate (δ), and the population
growth rate (n) raise the unemployment rate so that the equilibrium interest and output
growth rates fall. The bubble expansion rate (µ), on the other hand, decreases theunemployment rate and increases the equilibrium interest and output growth rates.
The effects of β, l, γ, δ, and n on unemployment, the interest rate, and the output growth rate
are all analogous to those in a non-bubble regime.
The bubble expansion rate (µ) has two positive effects on the output growth rate: The
first occurs directly by means of an increase in asset holdings (asset effect) (Futagami andShibata, 2000); this effect shifts the g BB = rBB + µ curve upward. Furthermore, this increasein economic growth raises wages growth, which in turn increases the number of operating
vacancies so that the unemployment rate falls. The second effect stems from the improvement
in employment; this induces economic growth through an increase in the interest rate. Thiseffect moves the positive bubble economy along the g BB curve at interest rate rBB . Figure 4
depicts how the presence of bubbles makes E BB (bubble economy) approach E NB (bubble-lesseconomy).

14

3.4. Comparison between regimes
The economy will be in a non-bubble regime (bubble regime) when the equilibrium
unemployment rate is higher (lower) than the threshold level. Thus, when policies orparameters are changed in such a way that decreases the equilibrium unemployment rate (e.g.,
a decrease in unemployment benefits) the economy will shift from a non-bubble regime to a
bubble regime. As shown in Figure 3, the output growth rate is always higher under a bubble
regime than under a non-bubble regime, even if bubbles occur.
Additionally, under a bubble regime, a steady state equilibrium can be achieved in either
the presence or the absence of bubbles (Tirole, 1985). As for equilibrium unemployment, therelationship u NB < u BB < u holds under bubble regimes. Thus, the conditions rNB > rBB > r
and g NB > g BB are satisfied. In regards to the growth rate, we find that bubbles create a
crowd-out effect by reducing capital accumulation (Grossman and Yanagawa, 1993;
Futagami and Shibata 2000).
The above properties can be formally restated as follows:
Proposition 4: The growth rate ( g NB ) in a non-bubble regime ( u > u ) is lower than thegrowth rate ( g ιB ) in a bubble regime ( u < u ), that is, g NB < g ιB . Under a bubble regime, thegrowth rate is lower when there are bubbles ( g BB ) than when there are not ( g NB ), that is,
g BB < g NB .

4. Conclusion
In this paper, we developed a continuous-time overlapping-generation model with labormarket friction and examined the conditions for bubbles. We showed theoretical relationships
between bubbles, economic growth, and unemployment. In contrast to previous studies that
only accounted for production technology in the calculation of the interest rate, we introduce
labor market friction into our endogenous growth model, so that the interest rate depends onlabor market conditions. This, in turn, determines the efficiency of labor in the production.
Allowing for unemployment, fluctuations induced by the labor market determine the type of
regime that the economy will be under. In particular, the lower the equilibrium
unemployment rate is (when the interest rate is relatively high), the more likely it is that there
will be a positive bubble equilibrium. Furthermore, labor market efficiency affects capital
15

accumulation by means of the marginal product of capital, so that economic growth is also
deeply contingent upon the employment situation.
Based on our finding that bubbles can (not) occur when the equilibrium unemployment
rate is low (high), and the interest and economic growth rates are high (low), we conclude
that policies that have a positive impact on the labor market (e.g., a decrease in
unemployment benefits) can improve employment and place the economy under a bubble
regime. This, in turn, will raise both the interest rate and the economic growth rate.

Appendix
Appendix A: The derivation of the flow budget constraint (8)We define bubble asset holdings in nominal terms as b(s, t). Then, the flow budget constraintof generation s at time t is[db(s, t)/dt]/p + dk(s, t)/dt = rk(s, t) + w(1 − u) + lwu + vI − τ − c(s, t).
Since the real value of the bubble asset is given by m(s, t) = b(s, t)/p, we obtain dm(s, t)/dt =[db(s, t)/dt]/p + rm(s, t) by using the arbitrage condition −( p /p) = r. Substituting this into theabove-equation yields
dm(s, t)/dt − rm(s, t) + dk(s, t)/dt = rk(s, t) + w(1 − u) + lwu + vI − τ − c(s, t).
Therefore, using z(s, t) = k(s, t) + m(s, t), we can obtain the flow budget constraint (8).
Appendix B: The derivation of the equilibrium condition in the goods market (16)
From Equations (13) and (14) and Z = K + M, we haveK = rK + [w(1 − u ) + γwv ]N − C .

(B1)

Because there is perfect competition in the market for production factors firms receive
zero profits. By using this information in conjunction with Equations (2) and (5), we find
Y = rK + [w(1 − u ) + γwv ]N ,(B2)

implying that total output is distributed among the capital income, the wage income, and thevacancy income. Substituting (B2) into (B1) yields the goods market equilibrium condition
(16).

16

Appendix C: The explicit expression for Γ(r)Using Equation (28), the quadratic equation for the growth rate (g) can be solved for the two
following solutions:g = (ρ + n − ρ + ρ / α ) ±(ρ + n − ρ + ρ / α )2 − 4((ρ + n − ρ)ρ / α − nρ) / 2 . However, since it is

required that g < r/α in order for c to take positive value (from Equation (23)), only one
solution may be used:g = Γ(ρ ) ≡1(ρ − ρ + n + ρ / α ) −2 

where Γ(0) ≡ (n − ρ) −(ρ − ρ + n + ρ / α )2 − 4((ρ − ρ + n)ρ / α − nρ) ,

(n − ρ)2 + 4nρ  / 2 < 0 .


17

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19

gg = Γ(r) (for bubble-less)g BB = r + µ (for positive bubble)

µ

1

0

r
r
Non-Bubble Regime

Bubble Regime

Figure 1. The threshold between a non-bubble regime and a bubble regime20ΦL(u)
ΦR(u)

ENB
δ+n−µ

0

uNB

u

1

u

Figure 2 (a). The steady state unemployment equilibrium under a non-bubble regime ( u > u )

ΦL(u)

ΦR(u)
E BB

δ+n−µ
E

0

B
N

u NB

u BB

u

1

u

Figure 2 (b). The steady state unemployment equilibrium under a bubble regime ( u < u )21

gg = Γ(r) (for bubble-less)g BB = r + µ (for positive bubble)
g NB

E NB

g BB

E BB

gNB

ENB

0

r
r

NB

r

B
B

r

Non-Bubble Regime

BNr

Bubble Regime

Figure 3. Comparison between regimes

22gg = Γ(r)

g NB

E NB

E BB

µ↑

g BB

g BB

g BB = r + µ

E BB

0

r
r

B
B

r

B
B

r

B
N

r

Figure 4. The effect of bubble growth (µ)

23