Asset Bubbles, Unemployment, and Financial Market Frictions
Ken-ichi Hashimoto Ryonghun Im
Takuma Kunieda
Akihisa Shibata
July, 2020
Discussion Paper No.2013

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANAsset Bubbles, Unemployment, and Financial
Market Frictions∗
Ken-ichi Hashimoto†Ryonghun Im‡
Akihisa Shibata¶

Takuma Kunieda§

July 31, 2020

Abstract
A tractable model with infinitely lived agents is constructed for the examination of bubbles and unemployment. It is demonstrated that the presenceof bubbles stimulates capital accumulation and reduces unemployment. The
presence of bubbles also changes the effects of government policies that target
unemployment and welfare conditions in the labor market. The main findings
are as follows: (i) the presence of bubbles is more beneficial to an economy withsevere credit constraints; (ii) the presence of bubbles mitigates the negative
effects of taxation and unemployment benefits on unemployment and welfare;and (iii) these mitigation effects decrease as credit constraints are relaxed.

Keywords: Asset bubbles, Unemployment, Labor-market matching frictions, Financial frictions.
JEL Classification Numbers: J64, O41, O42.
∗

This work is financially supported by the joint research program of KIER and the Japan Society
for the Promotion of Science: Grants-in-Aid for Scientific Research (Nos. 16H02026, 16K03685,
17H02516, 20K01631, and 20K01647) and Grant-in-Aid for JSPS Fellows (No. 16J04950). The
authors would like to thank Koichi Futagami and Toshihiko Mukoyama for their invaluable comments
and suggestions. The authors are also grateful to the participants at the JEA Spring Meeting in2018 for their comments. All remaining errors, if any, are ours.
†
Address: Graduate School of Economics, Kobe University, Rokko-dai 2-1, Kobe 657-8501,
JAPAN; Fax: +81-78-803-7293; E-mail: hashimoto@econ.kobe-u.ac.jp‡
Address: Institute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto
606-8501, JAPAN; E-mail: ryonghunim@gmail.com§
Corresponding author. Address: School of Economics, Kwansei Gakuin University, 1-155 Uegahara Ichiban-cho, Nishinomiya 662-8501, Hyogo, JAPAN; Fax: +81-798-51-0944; E-mail: tkunieda@kwansei.ac.jp¶
Address: Institute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto
606-8501, JAPAN; Fax:+81-75-753-7158; E-mail: shibata@kier.kyoto-u.ac.jp1

1

Introduction

Upward deviations of asset prices from their trend values are often followed by a sharp
drop in prices and then a recession. As a result, several researchers have developed dynamic general equilibrium models that characterize these phenomena as the emergenceand subsequent collapse of asset price bubbles. These studies focus on the positive effect of bubbles on investment and output and propose several mechanisms that might
drive this effect.1 It should be noted here that these large swings in asset prices areoften accompanied by corresponding changes in employment and investment. In fact,
Phelps (1999), Fitoussi et al. (2000), and Pan (2020) statistically confirm that highasset prices reduce unemployment. Motivated by these observations, we construct amodel with infinitely lived agents that incorporates unemployment. Then, we use thismodel to investigate how asset bubbles affect unemployment, capital accumulation,
and welfare.
In our model, there are three types of economic agents: entrepreneurs, workers,
and firms. We also include a representative financial intermediary, which is a veil in
our model. Entrepreneurs are potential capital goods producers. In each period, theyreceive idiosyncratic productivity shocks. Entrepreneurs who draw higher productivity shocks borrow from the financial intermediary and undertake an investment projectfor capital goods production. Those who draw lower productivity shocks deposit theirsavings with the financial intermediary without engaging in capital goods production.
In other words, depositors (effective lenders) and borrowers appear endogenously ineach period. Moreover, entrepreneurs are assumed to face credit constraints in that1

According to this growing stream of literature, financial market imperfections and productivity
differences across agents are key factors that produce situations in which asset bubbles enhancecapital accumulation. For example, Farhi and Tirole (2012), Martin and Ventura (2012), Carvalho etal. (2012), and Kunieda (2014) apply overlapping generations (OLG) models. Furthermore, despite
the assumption of infinitely lived agents, in the dynamic general equilibrium models of Kocherlakota(2009), Kiyotaki and Moore (2012), Aoki and Nikolov (2015), Hirano et al. (2015), Kunieda andShibata (2016), and Hirano and Yanagawa (2017), the presence of bubbles promotes economic growththrough a mechanism similar to that in Mitsui and Watanabe (1989), which is the earliest study toshow the capital-enhancing effect of bubbles.

2

they can borrow only up to a certain proportion of their net worth. Therefore, thedemand for borrowing is smaller than that in the case of no credit constraints. As aresult, in the presence of such constraints, the equilibrium interest rate can be lower
than the growth rate of the economy, which makes it possible for an asset bubble toexist.2
Workers and firms are subject to labor-market matching frictions. We employ thesame matching function as that in Diamond (1982), Mortensen and Pissarides (1999),
and Pissarides (2000). A worker who successfully matches with a firm inelasticallysupplies one unit of labor to the firm and earns a wage income in each period. However, workers may fail to match with a firm owing to labor-market matching frictions.
These workers are unemployed and receive unemployment benefits from the government. Both employed and unemployed workers consume all their earnings in eachperiod; that is, they are hand-to-mouth consumers because they cannot borrow in thefinancial market, and their subjective discount factor is so small that the borrowing
constraints are always binding. Firms are endowed with identical constant-returnsto-scale technology. Each firm must hire a worker to operate its business. However,
the aforementioned frictions can result in firms failing to hire a worker. Such firms
are not engaged in production activities.
Several studies are related to ours. The seminal paper by Miao et al. (2016)
investigates the relationship between unemployment and stock market bubbles in an
economy with both labor- and financial-market frictions. They derive the implications of labor-market policies such as unemployment benefits and hiring subsidies for
macroeconomic variables.3 Although we share numerous research interests with Miaoet al., our research departs from theirs in several respects. First, in their model,
2

A necessary condition for asset bubbles to appear in growth models is that the equilibrium
interest rate is lower than the economic growth rate. See, for example, King and Ferguson (1993).
3
Vuillemey and Wasmer (2020) study the effect of nonfundamental shocks on unemployment byapplying a standard search-and-matching model. However, they derive bubbles from the model
without rationality. In contrast, rational bubbles are derived in Miao et al. (2016) and our paper.

3

bubbles play the key role in increasing a firm’s fundamental value through the relaxation of the collateral constraint such that the firm increases its production andemployment being stimulated by extrinsic uncertainty. In contrast, in our model,
the presence of intrinsically useless assets á la Tirole (1985) can correct allocative
inefficiency regarding production resources, and as a result, capital accumulation is
promoted so that firms are incentivized to increase employment. Second, whereas
Miao et al. do not examine how the extent of financial frictions changes the effectsof labor-market policies on unemployment (and other macroeconomic variables), we
demonstrate that the effects of labor-market policies on unemployment (and othermacroeconomic variables) are crucially dependent upon the extent of these frictions.
Moreover, we provide a welfare analysis of the policies. Third, whereas in the modelof Miao et al., agents are risk-neutral and thus consumption dynamics does not arise,in our model, agents are risk-averse, so consumption and asset dynamics can be obtained. Furthermore, whereas Miao et al. adopt a Leontief production technology inwhich there is no substitution between capital and labor, we employ a general neoclassical production technology that exhibits factor substitution. Kocherlakota (2011)
investigates the effect of asset bubbles on unemployment, assuming away capital accumulation. Hashimoto and Im (2016, 2019) and Hashimoto et al. (2020) introduce
labor-market frictions into OLG models to study the effect of bubbles on both capital
accumulation and unemployment. In the models of Hashimoto and Im (2016, 2019),
because the financial market is perfect, bubbles have only crowding-out effects on capital accumulation.4 In the model of Hashimoto et al. (2020), although the financial
market is imperfect and the presence of bubbles promotes capital accumulation, how
the extent of financial constraints affects unemployment when bubbles are present
cannot be investigated because agents cannot borrow at all in the financial market,
4

Many previous studies have examined the crowding-out effect of asset bubbles on capital accumulation using OLG models (e.g., Tirole, 1985; Weil, 1987; Grossman and Yanagawa, 1993; Kingand Ferguson, 1993; Futagami and Shibata, 1999, 2000; Kunieda, 2008; Mino, 2008; Matsuoka andShibata, 2012).

4

unlike in the current paper. Introducing downward wage rigidity, Hanson and Phan(2017) and Biswas et al. (2020) show that collapses of bubbles may cause a large andprolonged recession with involuntary unemployment. In contrast with their models,
we introduce search matching frictions into a dynamic general equilibrium model andanalyze the effects of labor-market environments on macroeconomic variables.
The remainder of this paper is organized as follows. In section 2, we develop
our model. Section 3 derives the existence condition for a bubbly steady state andexamines the stability of the bubbly and bubbleless equilibria. Section 4 compares unemployment, capital accumulation, and aggregate consumption in the bubbly steadystate with their counterparts in the bubbleless steady state. By means of numerical simulations, section 5 investigates how changes in the degree of financial frictions
affect unemployment, capital accumulation, and welfare in both the bubbly and bubbleless steady states. Section 5 also investigates the effects of government policies on
unemployment and welfare. Section 6 concludes the paper.

2

The model

In this section, we develop a dynamic general equilibrium model with infinitely lived
agents. The basic structure of our model follows that of Kunieda and Shibata (2016),
but it differs from their model in some respects. First, in their model there are only
entrepreneurs and a financial intermediary, whereas our economy also includes workers
and firms. Second, we introduce labor market frictions into our model, whereas thelabor market in their model is perfect. The economy is measured in discrete time,
ranging from period 0 to ∞.

2.1

Entrepreneurs

Entrepreneurs are infinitely lived and are uniformly distributed on [0, 1]. Entrepreneurj ∈ Ω is endowed with a linear investment technology such that kt (j) = AΦt−1 (j)it−1 (j),5where Ω is the set of all entrepreneurs, kt (j) represents the capital goods produced inperiod t, Φt−1 (j) is an individual-specific productivity shock in period t − 1, it−1 (j) is
the investment undertaken in period t−1, and A is a positive constant. Entrepreneurs
who invest one unit of funds in a project in period t − 1 produce AΦt−1 (j) units ofcapital goods, which are sold to firms at price ρt in period t. Capital goods depreciate
in one period. Note that Φt−1 (j) is a random variable realized in period t − 1. andeach entrepreneur has information about Φt−1 (j) before it−1 is undertaken. AlthoughΦ(j) is idiosyncratic, there is no insurance market for the productivity shocks and,thus, the realization of low productivity cannot be insured. It is assumed that theidiosyncratic productivity shocks Φ0 (j), Φ1 (j), ... are independently and identically
distributed (i.i.d.) across both time and entrepreneurs. Specifically, we assume that
Φ(j) has support over [0, η] and its cumulative distribution function is G(Φ(j)), whichis continuous and differentiable on the support.
Entrepreneur j solves the following maximization problem:

max Et"∞X#β s−t ln cs (j) ,s=t

subject tois (j) + bs (j) = (ρs AΦs−1 is−1 (j) + rs bs−1 (j))(1 − τ ) − cs (j) for s ≥ 1,

(1)

bs (j) ≥ −λas (j),(2)

is (j) ≥ 0,

(3)

where β ∈ (0, 1) is the entrepreneur’s subjective discount factor, cs (j) is her consumption in period s, bs (j) is a deposit if positive and a debt if negative, ρs is thecapital price, rs is the gross interest rate, and τ is a tax on the entrepreneur’s income, which is constant over time. In inequality (2), as (j) := (ρs AΦs−1 is−1 (j) +6

rs bs−1 (j))(1 − τ ) − cs (j) for s ≥ 1 is her saving (or the net worth remaining after
she consumes in period s). In period s = 0, the flow budget constraint is given byi0 (j) + b0 (j) = e0 (1 − τ ) − c0 (j), where e0 is the initial endowment of the entrepreneur
at birth, which is common to all entrepreneurs. Inequality (2) is the credit constraint
faced by entrepreneur j.5 λ ∈ (0, ∞) measures the extent of the credit constraint.
Inequality (2) implies that an entrepreneur can borrow in the financial market only
up to λ times her net worth. It follows from Eq. (1) that as (j) = is (j) + bs (j). Then,the credit constraint is rewritten asbs (j) ≥ −µis (j),(4)

where µ := λ/(1+λ) ∈ (0, 1) also measures the extent of the credit constraint. Finally,
inequality (3) is the nonnegativity constraint on investment.

2.2

Optimal behavior

Let us define
φt := rt+1 /Aρt+1 .

(5)Then, it is optimal for entrepreneurs who are more productive (i.e., Φt (j) > φt )to produce capital goods by borrowing up to the limit of their credit constraints.
However, it is optimal for entrepreneurs who are less productive (i.e., Φt (j) ≤ φt )to deposit their net worth with the financial intermediary in order to obtain the
gross interest rate, rt+1 , without engaging in capital goods production. Henceforth,the former are called capital producers (borrowers) and the latter are called lenders.
The cutoff, φt , divides entrepreneurs into capital producers and lenders. Hence, after
observing the idiosyncratic productivity shock, Φt (j), entrepreneur j with net worth
5This formulation of credit constraints is standard in the literature (e.g., Aghion et al., 1999;
Aghion and Banerjee, 2005; Aghion et al., 2005; Antrás and Caballero, 2009).

7at (j) in period t plans to invest, borrow, or lend, as follows:

it (j) =0

if Φt (j) ≤ φt


 at (j)if Φt (j) > φt ,

1−µ

and
bt (j) =at (j)

if Φt (j) ≤ φt
 − µ at (j)if Φt (j) > φt .

1−µ

(6)

(7)

(j)−rs µ
By defining Rs (j) := max{rs , ρs AΦs−1}, and from Eqs. (6) and (7), the flow
1−µbudget constraint of entrepreneur j in period s is expressed as

as (j) = (1 − τ )Rs (j)as−1 (j) − cs (j).

(8)

Entrepreneur j solves the intertemporal maximization problem subject to Eq. (8).
The Euler equation for all t ≥ 0 is given by11= β(1 − τ )Et Rt+1 (j).
ct (j)
ct+1 (j)(9)

Because the lifetime utility function is log-linear, it follows from Eqs. (8) and (9) andthe transversality condition lims→∞ β s Et [at+s (j)/ct+s (j)] = 0 thatat+1 (j) = β(1 − τ )Rt+1 (j)at (j).

2.3

(10)

Financial intermediary

As in Grandmont (1983) and Rochon and Polemarchakis (2006), we assume a representative financial intermediary. The financial sector is competitive and, thus, therepresentative financial intermediary does not earn a profit. The financial intermediary accepts deposits from lenders and lends funds to capital producers. The financial

8intermediary purchases an intrinsically useless asset using the excess total savings.
Therefore, the intermediary’s balance sheet is given by

Lt + Bt = Dt ,(11)

where Lt and Dt are aggregate loans and deposits, respectively. The nominal supply of
the intrinsically useless asset is constant, and denoted by M . It follows that pt M = Bt ,where pt is the price of the asset. Because this asset is freely disposable, Bt cannot be
negative. When Bt is strictly greater than zero, a bubble on the asset occurs because
a bubble is defined as the difference between the fundamental and market values of anasset. Because there is no opportunity for the financial intermediary to earn a profit,
it follows that pt /pt−1 = rt in equilibrium. As such, a dynamic equation with respect
to Bt is obtained as
Bt = rt Bt−1 .
2.4

(12)

Final goods sector

To produce final goods, a firm hires a worker. However, workers and firms facesearch-matching frictions in the labor market. A firm that matches with a worker
purchases capital goods as input. We denote such a firm as firm h. Firm h produces final goods, Yt (h), in period t. Its production technology is represented by
Yt (h) = F (Kt (h), Nt (h)), where Nt (h) and Kt (h) are the labor and capital employedby the firm, respectively. As previously assumed, capital depreciates entirely in
one period. The production function is at least twice continuously differentiable,
concave, homogeneous of degree one, and increasing with respect to both Kt (h)and Nt (h). It is assumed that F (0, Nt (h)) = 0 and F (Kt (h), 0) = 0. We definef (kt (h)) := F (Kt (h)/Nt (h), 1), where kt (h) := Kt (h)/Nt (h) is the capital–labor ratioof firm h and f (kt (h)) satisfies f (0) = 0. Because a firm hires only one worker, it holds

9that Nt (h) = 1 and the capital–labor ratio is equal to the capital per firm. Becausef (kt (h)) is increasing and concave, it follows that f 0 (kt (h)) > 0 > f 00 (kt (h)).
Under perfect competition in the capital market, the marginal productivity ofcapital is equal to its price:ρt = f 0 (kt (h)).
Because every firm faces a common capital price, they employ the same amount of
capital. Thus, we can drop the index h in the above equation, yieldingρt = f 0 (kt ).

(13)

The remainder of the output allotted between a firm and its worker is given by

πt := f (kt ) − f 0 (kt )kt .

(14)

For simplicity, we impose Assumption 1 in the following analysis.
Assumption 1 [f 0 (kt )kt ]0 > 0.
Assumption 1 holds when F (Kt (h), Nt (h)) is of the Cobb-Douglas class.
2.5

Workers

Each worker lives forever and is endowed with one unit of labor at the beginning ofeach period. The population of workers is equal to N . If a worker matches with afirm, she is hired by the firm and earns a wage income, wt . Otherwise, she becomes ajobless person and receives the unemployment benefit γt from the government. The
government taxes income at rate τtw and the tax revenue covers the unemployment
benefit. Thus, after-tax income is denoted by ωtι (1−τtw ), where ι represents a worker’s
employment status and ωtι is a worker’s income: ι = e and ωte = wt if employed, and10ι = u and ωtu = γt if unemployed. The workers are hand-to-mouth consumers; thatis, they consume their current income entirely. Thus, their consumption iscw,ι= ωtι (1 − τtw )t(15)

for all t ≥ 0, where cw,ιis a worker’s consumption. A worker’s expected lifetime utilitytis given by

Utw,ι

=

cw,ι
t + Et∞ hXβ˜s cw,ι
t+s

i

s=1

=

ωtι (1 − τtw ) + Et

∞ hiXβ˜s cw,ι
t+s ,

(16)

s=1

where β̃ ∈ (0, β) is the worker’s subjective discount factor.6 We assume that theprobability of a worker matching with a firm from period t + 1 onward is independentof her current status ι.
2.6

Government

The government runs a balanced budget to provide unemployment benefits for workers:Zτ(ρt AΦt−1 (j)it−1 (j) + rt bt−1 (j))dj + τtw (wt (1 − ut ) + γt ut ) N = γt ut N,

(17)

j∈Ω

where ut is the unemployment rate. The left-hand side of Eq. (17) is equal to theaggregate tax revenue and the right-hand side represents the total payments for unemployment benefits. The tax rate on entrepreneurs’ income, τ , is assumed to beconstant over time, and γt is assumed to be a constant fraction of wages, γwt . Thus,
the tax rate on workers’ income, τtω , is determined endogenously to satisfy Eq. (17).
6

If a worker’s subjective discount factor is sufficiently small and she cannot borrow in the financial
market, she behaves in a hand-to-mouth manner. King and Leape (1998) and Guiso et al. (2003)
provide empirical evidence supporting the existence of hand-to-mouth consumers.

11

2.7

Labor market

We introduce labor-market matching frictions into our model. That is, we assume theexistence of search costs and a matching function between firms and workers. A firmthat matches with a worker begins operating without any time lags.

2.7.1

Matching mechanism

Each worker is endowed with one unit of labor in each period. Because firms must pay
a fixed cost to search for a worker and workers and firms face matching frictions, unemployment occurs in equilibrium. The match between a worker and a firm is assumed
to be broken in one period. Because of free entry into the labor market, each firm is
indifferent between entry and no-entry into the labor market and the equilibrium entry rate of firms is determined by macroeconomic conditions. In contrast, all workers
search in every period because they can enter the labor market without incurring acost. We denote the entry rate of firms by υt . Thus, we refer to υt as the number offirms with vacancies in the labor market. The number of matches is a function of the
population of workers, N , and the number of firms with vacancies, υt , which is given bym(N, υt ). We assume that 0 ≤ m(N, υt ) ≤ min{N, υt }, for N ∈ [0, ∞) and υt ∈ [0, ∞),m(0, υt ) = 0, and m(N, 0) = 0. The matching function, m(N, υt ), is continuously
differentiable, concave, homogeneous of degree one, and increasing with respect toboth N and υt . The tightness of the labor market is measured by θt := υt /N ∈ (0, ∞),which is the jobs-to-applicants ratio. The probability that a firm with a vacancy
matches with a worker is given by m(N, υt )/υt = m(1/θt , 1) =: q(θt ). q(θt ) is continuously differentiable in (0, ∞), where q 0 (θt ) < 0, for θt ∈ (0, ∞), limθt →0 q(θt ) = 1,
and limθt →∞ q(θt ) = 0. The number of employed workers is equal to the number ofmatches. Thus, it follows that (1 − ut )N = m(N, υt ) or, equivalently,

1 − ut = θt q(θt ).
12

(18)

Eq. (18) derives the so-called the Beveridge curve, which represents a negative relationship between the unemployment rate and the labor market tightness. More specifically, from Eq. (18), we obtain the unemployment rate, ut , as a decreasing functionof θt , ut = u(θt ). It holds that u0 (θt ) < 0 because ∂[θt q(θt )]/∂θt = ∂m(1, θt )/∂θt > 0.
A firm that matches with a worker produces final goods. The value of a firm that
produces final goods, Jte , and the value of a firm that does not match with a worker,
Jtu , are given by
Jte = πt − wt − ζ +


1 
e
u
q(θt+1 )Jt+1+ (1 − q(θt+1 ))Jt+1

ρt+1

and
Jtu = −ζ +

1 
ρt+1


e
u
q(θt+1 )Jt+1+ (1 − q(θt+1 ))Jt+1,

respectively, where ζ is the search cost that the firm incurs when searching for a worker
in the labor market.7 If the actual revenue πt − wt is less than ζ, no firms operate.
We proceed with our investigation assuming the nontrivial case in which πt − wt ≥ ζ,
unless otherwise stated.8 We can write a firm’s expected entry value, Qt , in the labormarket as follows:

Qt = q(θt )Jte + (1 − q(θt ))Jtu ,= q(θt )(πt − wt ) − ζ +1
ρt+1

Qt+1 .

(19)

The free-entry condition for the labor market leads to zero profit for each firm,
7

Since the capital depreciation rate is 1, the net rental rate of capital is equal to ρt+1 − 1 in periodt + 1. Therefore, a firm’s expected entry value is discounted by ρt+1 . We assume ρt+1 > 1 for allt ≥ 0.
8
Each firm incurs a search cost because of recruitment activities such as job interviews andevaluations of reference letters. The firm’s operating resources cover the search cost, which is an
implicit opportunity cost.

13

that is, Qt = 0 for any t or, equivalently,

πt − w t =

2.7.2

ζ
.
q(θt )(20)

Nash bargaining

A firm and its worker divide πt , which is equal to the output minus the capital
payments. according to Nash bargaining: that is, the shares for the firm and theworker are obtained from the following maximization problem:

wt = arg max(Utw,e − Utw,u ) (Jte − Jtu )1−wt

= arg max(1 − τtw ) (wt − γt ) (πt − wt )1− ,wtwhere  ∈ (0, 1) is the worker’s bargaining power. When bargaining they solve thisproblem by viewing γt as exogenous because they have no information how γt isformed. The Nash bargaining solution is given by

wt = (1 − )γt + πt .

(21)

As stated previously, unemployment benefits are paid to unemployed workers in sucha way that γt = γwt , where γ ∈ [0, 1). Substituting γt = γwt into Eq. (21), we obtainwt = Θπt ,(22)

where Θ := /{1 − (1 − )γ} ∈ (0, 1) is the worker’s output share of πt . Note thata larger outside option, γwt , and greater Nash bargaining power, , lead to a greater
share for the worker, Θ. Substituting Eqs. (14) and (22) into Eq. (20) yields(1 − Θ)(f (kt ) − f 0 (kt )kt ) =14ζ.
q(θt )(23)

This equation determines the equilibrium value of θt , the entry rate of firms into thelabor market, υt , and the unemployment rate, ut = u(θt ).
Define k̄ such that f (k̄) − f 0 (k̄)k̄ = ζ/(1 − Θ). Then, it is easy to show from
Eq. (23) that for the economy to be feasible, it must hold that kt > k̄ for all t ≥ 0.
Otherwise, no firms can cover the search cost, ζ, and hire a worker. In this case, theeconomy becomes infeasible and does not produce final goods. In what follows, we
focus on the case in which kt > k̄, for all t ≥ 0.
2.8

Aggregation

We can aggregate variables in the same manner as in Kunieda and Shibata (2016),
in which the i.i.d. assumption simplifies the aggregation. The before-tax aggregate
income over all entrepreneurs is equal to the total capital income plus the total income
from the intrinsically useless asset. Because each entrepreneur’s marginal propensity
to save is β, the aggregate saving by entrepreneurs is given byZat (j)dj = β(1 − τ )(ρt Zt + rt Bt−1 ),

(24)

j∈Ω

where Zt :=Rj∈ΩAΦt−1 it−1 (j)dj is the total of capital goods produced by high-productivity entrepreneurs. Because the number of firms that match with a worker inperiod t is (1 − ut )N , the capital goods market clearing condition is given as follows:

Zt = kt (1 − ut )N.

(25)

In period t, an entrepreneur becomes a lender with probability G(φt ) and a capital
producer (borrower) with probability 1 − G(φt ). The i.i.d. assumption allows us toapply the law of large numbers to our economy. Thus, the population of lenders
is equal to G(φt ) in period t and the population of capital producers is equal to
1 − G(φt ). Then, Eqs. (7) and (24) yield Dt = β(1 − τ )(ρt Zt + rt Bt−1 )G(φt ) and15

Lt = β(1 − τ )(ρt Zt + rt Bt−1 )µ[1 − G(φt )]/(1 − µ). From Eq. (11), the total value ofthe intrinsically useless asset is obtained as

Bt = β(1 − τ )(ρt Zt + rt Bt−1 )G(φt ) − µ
.
1−µ

(26)

Eq. (26) shows that the intrinsically useless asset has a positive value if G(φt ) > µ.
In what follows, we focus exclusively on the case in which G(φt ) ≥ µ, such that Bt isnonnegative. It follows from Eqs. (6) and (24) that the total of capital goods, Zt+1 ,is given by
Zt+1 =
where H(φt ) :=
2.9

Rη
φt

β(1 − τ )AH(φt )(ρt Zt + rt Bt−1 ),
1−µ

(27)

Φt (j)dG(Φ).

Dynamical system

The dynamical system of our economy is obtained from Eqs. (5), (12), (13), (26), and(27). From these equations, we can derive the dynamical equations of the cutoff, φt ,and the intrinsically useless asset, as follows:
φt−1 (G(φt−1 ) − µ)G(φt ) − µ=,
1 − µ − β(1 − τ )(G(φt ) − µ)β(1 − τ )H(φt−1 )(28)

Bt = Aφt−1 f 0 (kt )Bt−1 ,

(29)

and

respectively.
From Eqs. (18) and (23), we haveζq −1
1 − ut =(1 − Θ)(f (kt ) − f 0 (kt )kt )ζ(1 − Θ)(f (kt ) − f 0 (kt )kt ):= Ψ(kt ). (30)

Using Eqs. (12), (13), (26), (27), and (30), we obtain the dynamical equation of16

capital, as follows:

kt+1 Ψ(kt+1 ) =β(1 − τ )H(φt )Af 0 (kt )kt Ψ(kt ).
1 − µ − β(1 − τ )(G(φt ) − µ)(31)

Using Eqs. (12), (13), (25), and (31), Eq. (26) is rewritten as

Bt =

3

β(1 − τ )(G(φt ) − µ)f 0 (kt )kt Ψ(kt )N.
1 − µ − β(1 − τ )(G(φt ) − µ)(32)

Steady states and stability

As stated previously, we focus on the case in which G(φt ) ≥ µ and restrict the domainof the dynamical equation, Eq. (28), to [G−1 (µ), η). Whereas Eq. (28) is solely
an autonomous difference equation with respect to φt , we consider an autonomous
dynamical system with respect to φt and kt that consists of Eqs. (28) and (31).
Proposition 1 Consider the dynamical system of Eq. (28) and (31). Then, thefollowing hold.
(i) There exist two steady states (k ∗ , φ∗ ) and (k ∗∗ , φ∗∗ ), where φ∗ > φ∗∗ , such that

1 − µ − β(1 − τ )(G(φ∗ ) − µ) = β(1 − τ )H(φ∗ )/φ∗ ,(33)

1 = φ∗ Af 0 (k ∗ ),(34)

G(φ∗∗ ) = µ,

(35)

and
1=

β(1 − τ )H(φ∗∗ ) 0 ∗∗Af (k ),
1−µ(36)

if and only if (1 − µ)φ∗∗ < β(1 − τ )H(φ∗∗ ). Moreover, the intrinsically uselessasset has a positive value in the steady state with φ∗ . Thus, asset bubbles exist
in this steady state.
17

(ii) There exists only one steady state, (k ∗∗ , φ∗∗ ), given by Eqs. (35) and (36), if andonly if (1 − µ)φ∗∗ ≥ β(1 − τ )H(φ∗∗ ). Moreover, the intrinsically useless asset
has no value in this steady state.
Proof: See Appendix A.1.
We call the steady state of (k ∗ , φ∗ ) a bubbly steady state and the steady state of(k ∗∗ , φ∗∗ ) a bubbleless steady state.
Proposition 2 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists in our economy(the first case of Proposition 1). Then, the bubbly steady state (k ∗ , φ∗ ) is a saddle
point and the bubbleless steady state (k ∗∗ , φ∗∗ ) is totally stable.
Proof: See Appendix A.2.
18

Figure 1 depicts the phase diagram of the economy. Because φ0 is a nonpredetermined variable and the bubbly steady state is a saddle point, the bubbly steady stateis locally determinate. However, the bubbleless steady state is totally stable. Thus,∗∗∗∗∗
any sequence of {kt+1 , φt }∞t=0 with (k1 , φ0 ) ∈ (k̄, ∞) × (0, φ ) converging to (k , φ )is an equilibrium. Therefore, the equilibrium is globally indeterminate. Note that∗any sequence of {kt+1 , φt }∞t=0 with (k1 , φ0 ) ∈ (k̄, ∞) × (φ , η] cannot be an equilibrium

because φt becomes greater than η or kt becomes less than k̄ in finite time.

4

Macroeconomic variables in the bubbly and bubbleless steadystatesIn general equilibrium models, the presence of asset bubbles can change the equilibrium allocation of resources and affect macroeconomic variables. In this section, we
analytically investigate the effects of bubbles on capital accumulation, unemployment,
and aggregate consumption by comparing the bubbly and bubbleless steady states.

4.1

Capital accumulation

We first investigate the long-run effect of bubbles on capital accumulation. Here, thefollowing lemma is useful.
Lemma 1 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists in the dynamical system consisting of Eqs. (28) and (31) (the first case of Proposition 1). Define thefollowing function:
S(φ) :=β(1 − τ )H(φ),
1 − µ − β(1 − τ )(G(φ) − µ)the domain of which is [0, η]. Then, S(φ) is maximized at φ = φ∗ , and thus S(φ∗ ) >S(φ∗∗ ).
Proof: See Appendix B.
19

From this lemma, we have Proposition 3.
Proposition 3 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists. Then, more
capital accumulates in the bubbly steady state than it does in the bubbleless steadystate.
Proof: From Lemma 1 and Eqs. (34) and (36), we have S(φ∗ ) > S(φ∗∗ ) ⇐⇒ φ∗ >β(1 − τ )H(φ∗∗ )/(1 − µ) ⇐⇒ f 0 (k ∗ ) < f 0 (k ∗∗ ) ⇐⇒ k ∗ > k ∗∗ . 
Although the presence of bubbles decreases the aggregate investment, the equilibrium interest rate increases. As a result, less productive entrepreneurs are ruled out
of capital production activities and the aggregate productivity in capital production
becomes higher. Then, the aggregate capital produced by entrepreneurs and, thus,
the capital per worker become greater in the bubbly steady state than those in the
bubbleless steady state.

4.2

Unemployment

We next consider the effect of bubbles on unemployment. From Eq. (23), we have∂θ/∂k > 0, and from Eq. (18), we have ∂u/∂θ < 0. Therefore, an increase in capital
reduces the rate of unemployment. Proposition 4 summarizes these results.
Proposition 4 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists. The unemployment rate in the bubbly stead state, u∗ , is lower than that in the bubbleless steady state,u∗∗ .
Intuitively, as more capital accumulates, the profits obtained by firms that havematched with a worker become higher. Thus, more firms enter the labor market.
As a result, the unemployment rate decreases (and the tightness of the labor market
increases) as shown in Figure 2.

20

21

4.3

Aggregate consumption

This subsection demonstrates that the aggregate consumption of entrepreneurs andthat of workers in the bubbly steady state are both greater than those in the bubblelesssteady state. The following lemma derives the aggregate consumption of entrepreneurs
and that of workers in the two steady states.
Lemma 2 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists. Then, the followinghold:(i) The aggregate consumption of entrepreneurs in the bubbly and bubbleless steadystates is given by

C p,∗ = (1 − β)(1 − τ )Γ(φ∗ )f 0 (k ∗ )k ∗ Ψ(k ∗ )N(37)

C p,∗∗ = (1 − β)(1 − τ )Γ(φ∗∗ )f 0 (k ∗∗ )k ∗∗ Ψ(k ∗∗ )N,

(38)

and

respectively, where Γ(φ) = (1 − µ)/[1 − µ − β(1 − τ )(G(φ) − µ)].
(ii) The aggregate consumption of workers in the bubbly and bubbleless steady statesis given by

C w,∗ = [Θ(f (k ∗ ) − f 0 (k ∗ )k ∗ ) + τ Γ(φ∗ )f 0 (k ∗ )k ∗ ] Ψ(k ∗ )N,(39)

C w,∗∗ = [Θ(f (k ∗∗ ) − f 0 (k ∗∗ )k ∗∗ ) + τ Γ(φ∗∗ )f 0 (k ∗∗ )k ∗∗ ] Ψ(k ∗∗ )N,

(40)

and

respectively.
Proof: See Appendix C.
Using this lemma, we can prove the following proposition.
22

Proposition 5 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists. Then, the aggregate consumption of entrepreneurs in the bubbly steady state, C p,∗ , is greater thanthat in the bubbleless steady state, C p,∗∗ .
Proof: Because φ∗ > φ∗∗ and Γ(φ) is an increasing function with respect to φ, itholds that Γ(φ∗ ) > Γ(φ∗∗ ). From the definition of Ψ(k) and Proposition 4, it follows
that Ψ(k ∗ ) > Ψ(k ∗∗ ). From Assumption 1, we have f 0 (k ∗ )k ∗ > f 0 (k ∗∗ )k ∗∗ . Then, from
Eqs. (37) and (38), the desired conclusion is obtained. 
In Eqs. (37) and (38), the presence of bubbles has two positive effects on theaggregate consumption of entrepreneurs. The first effect is produced by an increasein the aggregate capital income, that is, f 0 (k ∗ )k ∗ Ψ(k ∗ )N > f 0 (k ∗∗ )k ∗∗ Ψ(k ∗∗ )N . The
second effect is the wealth effect of the bubbles, which is reflected by Γ(φ∗ ) > Γ(φ∗∗ ) =1. The wealth effect increases entrepreneurs’ consumption.
Proposition 6 Suppose that the bubbly steady state, (k ∗ , φ∗ ), exists. The aggregate
consumption of workers in the bubbly steady state, C w,∗ , is greater than that in the
bubbleless steady state, C w,∗∗ .
Proof: Θ(f (k) − f 0 (k)) is an increasing function with respect to k. Thus, f (k ∗ ) −f 0 (k ∗ )k ∗ > f (k ∗∗ )−f 0 (k ∗∗ )k ∗∗ . As in Proposition 5, it follows that Γ(φ∗ )f 0 (k ∗ )k ∗ Ψ(k ∗ )N >Γ(φ∗∗ )f 0 (k ∗∗ )k ∗∗ Ψ(k ∗∗ )N . Then, from Eqs. (39) and (40), the desired conclusion is
obtained. 
In Eqs. (39) and (40), the first term, Θ(f (k) − f 0 (k)k)Ψ(k ∗∗ )N , is the aggregate
wage income in which Ψ(k)N reflects the job-matching effect from Proposition 4. The
second term, τ Γ(φ)f 0 (k)kΨ(k)N , represents the effect of the redistribution policy fromentrepreneurs to workers through the unemployment benefits (see Eq. (17)). These
effects mean that workers can receive more in unemployment benefits in the bubbly
steady state than they can in the bubbleless steady state.
23

5

Numerical analysis

In this section, we numerically analyze the effects of changes in the credit constraints,
labor market conditions, and redistribution policy on the macroeconomic variables.

5.1

Specification and parameterization

Following Den Haan et al. (2000), we specify the matching function as m(N, vt ) =N vt (N σ + vtσ )−1/σ .9 The individual-specific productivity, Φ, is uniformly distributed
in [0, 1]. For the final good production, the Cobb–Douglas production function isassumed, that is, F (Kt , N ) = ĀKtα N 1−α , where Ā is the productivity parameter.
Then, the production function per worker is given by f (kt ) = Āktα . Under these
specifications, we obtain macroeconomic variables such as the capital accumulation,jobs-to-applicants ratio (tightness of the labor market), unemployment rate, and aggregate consumption of entrepreneurs and of workers in the bubbly and bubblelesssteady states. We also compute the expected lifetime utility of an entrepreneur andthat of a worker in the two steady states. See Appendices D and E for the derivations.
We set α = 0.33, β = 0.98, and β̃ = 0.8. Note that the workers are more
impatient than entrepreneurs and are hand-to-mouth consumers, as explained earlier.
We normalize the population of entrepreneurs to one and that of the workers to
N = 100. Under this parameter setting, we examine the effects of changes in the credit
constraints, µ, labor market conditions (including the worker’s bargaining power), ,
search cost, ζ, unemployment benefit, γ, and tax rate on the entrepreneur’s income,τ.
If  and γ are close to one, Θ is also close to one and the economy becomes
infeasible because production never occurs. Thus, we must impose a ceiling on and γ. According to the Organisation for Economic Co-operation and Development(OECD, 2006), unemployment benefits in the 26 OECD countries in 2004 ranged
9This matching function satisfies the conditions imposed in Section 2.

24

between 45% and 83% of net earnings. Thus, we vary γ from 0.65 to 0.8, which isa subset of [0.45, 0.83], in order to examine the effect of γ. Because Den Haan et al.
(2000) and Shimer (2005) set  = 0.5 and 0.72, respectively, we vary  from 0.5 to
0.7. As in the case of  and γ, if the search cost, ζ, is high, the economy is infeasible.
Therefore, we use a relatively low value of ζ, ranging between 0.12 and 0.14. We set
the remaining parameter values as A = 1, Ā = 2.5, and σ = 4 to produce plausible
values for the tightness of the labor market, θ, and the unemployment rate, u. We fix
τ = 0.01,  = 0.6, γ = 0.7, and ζ = 0.12 when varying one of the parameter values.
Table 1 summarizes our parameter settings.

Table 1: Parameterization
α = 0.33 β = 0.98 β̃ = 0.8 A = 1
Ā = 2.5 σ = 4
γ = 0.7  = 0.6
ζ = 0.12 τ = 0.01

5.2

Credit constraints

5.2.1

Effects on macroeconomic variables

Figure 3 shows the effects of µ on the macroeconomic variables in the bubbly andbubbleless steady states. The dotted vertical line in each panel shows µ = 0.94,
beyond which bubbles cannot exist. As shown in the panel of bubbles, we can computethe value of the bubbly assets, B, for µ > 0.94. However, it becomes negative, which
is ruled out by the free disposability of the asset.
Before discussing the effects of µ, we numerically confirm the outcomes obtainedin Propositions 3–6. Figure 3 indicates that the capital per worker in the bubbly
steady state, k ∗ , is always higher than that in the bubbleless steady state, k ∗∗ . This isbecause the presence of bubbles always promotes capital accumulation (Proposition
3). Moreover, the jobs-to-applications ratio in the bubbly steady state, θ∗ , is higher25

26

than that in the bubbly steady state, θ∗∗ . Thus, the unemployment rate in the bubbly
steady state, u∗ , is lower than that in the bubbleless steady state, u∗∗ (Proposition 4).
The aggregate consumption of entrepreneurs and that of workers in the bubbly steadystate, C p,∗ and C w,∗ , respectively, are higher than those in the bubbleless steady state,
C p,∗∗ and C w,∗∗ , respectively (Propositions 5 and 6).
As shown in Figure 3, relaxing the credit constraints promotes capital accumulation in both steady states. However, the marginal effect of µ on the capital per worker
in the bubbly steady state is always smaller than that in the bubbleless steady state.
Furthermore, the capital per worker in the bubbly steady state is equal to that in the
bubbleless steady state at µ = 0.94. The presence of asset bubbles (partially) corrects
allocative inefficiency and promotes capital accumulation, reallocating production resources from lower-productivity entrepreneurs to higher-productivity entrepreneurs.
However, as µ increases, the value of the bubbly assets, B, decreases. Thus, theallocative-efficiency effect of the asset bubbles shrinks. Therefore, the marginal effectof µ on the capital per worker is always smaller in the bubbly steady state than it is in
the bubbleless steady state. Eventually, the capital per worker in the bubbly steadystate coincides with that in the bubbleless steady state at µ = 0.94.
The graphs of the jobs-to-applicants ratio, θ, in the two steady states are similar
to those of the capital per worker, k. This is because the jobs-to-applicants ratio has a
one-to-one positive relationship with the capital per worker. As µ increases, the jobsto-applicants ratio, θ, increases, and accordingly, the unemployment rate decreases
in both steady states. Because more capital accumulates in both steady states as µincreases, the profits of firms that match with a worker increase. As a result, more
firms enter the labor market. Thus, the jobs-to-applicants ratio, θ, increases and theunemployment rate, u, decreases in both steady states.
As in the case of the jobs-to-applicants ratio, the graphs of per worker consumption, C w /N , in both steady states are similar to those of the capital per worker, k.

27

This is again because the per worker consumption has a one-to-one positive relationship with the capital per worker. On the other hand, as µ increases, the aggregate
consumption of entrepreneurs in the bubbly steady state, C p,∗ , decreases, whereasthat in the bubbleless steady state, C p,∗∗ , increases. As noted previously, when thecredit constraints are severe, the value of the bubbly assets is high. In this case,the aggregate net worth that the entrepreneurs hold is high. Because the marginal
propensity to consume is constant over time, the larger aggregate net worth increases
the aggregate consumption. However, as the credit constraint is relaxed, the wealtheffect of the bubbles shrinks and the aggregate consumption of entrepreneurs in the
bubbly steady state decreases. Thus, the decreasing trend of C p,∗ is produced in Figure 3. The wealth effect is indicated by the fact that the graph of C p in the bubbly
steady state has a similar feature to that of B.

5.2.2

Welfare effectHere, we compare the welfare effects of changes in the credit constraints in the bubblyand bubbleless steady states. Suppose that the economy is in one of the steady states.
Then, as shown in Appendix E, the expected lifetime utility of an entrepreneur withincome, It , is given by





ln (1 − β)1−β β βln(1 − τ )β
1 − µφ1ln It +++
(1 − µφ) ln
V (It ) :=1−β
(1 − β)2(1 − β)2(1 − β)2
1−µβ+[ln(ρA) + µφ ln φ − (1 − φ)] ,(41)(1 − β)2where φ = φ∗ or φ∗∗ and ρ is the steady-state value of ρt .
Figure 4 provides the expected lifetime utilities of ex-ante homogeneous entrepreneursin period zero in the bubbly and the bubbleless steady states.

We set the en-

trepreneur’s initial endowment at I0 = e0 = 20. As seen in the northwest panel,
the lifetime utility in the bubbly steady state is higher than that in the bubbleless

28

29

30

steady state. In order to see the detail of the effect of the credit constraint on welfare
in the bubbly steady state, we depict an enlarged graph of the lifetime utility in thissteady state in the northeast panel (the lifetime utility in the bubbleless steady state is
depicted in the southwest panel). As shown in the northeast panel, the lifetime utility
in the bubbly steady state decreases monotonically as µ increases from 0 to 0.94. Thisis because as µ increases, the wealth effect of the bubbles on the entrepreneurs’ consumption shrinks, as discussed in the previous section. In contrast, the lifetime utility
in the bubbleless steady state increases, as in the case of aggregate consumption.
Next, we consider the expected lifetime utilities of both employed and unemployedworkers in the bubbly and bubbleless steady states. As shown in Appendix E, theemployed and unemployed workers’ expected utility functions are given by#β̃β̃θq(θ) +
(1 − θq(θ))γ w(1 − τ w ),U w,e := 1 +1 − β̃
1 − β̃
"

and
"

#
β̃
β̃
U w,u := γ +θq(θ) +
(1 − θq(θ))γ w(1 − τ w ),1 − β̃
1 − β̃
respectively, where w = Θ(1 − α)Āk α , Θ := /{1 − (1 − )γ}, and τ w = [γ(1 −Ψ(k)) − ατ Γ(φ)Ψ(k)/(Θ(1 − α))]/[(1 − γ)Ψ(k) + γ]. Figure 5 shows the lifetime
utilities of employed and unemployed workers in the bubbly and bubbleless steadystates. The lifetime utility in the bubbly steady state is always higher than that in
the bubbleless steady state for all workers, and the lifetime utilities in both steadystates increase monotonically as µ increases. Note that the effects of an increase inµ on the lifetime utilities of employed and unemployed workers (Figure 5) are similar
to those on the capital per worker (Figure 3). This is because the lifetime utilities of
employed and unemployed workers have a positive one-to-one relationship with thecapital per worker, as in the case of per worker consumption.

31

32

33

34

5.3

Labor market conditions

Figure 6 shows the effects of the unemployment benefit, γ, on θ, u, U w,e , and U w,uin both the bubbly and bubbleless steady states when µ takes values of 0, 0.4, and0.8. In our model, the labor market conditions (e.g., , γ, and ζ) have no impact on
the capital per worker because the tax rate imposed on the entrepreneur’s income isfixed. Thus, the optimal behavior of entrepreneurs is not affected by the labor market
conditions.
As shown in Figure 6, the jobs-to-applicants ratio in the bubbly steady stateis always higher than that in the bubbleless steady state, and the unemployment
rate in the bubbly steady state is always lower than that in the bubbleless steadystate, for any value of γ. Additionally, the lifetime utilities of the employed andunemployed (i.e., U w,e and U w,u , respectively) in the bubbly steady state are higher
than those in the bubbleless steady state. However, from the three cases of µ=0, 0.4, and 0.8, we find that although asset bubbles mitigate the negative effects of
the unemployment benefit on the jobs-to-applicants ratio, unemployment rate, andwelfare, this mitigation becomes limited as the credit constraint is relaxed.
Figure 6 also indicates that the jobs-to-applicants ratio decreases and the unemployment rate increases in both steady states as γ increases from 0.65 to 0.8. Whenthe unemployment benefit, γ, increases, the worker’s output share, Θ, also increases.
Then, the profits yielded by a match between a firm and a worker decrease. Thus,
the firms are incentivized to not enter the labor market. As a result, the jobs-toapplicants ratio decreases and the unemployment rate increases in both steady states.
The graphs of U w,e and U w,u in Figure 6 have an inverted U-shape. As γ increases,
the employed and unemployed workers obtain a higher wage income and higher unemployment benefits, respectively, which positively affect U w,e and U w,u , respectively.
On the other hand, as γ increases, the probability of a worker being unemployed ishigher, which means that γ has a negative effect on both U w,e and U w,u . Moreover, γ
35negatively affects U w,e and U w,u indirectly through τ w . These effects are mixed, andthus, U w,e and U w,u have an inverted U-shape.
Figure 7 examines the effects of the worker’s bargaining power, , on u, θ, U w,e ,and U w,u in both steady states when  varies from 0.5 to 0.7. As seen in the figure,’s effects on each variable are similar to those of γ because  affects each variable inalmost the same manner as γ does.
Figure 8 shows the effects of the search cost, ζ, on u, θ, U w,e , and U w,u in both
steady states. As in the cases of γ and , the presence of an asset bubble mitigates
the negative effects of the search cost on the jobs-to-applicants ratio, unemployment
rate, and welfare, but this mitigation becomes limited as the credit constraint isrelaxed. As ζ varies from 0.12 to 0.14, the jobs-to-applicants ratio, θ, decreases andthe unemployment ratio, u, increases in both steady states because the increase in ζ
impedes the entry of firms with vacancies into the labor market. Moreover, an increasein the search cost monotonically reduces the lifetime utilities of the employed andunemployed in both steady states. This is because the unemployment rate increases
as ζ increases, which has a negative effect on the lifetime utilities of all workers.

5.4

Tax on the entrepreneur’s income

Thus far, we have fixed the tax rate imposed on the entrepreneur’s income. In thissubsection, we examine the effects of changes in τ . The tax revenue collected fromentrepreneurs is used to pay for the unemployment benefits. Thus, the taxation that
we analyze is a redistribution policy from entrepreneurs to workers. As shown in
Figure 9, the negative effects that the income tax has on u, θ, U w,e , and U w,u are
weakened by the presence of asset bubbles. However, as in the previous cases, thismitigation becomes weaker as the credit constraint is relaxed.
Figure 9 shows that as τ increases from 0 to 0.1, the jobs-to-applicants ratio,
θ, decreases and the unemployment rate, u, increases in both steady states. This

36

37

is because as τ increases, the investment by entrepreneurs decreases, and thus, thecapital per worker, k, decreases. As a result, the profit yielded by a match between
a firm and a worker decreases and the number of firms that enter the job market
decreases. The graphs of U w,e and U w,u are downward sloping when µ = 0 and 0.4,
but the graph when µ = 0.8 has an inverted U-shape in both steady states. As
τ increases, the probability of a worker being unemployed becomes higher, which
negatively affects both U w,e and U w,u . On the other hand, an increase in τ promotes
the redistribution from entrepreneurs to workers. Then, both U w,e and U w,u increaseas τ increases. Owing to these conflicting effects of τ, the graphs of U w,e and U w,uhave an inverted U-shape.

6

Conclusion

Our analysis has shown that the presence of asset bubbles mitigates the negative
effects of taxation and unemployment benefits on unemployment rates and welfare.
As the credit constraint is relaxed, this mitigation becomes limited. This means thatthe presence of asset bubbles is more beneficial in an economy with severe credit
constraints.
Although the presence of asset bubbles increases the aggregate productivity in theeconomy by excluding less productive entrepreneurs from production activities, only
the second-best outcome can be attained, as in the model of Bewley (1980). Thisis because, in our model, not only the most productive entrepreneurs but also therelatively less productive entrepreneurs engage in production when asset bubbles arepresent. As a result, the unemployment rate when asset bubbles occur is not as low
as that in the first-best outcome. Therefore, a government policy is necessary for the
economy to be Pareto-improved, even though the presence of asset bubbles reduces
the unemployment rate. An analysis of such a government policy is left for future
research.
38

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Appendices
Appendix A: Proof of Propositions 1 and 2
To prove Propositions 1 and 2, we obtain the following useful lemma.
Lemma 3 Define φ∗ and φ∗∗ such thatβ(1 − τ )H(φ∗ )= φ∗ ,∗
1 − µ − β(1 − τ )(G(φ ) − µ)andG(φ∗∗ ) = µ.
Then, both φ∗ and φ∗∗ are uniquely determined.

43

Proof: Because G(Φ) is a strictly increasing function over the support of Φ, φ∗∗uniquely determined. With regard to φ∗ , note that T (φ) := β(1 − τ )H(φ)/φ − [1 − µ −β(1 − τ )(G(φ) − µ)] is strictly decreasing in (0, η) because over the support, T 0 (φ) =β(1 − τ ) [−H(φ)/φ2 − dG(φ)/dφ] + β(1 − τ )dG(φ)/dφ = −β(1 − τ )H(φ)/φ2 < 0, andin the area other than the support, both H(φ) and G(φ) are constant. In addition,
limφ→0 T (φ) = ∞ and limφ→η T (φ) = − [β(1 − µ)(1 − τ )] < 0. Hence, φ∗ , which isthe solution of T (φ) = 0, is uniquely determined. 
Appendix A.1: Proof of Proposition 1
As shown by Lemma 3, φ∗ is given by the solution of T (φ) = 0, where T (φ) =β(1 − τ )H(φ)/φ − [1 − µ − β(1 − τ )(G(φ) − µ)] is a decreasing function with respect
to φ. Eq. (28) has two steady-state equilibria, φ∗ and φ∗∗ , if and only if φ∗ is strictly
greater than φ∗∗ . This is because the domain of the dynamical system in Eq. (28) is[φ∗∗ , η] because of the free disposability of the bubbly assets. φ∗ is strictly greater thanφ∗∗ if and only if T (φ∗∗ ) > 0 or, equivalently, (1 − µ)φ∗∗ < β(1 − τ )H(φ∗∗ ) because
0 = T (φ∗ ) < T (φ∗∗ ) and T (φ) is a decreasing function with respect to φ. In this case,the asset bubble has a positive value in the steady state φ∗ . At the same time, Eq.
(28) has only one steady-state equilibrium if and only if (1 − µ)φ∗∗ ≥ β(1 − τ )H(φ∗∗ ).

Appendix A.2: Proof of Proposition2
The linear approximation of the dynamical system around a steady state is computedfrom Eqs. (28) and (32), as follows:










 kt+1 − k̂   κ1 (k̂, φ̂) x(k̂, φ̂)   kt − k̂ 

=

,
φt+1 − φ̂
0κ2 (φ̂)φt − φ̂44(A.1)

where (k̂, φ̂)=(k ∗ , φ∗ ) or (k ∗∗ , φ∗∗ ),

κ1 =



Aβ(1 − τ )H(φ̂)
 f 00 (k̂)k̂Ψ(k̂) + f 0 (k̂)Ψ(k̂) + f 0 (k̂)k̂Ψ0 (k̂) ,Ψ(k̂) + k̂Ψ0 (k̂) 1 − µ(1 − β) G(φ̂) − µ1G0 (φ̂)(1 − µ)
κ2 = 
2 ,1 − µ − β(1 − τ ) G(φ̂) − µandβ(1 − τ )H (φ̂) 1 − µ − β(1 − τ )(G(φ̂) − µ) + β 2 (1 − τ )2 H(φ̂)G0 (φ̂)
1
x=
.

2
Ψ(k̂) + k̂Ψ0 (k̂)
1 − µ − β(1 − τ )(G(φ̂) − µ)0The eigenvalues of the local dynamical system associated with Eq. (A.1) aroundthe bubbly steady state, (k ∗ , φ∗ ), are given by

κ1 (k ∗ , φ∗ ) =φ∗ A(f 00 (k ∗ )k ∗ Ψ(k ∗ ) + f 0 (k ∗ )Ψ(k ∗ ) + f 0 (k ∗ )k ∗ Ψ0 (k ∗ )) ,∗∗0∗
Ψ(k ) + k Ψ (k )∗

κ2 (φ ) =G(φ∗ ) − µ+1 .
φ∗ G0 (φ∗ )H(φ∗ ) + (φ∗ )2 G0 (φ∗ )(G(φ∗ ) − µ)Using [kt f 0 (kt )]0 > 0 and 1 = φ∗ Af 0 (k ∗ ), it follows that 0 < κ1 (k ∗ , φ∗ ) < 1 <κ2 (k ∗ , φ∗ ). Thus, the bubbly steady state, (k ∗ , φ∗ ) is a saddle point.
The eigenvalues of the local dynamical system associated with Eq. (A.1) aroundthe bubbleless steady state, (k ∗∗ , φ∗∗ ), are given by
Aβ(1 − τ )H(φ∗∗ )κ1 (k , φ ) =(f 00 (k ∗∗ )k ∗∗ Ψ(k ∗∗ ) + f 0 (k ∗∗ )Ψ(k ∗∗ )
∗∗
∗∗
0
∗∗
(1 − µ)(Ψ(k ) + k Ψ (k ))∗∗∗∗+ f 0 (k ∗∗ )k ∗∗ Ψ0 (k ∗∗ )),κ2 (φ∗∗ ) =(1 − µ)φ∗∗.
β(1 − τ )H(φ∗∗ )From proposition 1, (1 − µ)φ∗∗ ≥ β(1 − τ )H(φ∗∗ ) is satisfied, and from the assumption

45

[kt f 0 (kt )]0 > 0, we obtain 0 < κ1 (k ∗∗ , φ∗∗ ), κ2 (k ∗ , φ∗ ) < 1. Therefore, the bubblelesssteady state (k ∗∗ , φ∗∗ ) is totally stable. 
Appendix B: Proof of Lemma 1
From Eqs. (33) and (35), S(φ∗ ) = φ∗ and S(φ∗∗ ) = β(1 − τ )H(φ∗∗ )/(1 − µ). The
differentiation of S(φ) is given by

S 0 (φ) = T (φ)β(1 − τ )G0 (φ)φ.
[1 − µ − β(1 − τ )(G(φ) − µ)]2

In the above equation T (φ) := β(1 − τ )H(φ)/φ − [1 − µ − β(1 − τ )(G(φ) − µ)], which
is obtained in the proof of Lemma 3. T (φ) is decreasing in (0, η), as shown by Lemma
3. It follows that limφ→0 T (φ) = ∞ and limφ→η T (φ) = − [β(1 − µ)(1 − τ )] < 0.
Moreover, it follows that S 0 (φ∗ ) = 0 holds because T (φ∗ ) = 0. Then, it follows thatS 0 (φ) is positive if 0 < φ < φ∗ and is negative if φ∗ < φ < η. Thus, S(φ∗ ) is a
maximum, and S(φ∗∗ ) < S(φ∗ ). 
Appendix C: Proof of Lemma 2
Because the total income over all entrepreneurs is the total capital income plus thetotal income from holding the bubbly asset, and because the marginal propensity toconsume is equal to 1 − β, the aggregate consumption of entrepreneurs is given as
follows:
Ctp := (1 − β)(1 − τ )(ρt Zt + rt Bt−1 ).

(C.1)

Using Eqs. (12) and (26), we obtain the after-tax aggregate income from savings in
the economy as (1−τ )(ρt Zt +rt Bt−1 ) = (1−τ )(1−µ)ρt Zt /[1−µ−β(1−τ )(G(φt )−µ)].
Using Eqs. (13), (25), and (30), we have ρt Zt = f 0 (kt )kt Ψ(kt )N . Thus, the aggregate46

consumption of entrepreneurs is obtained as follows:

Ctp =(1 − β)(1 − τ )(1 − µ)f 0 (kt )kt Ψ(kt )N
1 − µ − β(1 − τ )(G(φt ) − µ))= (1 − β)(1 − τ )Γ(φt )f 0 (kt )kt Ψ(kt )N,(C.2)

where Γ(φ) = (1 − µ)/[1 − µ − β(1 − τ )(G(φ) − µ)]. Substituting the steady-statevalues into Eq. (C.2) yields Eqs. (37) and (38). Then, it is straightforward to obtain
the aggregate consumption of workers, as follows:

Ctw = (1 − τtw )(wt (1 − ut )N + γt ut N ).

(C.3)

The aggregate tax revenue from entrepreneurs is given by τ (ρt Zt + rt Bt−1 ) = τ (1 −µ)ρt Zt /[1−µ−β(1−τ )(G(φt )−µ)]. From Eqs. (14), (17), (22), ρt Zt = f 0 (kt )kt Ψ(kt )N ,and the aggregate tax revenue from entrepreneurs, Eq. (C.3) can be rewritten as

Ctw =


Θ(f (kt ) − f 0 (kt )kt ) +τ (1 − µ)0f (kt )kt Ψ(kt )N
1 − µ − β(1 − τ )(G(φt ) − µ)= [Θ(f (kt ) − f 0 (kt )kt ) + τ Γ(φt )f 0 (kt )kt ] Ψ(kt )N.

(C.4)

Substituting the steady-state values into Eq. (C.4) yields Eqs. (39) and (40). 
Appendix D: Steady states in the numerical analysis
Under the functional form setting in Section 5, each macroeconomic variable can becomputed as follows. In the bubbly steady state, we have1k ∗ = (φ∗ AαĀ) 1−α ,u =1− 1−∗ζ
(1 − Θ)(1 − α)Ā(k ∗ )α47

σ  σ1
,



∗

θ =

(1 − u∗ )σ1 − (1 − u∗ )σ1/σ,β(1 − τ )(φ∗ − µ)αĀ(k ∗ )α (1 − u∗ )N,B =∗
1 − µ − β(1 − τ )(φ − µ)∗C p,∗ = (1 − β)(1 − τ )Γ(φ∗ )αĀ(k ∗ )α (1 − u∗ )N,andC w,∗ = [Θ(1 − α)Ā(k ∗ )α + τ Γ(φ∗ )αĀ(k ∗ )α ](1 − u∗ )N,
where k ∗ , u∗ , B ∗ , C p,∗ , and C w,∗ are the capital per worker, unemployment rate, total value of bubbly assets, aggregate consumption of entrepreneurs, and aggregate
consumption of workers, respectively.
Additionally, in the bubbleless steady state, we have

k

∗∗

u

∗∗


=

β(1 − τ )(1 − µ2 )AαĀ
2(1 − µ)
=1− 1−

θ

∗∗


=

1
 1−α

,

ζ
(1 − Θ)(1 − α)Ā(k ∗∗ )α(1 − u∗∗ )σ
1 − (1 − u∗∗ )σσ  σ1,

1/σ,

C p,∗∗ = (1 − β)(1 − τ )αĀ(k ∗∗ )α (1 − u∗∗ )N,andC w,∗∗ = [Θ(1 − α)Ā(k ∗∗ )α + τ αĀ(k ∗∗ )α ](1 − u∗ )N.

Appendix E: Derivation of the indirect lifetime utility
The income of an entrepreneur is given by It = ρt AΦt−1 it−1 + rt bt−1 . From theoptimization problem, we obtain the law of motion of It as It+1 = β(1 − τ )Rt+1 It and,thus, the entrepreneur’s optimal consumption is ct = (1 − β)(1 − τ )It . We derive theexpected indirect lifetime utility by the guess-and-verify method. We guess the form48

of the expected indirect utility function as V (It ) = m ln It + n, where we assume thatm and n are certain constants because the economy is in a steady state. Using the
Bellman equation, V (It ) = ln[(1 − β)It ] + βEt V (It+1 ), and It+1 = β(1 − τ )Rt+1 It , weobtain the following expression:m ln It + n = ln[(1 − β)(1 − τ )It ] + mβ ln It + mβ ln[(1 − β)(1 − τ )] + nβ + mβEt ln Rt+1 .
(E.1)
Et ln Rt+1 can be calculated as follows:
Z 1



ρt+1 AΦt − rt+1 µln max rt+1 ,dΦt
Et ln Rt+1 =1−µ0
Z 1 
ρt+1 AΦt − ρt+1 Aφt µ
ln max ρt+1 Aφt ,
=
dΦt
1−µ
0

Z 1 
Z φt
ρt+1 AΦt − ρt+1 Aφt µlnln [ρt+1 Aφt ] dΦt +
=
dΦt
1−µ
φt
0


1 − µφt.
= ln(ρt+1 A) + µφt ln φt − (1 − φt ) + (1 − µφt ) ln
1−µ

(E.2)

Using Eqs. (E.1) and (E.2), we have

m=

1
,
1−β

and


ln (1 − β)1−β β βln(1 − τ )β++[ln(ρA) + µφt ln φt − (1 − φt )]n =2
2(1 − β)(1 − β)(1 − β)2β
1 − µφt+(1 − µφt ) ln,
(1 − β)2
1−µ49

where φt = φ∗ or φ∗∗ and ρ is the steady-state value of ρt . Using m and n above, wehave the expected indirect lifetime utility of the entrepreneur, V (It ), as follows:





ln (1 − β)1−β β β
1 − µφt1ln(1 − τ )βln It +
(1 − µφt ) ln
V (It ) =++1−β
(1 − β)2(1 − β)2(1 − β)2
1−µβ+[ln(ρA) + µφt ln φt − (1 − φt )] .
(E.3)
(1 − β)2
By omitting the time subscripts from Eq. (E.3), we have Eq. (41), with φt = φ∗ orφ∗∗ .
Next, we derive the lifetime utility functions of the employed and unemployedworkers in a steady state. Using Eqs. (15) and (16), we obtainUtw,e = wt (1−τtw )+

∞ 
X


β˜s [θt+s q(θt+s )wt+s (1 − τtw ) + (1 − θt+s q(θt+s ))γt+s (1 − τtw )] ,

s=1

and

Utw,u = γt (1−τtw )+

∞ 
X


β˜s [θt+s q(θt+s )wt+s (1 − τtw ) + (1 − θt+s q(θt+s ))γt+s (1 − τtw )] .

s=1

From γt = γwt , in a steady state, the above equations can be written as follows:

U

w,e

U

w,u


= 1+


β
β
θq(θ) +
(1 − θq(θ))γ w(1 − τ w ),
1−β
1−β

and



β
β
θq(θ) +
(1 − θq(θ))γ w(1 − τ w ),
= γ+
1−β
1−β

where γ, θ, τ w , and w are the steady-state values.

50