Discussion Paper No.314

Wage Profiles and Income Inequality among
Identical Workers: A Simple Formalization

Akiomi Kitagawa

March 11, 2014

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANWage Profiles and Income Inequality among
Identical Workers: A Simple Formalization
Akiomi KITAGAWA∗

Abstract
This paper explores the implications of the possible bankruptcies offirms for their own wage schemes and the structure of the labor market using a two-period general equilibrium model. The bankruptcy
risk flattens the wage profile of each firm, weakening its incentive effect and thereby making room for the eﬃciency wage to be used asa supplementary incentive device. This, in turn, stratifies the labormarket into the primary market, in which job rationing is observed,
and the secondary market, in which job rationing is not observed. A
substantial utility diﬀerential emerges between those who have founda job in the primary market and those who have not, even though
there is no diﬀerence in their innate abilities. This diﬀerential widens
as bankruptcies become more likely. Moreover, the dual structure of
the labor market renders the employment size in the primary market
too small to attain a social optimum.
Keywords: wage profile, eﬃciency wages, dual labor markets.
JEL classification: J31, J41, J65∗

Graduate School of Economics and Management, Tohoku University, 21-1, Kawauchi,
Aoba-ku, Sendai, 980-8576, JAPAN. E-mail: kitagawa@econ.tohoku.ac.jp. I thank Yuji
Genda, Akihisa Shibata, Makoto Saito, Ryo Horii, and the other seminar participants at
Hitotsubashi, Osaka and Nihon Universities and the 2006 autumn meeting of the Japanese
Economic Association for their helpful comments on earlier versions of this paper. This
work was supported by JSPS KAKENHI Grant Number 23330060.

1

1

Introduction

Motivating hired workers is one of the most important tasks assigned to thosein charge of business management. The diﬃculty in accomplishing this task
depends on various factors, but when workers’ performance is either directly
or indirectly observable, an employer can motivate workers by paying wages
according to their observed performance. This system is simple and eﬀective,
and thus has been used in many workplaces. The piece rate system applied
to insurance agents and taxi drivers is a well-known example of this practice;others include the systems of seniority wage and severance pay applied tosalaried workers. As Lazear (1979, 1981) argued, in eﬀect, the latter systems
force workers to deposit part of the wages they earn in the early stages of
their career with the employer, with the condition that it will be refunded
if their performance is satisfactory.
For these systems to work well, some conditions must be satisfied. First,
workers’ performance must be observable by the employer since these systems make the wage payment dependent on it. Their performance need not
be observable at any moment, but must become observable by the end of
the employment period. Second, the employer must not go bankrupt. Whenthe employer goes bankrupt with some positive probability, the workers inescapably bear the risk of not receiving their wages, which seriously weakens
the motivating eﬀect of these systems. If one of these conditions is not satisfied, neither the seniority wage system nor the severance pay system caneﬀectively motivate workers any longer. In such a situation, the employer isforced either to implement an additional incentive device or to switch fromthese systems to a totally diﬀerent one.
This paper explores the implications of the possible bankruptcies of firms
for their own wage schemes and the structure of the labor market in theframework of a two-period general equilibrium model. In this model, theeconomy consists of “capital good” and “consumption good” sectors. In thecapital good sector, firms produce the capital good from labor services on a
one-to-one basis. They can observe the performance of hired workers withoutcost, thus paying a competitive wage only to workers who have delivered thedesired performance. In the consumption good sector, firms produce theconsumption good from the capital good and labor services subject to thefollowing technological constraints. First, they must input labor services
in the first period, although the output is obtained in the second period.
Second, they cannot observe the performance of hired workers until thesecond period. Third, a substantial percentage of the firms that have hired
workers cannot output the consumption good in the second period; however,in the first period, no one knows which firms will become unproductive in
the next period. The third constraint means that with a positive probability,each firm in the consumption good sector goes bankrupt, and thus, its hired
workers cannot receive their wages. No insurance against this “bankruptcy
2

risk” is sold in this economy.
The presence of the bankruptcy risk gives special weight to the second
constraint that the performance of hired workers becomes known with a
one-period lag. When the risk is absent, this constraint does not matter tofirms in the consumption good sector since they can choose to pay the entire
amount of wages in the second period, thereby motivating hired workers. In
contrast, when the risk is present, it poses the following agency problem tothese firms: they are forced to “flatten” the wage profile—to pay a portion of
the wages in the first period—because their workers desire to save it to protect themselves against the bankruptcy risk. Under the second constraint,
however, this payment gives a moral hazard incentive to the workers. Since
their performance is not observable in the first period, the wage payment inthat period must be made unconditionally. They may make bad use of it,
working not for their original employer but for a firm in the capital good
sector, thereby receiving wage revenues from both employers. Of course,such behavior is known to the original employer in the second period, andthus, the cheating workers cannot receive severance pay, that is, the wageotherwise paid in the second period. However, if the lifetime utility attainedby this cheating behavior is higher than that by honest behavior, the workers choose to cheat their original employer. To prevent this moral hazard,
the firms must pay a suﬃcient amount of severance pay, thereby keeping
the lifetime utility attained by honest behavior higher than that by cheating
behavior.
Such responses by the firms in the consumption good sector cause asegmentation of the labor market. The bankruptcy risk induces them to
pay more wages, which makes the lifetime utility of their workers far higher
than that of those working in the capital good sector while reducing thetotal employment of the consumption good sector. As a result, jobs in the
consumption good sector are rationed among applicants, and those who failto obtain these jobs must then work for firms producing the capital good.
Put another way, the labor market is stratified into the “primary” market,
where job rationing is observed, and the “secondary” market, where job
rationing is not observed. The utility diﬀerential between those who havefound jobs in the consumption good sector and those who have not widens
as bankruptcies become more likely. It is also shown that the dual structureof the labor market reduces the total employment of the consumption good
sector to an ineﬃciently low level. Given the utility diﬀerential mentioned
above, the hiring decision of a firm in that sector has a substantial eﬀect
on the lifetime utility of the hired worker since, if not hired by this firm,
she has no choice but to work in the capital good sector, accepting a low
level of lifetime utility. However, no firms in the consumption good sector
consider this external eﬀect when making a hiring decision, which renders
their employment size too small to attain a social optimum.
It must be emphasized that the model of this paper bears characteristics3

of both the incentive theory of wage profiles developed by such authors
as Lazear (1979, 1981) and the eﬃciency wage theory developed by such
authors as Shapiro and Stiglitz (1984); these have been viewed as opposing
theories on motivating workers in the literature. On the one hand, themodel of this paper yields the result that severance pay is used to motivate
workers, which is a characteristic of the incentive theory of wage profiles.
On the other hand, it yields the result that the total wage payment by
a firm in the consumption good sector substantially exceeds the marketclearing level, which is a characteristic of the eﬃciency wage theory. The
bankruptcy risk unites within the model the characteristics of these two
distinct theories in such a manner that it flattens the wage profile, weakening
its incentive eﬀect and thereby making room for the eﬃciency wage to be
used as a supplementary incentive device. Such an eﬀect of the bankruptcy
risk immunizes the present model against the criticism of Carmichael (1985).
He criticizes the eﬃciency wage theory for neglecting the possibility thatworkers can buy a job in the primary labor market by paying an “entrancefee.” However, such an oﬀer is never accepted by firms in the present model
since it is equivalent to reducing the wage payment in the first period, which
seriously weakens the morale of workers.
It should also be noted that the flatter wage profile (which, in the modelof this paper, results from an increased bankruptcy risk) is not merely a theoretical possibility, but a reality for some economies. For example, Hamaaki,
Hori, Maeda, and Murata (2012) analyze a dataset of the wage structure
in Japan and find that the age–wage profile has been gradually flattening
since the burst of the Japanese economic bubble in the 1990s, especially for
employees in the middle or final phase of their career. Hori and Iwamoto(2012) confirm this finding using a diﬀerent dataset.1 Behind their findings, the mechanism stated above might play a role because there is no
doubt that after experiencing the burst of that bubble, increasingly fewer
Japanese businesspeople can have good future prospects.
The rest of this paper is organized as follows. Section 2 presents themodel. Section 3 considers an optimal wage scheme that solves the agency
problem for firms in the consumption good sector. Section 4 examines thegeneral equilibrium of this model using analytical and numerical methods.
Section 5 adduces a Pareto-improving policy to establish an allocative inefficiency of the general equilibrium. Section 6 concludes.

1
Kambourov and Manovskii (2009) and Kong and Ravikumar (2012, 2013) also reportthat in the US economy, workers who have recently entered the workforce experiencea slower increase in wages than those who entered it many decades ago, although they
investigate not only the wage profiles of those who are guaranteed lifelong employment
but also the earnings profiles of those who switch jobs several times during their careers.

42

The Model

Time is discrete. The economy operates during two periods. At the beginning of the first period, a continuum of identical two-period-lived workers ofmeasure one are born. For these workers, the first and second periods are,respectively, their young and old ages. They are endowed with one unit ofindivisible labor service in the first period and want to consume in the second
period. Their lifetime utility depends solely on their old-age consumption,
which is formalized as
U = log C,(1)where U and C denote, respectively, the ex-post level of lifetime utility andthe amount of old-age consumption. Thus, in the first period, every worker
hires out the endowed labor service and saves all the received income for her
old age.
In both periods, a single kind of final good is competitively produced.
The final good can be consumed in the same period in which it is produced.
If it is carried over to the next period, it can be also used as “physicalcapital.” Under the assumption that workers consume only in the second
period, all of the final good produced in the first period is carried over to thesecond period and used as physical capital, whereas all of the good producedin the second period is consumed within that period. Thus, we call the final
good produced in the first period the “capital good,” and that produced in
the second period the “consumption good.”
There are two technologies available for the production of the final good.
The first technology turns labor services into the final good in the first period
on a one-to-one basis. It is formulated as
Y S = LS ,(2)where Y S and LS denote, respectively, the output of the final good andthe input of labor services. Firms using this technology can monitor the
performance of hired workers without cost, so hired workers receive wages
equal to the marginal product of labor:
wS = 1.

(3)

These firms constitute the “capital good sector” of this economy.
The second technology turns capital and labor services into the final
good in the second period. It is formulated as( )1−α,Y P = AK α LP

(4)where Y P denotes the output of the final good in the second period, A the
total factor productivity, K the input of physical capital, LP the input of5

labor services, and α a constant satisfying α ∈ (0, 1). We assume that the
input of labor services must be done in the first period, while that of physicalcapital must be done in the second period, which diﬀerentiates (4) from thenormal type of Cobb–Douglas technology that assumes the simultaneous input of capital and labor services. We make the following assumptions about
firms that use this technology, which constitute the “consumption good sector” of this economy. First, the input level of physical capital that a firm
can choose is either 0 or 1, which ensures that firms yielding the final goodin the second period are of the same size. It is also assumed that these firms
can procure physical capital at a price of r in the competitive market that
opens in the second period and that physical capital fully depreciates once
inputted into the production process. Second, firms using this technology
cannot know the performance of hired workers until the second period. Thisimplies that if a firm and a worker have written a contract in which wage
payments are made in both periods, then the payment in the first period
must be made unconditionally. Third, a percentage θ (∈ (0, 1]) of firms using
this technology find at the beginning of the second period that their totalfactor productivities take a unitary value, that is, A = 1, while the others
find themselves unproductive, that is, A = 0 (although, in the first period,no one knows which firms become unproductive in the next period). In thefollowing, we express the state of a firm that turns out to be unproductivein the second period as “bankruptcy.” The possibility of bankruptcy forces
the workers hired by such firms to bear the risk of not receiving the portionof their wages that is supposed to be paid in the second period. We callthis the “bankruptcy risk,” and we assume that no insurance against thebankruptcy risk is sold in this economy.2
Next, consider a young worker who has received a certain amount of the
final good in exchange for the endowed labor service. She can save for her
old age by exercising either of the following two options. The first option
is to carry the received good over to the next period and then to sell it as
the capital good in the competitive market. The gross rate of return on
this option is r. The second option is to lend it at the gross interest rateof R to firms in the consumption good sector, which need the final good forfirst-period wage payments. Although each firm in the consumption good
sector goes bankrupt and defaults on its debts with probability 1 − θ, theworker can secure the gross rate of return on this option at θR by suﬃciently
diversifying among borrowers. As a result of the interest arbitrage behavior,

2

Some readers may think that this assumption is not justifiable because the bankruptcy
risk is idiosyncratic and thus insurable, at least in theory. However, there are someconceivable situations in which this assumption can be justified. For example, wheninsurance companies cannot at all track the working records of insured workers, they
never provide insurance against the bankruptcy risk since they cannot determine whoshould receive insurance money.

6

these rates of return are equal in equilibrium:r = θR.

(5)

Thus, if this worker is hired by a firm in the capital good sector, then her
lifetime utility is given by
U S = log rwS = log r.

(6)

If she is hired by a firm in the consumption good sector, then her (expected)
lifetime utility is given by
U P = θ log(rw1P + w2P ) + (1 − θ) log rw1P

(7)

where w1P and w2P denote, respectively, the wage paid in the first period andthat paid in the second period (i.e., severance pay).
When the bankruptcy risk is present, that is, θ < 1, both w1P and w2P
take a positive value, which can be explained as follows. In such a case,workers hired by a firm in the consumption good sector eagerly desire wage
payment in the first period. Although they are forced to bear the risk of notreceiving severance pay with probability 1 − θ, no insurance against this risk
is provided in this economy. All they can do to protect themselves against
this risk is to save the wage paid in the first period, thereby securing theirold-age consumption in the case of bankruptcy. Put another way, if a firmin the consumption good sector decides to make no wage payment in thefirst period, that is, w1P = 0, then the expected utility of a worker employedby this firm becomes a negative infinity, that is, U P = −∞ (See (7)). Since
the lifetime utility of a worker employed in the capital good sector is strictly
higher than a negative infinity, that is, U S = log r > −∞ (See (6)), this firm
finds no job applicant as long as it maintains a zero wage payment in thefirst period. For a firm in the consumption good sector, it is not optimal to
pay the whole amount of wage in the first period either. As stated above,
the wage payment in the first period is unconditionally made since the firmcannot observe the performance of hired workers in that period. Thus, if thelabor contract stipulates that all of the wage must be paid in the first period,
hired workers have an incentive to work not for their original employer, but
for a firm in the capital good sector, and to thus receive wages from both
employers. To prevent such a moral hazard, some portion of wages must be
paid in the second period. We will explore these points further in the next
section.

3

Optimal Design of the Wage Scheme

As pointed out in the previous section, the wage payment to a worker employed in the consumption good sector is usually made in installments. This7

section examines how the levels of payment in the first period, w1P , and thesecond period, w2P , are determined.
Let us first consider the problem of moral hazard on the part of hired
workers. As described in the previous section, workers hired by a firm producing the consumption good have an incentive to make bad use of the contract stipulation that the wage payment in the first period must be madeunconditionally. Specifically, they work not for their original employer but
for a firm producing the capital good, thereby receiving not only the wage
payment from the original employer, w1P , but also extra income from thefirm in the capital good sector, wS = 1. Of course, such behavior becomes
known to the original employer in the second period, and thus, severance
pay is not paid to the cheating workers. However, by saving this income,the cheating workers can attain a lifetime utility equal toU M = log r(1 + w1P ).

(8)

If this utility dominates that attained by honest behavior, U P , all workers
choose to cheat their original employer. In other words, firms in the consumption good sector must design their wage scheme in such a manner thatU P ≥ U M or, equivalently,
θ log(rw1P + w2P ) + (1 − θ) log rw1P ≥ log r(1 + w1P ).

(9)Otherwise, they cannot prevent the moral hazard of their hired workers.
Given the interest rate R and the market price of the capital good r,
firms in the consumption good sector maximize their profit subject to (9)by optimally choosing the number of workers, LP , the wage paid in the first
period, w1P , and severance pay, w2P . This problem can be formulated asmax

LP , w1P , w2P(LP )1−α − r − (Rw1P + w2P )LP(10)s.t. θ log(rw1P + w2P ) + (1 − θ) log rw1P ≥ log r(1 + w1P ).
In solving this problem, we should bear the following two points in mind.
First, the above formulation does not reflect the possibility of bankruptcy.
Firms need not consider this possibility because in the case of bankruptcy,
they earn zero profit, being exempted from the repayment of debts becauseof limited liability. Second, in the formulation of (10), the input of physicalcapital is preset at K = 1. Indeed, it is conceivable that profit takes anegative value for any possible combination of LP , w1P , and w2P , and thus,
productive firms may decide to set the input of physical capital at zero.
However, this cannot be an equilibrium phenomenon because it means zeroproduction of the consumption good.

8

The first-order conditions of (10) are derived as(1 − α)(LP )−α − Rw1P − w2P = 0,
]
r
r
r
P
−RL + λ θ P+ (1 − θ) P −≤ 0,
rw1 + w2P
rw1
r(1 + w1P )
{
[
]}
r
r
r
−RLP + λ θ P
+
(1
−
θ)
−
w1P = 0,
rw1 + w2P
rw1P
r(1 + w1P )[−LP + λθ/(rw1P + w2P ) = 0,

(11)
(12)
(13)
(14)

where λ is the Lagrangian multiplier associated with the incentive compatibility constraint (9).3 The optimal employment size and optimal wage
scheme are obtained by solving (9), (11)–(14) with respect to LP , w1P , w2P ,and λ.
Conditions (9), (11)–(14) suggest that the optimal wage scheme varies
depending on whether or not the bankruptcy risk is present. When thebankruptcy risk is absent, that is, θ = 1, conditions (9), (11)–(14) are easily
solved as
w1P = 0,w2P = r,LP = [(1 − α)/r]1/α ,(15)which imply that the whole amount of the wage is paid in the second period.
This result is not surprising since the wage payment in the first period is
made as insurance against the bankruptcy risk. In this case, the lifetime
utility of a worker employed in the consumption good sector is derived as
U P = log r, and thus, there is no diﬀerence in lifetime utility between theworkers employed in the capital good sector and those working in the consumption good sector. In contrast, when the bankruptcy risk is present,that is, θ < 1, w1P never takes a zero value because if w1P = 0, the incentive compatibility constraint (9) does not hold. In this case, the workers
employed in the consumption good sector are better oﬀ than those working
in the capital good sector. As implied by (9), their lifetime utilities areequal to log r(1 + w1P ), which is strictly higher than U S = log r. Naturally,
all of the workers desire to be hired by a firm producing the consumption
good, although this desire is fulfilled for only a limited number of workers.
In practice, jobs in the consumption good sector are rationed among applicants, and those who fail to obtain these jobs must then work for firms
producing the capital good.4 In other words, the presence of the bankruptcy
The first-order condition with respect to w1P must be given by (12) and (13) because
the optimal choice of w1P can be a corner solution, that is, zero, in the absence of thebankruptcy risk. No such possibility arises with LP or w2P .
4
In contrast to the argument by Carmichael (1985), the workers in this economy cannot
buy a job in the consumption good sector by paying an “entrance fee.” In this model, thepayment of an entrance fee is equivalent to the reduction of the wage payment in the first
period, w1P , which makes it diﬃcult for the incentive compatibility constraint (9) to hold.
Thus, every firm in that sector rejects such an oﬀer.
3

9

risk stratifies the labor market of this economy into the “primary” market,
where job rationing is observed, and the “secondary” market, where job
rationing is not observed.

4

General Equilibrium

To define the general equilibrium of this economy, we need to evaluate thefirst-order conditions for the firms in the consumption good sector from an
economy-wide perspective. Let l denote the percentage of workers who are
hired by firms in the consumption good sector. Using this notation, we can
express the capital–labor ratio of the consumption good sector as (1 − l)/θl.
Since all of the firms in this sector are identical, their capital–labor ratios
coincide with that value:
1/LP = (1 − l)/θl.

(16)

Using (5) and (16), we can rewrite (11)–(14) as(1 − α)[(1 − l)/θl]α − (r/θ)w1P − w2P = 0,

(17)

w2P (1 − θ − θw1P ) − r(w1P )2 = 0.

(18)

Then, we define the general equilibrium of this economy as follows.
Definition 1. Given the values of α and θ, the general equilibrium of thiseconomy is a combination of w1P , w2P , l, and r that satisfies (9), (17), (18),andα[(1 − l)/θl]α−1 − r = 0.

(19)

Condition (19) requires that the marginal product of physical capital isequal to its market value in equilibrium. This equalization is attained byfree entry to the consumption good sector. As long as the marginal productof physical capital dominates its price, new entry to this sector continues
since it is profitable to set up a consumption good-producing firm. Suchcontinuous entry decreases the level of l, narrowing the diﬀerence between
the marginal product and the price to zero.5
We examine the general equilibrium of this economy by taking the following steps. First, we analyze a case in which the bankruptcy risk is absent. This case serves as a benchmark in analyzing the cases in which thebankruptcy risk is present. Then, we move to risky cases in which each firmin the consumption good sector goes bankrupt with some positive probability, focusing on the responses of the equilibrium values of w1P , w2P , l, and r
to a change in θ.
5

As the consumption good sector contains increasingly more firms, increasingly fewer
workers are hired by that sector. This paradoxical result can be explained by the fact thatan increase in the number of consumption good-producing firms necessitates the increased
production of physical capital, and thus, employment increases in the capital good sector.

10

4.1

The Riskless Case

When the bankruptcy risk is absent, that is, θ = 1, firms in the consumption
good sector set their wage scheme as w1P = 0 and w2P = r, as seen in theprevious section. Under this wage scheme, conditions (17) and (19) jointly
produceα[(1 − l)/l]α−1 = (1 − α)[(1 − l)/l]αor, equivalently,
l = 1 − α.

(20)

By substituting (20) into (19), we obtainr = w2P = αα (1 − α)1−α .

(21)In this case, there is no diﬀerence in lifetime utility among workers, that is,
U P = U S = log αα (1 − α)1−α .

4.2

Risky Cases

When the bankruptcy risk is present, that is, θ < 1, the general equilibrium
can be obtained through the following procedure. Equations (9) and (18)
produce()θ−1 ()θ(1 − θ)θ w1P 1 + w1P/ 1 − θ − θw1P = 1,(22)which uniquely determines the equilibrium value of w1P .6 On the other hand,equations (17) and (19) produce
α
1−θ
w1P
1−l
=
·
·
.
θl
1−α
θ
1 − θ − θw1P

(23)

By substituting the equilibrium value of w1P into (23), we can obtain thevalue of l, which, combined with (19), determines the value of r. Finally,by substituting the equilibrium values of w1P and r into (18), we can obtainthe value of w2P .
Proposition 1. Let w1P (θ), l(α, θ), and r(α, θ) denote the equilibrium valuesof w1P , l, and r, respectively. Then, for ∀(α, θ) ∈ (0, 1) × (0, 1),
dw1P /dθ < 0,∂l/∂θ > 0,∂r/∂θ > 0,(24)

The uniqueness of the equilibrium value of w1P is established by the fact that the LHSof (22) is an increasing function of w1P ∈ [0, (1 − θ)/θ), which takes a zero value at w1P = 0
and approaches a positive infinity as w1P → (1 − θ)/θ.
6

11

1
0.9
0.8
0.7

∂w2P
<0
∂θ

0.6

α

0.5
0.4
0.3
0.2

∂w2P
>0
∂θ

0.1
0

0

0.2

0.4

θ

0.6

0.8

1

Figure 1: The sign of ∂w2P /∂θ
withlim w1P (θ) = +∞,θ→0lim l(α, θ) = 0,

θ→0lim r(α, θ) = 0,θ→0lim w1P (θ) = 0,(25)

lim l(α, θ) = 1 − α,(26)

lim r(α, θ) = αα (1 − α)1−α .

(27)

θ→1

θ→1
θ→1

Proof. See Appendix.
This proposition asserts that the equilibrium value of w1P is decreasing
with θ from a positive infinity to zero, and that those of l and r are increasing
with θ from zero to 1−α and αα (1−α)1−α , respectively. As θ takes a smaller
value or, equivalently, the production of the consumption good becomesriskier, firms producing that good choose to hire fewer workers and pay
them more in the first period. A smaller value of θ also lowers the price ofthe capital good since the employment reduction in the consumption good
sector necessarily leads to the same scale of employment expansion in thecapital good sector, and thus to the increased production of that good.
While the signs of dw1P /dθ, ∂l/∂θ, and ∂r/∂θ are determined as in (24),that of ∂w2P /∂θ may take either a positive or negative value depending on
α and θ, as illustrated in Figure 1. The line drawn in that figure, on which∂w2P /∂θ takes a zero value, divides the unit square of the possible combinations of α and θ into two regions: w2P is decreasing with θ in the upper
region and increasing with θ in the lower region. This contrast between the
12

1

0.5

0.9

1

0.8

0.7

0.6

α

0.5

2

0

3

0.4

4

0.3

5

−1

0.2

−2
0.1

0

−5

−10
0

0.1

0.2

10

−3

50
0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

Figure 2: Contours for wage profiles
two regions can be explained by the diﬀerential eﬀects of a given increase inθ on U P and U M . As shown in Proposition 1, such an increase in θ lowers
the level of w1P and raises that of r, thereby changing the values of U P andU M in such a manner that their equality cannot be maintained unless thelevel of w2P is changed. Because the level of w2P is set to restore that equality,
it is decreased to reduce the value of U P if the increase in θ makes U P larger
than U M , and it is increased if the increase in θ makes U P smaller than U M .
The first type of response is observed in the upper region of Figure 1, andthe second type is observed in the lower region.
Numerical methods also enable us to examine how a wage profile for theworkers employed in the consumption good sector, which is measured by thediﬀerence between the wages paid in the first and second periods—that is,
w2P − w1P —is determined for a given pair of α and θ. Figure 2 depicts thecontours for the wage profiles, each of which consists of combinations of αand θ that produce a given value of w2P − w1P . Specifically, the contour nconsists of combinations of α and θ that support w2P − w1P = n. As shownin Figure 2, the wage profiles for the workers employed in the consumption
good sector are front-loading—that is, w2P − w1P < 0—when both α and θ
are small enough, and they are back-loading—that is, w2P − w1P > 0—in
other cases. The value of w2P − w1P is not always increasing with θ, but
13

1
0.9

∂U P
<0
∂θ

0.8
0.7
0.6

α 0.5
∂U P
>0
∂θ

0.4
0.3
0.2
0.1
0

0

0.2

0.4

0.6

0.8

1

θ
Figure 3: The sign of ∂U P /∂θ
when θ is as large as θ > 0.5, a small increase in θ leads to an increased
value of w2P − w1P for any value of α. Put another way, if the firms in the
consumption good sector perceive a small reduction in θ from such a highlevel, they oﬀer a flatter wage profile to their job applicants.
Proposition 2. Let U P (α, θ) and U S (α, θ) denote, respectively, the equilibrium values of U P and U S . Then, for ∀(α, θ) ∈ (0, 1) × (0, 1), the following
are true:[]∂U S /∂θ > max ∂U P /∂θ, 0 , U P (α, θ) > U S (α, θ).
(28)
Proof. See Appendix.
This proposition asserts that when the bankruptcy risk is present, job
rationing occurs at the entrance of the consumption good sector, and thus,
the workers employed in the consumption good sector are better oﬀ thanthose working in the capital good sector.
Proposition 2 also implies that the bankruptcy risk diﬀerentially aﬀects
the welfare of the workers employed in the capital good sector and that ofthose working in the consumption good sector. Specifically, a smaller valueof θ invariably worsens the welfare of those working in the capital good
sector while worsening less (or even improving in some cases) the welfareof those working in the consumption good sector, as illustrated in Figure
3. This result is paradoxical since the bankruptcy risk is more detrimental
to those working in the capital good sector, who seem less subject to its
14

influence, than to those working in the consumption good sector, who seem
more subject to its influence. This paradox can be explained by the factthat the risk has two opposing eﬀects on the welfare of those working in the
consumption good sector, but only a negative eﬀect on the welfare of thoseworking in the capital good sector. When the risk is present, the firms in the
consumption good sector are compelled to pay a portion of the wage in thefirst period. This gives a moral hazard incentive to their employed workers,
which, in turn, induces these firms to oﬀer a suﬃcient amount of severance
pay to prevent it. Undoubtedly, this sequence contributes to improving thewelfare of those working in the consumption good sector, although it reduces
that sector’s total employment. This reduction expands employment in thecapital good sector on the same scale, increasing the production of that
good and lowering its market price. The lowered price of the capital good
contributes to the worsening of the welfare of all workers since it implies adecreased rate of return on their savings. Thus, those working in the capital
good sector are more heavily damaged by the bankruptcy risk than thoseworking in the consumption good sector.

5

A Pareto-improving Policy

Our final task is to demonstrate that the general equilibrium of this economy fails to attain a constrained eﬃcient allocation by adducing a Paretoimproving policy.
In the previous section, we saw that an increase in the bankruptcy risk
may improve the welfare of those working in the consumption good sector
while invariably worsening the welfare of those working in the capital good
sector. However, on the balance, an increase in the bankruptcy risk negatively aﬀects economic welfare, which is measured by the average value of
lifetime utilities, lU P + (1 − l)U S . This measure coincides with the expectedvalue of the lifetime utility each worker estimates before entering the labormarket. For any value of α, this measure is increasing with θ, implying thatthe ex-ante economic welfare deteriorates as bankruptcies become more andmore likely.
Proposition 3. Let l(α, θ), U P (α, θ), and U S (α, θ) be defined as in Propositions 1 and 2. Then, for ∀(α, θ) ∈ (0, 1) × (0, 1), the following is true:}∂ {l(α, θ)U P (α, θ) + [1 − l(α, θ)]U S (α, θ) > 0.
∂θ

(29)

Proof. See Appendix.
We must note that part of the observed deterioration in economic welfare is unavoidable. This deterioration reflects the reduced production of
the consumption good, which is caused by an increase in the bankruptcy
15

risk. Note that the aggregate production of the consumption good can be
expressed as (1 − l)α (θl)1−α . As is obvious from this expression, a decreasein θ reduces the aggregate output for any given value of l. We call thisthe “direct” eﬀect. In addition, it reduces the aggregate output by lowering
the level of l. As is easily verified, the aggregate output is maximized at
l = 1 − α, although the realized level of l cannot take this value unless θ = 1.
As θ takes a smaller value, the realized level of l deviates from 1 − α, whicheﬀectively reduces the aggregate output. We call this the “indirect” eﬀect.
Of these eﬀects, the direct one is obviously unavoidable. Part of the indirect eﬀect is also unavoidable since it arises from the necessity of providing
hired workers with a chance of self-insurance and preventing their moral
hazard. Therefore, the question is whether the realized level of l is too low
to be justified by these necessities. The answer is in the aﬃrmative because
the following policy can increase the employment of the consumption good
sector, thereby improving economic welfare.
Suppose that the government decides to subsidize firms in the consumption good sector by s per worker. If this subsidy is financed by the lump-sum
tax on the consumption good-producing firms, then the subsidy rate, s, andthe lump-sum tax rate, t, must satisfy
sLP = t,(30)

where LP denotes the number of workers per firm. Under this policy, theprofit of a firm can be written as(LP )1−α − r − (Rw1P + w2P − s)LP − t,which modifies the first-order condition (11) as(1 − α)(LP )−α − Rw1P − w2P + s = 0(31)

and leaves the other conditions unchanged. This implies that the policy is neutral to the equilibrium value of w1P since it is obtained fromthe unchanged conditions (18) and (22). In equilibrium, r = θR and
LP = θl/(1 − l), so (31) can be reduced to(1 − α)[(1 − l)/θl]α = (r/θ)w1P + w2P − s.

(32)

The policy also modifies the equilibrium condition (19) asα[(1 − l)/θl]α−1 = r + t.

(33)

Using k ≡ (1 − l)/θl, we can rewrite (30), (32), and (33) as, respectively,t = s/k,(1 − α)k
αk

α

α−1

=

(r/θ)w1P + w2P − s,

= r + s/k.
16

(34)
(35)
(36)

Combined with (18), conditions (35) and (36) produce(1 − α)k α /s = B/[αB − (1 − α)k] − 1,

(37)

α

(38)

r = k /(B + k),

where B ≡ (1 − θ)w1P /θ(1 − θ − θw1P ). We can treat B as a constant since
the equilibrium value of w1P depends solely on θ.
When the value of s is suﬃciently close to zero, equation (37) has two
distinct roots, k ∗ (s) and k ∗∗ (s)(> k ∗ (s)). As is easily verified, these roots
satisfy
0 < k ∗ (s) < k ∗∗ (s) < [α/(1 − α)]B,∗

dk /ds > 0,
dk ∗∗ /ds < 0,

∗

(39)

lim k (s) = 0,(40)

lim k ∗∗ (s) = [α/(1 − α)]B,

(41)

s→0

s→0

where [α/(1 − α)]B is the capital–labor ratio when s = 0 (See (23)). Whichvalue of k is chosen in equilibrium depends crucially on the value of t. Thatis, if t is set as t = s/k ∗ (s), then k ∗ (s) is realized. However, if t is set as
t = s/k ∗∗ (s), then k ∗∗ (s) is realized. In what follows, we assume that t is
set as t = s/k ∗∗ (s) because it is preferred by all workers.7
When t = s/k ∗∗ (s), the average value of lifetime utilities can be written
as follows:
lU P + (1 − l)U S= lU M + (1 − l)U S= l log r(1 + w1P ) + (1 − l) log r= log r + l log(1 + w1P ))( ∗∗1(k (s))α+ ∗∗log(1 + w1P ).
= log
B + k ∗∗ (s)θk (s) + 1

By diﬀerentiating this with respect to s and then evaluating it at s = 0, we
can obtain
d
θ
dk ∗∗
[lU P + (1 − l)U S ]=−log(1 + w1P )
> 0,
∗∗
2
ds
(θk (s) + 1)ds
s=0
which implies that the government can improve economic welfare by slightly
subsidizing firms in the consumption good sector, and thus, the laissez-faireequilibrium fails to attain a constrained eﬃcient allocation. The above policy
has two conflicting eﬀects on the average value of lifetime utilities. On the
one hand, it lowers the capital good price, thereby reducing the values of U P
and U S . This contributes to decreasing the average value of lifetime utilities.
Workers are better oﬀ when k∗∗ (s) is realized than when k∗ (s) is realized. This isbecause both U P (= log r(1 + w1P )) and U S (= log r) are increasing with r and because
under this policy, r is increasing with k ∈ (0, [α/(1 − α)]B) (See (38)).
7

17

On the other hand, it augments employment in the consumption good sector,
thereby enhancing the probability of a worker being hired by that sector.
This contributes to increasing the average value of lifetime utilities. Whenthe subsidy rate, s, is suﬃciently close to zero, the former negative eﬀect
becomes negligible, while the latter positive eﬀect remains substantial. As
implied by (38), the level of r is maximized at k = [α/(1 − α)]B. Sincek ∗∗ (s) satisfies (41), a suﬃciently small subsidy hardly changes the level ofr, while it enhances the level of l in the first order. This is why this policy
can increase the average value of lifetime utilities and thus improve economic
welfare.
By resorting to a numerical method, we can compute the optimal levelsof tax and subsidy and their eﬀects on employment and economic welfare. In
the form of contours, Figures 4 and 5 summarize, respectively, the levels oftax (i.e., t) and subsidy (i.e., s) that maximize the average value of lifetime
utilities for possible pairs of α and θ. In Figure 4, contours are drawn inincrements of 0.01; the figure suggests that the optimal tax rate is decreasing
with θ unless α takes a suﬃciently small value. When α is suﬃciently small,the optimal tax rate is not a monotone function of θ. It is increasing in somedomains of θ and decreasing in others. In Figure 5, contours are drawn inincrements of 1 because the optimal subsidy rate can take a broad range of
positive values: it becomes as small as 0.0001 when α = 0.99 and θ = 0.01,
and as large as 292.38 when α = 0.01 and θ = 0.99. As suggested by thesecontours, the optimal subsidy rate is decreasing with α and increasing withθ.
The eﬀects of the policy with the optimally chosen tax and subsidy rates
are summarized in Figures 6 and 7, which respectively report the populationof workers who are additionally employed in the consumption good sector
and the increments of the average value of the lifetime utilities attained bythat policy. As shown in these figures, the policy has the biggest eﬀect whenboth α and θ take a suﬃciently small value. To understand why, we need torecall Proposition 1—(26) in particular—which has demonstrated that thepotential number of workers to be hired in the consumption good sector,
1 − α, increases as α takes a smaller value, and that the actual number ofworkers hired in that sector, l(α, θ), decreases as θ takes a smaller value.
These imply that when both α and θ are suﬃciently small, there is greatroom for improvement in the economy since the actual employment in the
consumption good sector is far below that sector’s employment capacity, andthe existence of such a large gap makes it possible for strong eﬀects to beobserved in these cases. Conversely, when either α or θ is suﬃciently large,
there is very limited room for improvement. When α is suﬃciently large, theemployment capacity of the consumption good sector is very small. When θis suﬃciently large, the actual employment in the consumption good sector
is extremely close to capacity. In both cases, the policy cannot be very
eﬀective because there is little room for a sizable increase in employment in
18

0.25

0.01

0.9

0.02
0.8

0.2

0.03

0.7

0.04
0.05

0.6

0.15

0.06

α

0.5

0.4

0.08
0.09
0.1

0.2
0.18
0.17

0.2

0.13
0.16

0.14

0.15

0.05

0.14
0.13

0.1

0.1

0.1

0.11
0.12

0.19
0.3

0.07

0.2

0.3

0.15 0.16
0.4

0.5

0.6

0.7

0.8

0.9

θ

Figure 4: Optimal levels of tax (t)

0.9
250
0.8

0.7

200

0.6

α

150

0.5

0.4
100
0.3

1

0.2

2
3

4

5

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

θ

Figure 5: Optimal levels of subsidy (s)
19

50

0.8

0.01
0.9

0.02

0.8

0.03

0.7

0.04

0.6

0.7

0.6

0.05

0.5

0.06

α

0.5

0.07

0.4

0.08
0.09
0.1

0.4

0.3

0.3
0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

θ

Figure 6: Increased employment in the consumption good sector

2

0.9

1.8

0.8

1.6

0.7

0.01
1.4

0.6

α

0.02
1.2

0.03

0.5

0.04

1

0.05
0.06

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

θ

Figure 7: Welfare gains from the policy
20the consumption good sector.
The argument presented in this section also reveals what makes the equilibrium value of l so small under laissez-faire policy. As a result of the positive spillover eﬀect arising from the dual structure of the labor market, when
there is a utility diﬀerential between those working in the capital good sector
and those working in the consumption good sector, the decision of a firmin the consumption good sector to hire an additional worker unambiguously
contributes to raising the average value of lifetime utilities. Nevertheless,no firms in the consumption good sector take into account this marginalcontribution to economy-wide welfare when making their hiring decisions
because it brings zero benefits to them. Such a gap between private andsocial gains from additional employment renders the total employment ofthat sector too small to attain a constrained eﬃcient allocation. The policy
considered here does a good job of filling that gap by subsidizing the firms
in the consumption good sector.

6

Conclusion

This paper demonstrated that the possible bankruptcies of firms can have asubstantial eﬀect on not only their own wage schemes but also on the structure of the labor market. The bankruptcy risk flattens the wage profile ofeach firm, weakening its incentive eﬀect, and thereby making room for theeﬃciency wage to be used as a supplementary incentive device. The use of
the eﬃciency wage, in turn, stratifies the labor market into the primary market, in which job rationing is observed, and the secondary market, in which
job rationing is not observed. As a result, a substantial utility diﬀerential
emerges between those who have found a job in the primary market andthose who have not, although there is no diﬀerence in their innate abilities.
This diﬀerential widens as bankruptcies become more likely. In addition,
the dual structure of the labor market renders the employment size of the
primary market too small to attain a social optimum.
The results of this paper have a wide range of applications. For instance,
they give an explanation for the observed decline in youth employment inJapan. As reported by Genda (2003), after the bursting of the economic
bubble in the early 1990’s, large companies in Japan became reluctant toexpand and hire, resulting in a substantial decline in youth employment.
While Genda identifies diminishing chances for youths to acquire skills aswell as severe legal restrictions on dismissal as major causes of their displacement from decent jobs, the results obtained here suggest that an increased
possibility of bankruptcy and informational asymmetry have jointly madehiring a new worker more costly and have thereby reduced job opportunities for youths. The results may also explain why large companies, the jobsof which constitute the primary labor market, choose not to substitute the21defined contribution system for the defined benefit system, but rather provide them together for their employees. As reported by Papke, Petersen,
and Poterba (1996), most of the American companies that adopted a 401(k)plan, a defined contribution type of corporate pension in the United States,
also continued to implement their pre-existing defined benefit pension plans.
From the viewpoint of the incentive theory of wage profiles, it is puzzling thatsome companies that seem to have a strong incentive to maintain lifetime employment provided defined contribution pensions for their employees. Since
this is equivalent to an upfront payment of severance pay, according to thistheory, it negatively aﬀects the morale of tenured workers.8 One possible
explanation is that these companies provided the defined contribution planfor their employees as insurance against the possible bankruptcy of their defined benefit plan while motivating them with high wages. This explanationis consistent with the observation of size–wage diﬀerential.

Appendix
Proof of Proposition 1
Proof. Equation (22) can be rewritten as
(

θ
1
−
P
1−θ
w1

)θ/(1−θ) (

1
+1
w1P

)
=1

or
(Ω − Θ)Θ (Ω + 1) = 1,

(42)where Ω ≡ 1/w1P and Θ ≡ θ/(1 − θ). First and most importantly, we needto demonstrate that (42) implicitly defines the function Ω(Θ) : R+ → R
and that Ω(Θ) has the following properties:
0 < Ω(Θ) − Θ < 1,

(43)

′

(Ω(Θ) − Θ) = Ω (Θ) − 1 > 0,(44)

lim (Ω(Θ) − Θ) = lim Ω(Θ) = 0,(45)

lim (Ω(Θ) − Θ) = 1.

(46)

Θ→0

′

Θ→0

Θ→+∞

Let Θ take a given positive value. Then, the LHS of (42) becomes zero
when Ω = Θ, and Θ + 2 when Ω = Θ + 1. Because it is continuously
8

A similar situation has also been observed in Japan. In July 2002, Toyota Motor
Corporation, one of Japan’s leading manufacturers, decided to implement defined contribution pensions together with defined benefit pensions by diverting part of the retirement
benefit toward contributions to this system. Toyota’s decision is puzzling since it seems
to contradict the corporate policy of maintaining lifetime employment, the importance of
which Toyota’s top management has repeatedly emphasized.

22

increasing with Ω, these facts imply that for any value of Θ, the value of Ωis uniquely determined between Θ and Θ + 1, establishing (43). To establish(44), diﬀerentiate both sides of (42) with respect to Θ. Then, we obtainlog(Ω − Θ) + ΘΩ′ (Θ) − 1 Ω′ (Θ)
+
= 0,
Ω−Θ
Ω+1

which can be reduced toΩ′ (Θ) =Θ− log(Ω − Θ) + Ω−Θ>01Θ
Ω−Θ + Ω+1

or, equivalently,Ω′ (Θ) − 1 =1− log(Ω − Θ) − Ω+1Θ1Ω−Θ + Ω+1

1

= Θ

1
log(Ω + 1) − Ω+1Θ1Ω−Θ + Ω+1,

(47)

the second equality of which is obtained from (42). Because (43) implies thatthe denominator of the RHS of (47) is positive, the sign of Ω′ (Θ)−1 coincides
with that of the numerator of the RHS of (47). Since that numerator isincreasing with Ω, we can say that the sign of Ω′ (Θ) − 1 is positive if thefollowing inequality is true:
]
[
1
1
1
1
log(Ω + 1) −= log(Θ + 1) −
> 0.
(48)
Θ
Ω + 1 Ω=Θ Θ
Θ+1
To prove (48), define function f as
f (x) = log(x + 1) −

x
.
x+1

This function satisfies f (x) > 0 for ∀x > 0 because f (0) = 0 and becausef ′ (x) = x/(x + 1)2 > 0 for ∀x > 0. Using function f , we can rewrite the
LHS of (48) as f (Θ)/Θ. Since Θ > 0, this means that (48) is true, thus
establishing (44). Properties (43) and (44) jointly imply that there exist
limit values of Ω(Θ) − Θ when Θ → 0 and when Θ → +∞, and that0 ≤ lim (Ω(Θ) − Θ) < lim (Ω(Θ) − Θ) ≤ 1.
Θ→0

Θ→+∞

To establish (45), suppose thatlim (Ω(Θ) − Θ) = lim Ω(Θ) > 0.

Θ→0

Θ→0

Then, when Θ → 0, (42) is simplified as[]()Θ
1 = lim (Ω(Θ) − Θ)lim Ω(Θ) + 1 = lim Ω(Θ) + 1,
Θ→0
Θ→0
Θ→0
|
{z
}
1

23

which implies that limΘ→0 Ω(Θ) = 0, contradicting our initial hypothesis.
To establish (46), rewrite (42) asΩ(Θ) − Θ = (Ω(Θ) + 1)−1/Θ .
Combined with (43), this equation implies that(Θ + 2)−1/Θ < Ω(Θ) − Θ < (Θ + 1)−1/Θ .
Thus, our desired result will be obtained if we can show thatlim (x + a)−1/x = 1

x→+∞or, equivalently,
lim −(1/x) log(x + a) = 0,

x→+∞

(49)

where a is a given positive number. Let b be another positive fixed number.
Then, for suﬃciently large values of x, the following inequalities must hold:[]log(x + a)
1 x−b
0≥−
≥−
+ log(a + b) ,xx a+b
which suggests that0 ≥ lim −x→+∞[]log(x + a)
1 x−b
1
≥ lim −+ log(a + b) = −.
x→+∞ x a + bx
a+bThis condition requires that (49) be true. Otherwise, the above condition
cannot be true for suﬃciently large values of b.
Now that (43)–(46) have been established, we can use them to prove(24)–(27). From the defining equations of Ω and Θ, we can obtain
(
)
dw1P
d
1
dΘ
Ω′ (Θ)
1
=
·
=−
·
< 0,
(50)
dθ
dΘ Ω(Θ)dθΩ2(1 − θ)2
which is the first condition of (24). In addition, (45) and (46) imply, respectively,
lim w1P = lim (1/Ω(Θ)) = +∞

θ→0

Θ→0

and
lim w1P = lim (1/Ω(Θ)) = lim (1/(Θ + 1)) = 0,

θ→1

Θ→+∞

Θ→+∞

both of which constitute (25). Using Θ and Ω(Θ), we can rewrite (23) as
1−l
α
1
=
·
.
l
1 − α Ω(Θ) − Θ
24

(51)

Diﬀerentiating both sides of (51) with respect to θ yields
1 ∂lα−Ω′ (Θ) + 1 dΘ
− 2·
=
·
·
l ∂θ
1 − α (Ω(Θ) − Θ)2 dθor∂lΩ′ (Θ) − 1
α
l2
·
> 0,
=
·
2
∂θ
1 − α (1 − θ) (Ω(Θ) − Θ)2which is the second condition of (24). In addition, (26) is established by(45), (46) and (51). Specifically, (45) implies thatαlim

Θ→0 1 − α

·

1
= +∞,
Ω(Θ) − Θwhich, combined with (51), means that limθ→0 l(α, θ) = 0. Likewise, (46)implies that
α
1
α
·
=
,
Θ→+∞ 1 − α Ω(Θ) − Θ
1−αlim

which, combined with (51), means that limθ→1 l(α, θ) = 1 − α. We canrewrite (19) as[r = α (1 − α)
α

1−α

]α−1
1+Θ
.
Θ(Ω(Θ) − Θ)Diﬀerentiating both sides of (52) with respect to θ yields[]
11Ω′ − 1 dΘ∂r/∂θ= −(1 − α)− −r
[ 1 + Θ Θ ′ Ω −]Θ dθ1Ω −1
1= (1 − α)+> 0,
Θ(1 + Θ) Ω − Θ (1 − θ)2

(52)

(53)

which is the third condition of (24). In addition, (27) is established by (45),(46) and (52). Specifically, (45) implies that
1+Θ
= +∞,
Θ→0 Θ(Ω(Θ) − Θ)limwhich, combined with (52), means that limθ→0 r(α, θ) = 0. Likewise, (46)implies that1+Θ= 1,
Θ→+∞ Θ(Ω(Θ) − Θ)limwhich, combined with (52), means that limθ→1 r(α, θ) = αα (1 − α)1−α .

25

Proof of Proposition 2
Proof. To establish this proposition, we need only recall thatU S = log r(α, θ)and that, in equilibrium,
UP − US = UM − US= log r(α, θ)(1 + w1P (θ)) − log r(α, θ)= log(1 + w1P (θ)).
The first condition implies that
∂U S
∂r/∂θ
=
> 0.
∂θ
r(α, θ)The second condition implies that
∂U P
∂U S
dw1P /dθ∂(U P − U S ) =
−
=
< 0.
∂θ
∂θ
∂θ
1 + w1P (θ)
These jointly mean that[]∂U S /∂θ > max ∂U P /∂θ, 0
and that the value of U P − U S is decreasing with θ. Moreover, when θ = 1,
U P − U S = log(1 + w1P (1)) = log 1 = 0.
Therefore, we can conclude that U P − U S > 0 for any value of θ ∈ (0, 1).
Proof of Proposition 3
Proof. The average value of lifetime utilities can be expressed aslU P + (1 − l)U S = U S + l(U P − U S )= log r(α, θ) + l(α, θ) log(1 + w1P (θ)).
Because 0 < l(α, θ) < 1 − α, we can obtain the following inequalities:
A(α, θ) < lU P + (1 − l)U S < B(α, θ),
where A and B are defined as
A(α, θ) ≡ log r(α, θ);B(α, θ) ≡ log r(α, θ) + (1 − α) log(1 + w1P (θ)).
26

(54)

As is easily verified,
∂A
∂r/∂θ
=
> 0,
∂θ
r
and, using Ω and Θ, both of which are defined in the proof of Proposition
1, we can show that
∂B
∂θ

=
=
=

∂r/∂θ
dwP /dθ+ (1 − α) 1 P
r
1 + w1[]1−α1Ω′ − 1Ω′+−(1 − θ)2 Θ(1 + Θ) Ω − Θ Ω(1 + Ω)
[
(
)]
1−α
1
Ω−Θ
1
Ω′ − 11−> 0,−+(1 − θ)2 Θ(1 + Θ) Ω(1 + Ω) Ω − ΘΩ(1 + Ω)

the second equality of which is obtained from (50) and (53). Define q(α, θ)as q(α, θ) ≡ l(α, θ)/(1−α). By construction, q(α, θ) is an increasing functionof θ, with limθ→0 q(α, θ) = 0 and limθ→1 q(α, θ) = 1. It is easy to show that
∂l/∂θ
∂q
=
> 0.
∂θ
1−α
Using q(α, θ), we can rewrite lU P + (1 − l)U S aslU P + (1 − l)U S = [1 − q(α, θ)] A(α, θ) + q(α, θ)B(α, θ).
By diﬀerentiating it with respect to θ, we can obtain
]
∂ [ P
∂A
∂B
∂q
lU + (1 − l)U S = (1 − q)+q
+ (B − A)
> 0,
∂θ
∂θ
∂θ
∂θ
which implies that the average value of lifetime utilities is an increasing
function of θ.

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