Discussion Paper Series

Graduate School of Economics and School of Economics

Meisei University

Discussion Paper Series, No.42
December 2018
(revised August 2020)

Optimal employment in frictional business cycles andintertemporal discontinuity of production and internal prices

Koji Yokota(Meisei University)

Hodokubo 2-1-1, Hino, Tokyo 191-8506
School of Economics, Meisei University
Phone:+81-(0)42-591-9479

Fax: +81-(0)42-599-3024

URL: https://keizai.meisei-u.ac.jp/econ/
E-mail: keizai@econ.meisei-u.ac.jp

Optimal employment in frictional business cycles
and intertemporal discontinuity of
production and internal prices∗
By Koji Yokota
Labor market with nonbinding contract has a particular pricemechanism that depends on bargaining. Can there be a case where
no autonomous mechanism is equipped that always brings the economy back to the long-term steady state? This paper shows thatexistence of convex hiring cost together with differentiated goods
market allows multiple prolonged inefficient paths out of long-term
steady states. The optimal behavior of firms in those inefficient
paths has peculiar characteristics expecially at the margin of firing
phases.
JEL: C6, E3, J2
Keywords: convex hiring cost, random matching, business cycles,
effective demand, optimal firing, labor hoarding, singular control,
costate discontinuity, periodic steady states

I.

Introduction

The present paper studies the optimal behavior of the firm facing labor friction
characterized by convex hiring cost and sufficiently differentiated output good.
Even though there are conflicting arguments on empirical applicability of convex
hiring cost, from the point of view that hiring cost is a variation of adjustment
cost, it should be natural to assume it from the theoretical perspective.1 The convexity in hiring cost makes the labor adjustment process time-consuming, which
in turn determines the level of income and demand for output at every moment
while the adjustment process is still going on. When output goods are sufficiently
differentiated, for such an adjustment process to be fulfilled, expectation on thesocial choice of the equilibrium path must be shared uninterruptedly among firmsthat the economy ultimately reaches to the long-term equilibrium. If expectation
is not well-coordinated, the investment in labor by a single firm according to theefficient path will cause a loss, since any discount in output prices will not retrieve the cost of investment. It raises multiple inefficient paths that depend on
the degree of coordination. Yashiv (2006, 2007) analyzed the efficient equilibrium∗ Revised in August 2020.
1 Yashiv (2000) and Blatter, Muehlemann and Schenker (2012) support convexity. On the other hand,
Abowd and Kramarz (2003) and Kramarz and Michaud (2010) finds concavity for long-term workers inFrance. However, the hiring cost in the latter is defined to be directly observable ones which differs fromour setup that the cost arises from the wage payment to workers internally allocated to the hiring sector
and arises from decreasing returns in both production and hiring sectors.

1

2

DECEMBER 2018

in an economy with convex hiring cost. The present paper extends them to thoseinefficient cases. In the sense that expectation is the main driving force of business cycles, the model shares the spirit with news-driven business cycle models
by Beaudry and Portier (2004) and others. However, in our model, the sourceof expectation is not limited to technological news on productivity fluctuation,
but it is more generalized shared view about future equilibrium characterized bynature of beauty contest.
A firm operating in an economy with labor friction has an intertemporal horizon. However, a model of a going concern with no firing cost easily falls in the
category of singular control problems. They are the cases in which Hamiltonian
becomes independent from some control variables and the (ordinal procedure)of the maximum principle does not allow to find the optimal control value.2
It occurs when the switching function becomes zero. Even though there werecases that singularity is artificially evaded in early times as Johnson and Gibson(1963) pointed out, obviously it is not guaranteed that those singularity should be“pathological”. The model studied in this paper shows that singularity is indeed
not only unexcludable but comprises important part of business cycles. Namely,
the firing phase is singular. Moreover, since the singular control is derived fromthe resulting state constraint on the boundary, the optimal control in the firing
phase is determined only in a derivative form, leaving determination of the initialcondition out of the firing phase. It brings intertemporal discontinuity in costatevariables and thus in the output level. This is a similar property to optimal
control problems with state variable inequality constraints (SVICs), which is basically brought by the truncatability of the problem into subperiods. It is known
that the occurrence time of costate discontinuity in state-constrained problems is
generally indeterminate between entering and leaving times.3 It will be provedthat our setup shows the discontinuity in both times.
The properties of the model with no firing cost gives a good implication for amodel with firing cost. It tells us that firing is optimal to visit in a chunk on
the first day of the firing phase in general. With firing cost, such a discontinuous
behavior is too costly. The firm is better to keep idle employment as far as thecost to hold it measured in its absolute value (which is negative) is less than thefiring cost. Thus, it brings labor hoarding.
Section II explains the model. Section III analyzes steady states both withunbounded and bounded demand constraints. Section V and VI study out ofsteady states. Section V examines firing phase and Section VI examines theentering and exiting from those phases. Section VII examines implication tolabor hoarding. Section VIII studies how labor hoarding behavior changes when
there is convex firing cost. Section IX shows that linear firing cost mixes resultsof the basic model and that of Section VIII. Section X concludes.
2 Rozonoer (1959).
3 This is true for many of problems. Hartl, Sethi and Vickson (1995) points out a case in which thediscontinuity occurs within the boundary intervals.

INTERTEMPORAL DISCONTINUITY

II.

3

The model with no firing regulation: a singular control problem

Consider an economy with differentiated output goods produced by labor. The
labor market is frictional characterized by convex hiring cost, which allows inefficient paths of the economy. Goods are perishable and normal. The goods market
is characterized by infinitely many producers, however, reflecting the differentiation of output goods, the market for each output good is monopolistic.4
A.

Households

The complexity of the demand side is minimized. There are infinitely manydifferentiated output goods. They belong to the same category so that they are
all symmetric in the utility function. Households are illustrated as a representative
consumer. She or he is endowed with the ownership of all firms with equal weights
as non-transferrable equities. The goods are perishable so that there is no room
for saving by goods. The problem for the representative consumer degenerates toinstantaneous utility maximization. At any moment, he or she can be in a stateof employed (state-0) or unemployed (state-1). The value function of state j as
(j)of time t is denoted by ut , the maximand of which is an additively seperable
utility function.
Z(j)u(qi ) di
ut = max{c}i∈ΩZpi qi di ≤ w(j) + π
s.t.
i∈Ω

where Ω is the set of firms the measure of which is fixed and normalized to one,u ∈ C 2 belongs to CRRA class satisfying u′ > 0 and u′′ < 0, qi is the demandfor i-th output good, pi is its price, w(j) is income in state j such that w(0) > 0
and w(1) = 0 and π is profits of own firms. Since firms are homogeneous, output
price is normalized to pi = 1 for any i in equilibrium so that qi = w(j) + π holds
for any i. Simple as the outcome is, the demand structure mainly affects firms in
off-equilibrium paths, which is analyzed in Section II.C.
B.

Firms

Firms are non-atomic in the output market having aggregate measure one.
They engage in two activities, production of goods and hiring of workers. Bothactivities require two inputs, labor and land where the latter is assumed to be
fixed throughout the analysis. The land usage in the production and hiring sectors
are fixed and total real land cost is c ≥ 0. The production function of goods is
4 Firms are monopolistic in the sense that they are allowed to manipulate output prices in offequilibrium context. However, nominal price manipulation will not be undertaken so that the samelevel of “normal” profits are shared among firms.

4

DECEMBER 2018

therefore denoted by f (ˆl) where fixed land input is implicitly embedded in thefunctional form of f , ˆl ≥ 0 is the measure of employment in the production
sector and f shows decreasing returns. More precisely, it is characterized by
f ′ > 0, f ′′ < 0, f (0) = 0 and Inada conditions. Another activity, hiring, requires
expense of internal resources to get hiring results. It is expressed by anotherproduction function g(˜l; θ) or simply g(˜l) in harmless places where land usage is
implicit again, ˜l ≥ 0 is the measure of employment in the hiring sector and θ is aparameter which represents vu ratio in the labor market. Utilizing the fact thatthe optimal firm efficiently utilizes existent employment when firing cost does notexist, we simplify our representation by assuming the sum of the employment in
the production and hiring sectors equates the total employment,5 i.e. ˆl + ˜l = lwhere l is the total employment of the firm. g satisfies similar assumptions as
f , i.e. g ′ > 0, g ′′ < 0, g(0; θ) = 0 and also Inada conditions. The decreasing
returns of g(·) induces convex hiring cost. The employees separate from the firm
at instantaneous natural seperation rate σt > 0 at time t. In the following, timeof variables is generally denoted by a subscript, however it may be expressed as
an explicit argument of a function where notation becomes cumbersome. Whenfurther reduction of labor is preferable, the firm retains an option to fire workers
with no cost. The firing rate is denoted by xt ≥ 0. The transition of labor
becomes(1)l˙t = g(˜lt ; θt ) − σt lt − xt .

The real wage rate is denoted by a right-continuous function of time wt > 0. Here,wage bargaining is assumed to be made at the industry level as is true in some
economies to get a simpler view of the model. Namely, the wage rate is exogenous
for the firm. Reflecting the characteristics of bargaining, the wage outcome isassumed to be less than the marginal productivity of labor in equilibrium, i.e.
wt ≤ f ′ (ˆlt ). Bargaining within the firm makes wt a function of the state variablel, as the value of the coalition is divided between the firm and workers, which
affects not only the distribution between them but also the production level when
the demand constraint is not binding. This additional source of inefficiency iseliminated here to clarify the discontinuous behavior of the firm on the margin of
firing phases. The firm discounts future real profits by instantaneous rate r > 0.
Denote the value of the representative firm at time t with employment l by Π(t, l).
Normalizing output price to one and taking the initial time to zero, it becomesZ ∞f (ˆlt ) − wt lt − c e−rt dt(2)
Π(0, l0 ) = maxl̂,x0

5 This assumption will be removed as we introduce the firing cost later.

INTERTEMPORAL DISCONTINUITY

5

which is subject to the labor transition (1) and a potentially binding demand
constraint(3)f (ˆlt ) ≤ yt

where yt > 0 is the demand directed to the firm which reflects the aggregate
demand in the equilibrium path. The condition in which constraint (3) becomes
effective will be argued in Section II.C. Control variables are subject to thefollowing constraints:(4)(5)

0 ≤ ˆlt ≤ lt
0 ≤ xtTwo points need to be mentioned. First, the demand constraint (3) appears since
its complement is excluded as a suboptimal domain in a larger problem where
off-equilibrium manuplation of output price is allowed. The assumption of differentiated outputs is only effective in this context to bring the emergence of theconstraint. In a word, when all firms cannot immediately adjust outputs, theoff-equilibrium attempt turns out to be unprofitable under fairly natural assumptions for a non-atomic firm to deprive of output demand from other firms via
reduction of nominal output price, even though the equilibrium output growth
is lower than the socially efficient level. It requires coordinated move by firmsof positive measure to increase output breaking the binding constraint. It will
be described below more precisely after establishing some optimality conditions.
Reflecting the optimal production of firms, y is assumed to be right-continuous
and right-differentiable in terms of time. Note that y is an equilibrium path which
can be interpreted as effective demand in the sense that the demand is not justplanned but endorsed by purchasing power in contrast to off-equilibrium paths.
Second, firing consraint (5) cannot have meaningful finite upper bound from an
economic view point. On the other hand, the linearity of firing x in the transition
equation indicates the possibility of bang-bang control. Namely, optimal firing
can diverge to infinity when x > 0 happens. It will turn out that such massive firing can only occur at the initial time and the time when binding y shows
discoutinuous downfall, both of which are reaction to nonautonomous move. Autonomous firing is always singular.6 Those at most countable discontinuities arehandled by partitioning the problem into subperiods. Thus, firing constraint (5)is left with no upper bounds.
The above properties of firing characterizes the dynamics of equilibrim employment and production. At any time where either there is no firing or firing issingular, the output is time-differentiable and so is equilibrium y. However, weneed to incorporate the costate discontinuity at the mergin of firing phases which
singularity of autonomous firing brings about, and also discontinuous downfall of
6 For a singular control problem, see Johnson and Gibson (1963) and Kelly, Kopp and Moyer (1967).

6

DECEMBER 2018

equilibrium output itself. Therefore, y is defined in the class of right differentiable
functions.
The proof of the following claim will be represented after optimal conditions
are characterized in Section V.
CLAIM 1: Suppose y is continuously differentiable except for the junction times
from/to the firing phase and for the time y shows discontinuity. Suppose x faces
a constraint x ∈ [0, x̄] where x̄ < ∞. If x̄ is set sufficiently large corresponding
to the maximum variation of y, the optimal x is never bound by x̄ except for the
initial time and the time where y shows downward discontinuity.
Costate variable of l is denoted by λ, Lagrangean of demand constraint by µ ≥ 0
and that of ˆl ≤ l by η ≥ 0. The optimal dynamics of the costate variable becomes
h
i
′
ˆ
(6)
λ̇ = − g (l − l) − (r + σ) λ + w − η.

for any time at which l is differentiable. Since y is assumed to be strictly positive and f satisfies Inada conditions, ˆl > 0 always holds. Then, the first ordercondition becomes(7)(1 − µ) f ′ (ˆl) = λ g′ (l − ˆl) + η

and for firing, by imposing arbitrarily large x̄ > 0,
if λ > 00(8)x = [0, x̄] if λ = 0x̄if λ < 0.

Together with (6) and (7), the optimal condition when ˆl ≤ l is not binding canbe expressed as(9)(r + σ) λ = (1 − µ) f ′ (ˆl) − w + λ̇.

Viewing the value of the firm as an asset, the above equation says its marginal costequates the sum of instantaneous effective marginal profitsand marginal capitalR
gain. Or, integrating (9) through l obtains (r + σ) Π = (1 − µ) f ′ (ˆl) dl − wl + Π̇,
the left-hand side of which is the corporate cost of ownership with separation
premium σ and the right-hand side is the instantaneous profits plus capital gain
where the revenue is discounted by the shadow price of the demand constraint.
The dynamics of shadow price µ of the demand constraint, the impact of unexpected variation of y on the value of the firm, and marginal value of labor
λ play significant roles especially on the mergin of firing phases. The following
proposition guarantees that they are time-continuous in the non-firing phases.
PROPOSITION 1: If λ > 0, then µ is time-continuous and λ is continuously
differentiable on the optimal path.

INTERTEMPORAL DISCONTINUITY

7

PROOF:
If demand constraint is not binding in any open domain in time, then µ ≡ 0 onthat domain so time-continuity of µ on the domain is obvious. Suppose demand
constraint is binding in any open domain in time. Perturbation specification of
the problem à la Bryson, Denham and Dreyfus (1963) would be
Z ti+1 ∂H∂H ˆ ∂HdΠ(ti , lti ) =
δl +
δl +
δx e−r(t−ti ) dt
∂l
∂x
∂ ˆl
ti


Z ti+1∂Π(T, lti+1 )−r(t−ti ) ˙− λti+1 e−r(ti+1 −ti ) δlti+1−
λe
δl dt +
∂lti+1tiwhich should be stationary at optimum for i-th subinterval (i = 0, 1, 2, . . . , N )after dividing [0, ∞) into N finite subintervals where H is the current value Hamiltonian, t0 = 0 and tN +1 = ∞. To minimize the subdivision of time, let N be
the number of indifferentiability of l in the time scale so that l is indifferentiable
on ti . This subdivision of time scale is necessary because the second term of theright hand side of the above equation needs to be integrable by parts to obtain(6). It requires the time derivative of l to exist. Namely, (6) is applicable only
to the interior time of each subinterval. Now, choose t ∈ {s : λs > 0}. Then,x = 0 and ˆl is differentiable from ˆl = f −1 (y) and differentiability of y, which
implies l is differentiable from (1). It implies t is interior time of subintervals
and indeterminate function λt can be safely defined as a differentiable function
via integration by parts of the second term of the above equation so that (6)holds. Then, the first-order condition (7) implies time-differentiability of µ on
the domain. Finally, suppose that the demand constraint is binding at time T
under consideration and there exists a point where the constraint is unbindingin any open neighborhood of T with radius ε > 0, which is denoted by Nε (T ).
Then, corresponding to a sequence of ε converging to zero, we can take converging
subsequence of time t ∈ Nε (T ) at any of which f ′ (ˆlt ) = λ g′ (lt − ˆlt ) holds fromthe first-order condition (7) when λ > 0 and also f (ˆlt ) ≤ yt holds. On the other
hand, if µT > 0, then (7) implies ˆlT < limt→T ˆlt from the continuity of l andλ. Since yT = f (ˆlT ) < limt→T f (ˆlt ) ≤ limt→T yt , it contradicts to the continuityof y. Therefore µT = 0. It implies that µ is continuously connected at time T
when transiting from domain µ > 0 to µ = 0 or vice versa. Finally, (6) and (7)obtain λ̇ = (r + σ)λ − [(1 − µ)f ′ (ˆl) − w]. If µ is continuous, λ is continuously
differentiable.
C.

Emergence condition of the demand constraint

To see the condition that breaking the demand constraint (3) becomes generally
unprofitable and thus it is excluded from the admissible controls, extend the
problem to allow the firm to set its individual nominal output price and focus

8

DECEMBER 2018

on the time immediately after entering the constraint. Namely, the optimand (2)becomesZ ∞pf (ˆlt ) − wt lt − c e−rt dt(2’)
Π(0, l0 ) = maxp,l̂,x0

with the same constraints (4) and (5) as the original reduced problem except that(3) does not exist. In the same line of argument as Kaneko (1982) and Massoand Rosenthal (1989), the action by a single firm is indiscernible by other firms
so that retaliation against deviation from a strategy is impossible.
The change of output price by the non-atomic firm producing good i alone does
not affect the budget constraint of a representative consumer. Its impact on i-th
firm revenue is


1
dpq
=q 1−
dp
Rr
where p is the output price the firm sets, q is the quantity of outputs, Rr is thecoefficient of relative risk aversion. Also, the impact on the cost is
wq 1
dwˆl
dq/dp
dwl
=−
≤
=w
ˆ
dp
dp
dq/dl
pf ′ (ˆl) Rr
where wage rate w is real against market output prices. Then, the effect of theprice change on the instantaneous profits becomes
!
"
#
dπ
1
w
(10)
≥ q 1 − 1 − ′ −1 dpRrf f (q)when p = 1. Defining the merginal share of profits by α := 1 − w/f ′ (ˆl) ≥1 − w/f ′ (l) ≥ 0, the condition(11)

Rr ≥ α,becomes the sufficient condition for dπ/dp ≥ 0 to hold as far as u′′ < 0. Obviously, (11) holds in all domain if w = f ′ (ˆl). Also, note that Rr = 0 when
there is no differentiation between output goods and Rr increases as outputs aredifferentiated so that the elasticity of substitution decreases.
The increased production capacity needs to be absorbed by increased demandbrought by ceaseless discount of the output price. From (10), the lower bound of
the profit change increases with q. Given condition (11) holds, the best deviation
is to infinitesimaly discount the price and to produce corresponding amount atleast for some time. Suppose the stopping time of the deviation is T and the firm
fires all additional employment used for the deviation at time T . Integrating theINTERTEMPORAL DISCONTINUITY

9

optimal condition (9) in off-equilibrium context gives
Z T
Z T

pf ′ (ˆl) dπ −(r+σ)t′ ˆ−(r+σ)tedt.
Rrpf (l) − w e
dt = −λ=f (ˆl) dp
00
The right hand side goes negative if (11) holds. Therefore, (11) is the sufficientcondition for profitable deviation from equilibrium becomes impossible.
It is worthwhile examining what happens in the case of non-differentiated output goods, i.e. the case of Rr = 0. The existnce of search friction and thelimitation of hiring technology indicate wdl/dp < ∞ which implies dπ/dp = −∞.
In such a case, reduced output price will attract infinite amount of demand which
grants the attempt of constraint violation. All firms will do so, which implies
that contemporaneous real wages settled by bargaining are unchanged, demand
constraint is broken, and ex-post interest rate rises only by ignorable degree. The
above argument suggests that the existence of demand constraint is peculiar todifferentiable goods such as industrialized ones. Homogeneous goods, represented
by agricultural goods for example, will not confront the constraint.
III.

Steady states

This section is for completion. When the state of coordination determines
the equilibrium path, steady states lose foundation for being a main focus. An
eternally unbinding path converges to a saddle-point steady state. Even though
the convex hiring cost prevents a jump to the steady state, it keeps a status asa good approximation in the long run. However, this is not the case for binding
paths. They need specify how they are bound. There can be multiple paths —very different ones— corresponding to the same binding steady state. One may bebinding forever after some point of time, one may become binding periodically.
The degree of binding in steady states is no more than statistics of how theconstraint binds in “transition”. By this reason, it is more useful that the degreeof coordination should be represented by a function of time, rather than by avalue in the terminal condition. y(t) defined in the previous sector is thus chosen
to represent the state of coordination over time and defined to be the on-path
output level in a given equilibrium. Whether the constraint is binding or not isonly distinguished by its shadow price in equilibrium since the output level always
coincides y(t). Note that the introduction of a coordination factor transforms thesystem into a non-autonomous one.
Nevertheless, steady states are worth studying since, with the additional assumption of stationary belief, they give simple ordering of equilibria within theclass and enables comparative statics in the long run. Both efficient and inefficient steady states are studied below. The demand constraint of the former isunbinding whereas that of the latter is binding. Steady states occur with λ > 0
and with no firing. They are ranked by the level of the demand constraint y,a representation of the degree of coordination among firms. It is shown that in

10

DECEMBER 2018steady states with demand constraint in effect, relaxation of the constraint implies
the increase of the effective marginal value of employment.7
1.

The efficient steady state

The efficient steady state has no binding demand constraint and thus characterized by µ = 0. λ̇ = 0 in the costate dynamics (6) gives
λss =w
g′ (lss − ˆlss ) − (r + σ)

> 0,which implies xss = 0 where subscript “ss” represents the value at the efficient
steady state. Inada conditions guarantee that the first-order condition (7) provides an interior solution when λ > 0 so we have f ′ (ˆlss ) = λss g′ (lss − ˆlss ). Together
with l̇ = 0 in labor transition (1), (lss , ˆlss ) is characterized to solveg′ (lss − ˆlss )w
g′ (lss − ˆlss ) − (r + σ)

(12)f ′ (ˆlss ) =(13)g(lss − ˆlss ) = σ lss ,the solution of which exists when r > 0.8 Note that (12) implies f ′ (ˆlss ) > w. The
marginal productivity of labor in the production sector is strictly greater thanthe marginal cost of labor simply because some labor is absorbed in the hiring
activity. If hiring efficiency rises, then the difference between f ′ and w shrinks.
2.

Inefficient steady states

Inefficient steady states are constrained steady states characterized by binding
demand constraint with µ > 0. They satisfy f (ˆliss ) = y where subscript “iss”
represents the value at inefficient steady states. For their existence, y must be
a constant function of time. With stationarity in labor transition, the inefficient
steady state for given y is characterized by
(14)
(15)

λiss =w

g′ liss − f −1 (y) − (r + σ)g liss − f −1 (y) = σ liss .

>0

7 As a benchmark, in spot production with demand constraint, the effective marginal value becomesthe present value of wages. It is derived from the fact that, in each moment, maxl f (l) − w l subject tof (l) ≤ y implies w = (1 − µ) f ′ (l) with µ ≥ 0. In this case, an increase of the effective marginal value of
employment leads to the rise of wages.
8 If r ≤ g ′ (L)−σ < 0 where L is the solution in g(L) = σL where all employment is dedicated to hiring,
the solution may not exist. However, since negative discount rate is unlikely to hold in steady states, we
can safely eliminate such a case. It implies g ′ (L) > r + σ. By decreasing returns of g, g ′ (l − l̂) > r + σ
holds for any unconstrained/constrained steady states.

INTERTEMPORAL DISCONTINUITY

11

The relation between inefficient and the efficient steady states is depicted in Figure 1. The curve labeled λ̇ = 0 shows the costate steady state condition for anlλ̇ = 0 with µ = 0
inefficient
steady statewith given ylssbbliss (y)efficientsteady statebbµ>0(shaded area)l˙ = 0f −1 (y)l̂ssl̂Figure 1. : The efficient and inefficient steady states

inefficient case (12). The curve labeled l̇ = 0 shows the labor steady state conditions (13) and (15). Since f −1 (y) ≤ ˆliss , stationarity conditions for labor (13) and(15) imply that liss ≤ lss the inequality of which is strict except for the obvious
case f (ˆliss ) = y. Therefore, the curve l̇ = 0 always locates below the curve λ̇ = 0
for any ˆl ≤ ˆlss . Efficient steady states locate on the curve l̇ = 0.
λiss given by (14) is expressed in another form λiss = [(1−µ)f ′ (ˆliss )−w]/(r +σ).
It reads the marginal value of employment for the firm is the discounted present
value of (1 − µ)f ′ (ˆlcss ) − w in which the discount rate is affected by separation
premium. In this place, (1 − µ)f ′ (ˆl) is termed the effective marginal productivityof the production sector. Facing the demand constraint, effective price of output
is discounted by the shadow price of the constraint. Hiring sector has costatevariable λ as its internal output price, which is also indirectly affected by present
and future shadow price of the demand constraint.
PROPOSITION 2: At an inefficient steady state, increase in demand y brings 1)
increase in total employment and in both production and hiring sectors, 2) increasein marginal value of employment, 3) increase in effective marginal productivity ofthe production sector and increase in marginal productivity of the hiring sector,
and 4) decrease in demand duals.
PROOF:
1) Applying the implicit function theorem to (15) obtains dliss /dy = g′ /[f ′ (g′ −σ)] > 0. On the other hand, dˆliss /dy = 1/f ′ > 0. They result in(16)

dliss dˆliss
σ 1d˜liss
=
−
= ′
> 0.
dy
dy
dy
g − σ f′

12

DECEMBER 20182) Similar application of the theorem to (14) derives dλiss /d˜liss = −g ′′ λ2iss /w,
from which (16) impliesg′′ λ2iss σ 1
dλiss
=−
> 0.
dy
w g′ − σ f ′3) From the above result,
dλiss g′ (˜liss )dλiss ′ d˜liss′′g= λiss g +dydyd˜liss


′
˜
dliss
g λiss
g′′ λ
= 1−
w
dy
dλiss
= (r + σ)>0dywhere λiss g ′ (˜liss ) is the marginal productivity of the hiring sector and from thefirst-order condition (7), it equates to the effective marginal productivity (1 −µ)f ′ (ˆliss ).
4) Since 1 − µiss = λiss g ′ (˜liss )/f ′ (ˆliss ),r + σ dλiss λiss g′ f ′′d(1 − µiss )
=
−
> 0,
dy
f ′ dy(f ′ )3
which brings dµiss /dy < 0.

The result that the effective marginal productivity and the marginal value of labor
increase may worth attention. Depending on the wage bargaining process, they
can imply the rise of wages. It means that increase in both wages and employment
coexist without technological progress. The last result of the proposition showsthat the impact of the unexpected variation of demand on the firm’s value becomes
smaller as the demand level becomes higher.
IV.

Effect of binding demand constraint

Let zt = (lt∗ , λ∗t ) be the optimal path. The next proposition says, if there arebinding and unbinding optimal paths which share common history after somepoint of time, the binding path has lower employment and lower marginal valueof labor before that point.
(1)

(2)

PROPOSITION 3: Consider two optimal paths zt and zt , and denote corresponding adjoint variable to the demand constraint by µ(1) and µ(2) , respectively.
(1)
(2)
(1)
Suppose that there exists t0 , t1 ∈ R such that zt1 = zt1 and zt accompanies
(1)

(2)

(2)

µt > 0 for ∀t ∈ [t0 , t1 ) whereas zt holds with µt = 0 for ∀t ∈ [t0 , t1 ). Then,(2)(1)
there exists tc < t1 such that zt < zt holds for any t ∈ [tc , t1 ).

INTERTEMPORAL DISCONTINUITY

13

PROOF:(2)(1)
Pick up time (T − ε) for an arbitrarily small ε > 0. Suppose λT −ε ≥ λT −ε .
From the first order condition (7),
(1)

(1)

(2)

1 − µT −εg′ (˜lT −ε )g′ (˜lT −ε )
1
=
<
=
(1)
(1)
(2)
(2)
f ′ (ˆlT −ε )
λT −ε
λT −ε
f ′ (ˆlT −ε )which implies
˜l(2)
˜l(1)
T −ε
> T(2)−ε
ˆl(1)
ˆl

(17)

T −ε

T −ε
(2)

(1)

by the decreasing returns of f and g. Then, ˜lT −ε > ˜lT −ε holds, since otherwise
(1)

(2)

(1)

(2)

l̇T −ε ≤ l̇T −ε holds which means lT −ε ≥ lT −ε . Together with (17), it implies
(2)
(1)
(2)
(1)
(2)
˜l(1) > ˜lT −ε , a contradiction. So, l˙T −ε > l̇T −ε and lT −ε < lT −ε hold. On the
T −ε
(2)

(1)

(2)

(1)

other hand, costate transition (6) implies λ̇T −ε > λ̇T −ε , i.e. λT −ε < λT −ε , which
is a contradiction to the first assumption.
(1)
(2)
(1)
(2)
(1)
(2)
Therefore, assume λT −ε < λT −ε which implies λ̇T −ε > λ̇T −ε from λT = λT .
(1)
(2)
(1)
is binding,. Since only z≥ l
It implies g ′ (˜l) > r + σ. Suppose l
T −ε

T −ε

T −ε

(2)
(1)
(2)
ˆl(1) < ˆlT −ε which means ˜lT −ε > ˜lT −ε . Since g′ (˜l) > r + σ > σ, it implies
T −ε
(1)

(1)

(2)

(1)

(2)

(2)

l̇T −ε > l̇T −ε , i.e. lT −ε < lT −ε , a contradiction. Therefore, lT −ε < lT −ε . By
continuity of l and λ, the statement of the proposition follows.

(1)

(2)

(2)

(1)

Note that if zt0 < zt0 and µt = µt = 0 for t < t0 , then tc < t0 by continuityof l and λ. Namely, if it is expected that the demand constraint becomes binding
in future, the firm refrains from hiring workers since their marginal value is low
and thus retains smaller number of employment.
V.

FiringAlthough section IV suggests that binding demand constraint today and in
future tends to bring a sluggish economic state, Proposition 1 guarantees thatthe presence of the demand constraint does not drastically change the structureof the model as far as the marginal value of labor is strictly positive. However,this is not true if it becomes zero, i.e. when workers are undergoing firing. Since
the transition of labor is linear in firing, precise treatment is necessary when
the switching function is on the boundary. Employment becomes redundant if
both current and future production requires smaller labor reflected in y and λ.
It is shown that, as far as that y is differentiable and the economy already hassome history, the adjustment of labor is fulfilled within continuous variation.
Namely, with feasible initial employment, firing is done with λ = 0 which implies

14

DECEMBER 2018

singularity of the problem, i.e. a non-standard procedure is required to obtain
the optimal path. On the other hand, the continuity of employment brings about
discontinuity in the costate variable. This can be observed by considering thefiring phase condition y = f (l) as a terminal surface in a subproblem where theterminal condition is to decide when and where to enter in the terminal surface.
The optimal terminal condition does not guarantee smooth entering.
In the following description, the following notations are used for subsets on thetime scale. Define the firing phase Xt the maximum connected set on time scale
including t where x∗s >S0 for any s ∈ Xt and the asterisk shows the optimal
path. Formally, Xt := i {Ii ∋ t : x∗s > 0, ∀s ∈ Ii } where Ii is a closed interval.
If x∗t = 0, then Xt = ∅. The closure of Xt writes in an interval form X̄t =[tE , tL ] where tE is the entering time in Xt and tL is the leaving time from Xt .
Although they are defined interval-wise, they are used without mentioning it if theobjective interval is obvious. Similarly, the non-hiring phase Λot is defined to be
the maximum
t where λ∗s = 0 for any s ∈ Λot ,S connected∗ set on time scale includingo∗i.e. Λt := i {Ii ∋ t : λs = 0, ∀s ∈ Ii }. If λt > 0, then Λot = ∅. In the interval
form, Λ̄ot = [te , tl ] where the entering and leaving time from/to Λot are denotedby te and tl , respectively.
Again, the same abuse of notation
applies as X̄t . Also,S∗ < 0, ∀s ∈ I } and W := t : l∗ < f ′−1 (w ) . Union ofdefine Λ−
:=
{I
∋
t
:
λ
i
i
t S
s
tS
t
i
o
o :=
X
,
Λ
those intervals
aredenotedwithout
subscripts
byX:=t Λt and
t t
S −
E
L
e
l
X
X
Λ
−
Λ := t Λt . Denote the set of all t ’s, t ’s, t ’s and t ’s by E , L , E andLΛ , respectively.
PROPOSITION 4: Xt ⊂ Λot for any t ∈ W .
PROOF:The result is obvious if Xt = ∅. Suppose Xt 6= ∅. Define x̄ = supt (−ẏt /f ′ (lt )).
Suppose that the demand constraint holds with equality, i.e. f (ˆlt ) = yt for some−−t ∈ Λ−t ∩ W such that Λt 6= ∅. Optimal xt = x̄ leads to µs = 0 for any s ∈ Λtsuch that s > t since f ′ (lt )l̇t < ẏt . Since µs is right-continuous, µt = 0 andλ̇s exists for such s and t + 0.9 From (6) and (7), costate transition becomesλ̇s = (r + σ)λs − f ′ (ls ) + ws < 0 for any s ≥ t. The relation is recursively justified
starting from s = t so that s ∈ Λ−t implies l̇s < 0 which means s + ε ∈ W∩
Wforarbitrarily
small ε ≥ 0. It ultimately causes
and thus s + ε ∈ Λ−tlims→∞ ls < 0. Therefore, any path which enters Λ−t ∩ W cannot be optimal. Ito ∪ Λ− for any X 6= ∅. Therefore,
implies that Xt ∩ W 6⊆ Λ−∩
Wwhereas
X
⊆
Λ
t
t
t
t
t
Xt ∩ W ⊆ Λot ∩ W .

The above proposition proves Claim 1 and firing x becomes a singular control,
not a bang-bang:
COROLLARY 1: Firing xt is a singular control if t ∈ W .
9 See Theorems 4.1 and 4.2 of Hartl, Sethi and Vickson (1995).

INTERTEMPORAL DISCONTINUITY

15

PROOF:
Since l is right-differentiable, |Xt | > 0 if Xt 6= ∅. From Proposition 4, it implies|Λot | > 0. Namely, if xt > 0, then λt = 0 and λt+ε = 0 for arbitrarily small ε > 0.
Since λt is a switching function of xt , which is zero on an interval with positive
measure whenever xt > 0, xt is a singular control.
The proposition suggests that, for a massive firing, i.e. discontinuous decrease of
employment, to happen, it should occur only once at the very beginning of theeconomy, which implies that we can safely separate such a phase from the analysis
and concentrate on the dynamics after the negative “big-bang”. Since x̄ is an
artificial boundary, if such a transition were to happen, it finishes instantaneously
by a discontinuous decrease of employment, which can be understood as the limitof dynamics when x̄ → ∞. Even if continuously large non-autonomous force
acts in the middle, it will not bring the system to the initial big-bang state since
sufficiently large continuous change of employment absorbs such a shock. Figure
2 shows a phase diagram for an unbounded autonomous case. The manifoldl̇ = 0λdomain of WbO

f ′−1 (w)

A

b
b

B

l
C

D

S

R
λ̇ = 0Figure 2. : Phase diagram for an unbounded autonomous casel̇ = 0 is an upward sloping curve stable in terms of l that passes through theneighborhood of the origin. That of λ̇ = 0 is downward sloping, unstable in termsof λ, so that λ → +∞ holds as l → 0 on it and passes through l = f ′−1 (w).
In the forth quadrant shown as a hatched area, there is “strong” leftward flows.
The flow instantaneously reaches to the goal as x̄ → ∞. There is a saddle pathto the unique non-zero steady state A in this particular unbounded autonomouscase. The saddle path in the forth quadrant is drawn as almost flat downward
sloping curve in the figure. However, since x̄ is arbitrary, it should be understood

16

DECEMBER 2018that it converges to the horizontal axis as x̄ → ∞, the optimal firing on the pathis x∗ → +∞. The same holds for any paths which reaches to any points on thehorizontal axis except interval OD. In non-autonomous case, the optimal pathis generally different from the one drawn in the diagram because phase diagram
itself transforms. If the optimal path in such a case were in the hatched lightgrey area S in Figure 2, the only way that it survives as optimal is to move tothe hatched dark grey area R. In other cases, the path trespasses on the negative
employment region due to λ̇ < 0. Move from S to R is generally impossible
in the autonomous case as the vector field in the figure shows. However, nonautonomous cases require a check whether sufficiently quick fluctuation of theboundary of domain of W caused by external forces does not actually allow sucha move. If allowed, the path may come back to the non-hatched area surviving
as optimal. The following proposition formalizes that it never happens even in
non-autonomous cases.
PROPOSITION 5 (Impossibility of a jump in the middle): If t ∈ W , then λs ≥0 for any s > t along the optimal path.
PROOF:
Suppose λv = 0 at v > t and λv+ε < 0 for arbitrarily small ε > 0. Then,from the transition (6) of λ, right-continuity of µ and continuity of w and l,(1−µv+ε )f ′ (lv+ε ) > wv+ε must hold for λv+ε < 0, which implies v +ε ∈ W . Then,since x is arbitrarily large in (1), l˙ is always smaller than any non-autonomouschange of f ′−1 (w), which implies the optimal path never enters W c from W ∩Λ−v+ε ,thus diverging to ls → −∞ as s → ∞. Thus, it is excluded from the optimalpath.
The proposition can be restated as follows with the optimal control (8).
COROLLARY 2: xt > 0 occurs only if λt = 0 for any t ∈ W along the optimal
path.
Based on the singularity of firing, the next proposition shows that it occurs only
when the demand constraint is binding and ẏ < 0.
PROPOSITION 6 (Optimal controls and binding demand constraint): ˆlt∗ = lt∗
and ˜lt∗ = 0 for t ∈ Λo ∩ W .
(18)

xt = −σlt −ẏtf ′ (lt )and µt > 0 for any t ∈ (Λ̊o ∪ E Λ ) ∩ W . Moreover, µt is differentiable for anyt ∈ Λ̊o ∩ W and right-differentiable for t ∈ E Λ . Also(19)

f (lt∗ ) = yt∗

for any t ∈ Λ̄o ∩ W and ẏt < 0 for any t ∈ Λo ∩ W .

INTERTEMPORAL DISCONTINUITY

17

PROOF:
λt = λ̇t = 0 for any t ∈ Λo ∩ W where λ̇ is right-derivative at t = te andleft-derivative at t = tl . Then, costate dynamics (6) implies η = w > 0 and alsoˆl = l and ˜l = 0 by complementarity. The first order condition (7) obtains(1 − µ) f ′ (l) = η = w.

(20)

It implies µt > 0 for any t ∈ Λ̊o ∩ W since f ′ (lt ) > wt for t ∈ W . (20) holds
for te ∈ W from the right-continuity of µ and the continuity of l and w, soµte > 0. Together with continuity of l, it implies that f (lt∗ ) = yt holds for anyt ∈ Λ̄o ∩ W . Since l̇t = −σlt − xt = ẏt /f ′ (lt ) holds for any t ∈ Λo ∩ W , optimal
firing is given by (18). Since xt ≥ 0, it implies ẏt ≤ −σlt f ′ (lt ) < 0. It also implies
differentiability of µt for any t ∈ Λ̊o ∩ W and right-differentiability for t ∈ E Λ bythe differentiability of l and w and f ∈ C 1 .
Suppose that Bt := Λot \ Xt 6= ∅ for some t and s ∈ Bt . If Bt consists of aninterval, then Proposition 6 implies l˙ = −σl and thus ẏ = −σlf ′ (l) need to holdon that interval. However, it is generically true that actual y will not satisfy thiscondition so that we can safely assume X̄t = Λot almost surely if both are not
empty and if y is considered to be randomly chosen by nature.
VI.

Entering and leaving from the non-hiring phase

Proposition 6 brings a similar situation as optimal control problems with statevariable inequality constraints (SVICs). A state constraint equivalent to (19) in
SVICs would have worked as a binding constraint on controls only in its derivative
form as f ′ l̇ = ẏ. By nature of the derivative form, it does not tell alone when theconstraint becomes binding or off-binding, which requires additional informationon the level. In the current problem, binding constraint (19) is not given but
derived from optimal conditions, however, the same property holds. The jumpcondition is derived from the truncatability of the problem into subperiods Λoand (Λo )c as do those in SVICs. Whenever E Λ 6= ∅, the truncated problem at
the initial time 0 6∈ Λo can be regarded as a problem with the terminal surface(19), where the terminal time te and the terminal state l(te ) are to be optimallydetermined. The problem brings the terminal costate variable to be λte > 0.
Since λ = 0 in Λo , it implies costate discontinuity at entering time. The same
holds for the leaving time.
Suppose E Λ 6= ∅ and 0 6∈ Λo . Let te ∈ E Λ be the first entering time to Λo .
Regarding te as the terminal time in discretion, the truncated problem as of time
zero rewrites as follows.
Z te 
ef (ˆl) − w l − c e−rt dt + Π(te , l(te )) e−rt(21)
Π(0, l0 ) = maxˆl,x,te0

The state transition and constraints remain the same as before. Note that the

18

DECEMBER 2018

firm retains an option to make a massive firing with no cost at the entering time.
Thus, the terminal condition is given by an inequality(22)
f l(te ) ≥ y(te ).

Define the terminal-time Lagrangean Φ(t, lt ) by

Φ(t, lt ) := Π(t, lt ) + µ(te ) (f (lt ) − yt )where µ(te ) is the Lagrange multiplier attached to the terminal surface. Define
general notations z(T− ) := limt↑T z(t) and z(T+ ) := limt↓T z(t) for any timedependent variable z. Then, the terminal condition on the costate variable becomes∂Π∂Φ=+ µ(te )f ′ .
λ(te− ) =∂l∂l
Since ∂Π/∂l = λ(te ) = 0, it implies(23)
λ(te− ) = µ(te ) f ′ l(te ) .

The condition on terminal time is given by
h
i


(24)
f ˆ
l(te− ) − y + λ(te− ) g(˜l(te− ) −σl = rΠ(te , l(te )) + µ(te ) ẏ.

(23) and (24) settle the relation between the entering time and the costate variable
as follows.




(25)
l̇− λ(te− ) − l̇+ λ(te− ) = y − f ˆl(λ(te− )) + rΠ(te , l(te ))
where l˙− λ(te− ) = g(˜l− (λ− ))−σl and l˙+ = ẏ/f ′ l(te ) . Note that l˙− is a functionof λ(te− ). This condition can be interpreted as∂Π∂Π= rΠ + (y − wl − c) + l˙+.
∂t∂l
Namely, the direct benefit of postponing the entering time equates the return of thefirm’s value plus instantaneous profit plus the increased value caused by change of
employment.
The following proposition guarantees that no massive firing occurs at the entering time. Namely, (22) does not hold with inequality. Also, it implies thatthe demand constraint must be unbinding before entering the firing phase. It is
possible for the demand constraint to be binding without accompanying firing,
however, it will not start firing without leaving the demand constraint once. Notethat the presence of labor search friction raises strictly positive rent in a match
between a firm and workers. As far as the distribution rule between the two parties is not dominated by workers, the firm receives strictly positive share of rent,

INTERTEMPORAL DISCONTINUITY

19

i.e. Π > 0. So the assumption in the proposition generally holds in an economy
with labor search friciton.

PROPOSITION 7: Suppose te − ε ∈ W for any ε > 0. If Π te− , l(te ) > 0, thenλ(te− ) > 0, µ(te− ) = 0, µ(te ) > 0 and f l(te ) = y(te ).

PROOF:λ(te− ) < 0 is impossible from (23). Suppose λ(te− ) = 0. Then, ˜l(te− ) = 0 andˆl(te ) = l(te ) from (7) and µ(te ) = 0 from (23). Then, (24) becomes
−
−

(24’)
f l(te− ) − y(te− ) = rΠ(te− , l(te )) > 0.

Then, f ˆl(te− ) ≤ y(te− ) < f l(te− ) implying ˜l(te− ) > 0, which contradicts the
optimality when λ(te− ) = 0. Therefore, λ(te− ) > 0.
Applying the above
result to (23) obtainsµ(te ) > 0. Also, applying to (25)


e
′
e
e
˙
implies l̇− λ(t− ) > l̇+ , namely f l(t ) l− (t− ) > ẏ. Since f (l(te )) = y(te ) from
Proposition 6 and ˆl(te− ) < l(te− ) hold, it implies f (ˆl(te− )) < f (l(te− )) = y, thusµ(te− ) = 0.

Since λ(te+ ) = 0, Proposition 7 implies general discontinuity of λ at entering time
for an economy with Π 6≡ 0. The second half of the proposition means thatthe path of l “bumps” into the demand surface as shown in Figure 3a. The
effect of the same costate discontinuity is reflected in the diagram of production
possibility set in Figure 3b. The optimal path goes into point A on the production
frontier where shadow price of hiring λ is strictly positive at the entering time andjumps to point B. Since B is bound by the demand constraint as Proposition 6
predicts, point A must be unbound because ˆl(te− ) < ˆl(te+ ). The entering behaviorg(l̃)
1λ(te− )ldemand constraint at te
optimalpath

y = f (l)path of l

production
possibilityset (jump)

negative
slope
te

t

(a) Kink in the employment path

bc

A

b B
y(te )O(b) Jump on production frontier

Figure 3. : Entering behavior

is summarized in a literate manner as follows.

f (l̂)

20

DECEMBER 2018

PROPOSITION 8: At entering time, production discontinuously increases sothat unbinding demand constraint beforehand becomes binding afterwards. Hiring discontinuously decreases from strictly positive to zero. Costate variable λjumps from strictly positive to zero. Demand dual µ jumps from zero to strictly
positive. Time path of l kinks so that l˙ jumps downwards.
Figure 4a draws how the jump condition (25) and the terminal surface (22) determine the first entering time. Starting from the initial employment l0 and the
hypothetical initial costate value λ0 , transitional equations (1) and (6) govern
the dynamics. The entering time must satisfy the jump condition (25) which
is drawn as a broken curve on the above plane. The time when the path of λencounters the surface of the jump condition is the entering time. At this time,
the production jumps from f (ˆl) to f (l) and the latter coincides with y. It only
holds for the correct initial costate value. If not, the initial hypothesis on λ must
be corrected. For the rest of entering times, if exist, λ0 and l0 should be replacedby λ(tl+ ) and l(tl ) where tl is the previous leaving time.

λλ0

jump condition (25)λ(te− )

ˆl0Ojump condition (29)λ(tl+ )

O
output
l0

λ

t

te

0
outputttl
y = f (l)y = f (l)

y
f (l)

f (l)
ˆl(tl )f (ˆl)te(a) Enteringtf (ˆl)0tlt(b) Leaving

Figure 4. : Determining junction time

The leaving behavior is analyzed in a similar fashion. Now, consider a production decision as of te . Again, the problem is truncated at tl ∈ Λ̄ote to check the
differentiability at the interface of the firing phase. As we have already obtained
the optimal policy on Λo in section V, the remaining problem is the choice of tl ,which is obtained from the terminal condition. The problem is the same as (21)except that the initial and terminal time is replaced by te and tl , respectively,
and the discounting period is modified to the interval starting from te . So, afterINTERTEMPORAL DISCONTINUITY

21

applying the optimal policy on Λo , we haveeeΠ(t , l(t )) = max
tl

Z tl
te


l
e
e
y − wf −1 (y) − c e−r(t−t ) dt + Π(tl , ltl ) e−r(t −t ) .

Transitions and constraints remain the same. At the leaving time, the following
terminal constraint must hold.
(26)

f (ltl ) = ytl

This is required before optimization is undertaken, since Proposition 6 tells thatthe effectiveness of the constraint (19) is derived from optimality only for t ∈Λ̊o ∩ W but it extends to t ∈ Λ̄o ∩ W by continuity of l. However, one may findsubtlety in inclusion of this terminal constraint in the problem. So let us check its
validity. First, we proceed with the constraint. The terminal condition becomes
h
i
h

i

(27)
− f ˆl(tl ) − f l(tl− ) − λ(tl ) g(˜l(tl ) − σl = rΠ tl , f −1 (ytl ) − ν l ẏλ(tl ) = ν l f ′ l(tl )(28)
where ν l is the Lagrange multiplier adjoint to the terminal constraint. If there
were not a terminal condition, we can set ν l = 0 which implies that λ(tl ) becomes
continuous at zero. By imposing ν l = λ(tl ) = 0, we have f l(tl− ) − f ˆl(tl ) =rΠ tl , f −1 (ytl ) . Since λtl = 0, we have ˆl(tl ) = l(tl ) implying Π tl , f −1 (ytl ) = 0.
This is impossible to happen as argued in the proof of Proposition 7. Therefore,
the terminal constraint (26) is required for the economy to exist within the framework of the maximum principle. Since ν = 0 is impossible in (28), it also showed
the following.
PROPOSITION 9: λ is discontinuous at the leaving time so that λ(tl− ) = 0 andλ(tl ) > 0 hold.
Next, derive the jump condition for leaving. From (27) and (28), we obtain



(29)
l̇+ (λ+ ) − l̇− λ+ = y − f (ˆl+ ) − rΠ(tl , l(tl )).


where l̇+ (λ+ ) = g ˜l+ (λ+ ) − σl and l̇− = ẏ/f ′ l(tl ) . Note that l̇(tl+ ) is a functionof λ(tl+ ). We can rewrite (29) asy − wl + l̇(tl− )λ(tl+ ) = rΠ(tl , l(tl )) + f ˆl(tl+ ) − wl + l̇(tl+ )λ(tl+ ).

It can be interpreted that, on the leaving time, the benefits of postponed and
immediate leave become equal. The left-hand side of the equation is the benefitof postponing the leave. By retarding the leave by dt, the firm receives boundinstantaneous profits and the value of employment change according to bound22

DECEMBER 2018

dynamics. The right-hand side is the benefit of immediate leave. By obtaining
the new state Π, the firm receives its return, and also instantaneous profits andthe value of employment change both according to the unbound path.
The leaving behavior is summarized symmetrically to the entering case.
PROPOSITION 10: At leaving time, production discontinuously decreases sothat binding demand constraint beforehand becomes unbinding afterwards. Hiring discontinuously increases from zero to strictly positive. Costate variable λjumps from zero to strictly positive. Demand dual µ jumps from strictly positive
to zero. Time path of l kinks so that l˙ jumps upwards.
Figure 4b draws how the jump condition (29) and the terminal surface (22) determine the entering time. Different from ordinary optimization problems, theinitial employment in the truncated problem after the leave is not given. The
choice of tl directly determines it according to the demand constraint (26). The
jump condition (25) drawn as as a broken curve on the above plane simultaneously determines λ(tl+ ) which corresponds to the hypothesis on λ0 . If the choiceof tl is correct, the dynamics of λ (6) provides the value of λ which satisfies thetransversality condition if tl is the last leaving time. If there is another enteringto Λo , the dynamics satisfies the entering conditions described above at the next
entering time. Figure 5 draws the kink in the employment path and the jump in
the production frontier at the leaving time.
g(l̃)

ly = f (l)1λ(tl+ )

path of l

production
possibility
set

negative
slopet(a) Kink in employment path

optimalpathb(jump)tl

demand constraint at tl

O

B

bc

A

y(tl )f (l̂)(b) Jump on production frontier

Figure 5. : Leaving behavior

VII.

Weak labor hoarding

A labor asset model can have labor hoarding within business cycles since its
demand depends on the value of the labor asset which does not necessarily meetthe demand for the spot labor expense. In the present model with no firing cost,INTERTEMPORAL DISCONTINUITY

23

it only occurs in a weak sense that firing will not take place even when the decline
in the spot labor demand is more than the natural separation. In this case, since
the value of labor is positive, they are instead hired in the hiring sector. Thisis easily observed by an example. Figure 6 shows a periodic steady state of a

B


λ
B

μ
B

Figure 6. : Periodic steady state of the toy economy

toy economy in which the demand has period 2π and the level of demand falls
below the unconstrained steady state only in t ∈ (0.61, 2.53) in the principal
domain [0, 2π).10 Studying the properties of periodic steady state is beneficial
to close this type of optimization problem with infinite horizon since specifying
external force y for infinite period is literally impossible and effects of faraway
future is discounted anyway. Instead of keeping y open-ended, we can safely
10 Note that this is a non-autonomous dynamical system. The periodicity is brought by that of thedemand constraint potentially binding. The toy economy has the production function f (l̂) = 5 l̂3/4 andg(l̃) = l̃3/4 . Separation occurs at the rate 0.03. Discount rate is set to 0.05. The demand constraint is
y = 10 + 2 sin(t − π) and wage rate is constant at w = 3. Note that, even though demand constraint isbinding only in some subperiod, it brings periodicity upon the whole optimal path. For the propertiesof periodic forced oscillation for linearized systems, see Kato, Naito and Shin (2005).

24

DECEMBER 2018

close it as a loop with sufficiently long period.11 The optimal path binds to theconstraint only in its subset, i.e. t ∈ (1.05, 2.07) =: B. As shown in the first figure,although ˆ
l = f −1 (y) in B, employment begins to increase right after entering in
B. Redundant labor is utilized to enforce the hiring sector in preparation for
future increase of production. In B c , the firm unboundedly chooses to operate ina lower production level than the overall unconstrained case in which the optimal
employment is constant at l = 2.17. Note that, in the constraint binding phase,the increase of employment is driven by the improvement of labor value shown inthe second figure. Since λ̇ = (r +σ)λ−(1−µ)f ′ (ˆl)+w, the rise of µ in the binding
phase more quickly improves the value of labor λ by adding external forces. Note
that, in the unbinding phase, the external force that affects λ is only throughthe change of the marginal productivity of labor. Drastic improvement of labor
value happens more easily in the binding phase in this sense. Also, note that theexistence of a small period of the binding phase can affect the whole dynamics. In
this example, the binding phase occupies only sixteen percent of the total period.
VIII.

Strong labor hoarding and firing cost

On the other hand, if there exists firing cost, strong labor hoarding can arise in
the sense that part of employment is put idle. Assume that there exists convex firing cost κ(x) where κ : R+ → R, κ′ , κ′′ > 0, κ(0) = κ′ (0) = 0 and κ′ (+∞) = +∞.
This specification implicitly assumes that firing activity does not consume internal human resources. This would be approximately true if sufficient information
on worker properties that is necessary for selection of firing target is alreadyaccumulated within everyday work, and if the main cost of firing is pecuniarycompensation. We also impose a moderate assumption that f ′ (ˆl) ≥ w reflecting
the bargaining outcome that the value
We modify
R ∞of profits is strictly positive.
the objective function to Π = maxl̂,x 0 f (ˆl) − w l − κ(x) − c e−rt dt, the labortransition to l̇ = g(˜l) − σl − x where ˜l is the employment in the hiring sector andadd conditions ˜l ≥ 0 and ˆl + ˜l ≤ l to allow for idle employment. Denote adjoint
variables to the last two constraints by η and θ, respectively. The demand constraint is unchanged. Also, we can safely omit the constraint ˆl ≥ 0 as far as y > 0
holds. Then, the first order conditions become
(30)
(31)

(1 − µ) f ′ (ˆl) = g ′ (˜l) λ + η = θ(
0
if λ ≥ 0x=′−1
κ (−λ) if λ ≤ 0.

Costate dynamics is unchanged from (6). Different from the previous model,
λ < 0 is required for firing to exist. Suppose λ < 0 so that x > 0. From (30) andµ ≥ 0, we get η > 0 and ˜l = 0. If µ = 0, (9) implies λ̇ < 0. If µ = 0 continues
11 This is true even when there is a long-run trend in y.

INTERTEMPORAL DISCONTINUITY

25

to hold, it finally violates the constraint l ≥ 0. So, there must be a period thatµ > 0 holds until λ ≥ 0 is achieved. Take such a period. Since µ > 0, it requiresl̇ = ẏ/f ′ (f −1 (y)) as far as we assume ˆl = l, which is generically impossible for ageneral function y. So, µ = 1 > 0, θ = 0 and thus the labor hoarding relation
ˆl = f −1 (y) ≤ l generically holds.
IX.

Implication to linear firing cost

The firing cost in the previous section is specified an exogenous factor represented by κ(x). Such specification minimizes alteration of the basic model but
may blur the actual origin of the cost. If the origin is viewed as that of humanresources directed to firing activities, it would be natural to assume κ(·) to be convex with the same reason as the hiring cost. A formal specification for such firing
cost should extend the basic model to add the firing sector. However, empiricalstudies such as Kramarz and Michaud (2010) show that there is quite different
firing cost structure among different countries, implying that empirical total firing cost may need to keep functional form of κ(·) more general. Kramarz andMichaud (2010) points out firing regulations in France and finds linear firing costfrom French data. If firing cost arising from the human resources is negligible,
the total firing cost may be indeed almost linear. In such a case, the singularity
results of the basic model apply, changing the definition of Λo being intervals of
λ = −k if κ(x) = kx. It assumes that firing cost does not accompany fixed cost.
Then, the optimal condition for firing (8) changes to
if λ > −k0x = [0, x̄] if λ = −kx̄if λ < −kfor arbitrarily large x̄, whereas other conditions are unchanged. It implies that
there exists no-firing and no-hiring interval in λ, i.e. −k < λ ≤ 0. In that interval,
η > 0 and θ ≥ 0 hold in (30). The same arguments in the previous section apply
and in general strong labor hoarding is observed in that interval.
X.

Conclusion

The basic assumptions added in this model are the existence of variety bothin goods and workers. Whereas variety of goods are observable, that of workers
are not by nature of human ability which causes search in face of hiring, which
combines monopolistic model in the output market with search in the labor market. Existence of convex hiring cost is critical for the existance of uncoordinatedpaths. It arises naturally from decreasing returns of individual hiring activity,
which prohibits jumps to the steady state. Namely, the “transition” process matters. Moreover, sufficiently high degree of differentiation in output goods does notalways allow simple transition to the unbounded steady state. Rather it confers

26

DECEMBER 2018a main role of determination of the whole path on coordinated expectation. In
those cases, the transition process is not a transition any more.
Viewing the problem from the perspective of the maximum principle, the present
model showed that the jump of costate variables can occur only with control constraints in singular problems. Jump condition of the state-constrained problems
have been well-known. However, singularity can bring indifferentiability of statevariables on boundary even when only control constraints are included in the
problem, which truncates the problem into subperiods in a similar fashion to thestate-constrained problems and brings about the costate jumps.
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