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Working Paper 8712
IMPLICIT CONTRACTS, ON-THE-JOB SEARCH
AND INVOLUNTARY UNEMPLOYMENT
by Charles T. Carlstrom
Charles T. Carlstrom is an economist at the
Federal Reserve Bank of Cleveland
Working Papers of the Federal Reserve Bank of
Cleveland are preliminary materials circulated to
stimulate discussion and critical comment. The views
stated herein are those of the author and not necessarily
those of the Federal Reserve Bank of Cleveland or of
the Board of Governors of the Federal Reserve System.
December 1987
Abstract
This paper extends t h e i m p l i c i t c o n t r a c t s framework t o a l l o w f o r on - the - job
search.
I t i s shown t h a t i n v o l u n t a r y unemployment can a r i s e i n such a
framework w i t h o u t p l a c i n g any a p r i o r i r e s t r i c t i o n s on e i t h e r wages o r
severance payments.
The model a l s o i m p l i e s t h a t firms w i l l p r a c t i c e a t w o - t i e r
system o f a d j u s t i n g t h e i r l a b o r f o r c e .
o u t s i d e j o b o f f e r s leave t h e f i r m .
I n t h e f i r s t stage, workers who r e c e i v e
The second stage c o n s i s t s o f f i r m s h i r i n g
a d d i t i o n a l workers d u r i n g good s t a t e s o f n a t u r e , and l a y i n g o f f workers d u r i n g
bad s t a t e s o f n a t u r e .
Furthermore, d u r i n g "bad enough" s t a t e s o f n a t u r e , f i r m s
w i l l o f f e r a severance payment o r bonus f o r those who want t o v o l u n t a r i l y
leave, and then l a y o f f workers w i t h o u t o f f e r i n g a l a r g e enough severance
payment t o compensate them f o r b e i n g unemployed.
INTRODUCTION
Traditional models of equilibrium unemployment have failed to explain why
some unemployment might be involuntary.
For example, sequential search models,
such as Lucas and Prescott's (1974) paper, imply that workers wi 1 1 become
unemployed when their expected present discounted value of future utility is
greater when they are unemployed than employed. Another objection to the
search model's explanation of unemployment is the assumption that unemployed
search is more productive than employed search. This assumption has been
frequently questioned and recent evidence suggests the opposite might be
true. '
The purpose of this paper is to provide a consistent story of involuntary
unemployment without placing any a priori restrictions on either wages or
severance payments. While the existence of involuntary unemployment is by no
means universally accepted, most economists accept it as a stylized fact of the
labor market.'
It therefore warrants an explanation within the traditional
framework of equilibrium economics. This paper imbeds a simple model of
on-the-job search in an implicit contracts framework. Implicit contracts
provided one of the first equilibrium attempts to explain involuntary
unemployment. In Azariadis' seminal work, involuntary unemployment results
from three assumptions: 1) Workers are risk averse while firms are risk
neutral. 2) Working is a 0 or 1 decision, that is, hours worked per worker is
not a choice variable.
3 ) Firms cannot make severance payments to unemployed
workers. Given these assumptions, involuntary unemployment results. Ex ante
the optimal contract calls for workers t o become unemployed during certain
states o f nature and t o consume the value of their leisure, thus truncating bad
states of nature. Since workers are risk averse, however, and desire a
constant consumption stream, it is not optimal to lower the wages of employed
workers in order to induce them to leave. Similarly, by assumption, firms
cannot make severance payments in order to induce workers to voluntarily leave.
Another characteristic of Azariadis' model is that there is
overemployment. That is, even though there is involuntary unemployment in the
sense that laid-off workers are worse off than their employed counterparts,
there is over-employment because there is more employment and less
unemployment than would occur in a pure Walrasian market. Workers remain
employed even though their marginal productivity of labor is less than their
reservation wage. Both involuntary unemployment and overemployment result
from the assumption that firms cannot make severance payments to laid-off
workers.
This inability to pay severance payments implies that firms will
partially insure workers against the risk of being laid-off by remaining
employed longer than they would in a pure Walrasian market. Once severance
payments are allowed, unemployment becomes purely voluntary and there is
production efficiency.
The goal of this paper is to integrate a simple model of on-the-job search
in an implicit contracts framework.
This paper investigates the conditions
under which involuntary unemployment will occur without placing any a priori
restrictions on severance payments. Like Azariadis' model, an explanation of
involuntary unemployment will necessitate their seeing overemployment.
This
is in contrast to Grossman and Hart who attempted to explain underemployment.
That is, Grossman and Hart attempt to explain the ex post regret on the part
of firms in the sense that they are laying off workers who ex post they would
want to remain employed. However, all unemployment was voluntary. A recent
paper by Oswald provides the first attempt to explain both involuntary
unemployment and underemployment, but to do so he exogenously assumed that
severance payments were zero. On the other hand, this paper attempts to
explain involuntary unemployment, i.e. the ex post rqgret of workers in the
sense that ex post they would rather remain employed with the firm given the
prevai 1 i ng wage rate.
In order to explain involuntary unemployment, it is promising to follow
the lines of Kahn (1985).
He showed that complete insurance is not possible
(or that wages will not be independent of the state of the world) when a firm
cannot monitor a worker's alternative wage offer. Arvan (1986) extended
Kahn's analysis and suggested this might explain why involuntary layoffs
occur. In Arvan's model, firms cannot insure against layoffs because of the
need to promote on-the-job search. However, Arvan implicitly constrains the
severance payment to laid-off workers to equal the severance payment offered
those who voluntarily quit their jobs. It is this assumption that is crucial
to explaining involuntary unemployment in his model.
This paper is similar to both Kahn's and Arvan's in that it integrates the
original implicit contract model with a simple model of on-the-job search.
The structure of the model differs from theirs by assuming that on-the-job
search may or may not result in a job offer, and by assuming that searching
does not affect the resulting wage offer. If a worker receives a job offer,
the present paper assumes the offer is exogenously given. These assumptions
are not necessary and are meant to simp1 ify the analysis. To explain
involuntary unemployment, no a priori restrictions wi 1 1 be placed on the
structure of severance payments.
The restrictions placed on severance
payments result from the incentive compatibility constraints. However, the
last section does assume that severance payments must be non-negative.
The structure of the paper is as follows: Section one considers the
symmetric information case when a firm can observe both a worker's search
intensity and whether or not the worker receives a job offer. The optimal
contract in this case implies complete insurance. Section two drops the
assumption that a firm can observe a worker's search intensity, but assumes
the firm can observe which workers receive job offers by making severance
payments conditional on the worker accepting an offer.
The section shows that
the inability of a firm to observe a worker's search efforts is not sufficient
to explain involuntary unemployment. However, the model results in incomplete
risk-sharing because firms trade off their desire to provide incentives for
on-the-job search and to insure workers against future wage changes. The
optimal contract is also characterized by production efficiency for laid-off
workers. However, workers who receive job offers are shown to leave more
often than would occur in a Walrasian world, implying that identical workers
(in terms of productivity) will leave the firm and commence working for the
firm simultaneously. The third section investigates the .conditions necessary
t o explain involuntary unemployment. It shows that when firms cannot observe
both a worker's search efforts and whether or not a worker receives a job
offer, the incentive compatible contract implies that laid-off workers will be
better off than their employed counterparts.
However, this result assumes
that firms can tax departing workers. If this assumption is dropped, the
optimal contract results in involuntary unemployment. This occurs in order t o
provide the proper incentive in "bad enough" states of nature for job finders
t o truthfully reveal that they received an offer. The section also discusses
how t h e preceding a n a l y s i s would change i f a worker c o u l d save o r borrow f o r
h i m s e l f r a t h e r than f i r m s a l s o a c t i n g as a bank f o r workers.
The l a s t s e c t i o n
concludes and discusses p o s s i b l e extensions f o r f u t u r e r e s e a r c h .
I. THE MODEL WITH SYMMETRIC INFORMATION
Consider an economy t h a t l a s t s f o r two p e r i o d s indexed by t
=
1 , 2.
Labor
i s h i r e d i n t h e f i r s t p e r i o d where p r o d u c t i o n takes p l a c e a c c o r d i n g t o a
deterministic production function, f ( N ) .
P r o d u c t i o n i n t h e second p e r i o d i s
s u b j e c t t o a random shock, 8, where t h e range o f 8 i s t h e c l o s e d i n t e r v a l
CO,
8"l,
w i t h d i s t r i b u t i o n f u n c t i o n s g(8) and G(B>, r e s p e c t i v e l y .
model may be i n t e r p r e t e d i n a s e c t o r a l s h i f t s framework.
The
Workers search f o r
a l t e r n a t e work i n t h e f i r s t p e r i o d i n case t h e demand f o r t h e i n d u s t r y ' s
o u t p u t f a l l s s u b s t a n t i a l l y i n t h e second p e r i o d o r , a l t e r n a t i v e l y , i f t h e r e i s
a bad shock t o p r o d u c t i o n i n t h e second p e r i o d .
I n t h e f i r s t p e r i o d , workers choose t h e i r search e f f o r t , A, where X
r e p r e s e n t s t h e p r o b a b i l i t y t h a t a worker w i l l r e c e i v e a j o b o f f e r .
A worker's
search i n t e n s i t y i s chosen i n t h e f i r s t p e r i o d b e f o r e t h e r e a l i z a t i o n o f t h e
random shock t o f i r m p r o d u c t i o n i s r e a l i z e d .
a f f e c t t h e p r o d u c t i v i t y o f a worker.
Searching i s assumed n o t t o
For s i m p l i c i t y , i t i s a l s o assumed t h a t
s e a r c h i n g does n o t r e q u i r e any monetary c o s t , b u t r e q u i r e s i n s t e a d a " p s y c h i c "
c o s t c(A>, which i s assumed n o t t o a f f e c t a w o r k e r ' s marginal u t i l i t y o f
income.
The assumption t h a t search e f f o r t e n t e r s s e p a r a b l y i n t h e w o r k e r ' s
u t i l i t y f u n c t i o n i s n o t c r u c i a l ; i t i s meant t o a i d comparison w i t h o t h e r
i m p l i c i t c o n t r a c t models.
I f a worker r e c e i v e s a j o b o f f e r , i t i s assumed t o
be for an exogenous amount, w ' .
Workers a r e assumed n o t t o be a b l e t o a f f e c t
t h i s wage o f f e r through searching.
I t i s t e m p o r a r i l y assumed t h a t t h e f i r m
cannot h i r e a d d i t i o n a l l a b o r i n t h e second p e r i o d .
T h i s assumption w i l l l a t e r
be dropped so t h a t a d d i t i o n a l l a b o r can be h i r e d i n p e r i o d two a t t h e market
wage r a t e ,
w'.
Firms compete f o r workers i n the f i r s t p e r i o d by o f f e r i n g an employment
contract.
C o m p e t i t i o n among f i r m s f o r workers i m p l i i e s t h a t t h e e q u i l i b r i u m
c o n t r a c t w i l l be chosen t o maximize t h e expected u t i l i t y o f a r e p r e s e n t a t i v e
worker s u b j e c t t o a z e r o e x p e c t e d - p r o f i t c o n d i t i o n f o r t h e f i r m .
C o n t r a c t s c o n s i s t o f wages, severance payments, l a y - o f f p r o b a b i l i t i e s and
search i n t e n s i t y .
s 1 ( 8 ) , s,(8),
A),
That i s , a c o n t r a c t c o n s i s t s o f { w l ,
w,(e),
where w l i s t h e f i r s t p e r i o d wage; w,(8>
1(8>, q(8),
i s the
second p e r i o d wage chosen a f t e r t h e r e a l i z a t i o n of 8; l ( 8 ) and q(8) a r e t h e
r e s p e c t i v e s e p a r a t i o n p r o b a b i l i t i e s i n t h e second p e r i o d f o r workers who d i d
n o t r e c e i v e a j o b o f f e r and d i d r e c e i v e a j o b o f f e r a f t e r
t h e end o f t h e f i r s t p e r i o d ; s , (8) and s q ( 8 > a r e t h e severance payments
( o r t a x e s ) g i v e n t o ( o r a p p l i e d t o ) workers who d i d n o t r e c e i v e j o b o f f e r s and
workers who d i d r e c e i v e j o b o f f e r s r e s p e c t i v e l y .
For t h e f u l l i n f o r m a t i o n
case considered below, one can t h i n k o f t h e f i r m as a l s o choosing t h e sear ch
i n t e n s i t y o f workers, A.
Assuming t h a t workers cannot save o r dissave ( t h i s assumption w i l l be
dropped l a t e r ) , so t h a t t h e i r income i n every p e r i o d i s i d e n t i c a l t o t h e i r
wage i n t h a t p e r i o d , t h e expected u t i 1i t y o f a r e p r e s e n t a t i v e worker e q u a l s :
where B i s t h e r e s e r v a t i o n wage o f a worker, o r t h e income e q u i v a l e n t of a
worker consuming h i s endowed l a b o r . I t i s a l s o assumed t h a t U " ( . >
e q u i v a l e n t l y , t h a t workers a r e r i s k averse.
< 0, -- o r
The i n t u i t i o n behind t h e above
equation i s as f o l l o w s : X ( l - q ( 8 > ) i s t h e p r o b a b i l i t y t h a t a worker
r e c e i v e s a j o b o f f e r , b u t remain employed a t t h e f i r m e a r n i n g w,;
Xq(9)
i s t h e p r o b a b i l i t y t h a t a worker r e c e i v e s a j o b o f f e r and accepts i t , i n w h i c h
case t h e worker earns w ' p l u s t h e severance payment s,;
(l-X>(1-1(8)>
is
the p r o b a b i l i t y t h a t a worker does n o t f i n d o t h e r employment and i s n o t l a i d
o f f , i n which case he earns w 2 ; (1-1)
i s t h e p r o b a b i l i t y t h a t the worker
does n o t r e c e i v e an o f f e r and i s subsequently l a i d o f f , i n which'case he e a r n s
t h e v a l u e o f h i s l e i s u r e , B, and t h e severance payment s , ( 8 > .
Assuming
t h e f i r m i s r i s k n e u t r a l , i t has p r e f e r e n c e s g i v e n by
The o p t i m a l employment- c o n t r a c t i s f o r t h e f i r m o r t h e p l a n n e r t o choose
{wl, w2(8), s , ( 8 ) ,
s 1 ( 8 ) , q(8),
1(8), A, N) t o s o l v e :
max E V ( w l ,
En(w
w~(€J),
s,(8),
s l ( B ) , q(B),
I ( @ ) , A)
w 2 (81, s , ( 8 ) , s , (B), q(@), I(@),
A , N) 2 0
The f i r s t - o r d e r c o n d i t i o n s f o r t h i s problem are:
w ' when 8 > 8'
va)
8f'([l-Xq(f3)1N)
vb)
8f'([(l-l(8))(1-X)lN>=B
=
where 8 < 8 ' = > q(8)=1,
when8<e1
0 < l ( 8 ) < 1 and 8 > 8' = > 1(8)=0, 0 < q(8) < 1 .
where 1 (Ot)=O and q ( e t > = l and y are the Lagrangian associated w i t h t h e
expected p r o f i t c o n s t r a i n t and y ,
= Ny.
The s o l u t i o n t o t h i s problem i s s t r a i g h t f o r w a r d .
Since there are no
i n f o r m a t i o n a l asymmetries, the optimal c o n t r a c t i n v o l v e s both p e r f e c t r i s k
sharing according t o Borch's r u l e and production e f f i c i e n c y .
Workers a r e
guaranteed the same income d u r i n g a l l s t a t e s o f the world, independent of both
the s t a t e o f nature and whether o r n o t a worker receives a j o b o f f e r .
Workers
successful i n t h e i r j o b search subsidize those who were unsuccessful.
Production e f f i c i e n c y i m p l i e s t h a t the f i r s t workers t o leave are those w i t h
the b e s t o u t s i d e o p p o r t u n i t i e s , i . e . the workers who r e c e i v e o f f e r s .
After
a l l the workers who have found jobs leave, f i r m s must a d j u s t the l a b o r f o r c e
by l a y i n g o f f workers.
Firms l a y o f f workers u n t i l the marginal p r o d u c t i v i t y
i s equal t o the r e s e r v a t i o n wage o f the marginal worker.
Workers are assumed
t o have non-market o p p o r t u n i t i e s t h a t g i v e the agent an income e q u i v a l e n t o f
B.
Firms then subsidize workers who a r e l a i d o f f by g i v i n g them a severance
payment so t h a t the worker i s i n d i f f e r e n t between s t a y i n g w i t h the f i r m o r
l e a v i n g the f i r m .
Firms a l s o f o r c e workers t o supply the optimum amount o f search i n t e n s i t y
g i v e n by v i i i ) .
One can t h i n k o f wages being s e t equal t o z e r o when workers
supply l e s s than the r e q u i r e d amount o f search e f f o r t .
searching i s equal t o the marginal b e n e f i t o f searching.
The marginal c o s t o f
The marginal b e n e f i t
o f searching i s the d i f f e r e n c e between what the worker w i l l earn i n an
a l t e r n a t e job, w ' , and what he would produce i n the c u r r e n t job, 8 f 1 ( . ) .
good s t a t e s o f nature (8 > 8 0 , t h i s d i f f e r e n c e i s zero from p r o d u c t i o n
efficiency,
w h i l e i n bad s t a t e s o f n a t u r e the d i f f e r e n c e i s w'-B.
The
In
marginal b e n e f i t o f s e a r c h i n g i s t h u s t h e p r o b a b i l i t y t h a t workers a r e l a i d
o f f , G ( B ' ) , m u l t i p l i e d by t h e v a l u e i n u t i l s o f e a r n i n g w ' versus B.
Since
the marginal cost o f searching i s i n u n i t s o f u t i l s t h i s q u a n t i t y i s
m u l t i p l i e d by a w o r k e r ' s marginal u t i l i t y o f income.
Not s u r p r i s i n g l y , t h e o p t i m a l c o n t r a c t w i t h f u l l i n f o r m a t i o n i m p l i e s
complete insurance and, hence, w i t h asymmetric i n f o r m a t i o n i t would n o t
p r o v i d e workers w i t h any i n c e n t i v e t o search.
The n e x t s e c t i o n c o n s i d e r s t h e
o p t i m a l c o n t r a c t when a f i r m cannot m o n i t o r a w o r k e r ' s search i n t e n s i t y and
can choose t h e s e p a r a t i o n p r o b a b i l i t i e s o f workers, t h a t i s , t h e f i r m can t a x
d e p a r t i n g workers.
11.
Both o f these assumptions a r e maintained u n t i 1 s e c t i o n 3.
IMPERFECT MONITORING
I n t h i s s e c t i o n , i t i s assumed t h a t a w o r k e r ' s search i n t e n s i t y i s p r i v a t e
i n f o r m a t i o n t o t h e worker.
However, f i r m s can observe which workers r e c e i v e
j o b s i n t h e second p e r i o d i n t h e f o l l o w i n g manner: severance payments can be
made c o n d i t i o n a l on t h e worker a c c e p t i n g a j o b o f f e r .
With asymmetric
i n f o r m a t i o n , f i r m s choose t h e o p t i m a l c o n t r a c t on t h e assumption t h a t workers
w i l l t h e n choose X t o maximize t h e i r u t i l i t y g i v e n t h i s c o n t r a c t .
g i v e n a c o n t r a c t {w, , w, ( 8 > , s, ( 8 ) , s , (€11, q ( 8 ) , 1(8), N) workers
w i l l choose t h e i r d e s i r e d search i n t e n s i t y , X* such t h a t :
A* =
X
argmax EV(wl, w,(8),
E C0,lI
s q ( 8 ) , s , (81, q(8), I(@),
1)
That i s ,
Replacing the above condition with the first-order condition for an
agent's search effort yields the reaction function for workers. It gives how
agents will choose X in response to the employment contract. This incentive
compatibility constraint is appended to the planner's (or the firm's) problem
in the previous section, so that the optimal contract is to choose {wl(B>,
w2(8),
sq(8), sr(8), q(f3),
max EV(w,, w2(8),
1(6), A , N) in order to solve the following problem:
s,(8),
s ,(8),q(8),
I(@),
The first-order conditions for this problem are
X)
va)
8f1(El-Xq(8)IN)
-
w'
=
{U(wl+sq> - U(w,>
-
U'(w1+sq>[(w'+s,> - w21)/U'(w1+s,>
vb)
8f1(C(1-l(e))(l-X)IN)
where
vi)
=
B
when 8 > 8 '
when 8 < 8'
l ( 8 ' ) = 0 and q ( 8 ' ) = 1.
f l ( N > = w,
Using va) and vb) t o s i m p l i f y ii) y i e l d s :
ii a >
U1(w,(8>) =
iib)
U'(w,(8>>, =
when 8 < 8 ' .
y , (1-1)
( 1-A-y2 1
when 8 > 8 '
y l ( 1-Xq(8)
C1-(X+y, )q(8)1
I n bad s t a t e s of n a t u r e when l a y o f f s occur, 8 < 8 ' , we have complete
insurance f o r l a i d - o f f workers, i . e . B
+ sl
= Wr.
Workers who r e c e i v e j o b
o f f e r s are subsidized and earn more than those who do n o t f i n d o t h e r
employment, i . e . w '
+
s,
> w,.
t h e o u t s i d e wage o f f e r w ' .
w ' i s l a r g e i t may be a t a x .
The s i g n o f sq depends on t h e magnitude o f
I f w ' i s small, i t w i l l always be a subsidy, w h i l e i f
I n good s t a t e s o f the w o r l d where 8 > 8 ' ,
no workers a r e l a i d o f f , and t h e workers r e c e i v i n g j o b o f f e r s earn more t h a n
those who d i d n o t f i n d a1 t e r n a t e employment.
smaller with better states o f nature.
However, t h i s d i f f e r e n t i a l g e t s
This i m p l i e s t h e paradoxical r e s u l t
t h a t the marginal p r o d u c t i v i t y o f l a b o r decreases w i t h b e t t e r s t a t e s o f n a t u r e
when 8 > 8 ' .
I n t h e l i m i t , when q(8)
=
0 , and thus when no workers
l e a v e t h e f i r m , workers w i l l earn equal wages i n b o t h t h e f i r s t p e r i o d and t h e
second p e r i o d .
I t should be noted t h a t t h i s s o l u t i o n i m p l i c i t l y assumes t h a t
f i r m s have t h e power t o e i t h e r s u b s i d i z e o r t a x workers who leave.
That i s ,
even though workers who f i n d a l t e r n a t e employment m i g h t earn more a t t h e i r new
jobs, w' >
WE,
f i r m s a r e assumed t o be a b l e t o t a x them ( o r sabotage t h e i r
f u t u r e j o b p r o s p e c t s ) t o p r e v e n t them from l e a v i n g , thus r e g u l a t i n g t h e number
o f workers who leave t h e f i r m .
Since workers respond o p t i m a l l y t o changes i n t h e c o n t r a c t o f f e r e d t o
them, e q u a t i o n v i i > . s t a t e s t h a t a w o r k e r ' s search i n t e n s i t y w i l l be chosen so
t h a t t h e change i n t h e marginal c o s t t o workers f r o m i n c r e a s i n g t h e i r search
e f f o r t i s equal t o t h e marginal b e n e f i t (expressed i n u n i t s o f u t i l i t y ) t o t h e
f i r m r e s u l t i n g f r o m workers i n c r e a s i n g t h e i r search e f f o r t .
The m a r g i n a l
b e n e f i t f r o m i n c r e a s i n g a w o r k e r ' s search i n t e n s i t y i s t h e d i f f e r e n c e between
what t h e worker is p a i d , w2 , and t h e sum o f what he produces, 8 f ' ( . > , and
t h e severance payment g i v e n t o d e p a r t i n g workers, s,(B).
This implies the
f a m i l i a r r e s u l t : t h a t t h e o p t i m a l c o n t r a c t w i l l s p e c i f y l e s s search e f f o r t
t h a n t h e f u l l i n f o r m a t i o n c o n t r a c t when y r > 0 .
That i s , 1 i s chosen
such t h a t t h e marginal b e n e f i t t o i n c r e a s i n g search e f f o r t i s s t r i c t l y
positive.
The p r o o f t h a t y z i s s t r i c t l y p o s i t i v e f o l l o w s f r o m t h e f i r s t -
order conditions.
effort.
I f y~ < 0 then workers would .not s u p p l y any search
A s u f f i c i e n t c o n d i t i o n for an i n t e r i o r s o l u t i o n t o o c c u r i s t h a t
~ ' ( 0 =) 0, ~ ' ( 1 )=
and w ' > 8, i . e . i t i s c o s t l e s s t o e x e r t a l i t t l e
search e f f o r t and t h e r e i s a p o s i t i v e b e n e f i t t o searching, w h i l e t h e m a r g i n a l
c o s t o f searching, so t h a t a j o b o f f e r i s c e r t a i n , i s s u f f i c i e n t l y c o s t l y s o
t h e p r o b a b i l i t y a worker w i l l r e c e i v e an o f f e r i s l e s s t h a n one.
N o t i c e , t h e r e i s p r o d u c t i o n e f f i c i e n c y when f i r m s l a y o f f workers.
This
i s n o t s u r p r i s i n g s i n c e t h e r e i s complete i n s u r a n c e f o r l a i d - o f f workers.
When t h e marginal worker t o leave, however, i s a worker who has r e c e i v e d a j o b
o f f e r , i . e . when no workers a r e l a i d o f f , t h e r e i s underemployment.
who f i n d j o b s l e a v e more o f t e n t h a n i n a Walrasian market.
Workers
T h i s i s symmetric
w i t h t h e r e s u l t i n A z a r i a d i s ' model t h a t t h e r e w i l l be overemployment when
t h e r e i s i n v o l u n t a r y unemployment.
The i n t u i t i o n behind t h e p r e s e n t r e s u l t i s
t h a t on- the- margin f i r m s f i n d i t o p t i m a l t o p r o v i d e a d d i t i o n a l i n c e n t i v e f o r
on- the- job search by a l l o w i n g workers t o earn more a f t e r t h e y f i n d a n o t h e r
j o b , and a l s o b y a l l o w i n g them t o l e a v e more o f t e n than t h e y would i n t h e f u l l
i n f o r m a t i o n case.
From va), t h e amount p r o d u c t i o n d i f f e r s f r o m t h a t which
would occur i n Walrasian market i s dependent on t h e c u r v a t u r e o f t h e u t i l i t y
f u n c t i o n , o r how r i s k averse workers are.
The more r i s k averse a r e workers,
t h e g r e a t e r t h e need t o i n s u r e a w o r k e r ' s income.
Because t h i s r e s u l t s i n
l e s s search e f f o r t , t h e r e i s a g r e a t e r need t o p r o v i d e i n c e n t i v e s f o r
on- the- job search by a l l o w i n g them t o leave more o f t e n than i n a w o r l d w i t h
symmetric i n f o r m a t i o n .
Given t h a t 8 f i ( . ) > w ' , t h e r e i s an i n c e n t i v e f o r workers who r e c e i v e
job o f f e r s t o recontract with the f i r m .
This i s n o t p o s s i b l e however, g i v e n
t h e assumption t h a t f i r m s can o n l y observe which workers r e c e i v e d j o b o f f e r s
a f t e r t h e o f f e r s were accepted.
I n a d d i t i o n , t h e r e i s t h e i m p l i c i t assumption
t h a t f i r m s cannot h i r e these workers back a f t e r t h e o f f e r has been accepted.
T h i s i s meant t o i m p l y t h a t o f f e r s cannot be c o s t l e s s l y observed.
If the f i r m
c o u l d c o s t l e s s l y observe a w o r k e r ' s o f f e r , t h e r e would always be p r o d u c t i o n
e f f i c i e n c y because f i r m s c o u l d b r i b e workers who f i n d j o b s t o c o n t i n u e
employment b y o f f e r i n g them a h i g h e r wage r a t e , w ' .
I f the marginal
p r o d u c t i v i t y o f l a b o r i s g r e a t e r t h a n w ' , t h e n t h e f i r m has an i n c e n t i v e t o
induce a worker who r e c e i v e d an o f f e r t o s t a y s i n c e t h e y can produce more a t
t h e i r p r e s e n t j o b t h a n t h e y can a t an a l t e r n a t i v e j o b .
Underemployment also results when 8 > 8' because firms, by assumption,
cannot hire workers in the second period at the market wage rate, w ' .
If
additional labor can be hired, then an interesting result occurs. Workers
will simultaneously leave the firm and accept employment with the firm.
Since
the marginal productivity of labor is greater than w ' , the firm will have an
incentive to hire additional workers at w'. Although ex post this seems
wasteful (although no mobility costs are built into the model), ex ante such
behavior is necessary in order to provide workers with the proper incentives
to search in the first period.
To formalize this, assume that the firm can hire n(8) workers in the
second period at a market wage rate of w ' .
choose { w , (8), w2(8),
s,(8),
s , (B) , q(8),
The optimal contract is then to
1(8), X , n(B),
the following problem:
max EV(w, , w,(8),
s,(8),
s ,(B), q(8),
1(8), A)
s.t.
E I I ( w , , ~,(8),
sq(8),
s.t.
~1(8),q(8),
I(@),
X , N) > 0
N) in order to solve
The first-order conditions for this problem are
vi)
{U(w8+s,) - U(w2) - U'(w'+sq)[(wl+sq) - wZI}/U'(w'+sq)
8f1(C1-Xq(B)IN + n(8))
= w' for 8 > 8'
vii)
8f1(C(1-l(B))(l-A)IN)
= B
for 8 < 8',where 1(8'>=
for 8 > 8'
0
and q(8')
Using v) and vi) to simplify i i ) yields:
The results of this exercise are as follows: Workers who stay with the
firm earn a wage rate, w,, which is independent of the state of the world.
=
1.
Workers who receive job offers receive a severance payment from the firms and
will always accept outside job offers. When workers are laid off by the firm,
8 < 8',
complete severance payments will be offered to them, thus there
will be neither under nor overemployment, the marginal productivity of labor
will be equal to B. No additional workers will be hired in these states of
nature. When 8 >
8' ;
however, so that no workers are being laid off, the
firm will hire additional workers at a wage of w ' until production efficiency
prevai 1 s.
This contract implies a two-tier system for adjusting a firm's work
force.
Firms first offer a severance payment to workers who wish to
voluntarily leave the firm. Every worker who has found another job will then
accept this offer. In more complex models, one can think of the severance
payment offered to departing workers as also consisting of possible early
retirement benefits, etc..
After workers accept this offer, the firm then
adjusts the labor force by laying off workers or hiring new workers until it
reaches the desired level of employment. This sort of two-tier system does
seem to have its counterpart in the world. The implication that workers, will
be induced to quit while the firm hires new workers also seems to occur.
Although the current analysis indicates that those who find jobs will always
leave the firm, this result is because there are no adjustment costs incurred
when hiring new workers. If there were adjustment costs (or firm specific
human capital), not all of the workers who found jobs would leave the firm.
It should be noted that since every worker who receives an outside job
offer is allowed to accept the offer, the assumption that firms have the power
to tax workers who leave is no longer necessary. Condition i i i ) assumes that
the severance payment to workers who receive job offers might be negative.
Dropping this assumption, however, would not change the nature of the results.
The c o n d i t i o n f o r the optimum search i n t e n s i t y can be determined b y
s u b s t i t u t i n g i i ) ,i v ) and v ) i n t o i x ) :
Since
YE,
c l ' ( X ) , y, > 0 the optimal contract implies t h a t s1(8) > s , ( B > .
The i n t u i t i o n behind t h i s r e s u l t i s s t r a i g h t f o r w a r d .
c o n t r a c t when workers a r e r i s k n e u t r a l .
Consider t h e o p t i m a l
I n t h i s case, p r o d u c t i o n e f f i c i e n c y
r e s u l t s and workers a r e p a i d t h e v a l u e o f t h e i r marginal p r o d u c t i v i t y i n e v e r y
s t a t e o f the world.
Assuming r i s k n e u t r a l i t y , workers would earn 0 , when 8
< e ' , and would earn w' when 8 > 8'. The f i r s t - p e r i o d wage would be
chosen so t h a t f i r m s earn z e r o expected p r o f i t s .
As workers become r i s k
averse, f i r m s t r a d e o f f t h e i n c e n t i v e s o r p r o v i d i n g on- the- job search f o r
i n s u r i n g workers a g a i n s t wage changes.
F i r s t - p e r i o d wages would be reduced i n
o r d e r t o reduce t h e d i s p e r s i o n i n second- period e a r n i n g s ; t h a t i s , s t ( @ ) >
s, (8).
Otherwise, i t would have been p r e f e r a b l e t o keep t h e c o n t r a c t t h a t r e s u l t e d
when workers were r i s k n e u t r a l as i t a l s o p r o v i d e c t h e p r o p e r i n c e n t i v e s for
on- the- job search.
The above c o n t r a c t must be m o d i f i e d when t h e assumption t h a t f i r m s can
observe which workers r e c e i v e j o b o f f e r s i s dropped, s i n c e t h e above c o n t r a c t
w i l l n o t be i n c e n t i v e compatible.
T h i s i s because t h e severance payment
o f f e r e d t o workers who f i n d a l t e r n a t e employment i s l e s s t h a n t h e one o f f e r e d
t o workers who a r e l a i d o f f (which j u s t compensates a worker f o r b e i n g
unemployed); workers who d i d n o t r e c e i v e j o b o f f e r s w i l l never want t o p r e t e n d
that they did receive a job offer. The opposite is not true, however, when a
large fraction of the labor force is being laid off. Workers who found other
jobs will wish to pretend they did not receive a job offer so they can be laid
off and thereby collect the larger severance payment offered laid-off
workers. The next section considers the optimal contract when the firm cannot
observe a worker's search intensity, or whether or not a worker receives a job
offer.
111. IMPERFECT MONITORING OF SEARCH EFFORTS
The assumption that firms can hire additional labor in the second period
will be maintained in this section, although this assumption is not necessary
for the following results. If firms cannot monitor who receives job offers,
then the optimal contract in the previous section is not incentive
compatible. The incentive-compatible contract will be characterized by either
involuntary unemployment or the opposite: involuntary employment, where
unemployed workers are better off than their employed counterparts. The
condition under which the first occurs, is if the firm cannot tax departing
workers. The second result occurs if the firm has the power to tax departing
workers, and thus the power to choose the separation rates for workers.
T o solve for the optimal contract, when firms cannot observe which workers
receive job offers, the following incentive compatibility constraint must be
placed on the problem:
The l e f t hand o f t h e above equation i s t h e expected u t i l i t y o f a worker if
he admits he r e c e i v e d a j o b o f f e r , w h i l e t h e r i g h t - h a n d s i d e i s t h e expected
u t i l i t y o f a worker i f he does n o t admit he r e c e i v e d a j o b o f f e r .
case, when he i s n o t l a i d o f f he earns w,
I n this
< w ' , since t h e f i r m can r e s t r i c t
his mobility.
The o p t i m a l c o n t r a c t w i t h t h i s r e s t r i c t i o n w i l l choose
{w, (8), w,(8),
s,(8),
max EV(w
w2(8),
s ( 8 ) , q ( 8 ) , I(@),
A , N, n(8)) t o s o l v e :
s,(8),
s I (8), q(8),
1 ( 8 ) , 1)
s. t.
EII(w,,
wZ(8), s q ( 8 ) , s l(B), q(8),
1 ( 8 ) , A, N) > 0
s.t.
.f{l-q(8) )U(w2 ( 8 ) + q(8)U(w1+s, ( 8 ) ))g(8)d8
-
J { ( 1 - - 1 ( 8 > > U ( ~ 2 ( 8 >- l(B)U(B+sl(B)))g(B)d8-
~'(1
=)
0
s.t.
q(e)u(wl+s,(e))
2 i ( e ) u ( w 8 + s 1 ( e ) ) + (l-I(e))u(w,(e))
s.t.
n(8) 2 0
The f i r s t - o r d e r c o n d i t i o n s f o r t h i s problem are:
vi)
Ut(w2(8)){8f'(C(l-1(8)(1-X)1N)CU(B+sl(8))
-
u(w~(8))l -
y3(8)/(1-X-yl-y2)[U(B+~1(8))
B
=
U'(W'+S,(B))CB+S~(B)-W~(~)It
- U(w1+s1(8))I
where l ( 8 ' ) = 0, and q ( 8 ' ) = 1.
Using va) and vb) t o s i m p l i f y ii) y i e l d s :
The s o l u t i o n t o t h i s problem i s i d e n t i c a l t o t h a t g i v e n i n t h e p r e v i o u s
s e c t i o n except f o r t h e i n c l u s i o n o f t h e c o s t a t e v a r i a b l e , y 3 ( 8 > , which
becomes b i n d i n g i n "bad enough" s t a t e s o f n a t u r e .
y3(8) >
I t can be shown t h a t
0, i m p l y i n g t h a t i n s t a t e s o f n a t u r e where a l a r g e f r a c t i o n o f a
g i v e n c o h o r t o f workers i s b e i n g l a i d o f f , t h e severance payment o f f e r e d t o
d e p a r t i n g workers i n c r e a s e s , w h i l e t h e wage o f f e r e d t o t h e j o b s t a y e r s and t h e
severance payment g i v e n t o l a i d - o f f workers decreases.
s u f f i c i e n t t o e x p l a i n i n v o l u n t a r y unemployment.
However, t h i s i s n o t
I n f a c t , l a i d - o f f workers a r e
b e t t e r o f f t h a n t h e i r employed c o u n t e r p a r t s .
can t a x d e p a r t i n g w o r k e r s .
T h i s o c c u r s because f i r m s
I f a worker r e c e i v e s a j o b o f f e r and p r e t e n d s t h a t
he d i d n o t f i n d o t h e r employment, t h e n t h e w o r k e r w i l l e a r n w,(8)
i s not l a i d off.
when he
To see t h e i m p o r t a n c e of t h i s assumption, c o n s i d e r t h e case
when f i r m s c a n n o t t a x d e p a r t i n g w o r k e r s .
I n t h i s case, t h e w o r k e r can q u i t
and e a r n w ' i n s t e a d o f s t a y i n g and e a r n i n g w,(8).
The e f f e c t o f t h e
i n c e n t i v e c o m p a t i b i l i t y c o n s t r a i n t w i l l be t o r e d u c e ~ ~ ( and
8 ) increase
s,(8>.
The severance payment t o l a i d - o f f w o r k e r s w i l l t h e n be l e s s t h a n
n e c e s s a r y t o compensate them f o r b e i n g l a i d o f f .
To m o d i f y t h e p r o b l e m so t h a t f i r m s c a n n o t choose t h e s e p a r a t i o n r a t e s o f
w o r k e r s , t h e i n c e n t i v e c o m p a t i b i l i t y g i v e n e a r l i e r must be m o d i f i e d :
The o p t i m a l c o n t r a c t w i 11 t h e n be t o choose { w , ( 8 ) , w 2 ( 8 ) ,
sl(8), yl(8),
y 2 ( 8 ) , I(@),
X, n ( 8 ) ,
N) t o s o l v e :
max EV(w,, ~ ~ ( 8 s1q,( 8 ) , s I (8), q ( 8 ) ,
I(@),
X)
s,(8),
To solve this problem, I will assume that w' is large enough so that
states of the world will exist such that w, < w'. Otherwise, the results
will be identical to that given.above and thus will not explain involuntary
unemployment. The first-order conditions for this problem are:
vii)
- B)
U'(w2(8)){8f'(C(1-1(B))(1-X)1N)
=
U1(wz(B))[w,(0)
+
U'(w,(B)>
- (B+s1(0))1
- U"(B+sl(B)>
-
CU(wz(0))
- U(B+S~(B))I
where l(0') = 0 and q(8')
=
1.
Using v) and vi) to simplify i i ) yields:
i ia)
U1(w2(8)>
=
y ,(1-1)
for all 8.
(1-1-yz-ys(8))
When no workers are laid off, the model produces a result similar to that
in the previous section; all the workers who receive job offers will be
permitted to leave. When workers are laid off, however, there is production
efficiency except in certain bad states of nature where there is
overemployment. This occurs because comparing iia) and iv) indicates that
when a large fraction of the labor force is being laid off, there will be
involuntary unemployment. The reason is that in order to get the job finders
to truthfully reveal that they have found jobs, the severance payment to
laid-off workers needs to be constrained. This differs from the result given
earlier in this section, because when firms cannot tax workers only severance
payments need to be constrained, rather than both severance payments and
second-period wages. Job finders will then truthfully reveal when they
receive offers.
The condition for involuntary unemployment to exist seems particularly
strong, since it requires a large fraction of the firm's labor force to be
laid off. However, the condition does not seem unreasonable if the condition
is reinterpreted as a plant closing, or where a large fraction of a given
cohort of workers is laid off. The latter might arise in more complex models
with firm-specific human capital which have a lay-off rule based on seniority.
The model predicts that severance payments to both quits and lay offs will
be state independent except during downturns. During severe downturns, the
severance payment o r bonus o f f e r e d i n t h e f i r s t phase o f t h e l a b o r f o r c e
adjustment w i l l a c t u a l l y i n c r e a s e .
t r u t h f u l l y reveal t h e i r job o f f e r s .
T h i s i s so t h a t workers who f i n d j o b s w i l l
I n a d d i t i o n , d u r i n g these downturns t h e
severance payments t o l a i d - o f f workers w i l l decrease so t h a t t h e y a r e
involuntarily laid off.
Because o f t h e c o m p l e x i t y of n o t a t i o n , I have assumed t h a t s,
> 0, o r
t h a t t h e n o n - n e g a t i v i t y c o n s t r a i n t on t h e severance payment o f f e r e d t o those
who f i n d o u t s i d e o f f e r s i s n o t b i n d i n g .
A l l o w i n g t h i s c o n s t r a i n t t o be
b i n d i n g does n o t a f f e c t t h e r e s u l t s .
A c r i t i c i s m o f t h e c u r r e n t model i s t h a t w h i l e i t extends A z a r i a d i s '
model, i t s t i l l p r e d i c t s t h a t severance payments should be p a i d t o l a i d - o f f
workers.
This i s i n l i g h t o f r e c e n t evidence by Oswald, t h a t r e l a t i v e l y few
i n d u s t r i e s a c t u a l l y o f f e r some f o r m of severance payments, a l t h o u g h o v e r 50
p e r c e n t o f manufacturing i n d u s t r i e s do o f f e r such payments.
I f t h i s model i s
extended, however, so t h a t workers can save, i n s t e a d o f a l l savings and
d i s s a v i n g s b e i n g p r o v i d e d by t h e f i r m , t h e p r e s e n t model w i l l p r e d i c t t h a t
severance payments m i g h t n o t be as p r e v a l e n t as e a r l i e r p r e d i c t e d .
Once savings a r e p e r m i t t e d one would n o t n e c e s s a r i l y p r e d i c t f i r m s t o save
f o r workers.
I f savings a r e p e r m i t t e d b e f o r e t h e r e a l i z a t i o n o f t h e shock t o
f i r m p r o d u c t i o n i n t h e second p e r i o d , then workers would save i n t h e f i r s t
p e r i o d an amount equal t o t h e severance payment o f f e r e d t o q u i t s i n t h e
previous analysis.
That i s , savings would be s , ( 8 > f o r 8 > 8 ' where
8' i s d e f i n e d t o be t h e c u t o f f a t which s, increases and s , decreases.
The r e s u l t would be severance payments o f f e r e d o n l y t o workers who want t o
v o l u n t a r i l y leave t h e f i r m d u r i n g bad shocks t o i n d u s t r y demand.
Casual
e m p i r i c i s m suggests t h a t indeed d u r i n g downturns f i r m s do o f f e r bonuses t o
those who want t o v o l u n t a r i l y leave t h e f i r m .
be v e r i f i e d by f u t u r e e m p i r i c a l work.
However, t h i s p r e d i c t i o n c o u l d
The model w i t h savings a l s o decreases
t h e amount o f severance payments t h a t i s o f f e r e d t o l a i d - o f f workers.
This
m i g h t h e l p e x p l a i n t h e seeming l a c k o f severance payments t o workers i n most
industries.
Of course, a l l o w i n g workers t o save w i l l change t h e f i r s t o r d e r
c o n d i t i o n s , since t h e r e would be another c o n s t r a i n t placed on t h e problem.
However, t h i s c o n s t r a i n t would be independent o f 8 and thus would n o t a f f e c t
t h e q u a l i t a t i v e r e s u l t s o f t h e paper.
I V.
CONCLUSIONS
The goal o f t h i s paper was t o i n v e s t i g a t e t h e c o n d i t i o n s under which
i n v o l u n t a r y unemployment can r e s u l t i n a standard i m p l i c i t c o n t r a c t s model
t h a t i n c l u d e s t h e p o s s i b i l i t y o f on- the- job search.
The r e s u l t s were
encouraging; i t was shown t h a t under c e r t a i n c o n d i t i o n s i n v o l u n t a r y
unemployment can e x i s t .
The c o n d i t i o n s necessary t o achieve t h i s r e s u l t were:
1) Firms cannot observe a w o r k e r ' s search i n t e n s i t y , 2 ) Firms cannot m o n i t o r
w h i c h workers r e c e i v e j o b o f f e r s , and 3 ) Firms cannot t a x d e p a r t i n g workers.
The q u e s t i o n o f whether o r n o t t h e c o n d i t i o n s necessary t o e x p l a i n i n v o l u n t a r y
unemployment occur o f t e n enough t o e x p l a i n t h e "observed" i n v o l u n t a r y
unemployment cannot be answered.
The paper a l s o showed t h a t f i r m s w i l l have a
t w o - t i e r procedure f o r a d j u s t i n g i t s l a b o r f o r c e t o c u r r e n t economic
conditions.
I n t h e f i r s t round, a model w i t h savings i m p l i e s t h a t workers
w i t h o u t s i d e o f f e r s would leave w i t h o u t t h e inducement of severance payments;
i n t h e second round, t h e f i r m a d j u s t s i t s l a b o r f o r c e by e i t h e r l a y i n g o f f
a d d i t i o n a l workers o r h i r i n g new workers.
The model i m p l i e s t h a t workers w i l l
leave t h e f i r m and new workers w i l l be h i r e d by t h e f i r m , a l t h o u g h a l l workers
a r e assumed t o be e q u a l l y p r o d u c t i v e .
This o c c u r s because, ex ante, f i r m s
have t o o f f e r c o n t r a c t s , which g i v e s workers t h e necessary i n c e n t i v e s t o
engage i n on- the- job search, which a l s o i m p l i e s s u b s i d i z i n g them when t h e y
leave t h e f i r m .
offers,
However, ex p o s t , s i n c e f i r m s cannot observe who r e c e i v e s j o b
they w i 11 f i n d i t p r o f i t a b l e t o h i r e new workers t o r e p l a c e those who
quit.
One frequent c r i t i c i s m o f t h e above a n a l y s i s i s t h a t i t i m p l i e s t h a t f i r m s
a r e s u b s i d i z i n g workers t o engage i n more on- the- job search.
Ex ante
c o n t r a c t s w i l l be chosen so t h a t workers w i l l f i n d i t o p t i m a l t o engage i n
such search a c t i v i t y , however, ex p o s t i t would n o t be s u r p r i s i n g t o t h i n k
t h a t f i r m s a r e i n some sense a n t a g o n i s t i c t o such a c t i v i t y .
Firms w i l l , of
course, wish t h a t none o f t h e i r workers a r e s u c c e s s f u l i n t h e i r j o b search.
S i m i l a r l y , another way o f t h i n k i n g about t h e problem i s t h a t f i r m s s i g n
c o n t r a c t s t h a t reduce worker m o b i l i t y i n o r d e r t o p a r t i a l l y i n s u r e workers
a g a i n s t income changes.
T h i s paper showed why complete insurance t o l a i d - o f f workers would n o t be
optimal, given the i n c e n t i v e c o m p a t i b i l i t y c o n s t r a i n t s .
Additional empirical
work i s necessary t o answer t h e q u e s t i o n o f whether t h e amount o f severance
payments p r e d i c t e d by models, such as t h e p r e s e n t one, occurs i n the w o r l d .
One reason t h e amount o f severance payments o f f e r e d by f i r m s m i g h t n o t be t h a t
e x t e n s i v e , i s because o f state- mandated unemployment b e n e f i t s .
Theory
suggests t h a t t h e two a r e s u b s t i t u t e s ; thus, i n c r e a s e s i n s t a t e - p r o v i d e d
unemployment i n s u r a n c e should decrease p r i v a t e severance payment programs.
F u t u r e e m p i r i c a l work can be conducted t o see if p r i v a t e l y f i n a n c e d
unemployment b e n e f i t s decrease w i t h increases i n s t a t e - p r o v i d e d unemployment
insurance.
Endnotes
1)
For example, B l a u (1986) f i n d s t h a t f o r l e s s e f f o r t , employed s e a r c h e r s
r e c e i v e more j o b o f f e r s t h a n unemployed j o b s e a r c h e r s .
However, due t o
unobserved d i f f e r e n c e s between employed and unemployed searchers, h i s d a t a
remains p u r e l y s u g g e s t i v e .
2)
For example, Lucas has argued:
" I n v o l u n t a r y unemployment i s n o t a f a c t o r
a phenomenon which i s t h e t a s k o f t h e o r i s t s t o e x p l a i n .
I t i s , on t h e
c o n t r a r y , a t h e o r e t i c a l c o n s t r u c t which Keynes i n t r o d u c e d i n t h e hopes i t
would be h e l p f u l i n d i s c o v e r i n g an e x p l a n a t i o n f o r a genuine phenomenon:
l a r g e - s c a l e f l u c t u a t i o n s i n t o t a l employment."
3)
The problem i s a c t u a l l y
t h e s o c i a l p l a n n e r ' s problem.
The " d u a l " p r o b l e m
where t h e f i r m maximizes i t s p r o f i t s s u b j e c t t o an i n d i v i d u a l r a t i o n a l i t y
c o n s t r a i n t f o r t h e worker, does n o t a f f e c t t h e r e s u l t s .
4)
To see t h i s , compare i i a ) w i t h i i i ) . Since U 1 ( w ' + s l ( B > > < U 1 ( B + s l ( B ) )
t h e n U 1 ( w 2 ) > U 1 ( B + s I ( B ) ) o r t h a t B + s l ( e ) > w,(B).
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