18

n(@; 5p) are positively related and both curves in figure 1 must be upward
sloping. By Prop. 6, as 1, (p) >1, n(8;,p) au and w(8; 5p) + we Thus (n; sw)
must be a point on the locus corresponding to state 6. By Prop. 4,

w(01,p) >w> W(855P)5 hence the locus for the bad state must lie above w and
the locus for the good state must lie below w. Note from figure 1 that it must
be the case that n(®,,p) > ny and n(@5,p) < My, as assertd in Prop. 4.

If actual observations were scattered uniformly on each curve we
would observe a positive correlation between wages and employment. However, we
have a negative correlation between p and @; hence observations should be
clustered around the high probability points at the bottom of the bad state
curve and the top of the good state curve. It is quite consistent with this
model to observe a negative correlation between wages and employment. The
degree of variation of wages depends on the flatness of the two curves, but it
does not seem inconsistent with our model to observe little cyclical variation
in wages,

The implications of the disequilibrium contract model are illustrated
in figure 2. In this model, the nominal wage is fixed in a contract before p
and @ are observed, and once p and 6 are observed, firms are allowed to
maximize profits. They hire (or lay off) along the relevant marginal
Productivity curve pictured in figure 2. The nominal wage is set at a level
that is expected to achieve some real wage target, we d/ If actual
observations were scattered uniformly on each curve we would observe a negative
correlation between wages and employment. However, the negative correlation
between p and @ implies that actual realizations should be clustered around the
lower portion of the lower curve and around the upper portion of the upper

curve. A positive correlation between wages and employment is consistent with


13

to assuming that the elasticity of expected utility in state (6) ,p) with
respect to wages is greater than or equal to the elasticity of expected utility
with respect to employment.) This condition is sufficient to insure that the
second-order conditions for a maximum are satisfied. It is assumed that the
unique solution to (13) is interior with respect to the feasibility
constraints. Such a solution satisfies the first-order conditions (12b)

and

(14a) ary (p) (0, '[n(0y,p)] - w(0y,p)} + (1-A)ny(p){ULw(0}5p)] ~ KI
~ yp(p) {@of '[n(014p)] - w(01,p)} = 0

(14) Lang(p) + yp(p) {oF 'En(@g4p)] - w(025p)}
+ (1 - A)no(p){ULW(Op,p)] - kK} = 0

(14c) -any(p) + yo(p) + (1 = A)my(p)U'Ew(01,p)] = 9
(144) -Ang(p) = vg(p) + (1 = a)ag(p)U'[w(02,p)] = 0.

While it is impossible to solve for closed-form expressions for w(8,p) and
n(@,p), many interesting results can be derived implicitely from the first
order conditions.

Prop 4. If w(0,p) and n(@, p) are the optimal wage and employment rules,

then
(i) w(y,p) > W > w(O2,p),
(ii) n(84y,p) > (<) ny as oof En(615p)] - w(81,p) < (>) 0

(iii) n(@2,p) < ng


Figure 4
Employment

+2515

~..2510 >

~ .2505 F—

9(6, »P)

n_=.2500[— —— _—
1

32495 J | | | | !
Se , , - JW, ()

oe

44500

24425 —

+4400 —

2(@ 5 5)

-4375

Lee
5
.
a
=)
€

+4350



shocks.

The production function of the typical firm is
(1) oy = ef(n),

where n is the fraction of the firm's work force that is actually working and 6
is a random variable. We assume that there is no job sharing; a worker is
either employed, or he is laid of t.2/ The shock @ can take two values, 8) and
855 with probabilities aT and Tos respectively. It is assumed that
Tm € (0, 1) for i = 1, 2, m +1) = 1, and 8, < By 8) is the "bad" state of
nature. The function f(+) is increasing, strictly concave, twice continuously
differentiable, and f'(n) += as n+ 0.

The utility of the typical worker is given by V(c,%), where c
represents consumption and & represents leisure. There is no job sharing, so &
can only take two values, 0 if the worker is employed and 1 if the worker is

laid off. Define
(2) U(w) = V(e4w,0)
(3) K = V(c, 1),

where w is the real wage and c is non-labor income. U(w) is the utility of an
employed worker earning wage w, and K is the utility of a laid-off worker. The
function V(c,%) is increasing in 2, so K > U(O). The function U(+) is
increasing, strictly concave, twice continuously differentiable, and U'(w) + 0
aswro,

At the beginning of each period the firm and its workers agree to a
contract specifying a wage rule, w(6,p), and an employment rule, n(@,p). The

rules are functions of the real shock and the aggregate price level. The


19

this paradigm. Little cyclical variation in wages is also consistent with this
framework.

To summarize, both the asymmetric-information contract model and the
disequilibrium contract model seem capable of explaining the stylized facts
about the cyclical behavior of real wages.

The Implications for Monetary Policy

Information constraints are like technology constraints; they limit
welfare. Prop. 5 states that the asymmetriceinformation structure we have
imposed results in too little employment in the bad state, 8) However,

Prop. 6 states that if the price level is a perfect signal of the productivity
shock, the symmetric-information contract is implementable. If p reveals @ to
the workers, then the information constraint is not binding. The normative
implication of this paradigm is that monetary policy should make p a good
signal of 6.

In equation (9), it is the unobserved disturbance v that makes pa
noisy signal of 6. If the monetary authority has superior information, it can

improve the quality of the signal. For example, the policy rule
(18) m= 1/v

would make p a perfect signal of 6. The monetary authority should offset
monetary disturbances (or fluctuations in aggregate demand).

In the islands model paradigm, the information constraint is that
local producers do not know the aggregate price level; local prices are a noisy
signal of p. This suggests that the monetary authority ought instead to use
its superior information (if it has any) to peg the price level, making it

predictable.


23

an(@1,p) 0
1

an(6asP) 4
we SO

Thus a velocity shock results in a rise in employment in the bad

state and a fall in employment in the good state.

Proceeding as in the text we can graph the locus of employment -wage

pairs for each 6.

real Figure 5
wages
4
81
w
1 I
J Bo S|
U

i » employment
ay ny

From the figure it is clear that n(6; 5p) < ny thus, employment is
always higher under symmetric information. The high probability points are
those on the bottom of the 8 curve and top of the 8 curve. Thus, the

negative relationship between wges and employment holds for this case.


~28

-27

” 26

we.25

+24

- 23

Figure 3

Real Wages
WC, »P)
w(®,,P)
| I ! i] ] l
al .2 3 5 7 8

| T @)
1


17

employment in each state decrease with an increase in p. The basic story that
seems to emerge from all of this is that a positive monetary disturbance raises
the price level and is taken as a noisy signal that the bad state cy may have
occurred. Employment and real wages fall, no matter what state has actually
occurred. The negative effect on employment runs counter to our first stylized
fact.

Prop. 7 can also be used to verify our earlier assertion that p and 6
are negatively correlated if velocity is not "too" large and "too" highly
correlated with @. To see this, suppose the reverse is true and high values of
Pp are associated with Bo. Then da, (p)/dp < 0 and in order for (7) to hold it
must be the case that an(8;.p)/ap = Can(6;,p)/am, (p) Lan, (p)/ap] <0. By
Prop. 7, this is a contradiction.

Cyclical Behavior of Real Wages

Fluctuations in the real wage and employment are generated by changes
in productivity in addition to unanticipated movements in velocity. As we
noted in the introduction, the stylized fact is that wages and employment are
negatively correlated in most countries other than the United States, and
either uncorrelated or mildly positively correlated in the United States. In
addition, there is a perception that there is not much cyclical variation in
wages, whatever the correlation. Here we draw out the implications of the
asymmetric-information contract model and compare them to the implications of
the disequilibrium contract model of Gray (1976) and Fischer (1977) 26/

The implications of the asymmetric-information contract model are
illustrated in figure 1. The higher curve represents the locus of wage and
employment levels for the bad state and the lower curve represents the locus of
wage and employment levels for the good state. By Prop. 7, as 11 (P) rises from

the 0 to l, w(0; 5p) and n(8;5p) fall for i = 1, 2. Thus w(8,5p) and


It should be noted that employment fluctuations are ex post, as well
as ex ante, optimal in the following sense. After the realization of 6, but
before the specification of which laborers are to be laid off, the contract
cannot be rewritten so as to make either the firm or any one worker better off

without some other worker or the firm being made worse off.

IV. CONTRACTING UNDER ASYMMETRIC INFORMATION

If the realization of 6 is the private information of the firm, and
workers never observe it directly, then the firm has a credibility problem.
The optimal symmetric-information contract, cs, is written in terms of the
unobservable 6, and the firm may have an incentive to report a lower value of 6
than has actually occurred. This is because the contract cS calls for a real
wage that is higher than the marginal product of labor. By understating 6, the
firm can lower employment and may increase profits. If the firm cannot be
relied upon to simply tell the truth, say for altruistic reasons, then cS is
not implementable.

In this section, we restrict our attention to contracts that are, in
Hurwicz's (1972) terminology, “incentive compatible". A contract is said to be
incentive compatible if the firm has no incentive to misreport its private
information. At first glance, this may appear to be an overly restrictive way
of dealing with the credibility problem; workers may be better off ina
contract where the firm is systematically misrepresenting 6. However, it turns
out that there is no loss of generality here; any contract that is not
incentive compatible can be rewritten in terms of one that is 2/

In this section, we also restrict our attention to contracts that
last only one period, and here we are being overly restrictive. Townsend

(1932) has shown there is a gain from enduring relationships that can be


26
References

Azariadis, Costas, "Implicit Contracts and Underemployment Equilibria,"
Journal of Political Economy, 83 (December 1975): 1183-1202.

Baily, Martin N., "Wages and Employment under Uncertain Demand," Review of
Economic Studies, 41 (January 1974): 37-50.

Barro, Robert J., "Rational Expectations and the Role of Monetary Policy,"
Journal of Monetary Economics, 3 (January 1976): 1-32.

_» “Long-Term Contracting, Sticky Prices, and Monetary Policy,"
Journal of Monetary Economics, 3 (July 1977) :305-16, ,

_» "Unanticipated Money, Output, and the Price Level in the United
States," Journal of Political Economy, 86 (August 1978): 549-80.

Bodkin, Ronald G., “Real Wages and Cyclical Variations in Employment: A
Reconsideration," Canadian Journal of Economics, 2 (August 1969): 353-
74,

Calvo, Guillermo and Edmund Phelps, "Employment and Wage Contingent

Contracts," Journal of Monetary Economics, 5 (supplement 1977): 160-
68.

Canzoneri, Matthew, “Labor Contracts and Monetary Policy," Journal of
Monet:ary Economics, 6 (April 1980): 241-55.

"The Returns to Labor and the Cyclical Behavior of Real Wages:
he Canadian Case," The Review of Economics and Statistics, 60
(February 1978): 19-24,

Canzoneri, M., D. Henderson, and K. Rogoff, "The Information Content of the

Interest Rate and Optimal Monetary Policy," Quarterly Journalof
Economics, 98 (November 1983): 545-566.

Chari, V.¥., “Involuntary Unemployment Implicit Contracts," Quarterly
Journal of Economics, 98 (supplement 1983): 107-22.

Cooper, Russell, "A Note on Overemployment /Underemployment in Labor
Contracts under Asymmetric Information," Economic Letters, 12 (1983):
81-87.

Fischer, Stanley, "Long-Term Contracts, Rational Expectations, and the
Optimal Money Supply Rule," Journal of Political Economy, 85 (February
1977): 191-205.

» “Retative Price Variability and Inflation in the United States
and Europe," European Economic Review, 18 (May/June 1982): 171-96.

Gordon, Donald F., "A Neoclassical Theory of Keynesian Unemployment,"
Economic Inquiry, 12 (December 1974): 431-59,


Expectations are taken with respect to the joint distribution of @ and p which
will be discussed in some detail below.

We examine two information structures in the sections that follow.
With symmetric information both the firm and its workers observe the real shock
8 ex post. With asymmetric information only the firm observes 6 ex post.
However, even if @ is private to the firm, workers can obtain some information
about it by observing movements in the aggregate price level, for p is assumed
to be a noisy signal of real disturbances. Before going on to define and
derive the optimal contract under each of the information structures, we want
to be more precise about the information embodied in p.

To do this, we close the model with the simple velocity equation
(7) mv = py = pof[n(6,p)],

where m is the money stock and v is a continuous random variable representing a
monetary disturbance (or a fluctuation in aggregate demand). In all of the
contracts we examine, it turns out that a decrease in 6 creates an excess
supply of money which can be eliminated by increasing p. So at a given p and
v, a decrease in @ creates an excess supply of money and p must rise. Thus, if
v is not "too" large and "too" strongly correlated with @, a higher p will be
interpreted by workers as an increase in the Probability that the bad state @
has occurred; letting TM% (p) and t2(p) be the probabilities of 8,and Bos
respectively, given that the price level is equal to p,

day (p) dao(p)

(8) > 0, < 0.
dp dp



reason for including the price level will be discussed later in this section.
Workers are assumed to be mobile, so each of the identical firms must offer the
same contract / Lay-offs are assumed to be random, so for a given state,
(0,p), a worker has probability n(6,p) of being employed.

Def. 1. A contract C = {n(e,p),w(6,p)} is said to be feasible if

ULw(e,p)] > K
(4) ef{n(6,p)] - n(e,p)w(e,p) > 0
0 <n(e,p) <1

for every possible (6,p).

The first condition asserts that workers prefer working to being laid
off at the specified wage. The second condition requires that the firm make a
positive profit. These conditions rule out contracts that give either the firm
or its workers an obvious incentive to renege if a bad state occurs. However,
it should also be noted that a laid-off worker will have an incentive to offer
to work for slightly less than the specified wage if U(w) > K. The firm will
be tempted to accept this offer and lay off someone else. Having ruled out the
most egregious cases, we still have to simply assume that the contract is
binding for the firm. It will be assumed that there exist constants n and w
such that U(w) > K, of(n) - wn > 0, and 0 <n <1. This will ensure that a
feasible contract exists.

Under the contract C, the firm's expected profit is
(5) P(C) = Ef{ef[n(e,p)] - w(e,p)n(6,p)},
and the worker's expected utility is

(6) 1(C) = E{n(o,p)ULw(6,p)] + [1 - n(e,p) IK}.


Asymmetric Information Contract Model

real
wages

4

—p employment

Figure 1

Disequilibrium Contract Model

> employment

Figure 2


given , there exists a unique solution to (9). We also assume that the

solution is interior, and thus, satisfies the first-order conditions.©/

410) A{9f'Cn(0;,p)] ~ w(0;4p)} + (1-a){ULW(0;.p)J - K} = 0 4 = 1, 25 ¥

=A + (1-A)U"EW(9; 4p)] = 0 i= 1, 25%.

Rearranging (10), the optimal symmetric-information contract, cS, is

characterized by

(lla) w(e;.p) =W where U'(w) = pA i=, 2;
(11b) n(0;,p) = ny where 0; f (nj) = w - [U(w)-KI/U' (w) i=1, 23 ¥p

Thus, by (lla), the real wage is constant across states; the firm
insures the workers against real wage variation. By (11b), the level of
employment is a monotonic function of 6. Sargent (1979, ch. 8) explains why
the firm does not insure workers completely against lay-offs. The reat wage
exceeds the marginal product of labor in each state. Workers accept a positive
Probability of unemployment in exchange for this premium.

It is apparent from (11) that neither the real wage nor employment
responds to fluctuations in the price level. A monetary disturbance,
therefore, has no real effects. Thus, contracting under symmetric information
cannot explain our first stylized fact. Employment does respond to
fluctuations in 6, so contracting with symmetric information Provides a real
business cycle model. Real wages are fixed at W so if business cycles are
indeed generated by real disturbances, then we have an explanation of our third

stylized fact.


21

APPENDIX
Proof of Prop 1. The inequality

Gof [n(62,p)] - wn(69,p) < Oof{n(0,,p)J - wn(01,p)
holds (by the Mean-Value theorem) if
oof (®) <W 5 ¥i © (n(0,4p)4n(60,p))-

<> - 2 um) - K1< (8 - 0%,

U (w)

where 3 = [w - Uiwl-K yy5'(g),
U (Ww)

Clearly 6 « (6), 85)3 hence this must be true for 8, - 6 sufficiently small.
Proof of Prop. 3. By contradiction.

Suppose Prop. 3 does not hold. Then ¥, (P) > Yo (P) = 0. The relevant

first-order conditions are (12a) and

(Al) Lany(p) + yy (p)]{@,f '[n(6,4p)) - w(0,,p)} +
(1 = A)my(p) {ULW(6),p)] - K} = 0

(A2) any (p) {@9f {n(@p,p)] - w(@p,p)} + (1 = A)ng(p) {ULW(e2.p) - K} -
¥1{0,f '£n(825p)] - w(@y,p)} = 0

(A3)—-any(p) = yy(p) + (1 = a)ny(p)U'Lw(0,,p)] = 0
(A4)-ang(p) + ¥y(P) + (1 = A)np(p)U'Lw(69,p)] = 0
By (A3) and (A4)

U"Ew(615p)] > U'LW(0,4p)]


International Finance Discussion Papers
Number 248

September 1984

THE MACROECONOMIC IMPLICATIONS OF LABOR CONTRACTING
WITH ASYMMETRIC INFORMATION

by

Matthew B. Canzoneri
Anne C. Sibert

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished materials)
should be cleared with the author or authors.


11

optimal contract follow directly from the incentive-compatibility restrictions.
Next, we characterize the optimal contract, illustrate the response of
employment to a monetary disturbance, and discuss the cyclical behaviour of
real wages. Finally, we discuss the implications for monetary policy.
Incentive Compatibility

Def. 3. A contract is said to be incentive compatible if

(12a) 6fEn(0;,p)] - w(@,,p)n(9;,p) > 6; F[n(62,p)J - w(89,p)n(69,p)
(1b) egf[n(@2,p)] - w(89,p)n(O5,p) > eof{n(e;,p)] - w(81,p)n(0,p).

The first equation says the firm is not tempted to report 85 when 8) has
occurred; the second equation says the firm is not tempted to report FT when
85 has occurred.

The optimal symmetric-information contract, cs, violates the second
condition if 8, is "close" to 856
Prop. 1 For @, - 8, sufficiently small, @f(n5) - wn, < 8of(n,) - wns
where ny ie 1, 2 and w are given by (11).
If the support for @ were continuous, no qualification would be necessary; the
reasoning given earlier would suffice. The proposition is proven in the
Appendix.

The incentive-compatibility constraints themselves determine several

characteristics of the contracts considered below.

Prop. 2 If C = {w(e,p),n(6,p)} is incentive compatible, then

(i) n(@,p) > n(6,,p) 4p
(44) w(@g,p)n(@p,p) > w(0,,p)n( 6, .p) ‘p
(i144) @gfCn(65,p)] - w(e,,p)n(@,,p) > e,f[n(6),p)] - w(0,,p)n(e,,p) ¥p


10

captured in miltiperiod contracts. Some insights as to why this is so can be
gotten by noting that firms may not be able to credibly understate 9 period
after period if workers know the probability distribution of 0.20/ Contractual
arrangements can make reporting a low © in the first period limit the options
of the firm in subsequent periods. We will not explore these possibilities
further, but we do note that Townsend's reason for the existence of long-term
contracts contrasts sharply with views coming from the disequilibrium contract
models. The latter stress costs of negotiation.21/

With asymmetric information and incentive-compatibility problems,
there is a need to use the information embodied in the price level when
designing optimal wage and employment rules. The price level provides a noisy
signal of 6; the signal is noisy because the price level also moves in response
to monetary disturbances (or fluctuations in aggregate demand). Note that if
contracts index wages and employment to the price level, then monetary
disturbances will pass to real economic activity. The asymmetric-information
contract model is a monetary model.

Both the new contract paradigm and the islands paradigm make money
matter by imposing an information constraint on the economy, and like a
technology constraint, these information constraints impose welfare costs. In
both paradigms, absent information constraints, agents would not let monetary
fluctuations affect real economy activity. The ultimate goal of monetary
policy is then to make the information constraint non-binding (by improving the
quality of local price signals in the isalnds paradigm and by improving the
quality of the aggregate price signal in the contract paradigm) and return to
an economy in which monetary fluctuations do not matter.

We will proceed as follows: First, we discuss incentive

compatibility in more detail; it turns out that several properties of the


28

Taylor, John B., "Recent Developments in the Theory of Stabilization
Policy," in Stabilization Policies: Lessons from the '70s and
Implications for the "80s, St. Louis: Center for the Study of American
Busness, Washington University, 1980: pp. 1-40.

Townsend, Robert M., "Optimal Multiperiod Contracts and the Gain from
Enduring Relationships under Private Information," Journal of
Political Economy, 90 (December 1982): 1166-1186.


27

Gray, Jo Anna, "Wage Indexation: A Macroeconomic Approach," Journal of

Monetary Economics, 2 (April 1976): 221-35.

> "On Indexation and Contract Length," Journal of Political
Economy, 86 (February 1978): 1-18,

Grossman, Sanford J., Oliver D. Hart, and Eric S. Maskin, "Unemployment
with Observable Aggregate Shocks," Journal of Political Economy, 91.
(December 1983): 907-28.

Harris, Milton and Robert M. Townsend, "Resource Allocation under
Asymmetric Information," Econometrica, 49 (January 1981): 33-64.

Hart, Oliver D., "Optimal Labor Contracts under Asymmetric Information,"
Review of Economic Studies, 50 (January 1983): 3-35,

Hurwicz, Leonid, "On Informationally Decentralized Systems," in Decision

and Organization, ch. 14, C.B. McGuire and R. Radner, eds., Amstercam:
North-Holland, 1972.

Lucas, Robert E., Jr., "Capacity, Overtime, and Empirical Production
Functions," American Economic Review, 60 (May 1970): 23-27.

» "Expectations and the Neutrality of Money," Journal of Economic
heory, 4 (April 1972): 103-24,

» " Some International Evidence on Output-Inflation Tradeoffs,"
American Economic Review, 63 (June 1973): 326-34.

» “Tobin and Monetarism: A Review Article," Journal of Economic
Literature, 19 (June 1981): 558-67.

McCallum, Bennett, "Rational Expectations and Macroeconomic Stabilization
Policy: An Overview," Journal of Money, Credit, and Banking, 12
(November 1980): 716-47.

Radner, Roy, "Monitering Cooperative Agreement in a Repeated Principal -
Agent Relationship," Econometrica, 49 (September 1981): 1127-48.

Sachs, Jeffrey, "Real Wages and Unemployment in the OECD Countries,"
Brookings Papers on Economic Activity (1983): 225-303.

Sargent, Thomas J., "A Classical Macroeconomic Model for the United
States," Journal of Political Economy, 84 (April 1976): 207-54,

» Macroeconomic Theory, New York: Academic Press, 1979, ch. 8.

Sims, Christopher A., "Comparison of Interwar and Postwar Business Cycles:
Monetarism Reconsidered," American Economic Review Papers and
Proceedings, 70 (May 1980): 250-57.



To simplify matters further it will also be assumed that the support of v is
large enough so that p is always an imperfect signal of 6, that is
0< 1, (p) <1 for i= 1, 2. Then equation (8) holds with strict inequality.
The asymmetric-information contract model of section IV rides on the
fact that the aggregate price level conveys some information about each
individual firm's productivity shock. The firm's private information must have
an aggregate component; this would appear to be a strong assumption. By
contrast, the private information in the islands paradigm is a "local" price

that has no aggregate component; this may be a more reasonable assumption.

IIIT. CONTRACTING UNDER SYMMETRIC INFORMATION

In the implicit contract models of Azariadis (1975) and Baily (1974),
both the firm and its workers observe the productivity shock 6. There is no
need to use the information embodied in the price level when determining
optimal wage and employment rules. As a result, models with a symmet ric-
information structure serve only as explanations of real business cycles. We
review such a model briefly in this section; the symmetric-information solution
serves as a benchmark in our discussion of the asymmetric-information case to
follow.

Def. 2. A contract is said to be Pareto optimal under symmetric information if

it solves

(9) Max {aP(C) + (1-a)1(C)}

subject to (4) (the feasibility constraints), where a e [0,1].

By assumption the set of feasible contracts is nonempty; hence, for a


20

Disequilibrium contract models that do not incorporate productivity
disturbances have policy implications that are similar to those of the islands
model paradigm; losses are associated with errors in Predicting p.18/

Models incorporating producitivity disturbances and Gray's (1976) indexing
scheme have policy implications identical to those of the asymmetric-
information contract model discussed here; the price level is used by Gray's

contractors as a noisy signal of the productivity disturbance.

V. EXAMPLE

In this section we give a simple example. Let

f(n) = K = 7/16
U(w) = Ww 8, = 3/16
A= 1/2 8 = 1/4.

Then the optimal symmetric-information contract is
w= 1/2 ny = 4/9
n= 1/4
This is easily seen to be non-incentive compatible.
It is not possible to solve analytically for the optimal asymmetric-

information contract. Numerical simulation gives Figures 3 and 4 which depict

wages arid employment for each 6 as a function of 1, ().


24
Footnotes

*We would like to thank G. Calvo, E. Green, D. Henderson, R. King, P. Minford,
and K. Rogoff for their helpful comments, and P. Decker who coaxed our
numerical solutions out of the computer. However, the views presented here are
our own; in particular, they do not necessari ly represent the views of the
Federal Reserve Board or any other member of its staff.

J/ See, for example the surveys of McCallum (1980) and Taylor (1980). .

2/ There is already an extensive literature discussing the microeconomic
implications of contracting with asymmetric information. See, for example,
Hart (1983) and the references therein. The Quarterly Journal of Economics
supplement Vol. 98, 1983, is almost exclusively devoted to this subject.
Grossman, Hart, and Maskin (1983) show that with asymmetric information an
observable aggregate shock can lead to a reduction in employment. But as far
as we are aware, ours is the only attempt to complete the analysis begun by
Calvo and Phelps (1977).

3/ This stylized fact is based upon emperical work by Lucas (1973), Barro
(1978), Sargent (1976) and others. Fischer (1982) and Sims (1980), using
vector autoregressions, find price impulses to be negatively correlated with
real economic activity; however, the vector autoregression technology confounds
lagged and contemporaneous correlations, and in addition, it is impossible to
tell whether the price impulses are due to real or monetary disturbances.

4/ Many studies have found a procyclical pattern for U.S. real wages; see
Bodkin (1969) and Lucas (1970) and the papers they reference. More recently,
Canzoneri (1978) found Canadian real wages to be countercyclical. Sachs (1983)
found real wages in Canada, Germany, the U.K., and Japan to be countercyclical,
but he found little or no cyclical pattern for U.S. and French real wages.

5/ This view seems to be widely recognized as a stylized fact, though its
documentation is more difficult. The old implicit contract models feature
fixed real wages as one of their major contributions. Taylor (1980) asserts
that there is not enough real wage variation in the U.S. for the Gray-Fischer
contract model to explain the data; he constructed his own contract model with
this fact in mind.

5/ The implicit contract models of Azariadis (1975) and Baily (1974) both
incorporate this assumption, as does the asymmetric-information model of Calvo
and Phelps (1977). Many of the more recent models of contracting under
asymmetric information do not; see for example the special supplement of the
Quarterly Journal of Economics (1983). As Chari (1983) points out, the
assumption has important implications for when there will be too little
employment in bad states of nature.



signal extraction problem. But the signal here is the aggregate price level,
which gives incomplete information about local productivity disturbances. In
the islands paradigm, local prices give incomplete information about the
aggregate price level (which is observed with delay). The islands paradigm
suggests that monetary policy should stabilize the aggregate price level to
improve the quality of the local price signals; the new paradigm suggests that
monetary policy ought to stablize aggregate demand to improve the quality of
the aggregate price signal.

Our paper proceeds as follows: In section II, we describe the basic
framework of implicit contract models. In section III, we review briefly the
symmetric-information contracts of Azariadis, Baily and Gordon. In section IV,
we derive the optimal asymmetric-information contract, completing the analysis
begun by Calvo and Phelps (1977), and we discuss its macroeconomic implications
when the model is closed with a simple velocity equation. Finally, in section

V, we present a numerical example.

II. THE BASIC FRAMEWORK OF IMPLICIT CONTRACT MODELS

Firms and workers enter into contractual relationships to share risk.
Workers are assumed to have little or no access to financial markets and are
thus unable to insure themselves against variations in income resulting from
business fluctuations. Firms on the other hand are assumed to have access to
financial markets and are better able to hedge their risks. Participation in
financial markets is not modelled directly (and this may be a shortcoming of
our analysis). Instead it is implicit in the assumption that workers are more
risk averse than firms. Here, for analytical convenience, we assume that firms
are actually risk neutral. Firms insure workers against variations in income

with labor contracts written prior to the realization of real and monetary


14

where w is the optimal symmetric-information wage and ny and Ny are the optimal
symmetric-information employment levels.

Proof (i) follows from (lla), (14c), and (14d). (ii) and (iii) are discussed
later in this section.

It may seem paradoxical that the real wage is higher in the bad state
than in the good state.23/ However, the reason is readily made apparent. To
keep the firm from understating 6 in the good state, the optimal contract must
make the bad state less attractive; one way it does so is by making
w(8,,p) > W(85,P)+
Prop. 5. Relative to symmetric information, the optimal contract under
asymmetric information is characterized by
(i) Underemployment in the bad state (6)

(ii) Efficient employment in the good state (85).

Proof To show (i), solve (14c) for yp(p) and substitute into (14a) to get
Uw(6;,p)] - K

U'Ew(e,,p)J

(15) af [n(8,5p)] w(@) 5p) -

A. - U' [we »p)] '
(0,-0,) 22" ¢'ta(o 01
U [w(e),p)]

+

ULw(6,,p)] - K

> w(8 4p) - ——
1 U Lw(@,p)J

where the inequality follows from (14c) and the fact that Yo(P) > 0. Thus, by
(11b), which gives the optimal tradeoff between labor and leisure when there is
no information asymmetry, the marginal product of labor is too high. To show

(ii), solve (14d) for Y2(p) and substitute into (14b) to get


16

n(@; 5p) +n, as 1, (P) al

The Real Effects of a Monetary Disturbance

An unanticipated and unobserved increase in the velocity v will
increase the price level p, and this will be used in the asymmetric information
contract as a signal that the bad state 8) is more likely to have occurred. In
other words, a positive shock to velocity will increase the conditional

probability  (p). In this section we discuss the impact of price and 1 (Pp)
1

changes on employment and wages. Some of the results depend on the sign of
apf [n( 8, p)] - w(@) 5p). In what follows we will assume that this quantity is
everywhere strictly negative. The other case will be considered in the
Appendix. A numerical example in section V will suffice to show that the

assumption is not vacuous.

Prop. 7.

aw(0,,p)

(i) —amteT <° i=1, 2

an(6.,p)

(ii) “amy < ° i

Proof. To derive (i) and (ii) required the incredibly tedious but

u
e
i)

straightforward total differentiation of the first-order conditions (12b) and
(14). Details are available from the authors. The assumption
'
Gof [n(6),p)] - w(e,,p) < 0
is a necessary and sufficient condition for an(6),p)/an, (p) < 0. 25/

By Prop. 7, in conjunction with (8), it follows that wages and


appears to have been borne out; the new contract models ride on the familiar
confounding of real and monetary disturbances. Here, the aggregate price level
(which is observed without delay) gives workers incomplete information about
the local productivity disturbances they are assumed not to observe. We leave
it to the reader to decide whether or not it is reasonable to assume that local
information that is private to the firm can have an aggregate componerit that is
partially revealed by the price level. The new paradigm rides on just such an
assertion.

Our purpose here is to draw out the macroeconomic implicaticns of
this new paradi gm.2/ We ask whether the “asymmetric-information contract
models" are consistent with a set of stylized facts, and we examine the
implications for monetary policy. Where possible, we compare the implications
of the new paradigm with those of the more familiar paradigms.

We take as stylized facts that: (1) An unanticipated increase in the
money supply (or aggregate demand) will produce an unanticipated increase in
the general price level that is associated with an increase in employment and
output 2/ (2) Real wages may move countercyclically (as in Canada, Japan and
most European countries), or they may show little cyclical pattern at all (as
in the us.) 4/ (3) Whatever the cyclical pattern, real wage variation is
sma11.2/ The asymmetric-information contract model presented below is
consistent with the second and third facts, but it does not appear to be
consistent with the first.

Our model also implies that employment will be too low (compared with
a complete information structure) in bad states of nature. The Policy
implications of the new paradigm are virtually identical to those of the Gray
(1976) contract-cum-indexing model, and rather at odds with the implications of

the islands paradigm. As Lucas conjectured, the new contract models ride on a


22

(AS) => w(8psp) > w(8),p)-
By (Al) and (A3)

(A6) 0, Gn(0,4p)] - w(o. yp) = - ee(ensP)d =
1 1 1 U'Cw(0,,p)]
By (12a)
6,f{n(6],p)] - n(0;.p)w(6,,p) = 84F{n(6,p)] - n(82,p)w(89,p)

=> 0, {f£n(@y,p)] - fLn(62,p)]} = n(@,,p)w(6,5p) - n(675p)w(6, 5p)
+ n(895p)w(0;,p) - n(€@2,p)w(62,p)

=> of ()En(0 ,p) - n(65,p)] = [n(@,,p) - n(62,p) Iw(6,,p)
+ n(@9sp)[w(0),p) - w(89.p)],

where fi € (n(6),p),n(@pp)) by the Mean-Value Theorem and Prop. 2(i). thus,
ayf (7) ~ w(8,p) = n(p,p)Ew( 5p) - w(0,5p)I/Lm(02,p) - n(01,p)] > 0
by (A5) and Prop 2(i). Thus,
o,f ‘[n(01,p)] - w(0;,p) > 0.

This contradicts (A6).

Case of eof [n(@,,p)] - w(0,,p) > 0

In this case we have by total differentiation of the first-order

conditions

mt?) <0


The Macroeconomic Implications of Labor Contracting with Asymmetric Information
Matthew B. Canzoneri and Anne Sibert*

I, INTRODUCTION AND SUMMARY

Monetary models of the business cycle are designed to explain certain
stylized facts and to make a case for a particular view about the nature of
employment fluctuations and optimal monetary policy. Two popular paradigms are
the “islands models" of Lucas (1972, 1973) and Barro (1976) and the
“disequilibrium contract models" of Gray (1976) and Fischer (1977). Both
paradigms have been developed extensively, and their macroeconomic implications
are widely known 2/

Lucas (1981) and Barro (1977) have been critical of the
disequilibrium contract models because the wage and employment rules postulated
in these models are not derived in a welfare-maximizing framework. Lucas
concludes that, "None of these models offers an explanation as to why people
should choose to bind themselves to contracts which seem to be in no one's
self-interest, and my conjecture is that when reasons for this are found they
will reduce to the kind of information difficulties already stressed in my 1972
article."

A new paradigm is now emerging out of the old “implicit contract
models" of Azariadis (1975), Baily (1974) and Gordon (1974). The old implicit
contract models are not monetary models; that is, monetary fluctuations (or
fluctuations in aggregate demand) have no effect on employment or output.
However, Calvo and Phelps (1977) have shown that the imposition of certain
informational asymmetries will convert these implicit contract models into
monetary models. Calvo and Phelps (1977) assumed that firms observe
productivity disturbances directly while workers do not; other asymmetries will
do as well. The new monetary paradigm that results addresses the criticism of
Barro and Lucas; contracts are efficient in the Paretian sense (given the

incomplete information structure). In addition, the conjecture of Lucas


25
7/ Most contracting models eschew the interesting aspects of search.

8/ Necessary and sufficient conditions for an interior solution are
(1) of (1) <0

(44) u'Cuml(K)] > a/(1-a)

(444) 9yfEF'72(a/0,)] > u'“Lay(a-ay JF ~1(a/0,)

(iv) od eu Lay(t-ay] - (l-aysaufu'Eay(a-a)] - kK}

9/ See, for example, Harris and Townsend (1981).
10/ See also Radner (1981).
l/ See Gray (1978) and Canzoneri (1980).

12/ Both constraints cannot hold with equality because this would imply
(94) = n(69,p), which is inconsistent with the first-order conditions which
follow.

13/_ Phelps and Calvo (1977) conjectured just the opposite, but their basic
point is correct; with asymmetric information, the real wage is not fixed.

14/ See for example, Hart (1983).

15/ If we use the quadratic production function y = 6n - hn@ where h > 0, then
the wage and employment rules are unambiguously strictly decreasing functions
of p. Sargent (1979) uses this production function.

16/ We would like to include the island paradigm here, but we are not sure
what to take as the representative model of wage determination. One might
simply identify the "local" price with the nominal wage, in which case the real
wage would move procyclically. However this Procedure is inconsistent with the
notion that the real wage is the relative price in a two good --labor and
output-- model. Sargent's (1979, ch. 8, section 4) model may be preferable
for such a comparison.

17/ The nominal wage may also be indexed in the way described by Gray (1976)
without affecting the basic discussion to follow.

tthe3) for example Canzoneri (1980) or Canzoneri, Henderson, and Rogoff


12

Employment, the wage bill and profits are all higher in the good state than in
the bad. These results follow directly from (12) and the fact that both (12a)
and (12b) cannot hold simultaneous ly 22/ The proof is left to the reader.
Optimal Contract

Def. 4. A contract is said to be Pareto optimal under asymmetric information

if its solves problem (9) subject to constraints (4) and (12).

It turns out that if the symmetric-information contract is not
implementable, then (12b) must be binding, but (12a) is not.

Prop. 3: 2 (P) > 1 (P) = 0, where y() and Y2(P) are the Lagrange miltipliers
for (12a) and (12b) respectively in the Pareto maximization problem.

This is, of course, what one would suspect from Prop. 1; the problem
is to keep the firm from understating 6. This proposition is also proven in
the Appendix.

The optimal contract is derived as follows. For every possible

realization of p, choose {n(;,p), w(0;,p), rol?) Har to maximize

2
(13) 4 wn 4; (P) (0; fEn( 0; p)] - w(0;.p)n(e; p)}

2
+ (1-a) J ay (p)n (0; op) {ULW( 0; ,p)] - K}
if

+ yo(p) {eof [n(62,p)J-w( 85p Jn (8o5p )~ Gof [n (64 .p) J+w( 0; 5p )n( 01,2 )}

subject to (4).
It was assumed that there exists a contract {n,w} which satisfies
(4). This contract clearly satisfies (12b); hence the set of feasible

contracts under asymmetric information is nonempty. We now further assume
Ulw(0,,p)] - kK
that w(6,,p) - Wnts, py) > 0 p at the solution. (This fs equivalent


15

ULw( 825p)] - k

(16) 9, F'En(e,4p)] = w(6,.p) - —;
a2 2 U'[w(62,p)]

Note that Prop. 4 states that n(@, 5p) may be greater than ny. The
import of Prop. 5 is that given w(815P)» n(6),p) is too low to represent an
efficient trade-off between labor and leisure (relative to full information).

The results in Prop. 5 are dependent upon our assumption about job
sharing and, in general, the specification of preferences. Suppose instead
that we model the representative worker with a divisible endowment of labor.
Employment is efficient in one state and inefficient in the other. If income
effects on leisure are negative, we again obtain underemployment in the bad
state. If income effects are positive, we obtain overemployment in the good
state. If there are no income effects on leisure, it turns out that the
optimal symmetric-information contract is incentive compatible. See Chari
(1983) and Cooper (1983) for a discussion of this. Many studies use the
utility function U[w-g(n)], which has no income effects, and assume the firm is
also risk averse; in this case the symmetric-information contract is again
incentive incompatible 24/

Prop. 6. As the price level becomes a perfect signal of the productivity
shock, the optimal asymmetric information contract approaches the optimal
symmetric-information contract.

Proof Note that by (14c) and (14d),
a7) J, (ov 'tw(o4,0)) = By

Thus as 1, (p) +1, w(0; 5p) +w, for i = 1, 2. Equations (15) - (17) imply that