Hidden effort, learning by doing, and wage dynamics.
Jarque, Arantxa
Many occupations are subject to learning by doing: Effort at the
workplace early in the career of a worker results in higher productivity
later on. (1) In such occupations, if effort at work is unobservable, a
moral hazard problem arises as well. The combination of these two
characteristics of effort implies that employers need to provide
incentives for the employee to work hard, possibly in the form of
pay-for-performance, (2) while taking into account at the same time the
optimal path of human capital accumulation over the duration of the
contract.
The recent crisis had a big impact on the labor market with high
job-destruction rates. If firm-specific human capital accumulation is
important, the effect of these separations on welfare may come from
several channels. A direct channel is through the loss of human capital
prompted by the exogenous separation, as well as the loss in welfare
from the decrease in wealth because of unemployment spells of workers. A
less direct channel, but potentially an important one, is the change in
the cost of providing incentives when the (exogenous to the incentive
provision) separation rate increases. However, we are far from being
able to understand and measure the importance of this cost, since little
is known so far about the structure of incentive provision in the
presence of learning by doing. (3) This article constitutes a modest
first step in this direction: Abstracting from separations and in a
partial equilibrium setting, this article studies the time allocation of
incentives and human capital accumulation in the optimal contract. This
simplified analysis should be a helpful benchmark in future studies of
the fully fledged model with separations and general equilibrium.
We modify the standard repeated moral hazard (RMH) framework from
Rogerson (1985a) to include learning by doing. In the standard
framework, a risk-neutral employer, the principal, designs a contract to
provide incentives for a risk-averse employee, the agent, to exert
effort in running the technology of the firm. Both the principal and the
agent commit to a long-term contract. The agent's effort is private
information and it affects the results of the firm stochastically: The
probability distribution over the results of the firm (the agent's
"productivity") in a given period is determined by the effort
choice of the agent in that same period only. We introduce the following
modification to this standard framework: We specify learning by doing by
assuming that the probability distribution over the results of the firm
in each period is determined by the sum of past undepreciated efforts of
the agent, as opposed to his current effort only. In other words, the
agent's productivity is determined by his "accumulated human
capital." More human capital implies higher expected output,
although all possible output levels may realize under any level of human
capital. In this specification, the agent determines his human capital
deterministically by choosing effort each period. Lower depreciation of
past effort is interpreted as "more persistence" of effort.
We present a model of two periods. The first period represents the
junior years, when the worker has just been hired and has little
experience. The second period represents the mature worker years, when
human capital has been potentially accumulated and there are no more
years ahead in which to exploit the productivity of the worker. A
contract contingent on the observed performance of the agent is designed
by the principal to implement the path of human capital accumulation
that maximizes the principal's expected profit (expected output
minus expected payments to the agent).
In our analysis, we find the following two main implications of the
presence of learning by doing. First, the principal does not find it
optimal to require a high level of human capital in the last period of
the contract, since there is not much time left to exploit the
productivity of the worker. Hence, the more experienced workers are not
the most productive ones, since they optimally are asked to let their
human capital depreciate. This implies that workers exert the most
effort in their junior years, and the least in their pre-retirement
years. In a comparison with the standard RMH problem, we find that the
frontloading of effort, as well as the low requirement at the end of the
worker's career, differ markedly from the optimal path of effort in
a context without learning by doing. Second, and in spite of this
difference in effort requirements over the contract length, we find that
learning by doing does not imply a change in the properties of
consumption paths; hence, the properties of consumption paths found by
previous studies, such as Phelan (1994), remain true in this context
(see also Ales and Maziero [2009]).
It is worth noting that in our analysis we assume perfect
commitment to the contract both from the employer and the employee, and
we do not allow for separations to be part of the contract. This means
we need to abstract from the usual career concerns that have been
explored in the literature (see Gibbons and Murphy [1992]). The
implications of the hidden human capital accumulation that we model here
should be viewed as complementary to the implications of career
concerns.
As pointed out above, the problem studied here differs from the
standard RMH in that the contingent contract needs to take into account
the persistent effects of effort on productivity. On the technical side,
this highly complicates solving for the optimal contract. The fact that
both past and current effort choices are not observable means that, at
the start of every period, the principal does not know the preferences
of the agent over continuation contracts (that is, the principal does
not know the true productivity of the agent for a given choice of effort
today). Jarque (2010) deals with this difficulty and presents a class of
problems with persistence for which a simple solution can be found. The
article studies a general framework in which past effort choices affect
current output, as opposed to other forms of persistence that one may
consider, such as through output autocorrelation (see, for example,
Kapicka [2008]). The learning-by-doing problem that we are interested
in, hence, constitutes a fitting application of the results in Jarque
(2010). We adapt the assumptions in Jarque (2010) to a finite horizon
and we show how this specification of learning by doing greatly
simplifies the analysis of the optimal contract.
In Section 1 we introduce the common assumptions throughout the
article. Section 2 presents, as a benchmark, the case in which the
principal can directly observe the level of effort chosen by the agent
every period, and hence can control his human capital at all times. For
reference, we also discuss the case in which the effort of the agent
does not have a persistent effect in time. The analytical properties of
the problem are discussed in both cases. Then we analyze the main case
of interest of this article, in which effort is unobservable and
contracts that specify payments contingent on the observable performance
of the agent are needed to implement the desired sequence of human
capital accumulation. In Section 3, we discuss the case without
persistence--a standard two-period repeated moral hazard problem. In
Section 4 we discuss the technical difficulties of allowing for effort
persistence in problems of repeated moral hazard, and the solutions
provided in the literature. Section 5 presents the framework of hidden
human capital accumulation, a particular case of effort persistence. As
the main result, we provide conditions under which the problem with
hidden human capital can be analyzed by studying a related auxiliary
problem that is formally a standard repeated moral hazard problem.
Hence, the discussion of the properties of the standard case in Section
3 becomes useful when deriving the properties of the case with
persistence. The numerical solution to an example is presented in
Section 6, together with a comparison to the standard RMH without
learning by doing, and a discussion of the main lessons about the
effects of hidden human capital accumulation on wage dynamics. Section 7
concludes.
1. Description of the Environment
The results in this article apply to contracts of finite length T;
however, in order to keep the exposition and the notation as simple as
possible, we discuss here the case of a two-period contract, T =2. We
assume that both parties commit to staying in the contract for the two
periods. For tractability, we assume that the principal has perfect
control over the savings of the agent. They both discount the future at
a rate ft. We assume that the principal is risk neutral and the agent is
risk averse, with additively separable utility that is linear in effort.
Assumption 1 The agent's utility is given by U ([c.sub.t],
[e.sub.t]) = u ([c.sub.t]) = v([e.sub.t]), where u is twice continuously
differentiable and strictly concave and c, and e, denote consumption and
effort at time t, respectively.
There is a finite set of possible outcomes in each period, Y =
{[y.sub.L], [y.sub.H]} Histories of outcomes are assumed to be
observable to both the principal and the agent. We assume both
consumption and effort lie in a compact set: [c.sub.t] [member of] [0,
[y.sub.t] and [e.sub.t], [member of] E = [e.bar],[bar.e] for all t.
We model the hidden accumulation of human capital by assuming that
the effect of effort is "persistent" over time, in a
learning-by-doing fashion. That is, we depart from the standard RMH
framework, which assumes that the probability distribution over possible
outcomes realizations at t depends only on [e.sub.t],. In our human
capital accumulation framework, the probability distribution at t
depends on all past efforts up to time t. Assumption 2 states this
formally for the two-period problem.
Assumption 2 The agent affects the probability distribution over
outcomes according to the following function:
Pr([y.sub.t]=[y.sub.H]|[s.sub.t]) [equivalent to] [pi]([s.sub.t]),
where
[s.sub.1]=[e.sub.1], (1)
[s.sub.2]=[[rho][s.sub.1]+[e.sub.2]] (2)
and [pi] (s) is continuous, differrentiable, concave, and [rho]
[member of] (0,1) In the human capital accumulation language, we could
equivalently write the law of motion for human capital as
[s.sub.1]=[e.sub.1],
[s.sub.2]=(1-[delta])[s.sub.1]+[e.sub.2,]
where [delta] = 1 - [rho] would represent the depreciation rate.
Then,
f([S.sub.t])={[y.sub.H] with probability [pi] ([S.sub.t]),[y.sub.L]
with probability 1-[pi] ([S.sub.t])
could be interpreted as the production function or technology of
the firm.
In the rest of the article, we loosely refer to Assumption 2 as
effort being "persistent," we refer to [s.sub.t], as the
accumulated human capital at time t, and we refer to [rho] as the
persistence rate.
The strategy of the principal consists of a sequence of consumption
transfers to the agent contingent on the history of outcome
realizations, c = {[c.sub.i], [c.sub.i,j] sub.i,j=1.,H']}. to which
the principal commits when offering the contract at time 0. The
agent's strategy is a sequence of period best-response effort
choices that maximize his expected utility from t on, given the past
history of output: e ={[e. sub.1],[e. sub. 2i]}.sub.i = L.H]} At the
beginning of each period, the agent chooses the level of current effort,
[e.sub.t]. Then output [y.sub.t], is realized according to the
distribution determined by all effort choices up to time t. Finally, the
corresponding amount of consumption is given to the agent.
A contract is a pair of contingent sequences c and e. For the
analysis in the rest of the article, it will be useful to follow
Grossman and Hart (1983) in using utility levels [u.sub.i] =
u[(c.sub.i)] and u = u [(c.sub.ij)] as choice variables. (4) To denote
the domain for this new choice variable, we need to introduce the
following set notation:
[U.sub.i]={u|u=u([c.sub.i]) for some ([c.sub.i]) [member of]
[0,[y.sub.i]],i=L,H}
[U.sub.ij]={u|u=u([c.sub.ij]) for some ([c.sub.ij]) [member of]
[0,[y.sub.j]]i,j=L,H}
The contingent sequence of utility is then denoted u =
[{[u.sub.i],[u.sub i,j]}i,j=L,H'] and we assume that [u.sub.i]
[member of] [U.sub.i],[u.sub.ij] [member of] [U.sub.ij]
In order to keep the expressions in the article as simple as
possible, and abusing notation slightly, we also introduce some notation
shortcuts. We denote [c.sub.i] = [u.sup.-1] for all i. We also write Pr
([y.sub.t] = [y.sub.H] | [S.sub.t] as [[pi].sub.H] and Pr([y.sub.t] =
[y.sub.L] || [[pi].sub.L] ([s.sub.t]).
The expected profit of the principal, denoted by V (u, e), depends
on the contract as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [s.sub.t], changes with [e.sub.t], as detailed in (1). In the
same way, we can write the agent's expected utility of accepting
to participate in the contract as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Within this environment we are now ready to set up the problem of
finding the optimal contract that will provide the right incentives for
human capital accumulation at the least expected cost. Before analyzing
the hidden human capital accumulation case, however, we go through a
series of related and simpler cases that will serve in clarifying the
main case of interest.
2. Observable Effort
The case of observable effort is often referred to in the
literature as first-best (FB) since it represents the maximum joint
utility achievable in the contractual relationship between the principal
and the agent. This is because, if effort is observable, the principal
can directly control the choice of effort of the agent and, hence, there
is no need for incentives. This implies that there is no need to impose
risk on the agent, which results in lower expected transfers from the
principal to the agent. Although we are interested in the case of
unobservable effort, it is useful to also analyze this simpler benchmark
to learn about the differences between the problem with effort
persistence (human capital accumulation) and the standard RMH problem
(in which human capital fully depreciates every period).
We will refer to the problem of the principal when effort is
observable as problem FB:
[MATHEMATICAL EXPRESSION NOT REPRODUCIABLE IN ASCII]
e [member of] [[[e.bar],[bar.e]].sup.3] (ED)
[u.sub.i] [member of] [U.sub.i], [u.sub.ij] [member of] [U.sub.ij]
[for all]i [for all]i, j (CD)
[w.sub.0] [less than or equal to] [W.sub.0] (u,e). (PC)
The solution to problem FB is a contract that consists of a pair of
contingent sequences of utility and effort that maximize the expected
profit of the principal subject to the participation constraint
(PC)--which assures that the agent expects as much utility from
accepting the contract than staying out--and the domain constraints for
consumption (CD) and effort (ED). Characterizing the solution to this
problem when considering all the possible combinations of (ED) and (CD)
binding constraints is very lengthy and tedious. In the interest of
space, we choose to discuss here only the case in which neither of the
constraints in (CD) or (ED) bind.
What are the properties of consumption and effort in the optimal
contract? We learn them from looking at the first-order conditions of
the problem. Let [lambda] [greater than or equal to] 0 be the multiplier
of the (PC). (5) We have:
([u.sub.i]): 1/[u.sup.']([c.sub.i]) = [lamda], for i = L,H
([u.sub.ij]): 1/[u.sup.']([c.sub.ij]) = [lamda], for i, j =
L,H
([e.sub.1]): [[[pi].sup.']([S.sub.1]) +
[[beta][rho][pi].sup.'] ([s.sub.2i])] ([y.sub.H]-[y.sub.L]) =
v[lambda]
([e.sub.2i]): [[pi].sup.']([S.sub.2i]) ([y.sub.H]-[y.sub.L]) =
v[lambda] for i = L,H (4)
We analyze in turn the case with and without persistence.
Full Depreciation
First we analyze the observable effort version of a standard
two-period RMH problem (see, for example, Rogerson [1985a]). This case
is nested in the common framework presented above, for a value of the
persistence parameter [rho] = 0. In this case, effort does not have a
persistent effect on the output distribution, that is, there is no
learning by doing. Hence, we can say that the human capital of the agent
fully depreciates every period.
Here and throughout the rest of the paper, we use stars to denote
the solutions to the problems. When necessary, we index the solutions by
two arguments: the first one takes a value P if [rho] [greater than] 0
(persistence) and a value N P if [rho] = 0 (no persistence). The second
one takes a value SB if we are in the case of
unobservable effort. Hence, here we denote the solution to problem
FB when [rho] = 0 as u* (N P, F B) and e* (N P, F B). Note that,
whenever it does not lead to confusion, we do not include these
arguments to keep the notation light.
Since the right-hand sides of all the first-order conditions for
utility are equal to [lambda], we conclude that the level of utility,
and hence consumption, should be the same independent of the output
realizations and the period: [u.sub.i.sup.*] = [u.sub.ij.sup *] for all
i, j. The first-order conditions of effort, in turn, imply that effort
requirements are independent of output realizations and the period:
[e.sub.1.sup.*] [e.sub.2i.sup.*] for all i. It is easy to see that'
given these properties of consumption and effort, the (PC) in problem FB
simplifies to [w.sub.0] = (l +[beta])(u*-ve*).
Hence, we can solve for the level of utility in the solution to the
FB problem:
u* [equivalent to] [w.sub.0] + v(1 + [beta]) e*/1 + [beta] (5)
Let c* = [u.sup.-1] (u*). Let [[pi].sub.j.sup.'] (e2) denote
the derivative of [[pi].sub.j] (e2). Noting that [[pi].sub.H.sup.']
(e) = [-[[pi].sub.L.sup.']](e) we can combine the first-order
conditions for consumption and effort to get
[u.sup.'] ([c.sup.*]) [[pi].sub.H.sup.'] ([e.sup.*])
([y.sub.H] - ([y.sub.L]) = v [for all]t. (6)
That is, the optimal effort level is such that the marginal benefit
from increased effort (the marginal increase in expected output times
the marginal utility of output) equals the marginal utility cost of
effort.
The following properties summarize our conclusions about the FB
problem with nonpersistent effort:
1A. We have that [c*.sub.1] = [c*.sub.2] =c*.
2A. We have [e*.sub.1] = [e*.sub.2] =e*.
The main property of the optimal consumption sequence of the FB
contract in the standard RMH problem is that the contract insures the
agent completely against consumption fluctuations whenever feasible. The
intuition for this result is straightforward: Since the agent has
concave utility in consumption, this is the cheapest way of providing
the agent with his outside utility. The main property of the optimal
effort sequence of the FB contract in the standard RMH problem is a
constant effort requirement over time. The tradeoff between increasing
the disutility suffered by the agent and increasing the expected output
is exactly the same in each period, and hence the solution is the same
each time.
It is worth noting that the solution in the observable-effort case
coincides with that of a repeated static problem ("spot"
contract) in which neither the agent nor the principal commit to the
two-period contract, and the outside utility of the agent is [w.sub.0]/2
each period. Hence, commitment has no value in the case of observable
effort and no persistence.
Table 1 Parameters of the Numerical Example
V Marginal effort 5.00
disutility
[beta] Discount factor 0.65
yH Output 30.00
realization,high
state
yL Output 20.00
realization, low
state
[w.sub.0] Outside utility 6.55
An example
Throughout this article, we illustrate the properties of each
particular case of the environment presented by solving a particular
numerical example. This makes it easy to compare across the different
cases presented. The common parameters of the example are listed in
Table 1.
We also assume u (c) = u(c) = 2[square root of (c)] and a
probability function
[pi](s) = [square root of (s,)]
as well as [e.bar] = 0.01 and [bar.e] = 0.99.
We now solve for c* and e*. Since we are in the case of full
depreciation of human capital, we use [rho] = 0 and the formulas derived
above. For our example, we have that (6) becomes
[1/[2[square root of (e*)]]](30 - 20) = 5[square root of
(c*)][1/[2[square root of (e*)]]] = [square root of (c*)] c* =
[0.25/e*].
Together with (5) this gives us the solutions listed in Table 2.
Observable Human Capital Accumulation
We now turn to analyzing the case in which the effects of effort
are persistent in time, with [rho] [greater than] 0. That is, we analyze
the optimal contract in the presence of human capital accumulation, or
learning by doing.
We established above that the main property of the optimal
consumption sequence of the FB contract in the standard RMH problem is
that the contract insures the agent completely against consumption
fluctuations. Here we will learn that this property remains true in the
case with effort persistence. The main property of the optimal effort
sequence of the FB contract in the standard RMH problem is also a
constant effort requirement over time. We will learn that when effort is
persistent this property no longer holds: Effort requirements will vary
over time even in the observable effort benchmark.
We now proceed to derive these results by formally analyzing the
problem of the principal FB for the case of [rho] [greater than] 0. She
chooses an optimal contract: a pair of contingent sequences [u.sup.*](P,
FB) and [e.sup.*](P, FB) that solve problem FB, i.e., they maximize the
expected profit of the principal subject to (PC) and the domain
constraints (CD) and (ED). We initially discuss the case in which
neither the (CD) nor the (ED) constraint bind. However, the lower (ED)
constraint (the non-negativity constraint on effort) may bind, with
persistence, in not-so-trivial cases. Because of its relevance, the case
of this constraint binding will be discussed in turn.
We can derive the properties of the solution by analyzing the
first-order conditions in (4) for the case of [rho] [Greater than] 0.
The first thing to note is that, as in the case without persistence,
neither consumption nor effort are contingent on output realizations.
However, effort recommendations will depend on the time period. We can
use the (PC) here as well to derive the optimal level of utility:
u* [equivalent to]
[[w.sub.0]+v([e*.sub.1+[[beta][e*.sub.2])]/[1+[beta]]
The optimal level of consumption will be [c.sup.*] [equivalent to]
[u.sup.-l] ([u.sup.*]). We can substitute the first-order condition for
effort [e.sub.2] into that for [e.sub.1], as well as the expression of
a. from the consumption first-order conditions, to get an expression for
the tradeoff determining the choice of [e.sub.1]:
u' (C*)[[[pi]'].sub.H]([s*.sub.1])([y.sub.H]-[y.sub.L])=v(1-[[beta][rho]]) (8)
Comparing this to the tradeoff determining the choice of [e.sub.2],
u' (C*)[[[pi]'.sub.H]([s*.sub.2i])([y.sub.H]-[y.sub.L])=v, (9)
we learn that the marginal cost of increasing effort in the first
period is different (smaller) than that in the second period. The
optimal choice takes into account that any effort [e.sub.1] exerted in
the first period persists into the second one, i.e., it
"saves" the agent the equivalent of the discounted disutility
of effort of exerting [rho][e.sub.1] in the second period. This
difference in the effective cost of effort that appears because of
persistence implies that the principal sets the effort requirements in a
way that implies a higher probability of observing [y.sub.H] in the
first period than in the second. We can see exactly how this difference
is determined by using the first-order conditions of effort to get the
following relationship:
[[[pi]'.sub.H]([s*.sub.1])]/[1-[beta][rho]] =
[[pi]'.sub.H]([s*.sub.2]) (10)
This implies .[s.sub.1] > [s.sub.2] since 1 - [beta][rho] is
always between 0 and 1. From the accumulation of human capital in (1) we
have that
[e*.sub.1] = [s*.sub.1],
[e*.sub.2] = [[s*.sub.2]] - [rho][[s*.sub.1] (11)
which implies a higher effort in the first period than in the
second, [e*.sub.1] > [e*.sub.2].
The following properties summarize our conclusions about the case
with persistence and observable effort:
1B. We have that [c*.sub.1] = [c*.sub.2] = C*.
2B. We have that [e*.sub.1] > [e*.sub.2].
That is, whenever [c.sup.*] is feasible in both states, the
principal provides complete consumption smoothing, both across states
and across time. As for effort requirements, the principal decreases the
requirement from the first to the second period. We repeat the intuition
for this result: In the first period, the effort disutility incurred by
the agent is a sort of "investment," since it improves the
conditional distribution not only in the current period but also in the
following one. At t = 2, however, there is no period to follow, so the
marginal benefits of effort are not as high, while the marginal cost is
the same as in the first period. (6)
An example
We now solve for the optimal contract with persistence and
observable effort. For this case with accumulation of human capital, we
use [rho] = 0.2 and the formulas derived above. We list the solution in
Table 2. Note that the level of [[s*.sub.2](P, FB) in this case is 0.16,
smaller than that of the second-period effort in the no-persistence case
of the previous section, which was [[e*.sub.2](NP, FB) = 0.17. Comparing
the equations that determine each ([6] for [[e*.sub.2] (NP, FB) and [9]
for [[s*.sub.2] (P, FB), we can see that C* (P, FB) < C* (NP,
FB)implies 1/u' (c* (P, FB)), and hence
[[pi]'.sub.H]([s*.sub.2](P, FB))>
[[pi]'.sub.H]([e*.sub.2](NP, FB)) Given the concavity of [pi] (.),
it follows that's [s*.sub.2](P, FB) < [e*.sub.2](P, FB).
The Nonnegativity Constraint on Effort
In light of this solution we can discuss the case of the lower
constraints in (ED) binding. As an introduction to why this case is of
particular relevance to the
Table 2 Solutions for the Numerical Example, FB Problem
FB [c*.sub.1] [c*.sub.2] [e*.sub.1] [e*.sub.2] [s*.sub.1]
Solution
NP 5.82 5.82 0.17 0.17 0.17
P 4.98 4.95 0.22 0.12 0.22
FB [s*.sub.2]
Solution
NP 0.17
P 0.16
Constraint (ED) is represented by the following set of
inequalities:
[s.sub.2i][less than or equal to] [rho][s.sub.1] + [-.e] + [e.-]
(12)
and
[s.sub.2i] [greater than or equal to] [rho][s.sub.1]+[e.-] (13)
Constraint (12) may be binding for some parametrizations. However,
we choose not to discuss this case explicitly here because it is easy to
impose ex ante conditions on the parameters that preclude it from
binding; for example, for the specification of the probability in (7),
it is easy to see that s [greater than or equal to] [-.e] is never
chosen in the optimal contract. The lower bound on s represented in
(13), however, is endogenous, and equation (13) cannot be checked
without having the solution [s.sub.1.sup.*] in hand. Fortunately, in the
case of observable effort that we are analyzing here, we are able to
include constraint (13) explicitly in the maximization problem FB. This
allows us to study how the solution properties differ from those in IB
and 2B discussed above when this constraint binds.
Let [[gamma].sub.i] [greater than or equal to] 0 be the multiplier
associated with constraint (13) in the version of problem FB for the
case [rho] > 0. We have that the first-order condition for [e.sub.2i]
is modified as follows:
[e.sub.2i]: [pi]' (s.sub.2i)([y.sub.H] - [y.sub.L]) =
v[lambda]-[[gamma].sub.i], for i=L, H (14)
Note that, again, the choice for effort in the second period is not
contingent on the first-period outcome, so we have [[gamma.sub.L]] =
[[gamma.sub.H]] = [gamma]. Then we can substitute (14) into the
unmodified first-order condition for first-period effort, ([e.sub.1]),
to get a general version of equation (8) that allows for the lower
domain constraint of effort to be binding:
[pi]'([s.sub.1])([y.sub.H] - [y.sub.L]) =
v[lambda](1-[[beta].sub.[rho]]) + [[beta].sub.[rho][gamma]] (15)
From the Kuhn-Tucker conditions, we know that whenever [gamma] >
0 we have [e*.sub.2] = 0 and, hence, [s*.sub.2] = [rho][s*.sub.1].
An example
In some special cases, we can check ex ante whether [gamma] = 0 is
a feasible solution to the FB problem, and hence we can restrict
ourselves to the simpler analysis without domain constraints. In
particular, with the specification for the probability function in (7)
that we are using for our example, equation (10) becomes
1/[1-[[beta].sub.[rho]] 2 [[square root of][s*.sub.1]] =
1/[2[[square root of[s*.sub.2*]]
Or, rewriting
[s*.sub.2] = [(1-[[beta].sub.[rho]]).sup.2] [s*.sub.1] (16)
This is the relationship that should hold between the level of
[s*.sub.1] and [s*.sub.2] whenever [gamma] = 0. Hence, the domain
condition [e.sub.2] [greater than or equal to] 0 is satisfied whenever
[s.sup.*.sub.2] [greater than or equal to] [rho][s.sup.*.sub.1], or,
substituting [s.sup.*.sub.2] from (16), whenever
[(1-[[beta].sub.[rho]]).sup.2] [greater than or equal to] [rho]
(17)
A closer inspection of condition (17) shows that, for [beta] [less
than or equal to] 0.5, it is always satisfied. For higher B values,
however, the condition is satisfied only for low enough [rho] values,
i.e., when effort is not "too persistent." In our example, for
[beta] = 0.65, we need to check whether (17) is satisfied: The left-hand
side is equal to 0.76, which is clearly greater than the right-hand
side, 0.2.
To summarize the findings of our analysis, we have shown that for
the numerical example presented here, we can provide ex ante conditions
(a functional form for the probability as in equations [7] and [17]) on
the parameters of the problem that assure us that the domain constraints
on (ED) do not bind. Under such restrictions, the characteristics of the
solution to the first-best problem 1B and 2B presented earlier in this
section are valid.
In relation to those characteristics, it is worth pointing out that
the properties of effort requirements depend strongly on our assumption
that the utility of the agent is linear in effort. Linearity implies
that there is no tradeoff between the efficient accumulation path of
human capital and smoothing effort disutility over time. In other words,
the smoothing of effort requirements over the duration of the contract
does not increase the overall utility of the agent, as is the case with
consumption smoothing; hence, the principal only takes into account the
effects that different accumulation paths have on the utility of the
agent and his own profit through the changes in expected output over
time. In the numerical example in Section 6, we will revisit the
solution to the observable-effort case discussed here, and we will see
the direct consequence of this: It is optimal to ask the agent to exert
effort earlier rather than later in the contract, since effort that is
done early improves the distribution over future output, holding
constant the level of future effort.
3. UNOBSERVABLE EFFORT WITH FULL DEPRECIATION
When effort is not directly observable, the principal must rely on
observed output realizations, which are imperfect signals about the
effort level of the agent, in order to implement the desired sequence of
human capital. Contrary to the case of observable effort, here
consumption in a given period will need to vary with the output
realization in order to provide incentives for the worker to choose the
recommended level of effort.
Formally, the problem of the principal, which we will refer to as
the second-best (SB), is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The incentive constraint (IC) ensures that the expected utility
that the agent gets from following the principal's recommendation
is at least as large as that of any other effort sequence.
In order to illustrate clearly the differences that derive from the
presence of effort persistence in this two-period problem, we analyze
first the version without persistence ([rho] = 0), that is, with full
depreciation of human capital every period, or no learning by doing.
Moreover, because the main result that we will derive when we study the
case with [rho] [Greater than] 0 is that, in some cases, the properties
for consumption in the optimal contract will be the same as those of the
optimal contract in a framework without persistence, it is useful to
analyze in detail the properties of the solution with observable effort.
Without persistence, the structure of the incentive constraints
simplifies considerably. This influences the solution, but also the ways
in which the problem can be studied. In particular, the standard RMH
problem has a simple recursive formulation that is not available with
persistence. In this section we provide an illustration of this
difference. Then, we discuss the difficulties of introducing
persistence, along with some potential solutions, in Section 4. In
Section 5 we discuss our example with human capital accumulation, a
particularly simple case with effort persistence for which a solution
can easily be found.
A Simplified Incentive Compatibility Constraint In the case without
persistence the structure of the incentive constraints simplifies
considerably. In particular, the expected utility of the agent in the
second period is independent of the first-period effort choice. Define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
as the expected utilities for the second period, contingent on the
first-period realization. This expression for the continuation utility
simplifies, when [RHO] = 0,to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(Note that, to distinguish the notation for continuation utilities
here from those of the general case that allows for persistence in (18),
we denote them here with a prime and we make explicit the independence
of [e.sub.1])
What is the simplification of the incentive constraints that
follows from this independence? As it turns out, all the sequences that
have the same choice of effort in the second period, regardless of the
first-period effort choice, provide the agent with the same expected
utility in the second period, conditional on the first-period output
realization being the same. In other words, the deviations of the agent
in the second period can be evaluated independently of the first-period
effort choice, and also independently at each node following the
first-period output realization. As a consequence, the number of
relevant incentive constraints for the agent is drastically decreased.
To see this formally, denote by [[omega].sub.1i] [equivalent to]
[[W.sub.1i].sup.'](u, [e.sub.2i]) the continuation utilities
evaluated at the effort requirement of the principal. Then all the
incentive constraints that involve deviations only in the second period,
or that have the same effort choice for the first period, simplify to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We refer to equation (20) as the "second-period incentive
constraints." (7)
Now note that the independence of [W.sub.1i.sup.'](u,
[.[e.sub.2i]]) on [e.sub.1] also implies the following: Imposing the
second-period incentive constraints in (20) serves to assure that all
potential deviations ([.[e.sub.1]], [.[e.sub.2L]], [.[e.sub.2H]]) that
consider effort choices in the second period that are not [e.sub.2H] and
[e.sub.2L] are dominated by a strategy ([.[e.sub.1]], [e.sub.2L],
[e.sub.2h]) that considers the same deviation in period 1 and none in
the second period. Formally, what we are saying is that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
trivially simplifies to the second-period incentive constraint in
(20). This is useful because it means that when we are evaluating
deviations in the first period we forget about potential deviations in
the second period as well, and simply substitute Wu into the second
period utility.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We refer to these constraints as the "first-period incentive
constraints."
The independence of second-period expected utility on first-period
effort choice not only decreases the number of IC constraints that we
need to consider, but also allows the problem of the principal to be
analyzed period by period. This is precisely because all future period
payoffs can be summarized through the promised utility [[omega].sub.1i],
without specifying the particular consumption transfers or effort
recommendations that will deliver [[omega].sub.1i], in the future. From
a practical point of view, it is important to note that the range of
values that [[omega].sub.1i] can take is independent of the agent's
action in the first period, and hence can be calculated by simply using
the domain restrictions for consumption and second-period effort,
together with the second-period IC in (20). This is a very useful
feature when we want to compute the solution for a particular numerical
example, as we will do in Section 6.
To summarize, the simplifications we just discussed are the reason
why the recursive formulation first introduced by Spear and Srivastava
(1987) is possible. In a finite two-period problem like the one
presented here, this also means that we can solve the problem backward
and characterize the properties of the solution. We proceed to do that
now.
A Backward Induction Solution to the Optimal Contract
As a first step, we use the fact that incentives in the second
period are independent of choices and utilities in the first period.
This allows us to split the problem of the agent in the IC into two
problems: a first-period problem and a second-period problem. The
second-period problem, [PIC.sub.2], is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and the first-period problem, [PIC.sub.1], is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[omega].sub.i], is the expected utility for the second
period in equilibrium.
If we want to characterize the optimal contract, first we need to
transform these maximization problems into an equality constraint that
we can include in the problem of the principal. Following the spirit of
the first-order approach (see Rogerson [1985b]), we establish concavity
of the maximization problems in [PIC.sub.1] and [PIC.sub.2]. Then we can
substitute them by their first-order conditions, which are necessary and
sufficient for a maximum. In our two-outcome example, this concavity is
fairly straightforward to guarantee. It is easy to see that, for any
positive first-period effort recommendation to satisfy the original
first-period IC in (21), we need [[mu].sub.H] + [beta][[omega].sub.H]
[Greater than] [[mu].sub.L] + [beta][[omega].sub.L]. Also, for any
second-period positive effort recommendation to satisfy the
second-period IC in (20), we need [[mu].sub.iH] [Greater than]
[[mu].sub.iL]. Since we have assumed that [[pi].sub.H](*) is a concave
function of effort, concavity of the expected utility of the agent in
effort follows. (8) Hence, we can substitute [PIC.sub.1] for its
first-order condition,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and we can substitute [PIC.sub.2]t by its corresponding first-order
condition,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Using these in place of the original IC allows us to derive some
properties for the optimal contract.
As a second step in characterizing the optimal contract, we appeal
to the same logic that we spelled out to show the independence of
second-period utility of the agent on his first-period actions, to argue
that the same independence holds for the expected profit of the
principal. The objective function in problem SB can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence, to solve problem SB subject to (PC) and (22) and
(23)-assuming the domain constraints are not binding-we can simply split
the problem across the two periods and solve it backward using subgame
perfection. First, we solve the second-period problem, [P.sub.2i], for
an unspecified value of [[omega].sub.1i].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let [[mu].sub.i] and [[lambda].sub.i] be the multipliers of the
first and second constraints, respectively. For each i = L, H, the
first-order conditions with respect to utility are
(u.sub.ij): 1/u' ([c.sub.ij]) = [[lambda].sub.i] +
[[mu].sub.i] [[[[pi]'.sub.j]([e.sub.2i])]/[[[pi].sub.j]([e.sub.2i])]], j=L, H (24)
This condition will be familiar to the reader acquainted with basic
contract theory: Since the second-period problem is, in fact, a static
moral hazard, we find that this first-order condition links consumption
to likelihood ratios in the same way as in a static contract (see
Prescott [1999] for a review of this textbook case). The likelihood
ratios capture the informational value of each possible output
realization. The same static intuition prevails in the case for effort.
The first-order conditions are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is easier to see the intuition when we substitute
[[pi].sub.L.sup.'](e) = [-[pi].sub.H.sup.'](e) in the
expression above and get
([e.sub.2i]): [[pi]'.sub.H] [(e.sub.2i)] [[y.sub.H] -
[y.sub.L] - ([c.sub.iH] - [c.sub.iL])] + [[mu].sub.i][[pi]".sub.H]
([e.sub.2i]) ([u.sub.iH] - [u.sub.iL]) = 0.
We see that the principal equates the marginal increase in the
expected net profit that comes from a higher probability of [y.sub.H]
with the change in the marginal increase in expected compensation
associated with it, given that [u.sub.iH] [Greater than] [u.sub.iL].
Note, however, that the solution for the second period is
contingent on the value of [[omega].sub.1i] (which plays the role of the
period outside utility in a static problem). With the solution to the
second-period problem in hand, we can calculate the value to the
principal of promising a level of utility of [[omega].sub.1i] to the
agent for the second period. Hence, we know the value of
[V.sub.1i]([[omega].sub.1i]) and we can substitute it in the
first-period problem, [P.sub.1]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let [mu] and [lambda] be the multipliers of the first and second
constraints, respectively. The first-order conditions for consumption
are
(u.sub.i): 1/u' ([c.sub.i]) = [lambda] +
[mu][[[pi]'.sub.i](e.sub.1)/[[pi].sub.i](e.sub.1)], i=L, H (26)
These mirror the conditions in (24) for the second period: The
ranking of consumption is again determined by the likelihood ratios,
although the dispersion is potentially different and depends on the
multiplier of the first-period incentive constraint, [mu]. The values of
[mu] and [[mu].sub.i] as well as [lambda] and [[lambda].sub.i],, are
difficult to get for generic utility functions. (To see this, note that
the first-order conditions give us information about
[[u.sup.'](c)], while the constraints of the problems [P.sub.1] and
[P.sub.2i], are written in terms of u (c); this makes for a highly
nonlinear system of equations that seldom has an explicit solution.)
This is why computing numerically the solution to particular problems is
a popular strategy in dynamic contract theory.(9)
Recall that in this first period the principal has an extra choice
variable with respect to problem [P.sub.2i],: the contingent levels of
expected utility of the agent in the second period, [[omega].sub.1i].
The importance of the value of [[omega].sub.1i] relative to that of
[u.sub.i], in the optimal contract is at the heart of dynamic
incentives. We can explore the optimal tradeoff between the two
variables by looking at the first-order condition for the continuation
utility:
(w.sub.1i): [V'.sub.1i](w.sub.1i) + [lambda] + [mu]
[[[pi]'.sub.i](e.sub.1)/[[pi].sub.i] (e.sub.1)] = 0, i = L, H. (27)
To interpret this condition we need to figure out the derivative of
the value function of the principal,
[V.sub.1i.sup.']([omega].sub.1i) = [-[lambda].sub.i]. We do this by
using the envelope theorem and the second-period problem [P.sub.2i],
that determines V ([omega].sub.1i):
[V'.sub.1i](w.sub.1i) = [-[lambda].sub.i]
Substituting this derivative into (27) we get [[lambda].sub.i] =
[lambda] + [mu] [[[pi]'.sub.i]([e.sub.1])/[[pi].sub.i]([e.sub.1])],
i = L, H.
Note that this, combined with (26), implies [[lambda].sub.i] =
1/[[u.sup.']([c.sub.i])]. What does the [[lambda].sub.i],
multiplier represent in the second period? It is the shadow value of
relaxing the "promise keeping" constraint of the principal in
the second period. The principal has committed to deliver a level of
expected utility of [[omega].sub.1i]. How costly this is for him depends
on the necessary spread of utilities in order to satisfy incentives in
the second period. This can be seen formally by multiplying the
first-order conditions for [u.sub.ij] in (24) for each j times
[[pi].sub.j]([e.sub.2i]), and then summing the resulting equations for j
= L and j = H; doing this we get that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The shadow value depends on the expected tradeoff between the
marginal value to the principal of increasing consumption, -1, and the
marginal increase in utility of spending this extra unit of consumption,
u' (c). Now we take this condition further: Since we had
established that [[lambda].sub.i] = [1/[u'([c.sub.i])]], we get the
following relationship of the inverse of the marginal utility of
consumption:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)
This is the so called "Rogerson condition," first derived
in Rogerson (1985a). It summarizes how the optimal dynamic contract with
commitment allocates incentives over time and histories. We now discuss
its implications for the choices of effort and consumption.
Effort and Consumption Choices Over Time
To illustrate the implications of the Rogerson condition, consider,
for the sake of comparison, a slightly different model to the one
presented here: Everything else equal, assume no commitment to long-term
contracts for both the principal and the agent. This is often referred
to as "spot contracting." For the purpose of our comparison,
set the per period outside utility for the agent to [w.sub.0]/2 in both
periods. It is easy to see that the solution to this problem without
commitment is the repetition of the one-period optimal contract. This
implies that the second-period consumptions would be independent of the
first-period realizations, and hence identical to those in the first:
[c.sub.H] = [c.sub.LH] = [c.sub.HH], as well as [c.sub.L] = [c.sub.HL] =
[c.sub.LL]. It is immediate that this solution to the spot contract
violates (28).
How is the contract with commitment different than the repetition
of the static contract? The main difference is that with commitment the
contract exhibits memory, i.e., the level of consumption in the second
period, contingent on a second-period realization, is different
depending on the first-period realization. Why is it optimal for the
contract with commitment to be different than the repetition of the
static contract? Because it allows incentives to be provided in a more
efficient way. The reason becomes clear if we consider how the principal
can improve on the repetition of the static contract once he has
commitment to a two-period contract. If the agent gets a [y.sub.H]
realization in the first period, his overall expected utility increases
if he trades off some of the consumption that the static contract
assigns him in the first period with some expected consumption in the
second. Because [c.sub.H] was high to start with, the decrease in his
first-period utility from postponing some consumption translates into a
bigger increase in expected utility in the second period, where he has
positive probability of facing low consumption whenever [y.sub.L]
realizes. This means the principal can, with this deviation from the
spot contract solution, keep some of the consumption for himself while
leaving constant the expected utility following the high realization
node in the first period, i.e., [u.sub.H] + [beta][w.sub.H]. In the same
way, if the agent gets a [y.sub.L] realization in the first period, he
is better off by trading some expected utility in the second period for
some consumption in the first, and this again saves resources for the
principal. Hence, in the optimal contract, we have that [w.sub.1H] >
[w.sub.1L]. It is worth noting that these optimal tradeoffs result in a
violation of the Euler equation of the agent, which is incompatible with
(28). (10)
The last first-order condition of problem [P.sub.1] left for
analyzing is that of effort in the first period:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This condition captures the same tradeoff discussed after deriving
the second-period effort first-order conditions in (25). Of course, the
values of the variables and multipliers will typically be different than
in the second period, implying a different solution across periods. To
gain some important insight in the properties of effort requirements
over time, it is again useful to compare the effort solution here to
that of the spot contract without commitment. It is easy to see that the
repetition of the static contract would imply [e.sub.1] = [e.sub.2H] =
[e.sub.2L]. (11) Here, instead, this is not the case. If we recall that
the optimal contract implies [w.sub.1H] > [w.sub.1L], a simple
inspection of the second-period problem [P.sub.2] tells us that, for the
principal, effort incentives are more expensive following a [y.sub.H]
realization than a [y.sub.L] realization. The continuation utility
[w.sub.1i] plays the role of the outside utility in a static contract.
It is immediate from the risk aversion of the agent that, for the same
spread of utility that would satisfy the IC in (23), a higher level of
outside utility translates into more consumption. Hence, the principal
will optimally choose [e.sub.2H] < [e.sub.2L]. Moreover, in the
second period the principal cannot provide incentives for effort as
efficiently as in the first period, since the intertemporal tradeoff of
consumption that we described above is not available (there are no
future periods after t = 2). This will typically imply a lower effort
requirement in the second period than in the first. We conclude that, in
contrast with the first-best property summarized in 2A, effort
requirements will fluctuate over time and across histories in the
unobservable effort case in order to provide incentives more
efficiently.
The solution to this version of our numerical example is presented
in Table 3 and Figures 1 and 2. We defer the discussion of this solution
example until Section 6, where we compare the solution to the unobserved
effort case both with full depreciation and without.
4. DEALING WITH PERSISTENCE
The simplifications outlined in the previous section, when effort
is not persistent, do not hold for the general case of [rho] > 0.
Before we go on to analyze a particular case of human capital
accumulation in Section 5 and illustrate the differences, we discuss
here the main particularities that persistence of effort introduces in
the analysis of the optimal contract.
Two main differences with respect to the standard framework appear
when effort is persistent. First, it is no longer the case that a given
choice for effort in the second period provides the agent with the same
expected utility [w.sub.1i] regardless of his first-period effort choice
[e.sub.1]. It follows that the number of relevant incentive constraints
is much higher in the problem with persistence. Second, the problem of
the principal cannot, in general, be written in the usual recursive form
in which the promised utility [w.sub.1i] summarizes all relevant
information about past periods. The relevant summary variable is the
original [W.sub.1i] (u, e), which depends on both the first- and the
second-period effort choices. The dependence of [W.sub.1i] (u, e) on
[e.sub.1] complicates the calculation of its possible values. In
particular, this state variable is not a number (like [w.sub.1i] was)
but a function: The principal needs to take into account all possible
choices for [e.sub.1], including those off the equilibrium path.
Finally, the conditions for concavity of the agent's problem in the
IC are difficult to establish, even in the two-outcome case presented
here.
These issues have so far been addressed in the literature with two
main strategies. The first strategy limits the effort choices to a
two-point set, and includes explicitly in the problem of the principal
the complete list of relevant incentive constraints for all possible
combinations of effort choices. The second strategy allows for a
continuum of effort choices, but puts restrictions on the functional
form of [pi] ([e.sub.1], [e.sub.2]) in order to simplify the set of
constraints. These approaches are now discussed in some detail.
A Hands-On Analysis of the Joint Deviations Problem
Within the first approach, the main contribution is Fernandes and
Phelan (2000). They provide a tractable setup in which an augmented
recursive formulation of the problem of the principal is possible.
Intuitively, this formulation has an increased number of state variables
with respect to the recursive formulation of the moral hazard problem
without persistence first presented in Spear and Srivastava (1987). The
simplified framework that allows for the recursive formulation limits
the effort choices and the output realizations to two. Also, the
contract lasts for an infinite number of periods but persistence lasts
only for one period; that is, effort at time t affects only the
probability distribution over outcomes at time t and t + 1. The
recursive formulation of the problem of the principal has three state
variables, one of which is the standard promised utility on Spear and
Srivastava's formulation. The two extra states allow the principal
to keep track of the marginal disutility of effort for the agent across
periods, as well as the set of utilities achievable by the agent off the
equilibrium path.
Still within the first approach, Mukoyama and Sahin (2005) limit
the effort choices and the output values to two and analyze a two-period
problem. They assume that high effort is optimal every period. They are
able to provide analytical conditions on the conditional probability
function under which the implications of persistence are drastically
different than those of no persistence: When the first-period effort
affects the second-period probability in a sufficiently stronger way
than the second-period effort, the optimal contract exhibits perfect
insurance in the initial period. Using a recursive formulation in the
spirit of Fernandes and Phelan (2000), Mukoyama and Sahin also analyze a
three-period problem numerically.
Kwon (2006) uses a very similar framework with discrete effort
choices (0 or 1), also assuming that high effort is implemented every
period. He imposes concavity of [pi] (*) on the sum of past effort
choices, so past effort is more effective than current effort. These
assumptions allow him to analyze a T > 2 period problem that shares
the same perfect insurance characteristic as in Mukoyama and Sahin
(2005).
A Particularly Simple Case of Persistence
The second approach, presented in Jarque (2010), allows for a
continuum of effort choices but assumes that the conditional probability
depends on past effort choices only through the sum of undepreciated
effort in the same manner as stated in Assumption 2. Note that, even for
a concave probability function [pi] (s), Assumption 2 implies that past
effort is less effective than current effort in contrast to what was
assumed in Mukoyama and Sahin (2005) or Kwon (2006). The article shows
that, for a subset of problems with this particular form of persistence,
the computation of the optimal contract simplifies considerably. For
these problems, an auxiliary standard repeated moral hazard problem
without persistence can be used to recover the solution to the optimal
contract. The linearity in effort of both variable s (which determines
the probability distribution) and the utility of the agent dramatically
simplifies the structure of the joint deviations across periods; in
practice, we can think of s as the choice variable, and the structure of
the resulting transformed problem is (under some conditions) equivalent
to that of a standard repeated moral hazard.
In the next section, a finite version of the model in Jarque (2010)
is presented and this result is explained in detail. The finite version
allows for the numerical computation of the optimal contract in an
example in which the stochastic structure is interpreted as unobservable
human capital accumulation.
5. HIDDEN HUMAN CAPITAL ACCUMULATION
The problem of the principal is again as in problem SB, but now we
consider the case [rho] > 0. We argued in Section 4 that this case is
more complicated because of the dependence of second-period utility and
optimal actions of the agent on first-period choices. In order to go
around some of these difficulties, here we adapt to our two-period
finite example the strategy presented in Jarque (2010) for solving
problems with persistence. Following this work we will show that, under
our assumptions, the structure of the problem simplifies to that of the
standard repeated moral hazard presented above, provided the domain
constraints in (ED) do not bind. This is an important qualification
since, as we learned when analyzing the case of observable human capital
accumulation in Section 2, in the presence of persistence the effort
domain constraints in (ED) will sometimes bind, especially for high
values of the persistence parameter [rho]. To deal with this issue, we
follow the approach in Jarque (2010): First, we find a candidate
solution assuming that the constraint in (ED) does not bind. Then we
need to check numerically that this constraint is indeed satisfied to be
sure that we have found a true solution. Unfortunately, a general
analysis of the optimization problem of the principal including the
inequality constraints for effort (again, as in Section 2) is more
difficult with unobserved effort. Hence, finding the properties of the
general case when constraint (ED) binds remains a question for future
research.
Rewriting the Problem
Jarque (2010) shows that, whenever the effort domain constraint
(ED) is not binding, we can find the solution to the problem with
persistence using a related RMH problem without persistence as an
auxiliary problem. The key observation for that result is that we can
write the expected utility of the agent, [W.sub.0] (u, e), as a function
of the s variable only. This is convenient because s is the variable
that effectively determines the probability distribution over outcomes
each period; different combinations of effort choices that give rise to
the same s are equivalent both for the principal and for the agent.
Hence, once we rewrite the problem with s as the choice variable, there
is no need to consider joint deviations across periods, the recursive
structure is recovered, and we can solve for the optimal contract as we
do with a standard repeated moral hazard.
Let [Tilde.W.sub.0] (u, s) = [W.sub.0] (u, e) for all the pairs of
s and e sequences such that s results from effort choices in e according
to the law of accumulation of human capital in (1). Writing the effort
in the second period as
[e.sub.2i] = [s.sub.2i] - [rho][s.sub.1],
we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Note that we have explicitly written the utility accrued in the
first period in the first row of this expression, and that of the second
period in the second row. With utility spelled out this way it is easy
to see that, although [s.sub.1] is all accumulated in the first period,
it appears both in the first- and second-period utility. Also, since
[s.sub.1] is not contingent on any realization, it appears in the second
period both after observing a first-period [y.sub.H] and a first-period
[y.sub.L]. Hence, we can group the [s.sub.1] terms of the second period
together with those of the first, to get an expression of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This allows us to interpret s as the variable being chosen by the
agent. In the first period, we can interpret v (1 - [beta][rho]) as the
"marginal disutility of exerting [s.sub.1]." In the second
period, the "marginal disutility of exerting [s.sub.2]" is
instead v.
This rearrangement of terms and thinking about s as the choice
variable is a useful trick. Note that in the second row the expression
inside the square brackets is independent of [s.sub.1]. Interpreting
[s.sub.2i] as the choice variable, we can see that we can do here as we
did in the case of no persistence and write the continuation utility of
the agent independently of the first period's choice for [s.sub.1]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence, we obtain expressions that parallel those of the standard
RMH formulation in (19). The expression in (29) can then simply be
rewritten as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Note also that the structure of the incentive constraints
simplifies as it did in the case of the RMH; in the second period, the
first-period choice of s drops out:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Again, all these changes of notation are simply aimed at pointing
to the following fact: The problem in which effort is persistent has a
similar structure to that of a standard RMH problem in which s is
interpreted as effort that is not persistent, but has marginal
disutility of v (1 - [beta][rho]) at t = 1 and of v at t = 2. To make
this explicit, using the intertemporal regrouping of [s.sub.1], the
problem of the principal in SB can be written as problem SB':
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
with [S.sub.1] = [[e.bar], [bar.e]] and [S.sub.2] = [[rho][s.sub.1]
+ [e.bar], [rho][s.sub.1] + [e.bar] + [bar.e]]. This rewriting leads to
the following observation: If problem SB' were in fact formally
equivalent to a standard RMH problem (with the modified structure of the
marginal disutility), this would help us enormously to find and
characterize the solution to SB, since we would know how to solve it (or
at least compute it numerically). However, a close inspection of
SB' points to a small but potentially important difference with a
standard RMH problem: In problem SB', the domain [S.sub.2] depends
on the choice of [s.sub.1], while in a standard RMH problem this domain
would be exogenously given.
Using a Related RMH Problem without Persistence as an Auxiliary
Problem
Following Jarque (2010), we now show that, in some instances, we
can work around the difficulty that an endogenous domain [S.sub.2] poses
by using a related auxiliary problem for our purposes instead of
SB'. Consider a problem S[B.sub.aux] that is equal to SB'
except for the domain [S.sub.2], which is substituted by an auxiliary
domain [[tilde.S].sub.2] = [[e.bar], [bar.e]]. Note that
[[tilde.S].sub.2] is exogenous so, interpreting s as effort, problem
S[B.sub.aux] is a standard RMH. We will now argue that, under some
conditions, the solution to SB' coincides with the solution to
S[B.sub.aux], and hence we can easily obtain a solution to our problem
with persistence.
The solution to problems SB' and S[B.sub.aux] coincides when
two conditions are satisfied: (i) [W.sub.0] (*) is concave in s, and
(ii) the resulting optimal choices for effort are interior. This is a
set of sufficient conditions because if the expected utility of the
agent is concave in his choice of s, then the relevant effort deviations
are those close to the optimal (interior) s, and not those at the limits
of the domain. This implies that using an auxiliary domain that does not
exactly overlap with the true domain is not changing the solution to the
problem, as long as this true solution is contained in the auxiliary
domain. Are each of these conditions satisfied in our framework?
(i) Concavity of [W.sub.0] (*) in s. In our particular example, it
is easy to argue that the problem of the agent is concave in [s.sub.t]
for all t. In fact, the argument is the same that we used earlier to
argue that problems PI[C.sub.1] and PI[C.sub.2] were concave: There are
only two outcomes, the probability of observing [y.sub.H] is concave in
[s.sub.t], and current and future utility assigned to [y.sub.H] is
always higher than current and future utility assigned to [y.sub.L].
(ii) Effort is interior. This is not satisfied trivially.
Constraint (ED) implies that two restrictions need to be checked to
establish that the true solution is contained in the proposed auxiliary
domain:
[s.sub.2i] [Less than] [rho][s.sub.1] + [e.bar] + [bar.e],i = L, H,
(30)
[s.sub.2i] [Greater than] [rho][s.sub.1] + [e.bar],i = L, H. (31)
Under the probability specification in (7), equation (30) is always
satisfied. Other specifications are easy to find for which the upper
bound of effort in (30) is not binding. The lower bound, however, is
endogenous, and equation (31) cannot be checked without having the
solution for s in hand. We conclude that the interiority cannot easily
be guaranteed ex ante. The strategy to go around this problem that is
proposed in Jarque (2010) is the following: Solve the problem assuming
that the domain constraint can be substituted--and hence the equivalence
to the RMH can be used--and then, with a candidate solution for s in
hand, check the constraint ex post. We follow this route in the
numerical computation of an example presented next. As it turns out, it
is easy to find parametrizations for which the ex post check on the
nonnegativity of effort is satisfied.
The Optimal Contract for Hidden Human Capital Accumulation
What do we conclude about the properties of the optimal contract in
the presence of hidden human capital accumulation? Denote as
[[tilde.c].sup.*] and [[tilde.e].sup.*] the solution to problem
S[B.sub.aux]. Whenever the sufficient conditions discussed above are
satisfied, we have that, in the optimal contract:
1. The optimal consumption sequence in problem SB, [c.sup.*] (P,
SB), is equal to Tilde.c*
2. The optimal human capital sequence in SB, [s.sup.*] (P, SB), is
equal to [[tilde.e].sup.*].
3. The optimal effort sequence in SB, e* (P, SB), can be recovered
from the effort solution to problem S[B.sub.aux] using
[e*.sub.1](P,SB) = [Tilde.e]*.sub.1]
[e*.sub.2](P,SB) = [Tilde.e.sub.2] - [rho][Tilde.e.sub.1].
Importantly, the optimal consumption sequence has the same
properties as in the solution to a standard RMH problem without
persistence. Also, the optimal human capital sequence has the same
properties as the effort sequence in a standard RMH problem. These
properties were discussed at length in Section 3. Using these
properties, we can reflect on the economic meaning of the ex post check
implied by equation (31).
Whenever the ex post check in (31) is satisfied, the optimal
contract asks the agent to increase human capital in every period. That
is, the remaining level of human capital from the previous period, after
depreciation, [rho][s.sub.1], is never sufficient to cover the
requirement of human capital for the current period, [s.sub.2i] for i =
L, H. In light of the properties of effort in a standard RMH problem, it
is easy to see that this condition may not be satisfied in some examples
since a decrease in the level of human capital from one period to the
next could be part of the optimal solution for the principal. In
particular, we learned in Section 3 that in an interior solution we will
typically have [e.sub.2H] < [e.sub.1], since the smoothing of
incentives that is present in the first period is not available in the
second, making effort in the second period relatively more expensive.
Given the results we just established for the case with persistence,
this means that we will typically have [s.sub.2H] < [s.sub.1] in the
optimal contract with hidden human capital accumulation. How does this
lead to a violation of the ex post check in equation (31)? For certain
parameters, we may have that [s.sub.2H] is so much smaller than
[s.sub.1] that, in fact, we have [s.sub.2H] < [rho][s.sub.1] +
[e.bar], violating the inferiority of effort choices. That is, if it
were feasible, the principal would choose to have [s.sub.2] lower than
[rho][s.sub.1] + [e.bar]. However, in the true problem with human
capital accumulation (problem SB), effort needs to stay within its
domain in each period, i.e., [e.sub.2i] > [e.bar] for all i, which
rules out the possibility of decreasing [s.sub.2] below [rho][s.sub.1] +
[e.bar]. Any adjustment should be made in the first period, when the
principal anticipates the added cost of future incentives. That is, the
solution for [s.sub.1] should differ from the one that was just
presented. Unfortunately, characterizing how exactly the solution for
[s.sub.1] changes is not easy. Solving for the optimal contract in this
case becomes more complicated. As we argued, the independence of
second-period choices from first-period choices breaks down, both for
the principal and for the agent. In practice, even the numerical
computation of examples is more involved, since all feasible
combinations of effort across the two periods (and choices contingent on
realizations of output) need to be tested for incentive compatibility.
The simple recursive structure with [w.sub.2i] as a state variable is no
longer valid, and the dimensionality of the computational problem is
similar to that of the strategy proposed in Fernandes and Phelan (2000).
The next section presents an example for which the ex post check in
(31) is satisfied, and hence solving for the optimal contract is simple.
Using the numerical solution, we discuss the implications of persistence
for consumption and effort paths by comparing the solution to that of
the case without persistence ([rho] = 0).
6. NUMERICAL EXAMPLE WITH UNOBSERVED EFFORT: A COMPARISON
For cases in which the equivalence to a RMH is valid, we can find
the solution to our problem with persistence using the usual numerical
methods to solve standard RMH problems without persistence.
Figures 1 and 2 illustrate the implications for effort and
consumption in the solution to an example with the parameter values
listed in Table 1. The example without persistence has [rho] = 0, while
the example with persistence has [rho] = 0.2. For the numerical examples
we use the functional forms u (c) = 2[square root of (c)] and the
probability specification in (7). We also set [e.bar] = 0.01 and [bar.e]
= 0.99 in order to restrict to cases with full support.
In Figure 1, the solution for s and e in the SB problem with
persistence is plotted with a solid line. As we can see in the top
panels, the level of [s.sub.1] in problem SB is always higher with
persistence than without persistence (dashed line). Since [s.sub.1] =
[e.sub.1], a higher level of [s.sub.1] with persistence reflects the
fact that human capital is accumulated in the first period with the same
cost as nonpersistent effort, but it lasts (partially) until the
following period. (12)
The solutions for the paths of optimal s in the FB model are also
represented in Figure 1 (dotted and dash-dotted line respectively for
the persistent and nonpersistent case). The comparison clearly shows
that human capital accumulation makes frontloading of s optimal. (This
also translates into front-loading of effort as shown clearly in the
bottom panels of Figure 1.) The main difference with the solutions to
the respective SB problem is the level (higher in the FB problem). A
second difference is that, even without persistence, in the second
period the requirement for s may decrease in the SB problem, for
incentive reasons, following both realizations (although the decrease
may be more pronounced after [y.sub.H]), and hence we have [s.sub.1]
> [s.sub.2i] for all i.
As we can see in the bottom panels of the solution to the SB
problem, both with persistence and without, effort is higher in the
initial period than in the second. However, the frontloading of effort
is much more pronounced with persistence. This is also true when
comparing the solutions for the FB problem: While effort stays constant
from one period to the next in the case without persistence, with
persistence it is frontloaded, as discussed in Section 2.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
Consumption, depicted in Figure 2, is, in the SB case, virtually
the same with and without persistence. It simply increases when the
realization is [y.sub.H] and decreases when it is [y.sub.L] for the
standard incentive provision reasons discussed in the earlier sections.
However, we can see in the FB case that consumption is slightly lower in
the case with persistence. Since the FB case is calculated numerically
but without using a grid, we conclude that most likely consumption is
also slightly lower with persistence in the true solution to the
unobservable effort case.
Table 3 Summary Statstics
[rho] (FB) [rho] (FB) [rho] .2 (SB) [rho]
= 0.2 = 0.0 = C =0
t = 1 t = t = 1 t = t = 1 t = 2 t = 1
2 2
E[c.sub.t.sup.*] 6.12 6.12 5.82 5.82 5.30 5.47 5.16
E[u(c.sub.t.sup.*)] 4.95 4.95 4.83 4.83 8.26 11.32 7.74
Var[c.sub.t.sup.*] 0 0 0 0 4.96 13.69 4.34
E[e.sub.t.sup.*] 0.22 0.16 0.17 0.17 0.14 0.05 0.11
Var[e.sub.t.sup.*] 0 0 0 0 0 0.00023 0
E[s.sub.t.sup.*] 0.22 0.16 0.17 0.17 0.14 0.0828 0.11
V [s.sub.t.sup.*] 0 0 0 0 0 0.00023 0
0 (SB)
t =2
E[c.sub.t.sup.*] 5.30
E[u(c.sub.t.sup.*)] 10.90
Var[c.sub.t.sup.*] 13.96
E[e.sub.t.sup.*] 0.084
Var[e.sub.t.sup.*] 0.OOO22
E[s.sub.t.sup.*] 0.0842
V [s.sub.t.sup.*] 0.00022
Table 3 reports the value of some simple statistics of the
comparison across the two models presented in Figures 1 and 2. The FB
model statistics are included for reference, since they correspond to
the solutions reported already in Sections 1 and 2. All expectations in
the first period are conditional on [s.sub.1.sup.*] and those in the
second are conditional on [s.sub.2i.sup.*] When comparing the statistics
for the SB problem, we see that persistence implies a higher level of
expected consumption, expected utility, and a slightly higher variance
of consumption in the first period. When looking at these three moments
across periods we see that persistence implies a steeper increase of
expected consumption in time. Again, the statistics on consumption need
to be interpreted with care since they are likely influenced by the use
of a grid.
As for expected effort, we see that the level is higher with
persistence in the initial period, but it drops below the no persistence
case in the second period (a much steeper decrease than without
persistence). The comparison of the expected accumulated human capital
explains this: The expected level of S] with persistence is much higher
than the level of [e.sub.1] without persistence, but the solution for
[s.sub.2]with persistence is similar (in this particular example,
identical) to the solution for [e.sub.2] without persistence.
7. CONCLUSIONS
When learning by doing is an important factor in a repeated agency
relationship, solving for the optimal contract is generally very
difficult. In the framework studied here, with linear disutility of
effort and the productivity of the agent being a distributed lag of past
efforts, we provide an example with a simple solution. This allows us to
numerically establish some properties of the optimal contract. On one
hand, the human capital of the agent in equilibrium and, hence, his
productivity tend to be higher with learning by doing than without.
Moreover, the optimal contract offered to the employee implies a lower
productivity in the final years of the contract. The human capital of
the agent is left to depreciate since, close to the end of the contract,
the cost of incentives of requiring a higher productivity is not
justified by the benefit of future productivity. This implies that, over
the contractual relationship, effort is frontloaded and follows a
steeper decreasing pattern than in the case without learning by doing.
On the other hand, we find that the properties of wage dynamics remain
unchanged with respect to those of the optimal contract without learning
by doing.
REFERENCES
Ales, Laurence, and Pricila Maziero. 2009. "Accounting for
Private Information." Mimeo.
Arrow, Kenneth J. 1962. "The Economic Implications of Learning
by Doing." The Review of Economic Studies 29 (June): 155-73.
Fernandes, Ana, and Christopher Phelan. 2000. "A Recursive
Formulation for Repeated Agency with History Dependence." Journal
of Economic Theory 91 (April): 223-47.
Gibbons, Robert, and Kevin J. Murphy. 1992. "Optimal Incentive
Contracts in the Presence of Career Concerns: Theory and Evidence."
Journal of Political Economy 100 (June): 468-505.
Grossman, Sanford J., and Oliver D. Hart. 1983. "An Analysis
of the Principal-Agent Problem." Econometrica 51 (January): 7-45.
Heckman, James J., Lance Lochner, and Christopher Taber. 1998.
"Explaining Rising Wage Inequality: Explorations with a Dynamic
General Equilibrium Model of Labor Earnings with Heterogeneous
Agents." Review of Economic Dynamics 1 (January): 1-58.
Jarque, Arantxa. 2010. "Repeated Moral Hazard with Effort
Persistence." Journal of Economic Theory 145 (November): 2,412-23.
Jewitt, Ian. 1988. "Justifying the First-Order Approach to
Principal-Agent Problems." Econometrica 56 (September): 1,177-90.
Kapicka, Marek. 2008. "Efficient Allocations in Dynamic
Private Information Economies with Persistent Shocks: A First-Order
Approach." Mimeo, University of California, Santa Barbara.
Kwon, Illoong. 2006. "Incentives, Wages, and Promotions:
Theory and Evidence." RAND Journal of Economics 37 (Spring):
100-20.
Lemieux, Thomas, W. Bentley MacLeod, and Daniel Parent. 2009.
"Performance Pay and Wage Inequality." Quarterly Journal of
Economics 124 (February): 1-49.
Lucas, Robert E., Jr. 1988. "On the Mechanics of Economic
Development." Journal of Monetary Economics 22 (July): 3-42.
MacLeod, W. Bentley, and Daniel Parent. 1999. "Job
Characteristics, Wages, and the Employment Contract." Federal
Reserve Bank of St. Louis Review (May): 13-27.
Mukoyama, Toshihiko, and Aysegul Sahin. 2005. "Repeated Moral
Hazard with Persistence." Economic Theory 25: 831-54.
Phelan, Christopher. 1994. "Incentives, Insurance, and the
Variability of Consumption and Leisure." Journal of Economic
Dynamics Control 18: 581-99.
Phelan, Christopher, and Robert M. Townsend. 1991. "Computing
Multi-Period, Information-Constrained Optima." Review of
Economic-Studies 58 (October): 853-81.
Prescott, Edward S. 1999. "A Primer on Moral-Hazard
Models." Federal Reserve Bank of Richmond Economic Quarterly 85
(Winter): 47-77.
Rogerson, William P. 1985a. "Repeated Moral Hazard."
Econometrica 53 (January): 69-76.
Rogerson, William P. 1985b. "The First-Order Approach to
Principal-Agent Problems." Econometrica 53 (November): 1,357-67.
Spear, Stephen E., and Sanjay Srivastava. 1987. "On Repeated
Moral Hazard with Discounting." Review of Economic Studies 54
(October): 599-617.
Wang, Cheng. 1997. "Incentives, CEO Compensation and
Shareholder Wealth in a Dynamic Agency Model." Journal of Economic
Theory 76 (September): 72-105.
(1) See Arrow (1962), Lucas (1988), and Heckman, Lochner, and Taber
(1998) for a complete discussion of this issue, as well as alternative
specifications of learning by doing.
(2) Lemieux, MacLeod, and Parent (2009) report that, for a Panel
Study of Income Dynamics sample of male household heads aged 18-65
working in private sector wage and salary jobs, the incidence of
pay-for-performance jobs was about 38 percent in the late 1970s and
increased to about 45 percent in the 1990s. They define
pay-for-performance jobs as employment relationships in which pan of the
worker's total compensation includes a variable pay component
(bonus, commission, piece rate). Any worker who reports overtime pay is
considered to be in a non-pay-for-performance job. See also MacLeod and
Parent (1999).
(3) The only articles dealing with effort persistence in a repeated
moral hazard problem are, to our knowledge, Fernandes and Phelan (2000),
Mukoyama and Sahin (2005). Kwon (2006), and Jarque (2010).
(4) If the reader is knowledgable about contract theory, he or she
may notice that this is not a simple change of notation. In fact, when
computing the soluation in numerical exaamples (see Section 6), we will
follow the two-step procedure proposed in Grossman and Hart (1983). This
procedures consists of splitting the expected profit-maximization of
implementing a given effort level (on a grid of efforts), and (2)
choosing the effort level (on a grid of efforts), and (2) choosing the
effort on the grid that implies the highest expected profit for the
principal Using utility as the choice variable, it is easy to show that
under the assumption of this article there will exist a unique minimum
in the cost minimization problem.
(5) Standard arguement for [lambda] [greater than] 0 hold in this
setup with presistence. The basic intuition is that V* (c, e; [w.sub.0])
is strictly decreasing in [w.sub.0]
(6.) In a T [Greater than] 2 framework with [s.sub.0 = 0, we would
have that [e.sub.1] [greater than or equal to] [e.sub.t] for t [Less
tahn] T, that [e.sub.t], = [e.sub.2] for t = 2, T--1, and [e.sub.T]
[less than or equal to] [e.sub.2]. Again, the intuition is that in all t
[Less than] T, effort improves the conditional distribution not only in
the current period, but also in the periods that follow. At t = 1, since
.[s sub 0 = 0, effort is higher than in any other period problem with
persistence, it is useful to consider the effect of changes in the
persistence parameter, [rho], on the effort solution just presented. For
a value of persistence [rho] = 0, effort equals accumulated effort
trivially, and its level is constant across periods. On the other hand,
if we instead substitute a value of persistence [rho] = 1,(1 -
[beta][rho]) takes its minimum value in (10) and the solution implies
the maximum difference between the level of [s.sub.1] and [s.sub.2],
with [s.sub.1] much higher than .[s.sub.2]. However, carefully
inspecting (11), we can already see that such high level of persistence
cannot be compatible with an interior solution for effort in period 2:
The principal would choose [[e.sub.2].sup.*] = 0. Since
[[s.sub.1].sup.*] [Greater than] [[s.sub.2].sup.*] for all values of
[rho] [Greater than] 0, effort [e.sub.2.sup.*] may not be interior for
other high enough values of [rho]. In other words, persistence implies
that, in many interesting cases, the lower domain constraint on effort
(ED) cannot be safely ignored.
(7.) For more concrete illustration, consider the case with
discrete effort and E = [[e.sub.L], [e.sub.L]]. Then the initial number
of IC constraints would be seven, and they would simplify to three: one
first-period constraint and two second-period constraints
(8.) For a higher number of output levels, the conditions on the
probability function that would assure concavity have not been
determined (see Rogerson [1985b] and Jewitt [1988] for a discussion of
these conditions in the context of a static contract)
(9.) For detail on these computation see for example, Phelan and
Townsend (1991) or Wang (1997).
(10) This follows from Jensen's inequality and the convexity
or 1/u' (c). For details, see Rogerson (1985a).
(11) Simply set [w.sub.1H] = [w.sub.1L] and note that
[[pi]'.sub.H] ([e.sub.1]) = -[[pi]'.sub.L] ([e.sub.1]).
(12) The level of [s.sub.2i] in this example coincides with and
without persistence for all i. This is particular to this example and is
violated if, for example, the level of [w.sub.0] is modified. Although
human capital in the second period is equivalent to nonpersistent effort
(because there are no further periods to exploit the persistence of
human capital), the optimal choice for [w.sub.2i] will typically be
different across the two models.
*I would like to thank Huberto Ennis, Juan Carlos Hatchondo. Tim
Hursey, and Pierre Sarte for helpful comments, as well as Nadezhda
Malysheva for great research assistance. Andreas Hornstein provided many
editorial suggestions that helped shape the final version of this
article. AH remaining errors are mine. The views presented in this
article do not necessarily represent those of the Federal Reserve Bank
of Richmond or the Federal Reserve System. E-mail:
arantxa.jarque@rich.frb.org.
