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Working Paper Series
A Theory of Political Cycles
WP 05-04R
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Leonardo Martinez
Federal Reserve Bank of Richmond
A Theory of Political Cycles∗
Leonardo Martinez†
Federal Reserve Bank of Richmond Working Paper No. 05-04R
Abstract
We study how the proximity of elections affects policy choices in a model in which
policymakers want to improve their reputation to increase their reelection chances. Policymakers’ equilibrium decisions depend on both their reputation and the proximity of the
next election. Typically, incentives to influence election results are stronger closer to the
election (for a given reputation level), as argued in the political cycles literature, and these
political cycles are less important when the policymaker’s reputation is better. Our analysis
sheds light on other agency relationships in which part of the compensation is decided upon
infrequently.
JEL numbers: C73, D72, D78, D82, D83.
Keywords: political cycles, career concerns, reputation, agency, dynamic games, elections.
∗
This paper presents results from my dissertation at the University of Rochester. I thank Per Krusell for
encouragement and advice. For helpful comments and suggestions, I thank an anonymous associate editor and
an anonymous referee, John Duggan, Huberto Ennis, Mark Fey, Borys Grochulski, Hugo Hopenhayn, Roger
Lagunoff, Uta Schoenberg, Francesco Squintani, Alex Wolman, and seminar and conference participants. Some
of the research for this paper was conducted during a visit to the I.I.E.S. Stockholm University, which I thank
for its hospitality. I thank Elaine Mandaleris-Preddy for writing guidance. Remaining mistakes are mine. Any
opinions expressed in this paper are those of the author and do not necessarily reflect those of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
†
Research Department. Federal Reserve Bank of Richmond. E-mail: leonardo.martinez@rich.frb.org.
1
1
Introduction
The literature on political cycles argues that the proximity of elections affects policy choices
(Alesina, Roubini, and Cohen [3], Drazen [10], and Shi and Svensson [30] present reviews of
this literature). More specifically, this literature argues that policymakers induce good economic
conditions just before an election. Why would policymakers prefer to influence economic conditions at the end of their term rather than at the beginning of their term? This paper addresses
this question. More generally, we discuss agency relationships in which an important part of the
compensation is decided upon infrequently. For instance, our framework could be used to discuss
incentives when a contract commits the principal to working with a certain agent for a number
of periods, but allows the principal to replace this agent after the contract ends. Consider, for
example, tenure-track positions. Stiroh [33] presents empirical evidence of a renegotiation cycle:
performance improves in the year before the signing of a multi-year contract, but declines after
the contract is signed. Renegotiation cycles resemble the cycles discussed in the political-economy
literature. Even though our analysis applies to other employment relationships, for concreteness,
this paper refers to voters and policymakers.1
We study political cycles in a standard political-agency model of career concerns. Each
period, the incumbent’s performance depends on his ability, his effort level, and luck. Voters do
not observe the incumbent’s ability, effort, and luck; instead they observe his performance. A
good current performance by the incumbent signals that he is capable of a good performance in
the future. Voters reelect the incumbent only if they expect his performance will be good in the
future. The incumbent wants to be reelected. Therefore, he exerts effort to improve his current
performance.
We show that the incumbent’s equilibrium effort choice depends on both the proximity of
1
Incentives generated by elections are similar to the ones generated by other compensation schemes that
imply a discontinuous increase in compensation when the agent’s reputation is good enough. Discontinuous
compensation schemes are widely observed in various occupations. An employee may be assigned to different
levels in a hierarchy according to his reputation, and these reassignments often imply a discontinuous change
in his compensation (see, for example, Kwon [18] , and Murphy [22]). Bernhardt [4] explains why a firm would
choose this compensation structure. Span-of-control models present a theory of why employees with higher ability
are assigned to higher levels in hierarchies (see, for example, Prescott [28]). Furthermore, capacity constraints
imply that as voters do with policymakers, employers replace incumbent employees who are expected to perform
worse than their replacements.
2
the next election and his reputation (which we refer to as the beliefs about his ability). Recall
that we want to study how the proximity of elections affects policy choices. Consequently, with
political cycles we refer to differences in the incumbent’s choices within a term in office for a
given reputation level.
For a given reputation level, why would the incumbent exert more effort closer to the election?
Consider, for example, an incumbent who starts his political career in period one with an average
reputation and can exert effort in periods one and two to increase his reelection probability. If
his reputation does not change after one period in office, why would the incumbent exert more
effort in period two than in period one?
The key insight to the answer of this question comes from the characterization of the incumbent’s effort-smoothing decision, which we show is such that he makes the marginal cost of
exerting effort in period one (roughly) equal to the expected marginal cost of exerting effort in
period two. This decision presents the typical intertemporal tradeoff in dynamic models: having
less utility in period one allows the incumbent to have more utility in period two. In this case,
a lower expected effort level in period two compensates a higher effort level in period one.
In period one, the incumbent (whose reputation is average) knows that his reputation is likely
to change and anticipates that this change will lead him to choose a lower effort level than the
one he would choose in period two if his reputation remains average—we show that extreme
reputations imply low efforts. Consequently, the expected marginal cost of exerting effort in
period two is lower than the marginal cost of the equilibrium period-two effort level for an average
reputation (the marginal cost is an increasing function). Thus, the incumbent’s effort-smoothing
decision implies that the marginal cost of the equilibrium period-one effort level—which is equal
to the expected marginal cost of exerting effort in period two—is lower than the marginal cost
of the equilibrium period-two effort level for the same (average) reputation. Therefore, for the
same reputation, the period-one equilibrium effort level is lower than that of period two. That
is, the model generates stronger incentives to influence the election results closer to the election,
as argued in the literature on political cycles.
In another context, consider an assistant professor who starts with an average reputation and
wants to exert effort writing papers in order to improve his reputation and be tenured. Our
3
model indicates that the optimal strategy for this professor is to wait until the end of his tenure
clock to see whether it is worth exerting a high effort level. At the beginning of his tenure clock,
he should choose an intermediate effort level. At the end of his tenure clock, if his reputation
remains average, he should choose a higher effort level. If his reputation became either very good
or very bad, he should choose a lower effort level. That is, for the same reputation level, the
assistant professor exerts more effort at the end of his tenure clock, and there is a “renegotiation
cycle.”
With the single-election example discussed above, we explained how a cycle may arise for the
average reputation (at the beginning of the term, the new incumbent has the average reputation).
The paper also analyzes a multiple-election version of the model in which the beginning-of-term
incumbent is not necessarily a new incumbent and, therefore, his reputation is not necessarily
the average reputation. We show that the insight described in the single-election example also
helps us understand political cycles for reputations different from the average reputation. We
also show that political cycles are less important when the incumbent’s reputation is very good.
In addition, we show that, in our framework, results from comparative-statics exercises (which
represent comparisons across different economic and/or institutional environments) are different
from findings in previous studies. In particular, we show that a change in the per-period value
that a policymaker assigns to being in office has no effect on the importance of the cycles, because
it strengthens the incumbent’s incentives to manipulate voters’ beliefs, not only in election times
but also between elections. This result contrasts with the findings presented by Shi and Svensson
[31].
1.1
Related literature
Previous theoretical studies of political cycles succeeded in showing that in environments with
asymmetric information about the incumbent’s unobservable and stochastically evolving ability
(as the one we study), cycles can arise with forward-looking and rational (as opposed to naı̈ve)
voters. The mechanism behind the cycles in these studies (see, for example, Rogoff [29], and Shi
and Svensson [31]) is different from the one we highlight. There, only the end-of-term incumbent’s
4
performance provides information about his post-election performance and only his end-of-term
action is effective in changing the election result. Thus, reelection concerns play a role only at
the end of a term, and, therefore, political cycles arise. These assumptions in previous studies are
motivated by the idea that the incumbent’s more recent performance may be more informative
and that, consequently, the incumbent’s actions closer to the election may be more effective in
influencing the election result. For expositional simplicity, previous studies model this intuition
in its most extreme form.
This paper contributes to the understanding of the relative effectiveness of the incumbent’s
actions across his term. As in previous studies, we assume that the incumbent’s more recent
performance provides more information about his future performance. That is, we allow for the
force highlighted in previous studies to play a role. However, in contrast with previous studies, we
consider that every performance (not only the last performance before the election) may provide
information.2 We show that the force driving cycles in previous studies may not survive when
weaker asymmetries across periods are assumed. That is, if we assume that the incumbent’s
more recent performance is more informative (as in previous studies), but his performance in all
previous periods is informative (in contrast with previous studies), more recent effort can be less
effective in manipulating voters’ beliefs about the incumbent’s future performance (in contrast
with previous studies).
A related literature studies how the alternation in power of politicians with different types
causes movements in the real economy. Partisan cycles are studied, for example, by Alesina [1],
and Hatchondo, Martinez, and Sapriza [14]. Besley and Case [6] and Hess and Orphanides [15,
16] study how the presence of term limits introduces electoral cycles between terms (while we
focus on cycles within terms).
The rest of the paper is structured as follows. Section 2 introduces the model. Section 3
2
Most empirical studies on economic voting do not discuss explicitly the time horizon considered by voters
(for a review, see for example, Lewis-Beck and Stegmaier [19]). However, some studies reject the hypothesis that
voters consider only the incumbent’s performance close to the election (see, for example, Brender [7], Fair [12],
Meloni [21], Panzer and Paredes [23], and Peltzman [24, 25]). In contrast, Eisenberg and Ketcham [11] find that
“only the most recent year of economic performance significantly determines the incumbent party’s vote share.”
Nevertheless, they consider four economic measures simultaneously and only 17 observations, so their analysis
has limited power. Moreover, when they consider county-level performance (21,368 observations), they find that
“voters appear to consider each of the three most recent years about equally.”
5
presents the results. Section 4 concludes and suggests possible extensions.
2
The model
Following the current consensus in the literature, we study political cycles in a political-agency
model of career concerns (see Shi and Svensson [30]). In most political models of career concerns,
the incumbent’s action is unobservable. But some models assume that the incumbent’s action
is observable and that voters are uninformed. In Besley and Case [6], the incumbent’s action is
unobservable effort. In Persson and Tabellini [26] the incumbent’s unobservable action is to allocate resources to less socially beneficial uses. Conversely, in Shi and Svensson [31] the incumbent
manipulates observable fiscal policy producing political budget cycles in an environment with
uninformed voters.
We focus on unobservable effort, but the incumbent’s effort decision can be reinterpreted
easily. For instance, suppose that the incumbent decides the resources he makes available for
providing a public service that voters appreciate (for example, education). Let τ − r denote
these resources, where τ denotes the total available resources and r denotes the resources the
incumbent reserves for his favorite interest group or himself (for a discussion of models of rentseeking see, for example, Persson and Tabellini [26]). Let u (r) denote the incumbent’s utility.
We can define a ≡ τ − r as the effort the incumbent exerts in public service (not every voter is
aware of the budget details) and c(a) ≡ −u (τ − a) as the cost of exerting effort.
We assume time is discrete and indexed by t ∈ {1, 2, ..., T }. In period t, the amount of public
good produced by the incumbent policymaker, yt , is a stochastic function of his ability, η t , and
his effort level, at ≥ 0. In particular,
yt = at + ηt + εt ,
(1)
where εt is a normally distributed random variable with expected value 0 and precision hε (the
variance is
1
).
hε
As it is standard in models of career concerns, we assume that the incumbent and voters
do not know the incumbent’s ability. This simplifies the analysis because, as described later,
6
it implies that voters and the incumbent have the same information on the equilibrium path.
This assumption also helps us understand situations where a policymaker in a new position may
be ignorant of his ability when met with new tasks, or where a policymaker’s success does not
only depend on his individual ability but also on the ability of others working with him. The
common belief about the ability of a new incumbent is given by the distribution of abilities in
the economy, which is normally distributed with mean m1 and precision hη .
Ability η t and εt are unobservable. Voters do not observe the effort level (which is, of course,
known by the incumbent). Voters and the incumbent learn about the incumbent’s ability using
Bayesian learning. From this point forward, belief refers to belief about the incumbent’s ability,
unless stated otherwise.
As in previous studies of political cycles, we assume that the incumbent’s ability evolves
over time. This assumption allows us to represent situations in which the incumbent’s tasks
are changing over time, and his ability depends on the tasks he is focusing on (consider, for
example, the president of a country that becomes involved in a war). In particular, we follow
Holmstrom’s [17] seminal paper and assume that ηt+1 = η t + ξ t , where ξ t is normally distributed
with mean 0 and precision hξ , and it is unobservable. Depending on hξ , the precision of beliefs
may be increasing or decreasing with respect to the number of performance observations—see
Holmstrom [17]. For simplicity, we assume that hξ is such that the precision of beliefs is constant.
That is, we assume that
hξ =
h2η + hη hε
.
hε
(2)
This assumption allows us to abstract from the tenure effect on the strength of career concerns
and, therefore, simplifies the analysis.3
Voters’ per-period utility is given by yt .4 They decide on reelection in order to maximize
their expected utility. In every election, the incumbent competes with a policymaker who was
3
As explained later, beliefs are Gaussian and, therefore, they can be characterized by their mean and their
precision. By making assumptions that guarantee that their precision is constant, we can keep track of the
evolution of beliefs by following the evolution of their mean. This simplifies the analysis. Note also that allowing
the precision of beliefs to change over time would introduce an additional asymmetry between the incumbent and
the challengers.
4
We assume that there are no conflicts among voters or that decisive voters care about future performance
and not about ideology. The model could be extended to include probabilistic voting as in Shi and Svensson [31].
7
not previously in office. A policymaker’s per-period utility is normalized to zero if he is not in
office. He receives R > 0 after winning an election and in a period without elections. The cost to
exerting effort is given by c (a), with c′ (a) ≥ 0, c′′ (a) > 0, and c′ (0) = 0. Let δ ∈ (0, 1) denote
the voters’ and the incumbent’s discount factor.
Elections occur every two periods starting in period three. That is, elections are held in all
odd periods starting in period three. The timing of events within each period is as follows. First,
in election periods, voters decide whether to reelect the incumbent. Second, after an election
or at the beginning of a period without elections, the incumbent decides on his effort level.
Third, ξ t−1 and εt are realized and yt is observed (in period 1, η 1 and ε1 are realized before y1
is observed). We assume that there are no term limits. Term limits would introduce additional
asymmetries between the incumbent and challenger policymakers, which would complicate the
analysis (for a discussion of term limits see, for example, Bernhardt, Dubey, and Hughson [5]
and the references therein).
3
Results
We first discuss equilibrium learning. Then, we describe equilibrium in the single-election version
of the model. Finally, we discuss equilibrium with multiple elections.
3.1
Equilibrium learning
Since beliefs are Gaussian and have a constant precision, their evolution is described by the
evolution of their mean. Let mvt and mit denote the mean of the voters’ and the incumbent’s
beliefs at the beginning of period t (from here on, at period t). We refer to a belief with mean m as
belief m. If the voters’ period-t belief coincides with the one of the incumbent, let mt = mvt = mit
denote their belief.
The voters’ and the incumbent’s strategies can be written as functions of their beliefs.5 The
incumbent’s equilibrium effort strategy is denoted by α̂t (mit , mvt ) (the incumbent understands
5
The proof of the sufficiency of the beliefs for characterizing the equilibrium strategies is straightforward and
follows exactly the proof in Martinez [20].
8
the game and therefore can compute the voters’ belief; t indicates the proximity of both the next
election and the end of the game). Let αt (x) ≡ α̂t (x, x) for all x denote the incumbent’s optimal
strategy if the voters’ and the incumbent’s beliefs coincide and are represented by x. If t is an
election period, ιt (mvt ) denotes voters’ equilibrium reelection strategy, which equals one if voters
want to reelect the incumbent, and zero otherwise—the incumbent’s belief computed by voters
is equal to the voters’ belief (see the discussion below).
Define st ≡ ηt + εt . We refer to st as the period-t signal of the incumbent’s ability. Eq. (2)
implies that for any t, the precision of the period-t + 1 beliefs about the signal st+1 is equal to
the precision of the period-t beliefs about the signal st . This precision is given by
H≡
hη hε
.
hε + hη
(3)
Since beliefs about the signal are Gaussian and have a constant precision, the evolution of these
beliefs is also summarized by the evolution of their mean, which is equal to the mean of the
beliefs about ability.
Bayes’ rule implies that the mean of the beliefs at t+ 1 is a weighted sum of the mean at t and
the period-t signal. Eq. (2) implies that the weight of the period-t mean belief in the period-t + 1
mean belief does not depend on the number of observations of the incumbent’s performance.
This weight is given by
hη
.
(4)
hη + hε
Observing yt allows voters and the incumbent to compute the signal st using their knowledge of
µ=
the effort exerted by the incumbent and the production function. The incumbent knows the effort
he exerted. Therefore, he can compute the true signal st = yt − at . Voters compute the signal
using the equilibrium effort level. They are rational and understand the game. In particular,
they know the incumbent’s equilibrium strategy. Thus, the signal computed by voters is given
by
svt (yt , mvt ) ≡ yt − αt (mvt ) = st + at − αt (mvt ).
Consequently,
mit+1 = µmit + (1 − µ)st ,
9
(5)
and
mvt+1 = µmvt + (1 − µ)svt (yt , mvt ) = µmvt + (1 − µ)[st + at − αt (mvt )].
(6)
Eq. (6) shows how exerting effort helps the incumbent to increase the reelection probability.
The expected ability in the voters’ belief is increasing with respect to effort, and we will show
that voters tend to reelect the incumbent if and only if they expect his ability to be good enough.
Recall that voters and the incumbent have the same period-one belief. Moreover, (5) shows
that in any period in which the incumbent exerts the equilibrium effort level (for example, on
the equilibrium path), voters and the incumbent compute the same signal. Consequently, on the
equilibrium path, the voters’ and the incumbent’s beliefs coincide (mvt = mit ).
3.2
The one-election version of the model
At the beginning of the game, a new incumbent is in office. He can exert effort in periods one
and two in order to affect the probability of reelection in period three. We solve the model using
backward induction.
3.2.1
Period three
In period three, both the period-two incumbent and a new appointee would exert zero effort.
Consequently, voters expect more output from the period-two incumbent if and only if mv3 > m1 .
Thus, the incumbent is reelected if and only if s2 >
m1 −µmv2
1−µ
+ α2 (mv2 ) −a2 (assuming that voters
replace the incumbent when they are indifferent between doing so or not).
3.2.2
Period two
Let φ(v; x, z) denote the density function for a normally distributed random variable V with mean
x and precision z, and let Φ(v; x, z) denote the corresponding cumulative distribution function.
The incumbent’s period-two maximization problem reads
m1 − µmv2
max δR 1 − Φ
+ α2 (mv2 ) − a2 ; mi2 , H
− c (a2 ) .
a2 ≥0
1−µ
10
(7)
In this paper, we characterize the incumbent’s equilibrium effort levels through the first-order
condition of his maximization problems. As in previous models of political agency, assumptions
are necessary to guarantee the concavity of these problems in which the reelection probability
may not be a concave function of the incumbent’s decision. For example, the first term in the
objective function in (7) may not be globally concave. In order to assure global concavity of the
incumbent’s problems, it is sufficient to assume enough convexity in the cost of effort function.
In order to find the equilibrium effort level, we solve a fixed-point problem. The effort level
that maximizes the incumbent’s expected utility in (7) depends on the effort level voters use
to compute the signal, α2 (mv2 ). On the equilibrium path, the incumbent’s effort level must be
equal to the effort level voters use to compute the signal. The following proposition shows that a
unique fixed point that solves for the period-two equilibrium effort strategy exists (see Appendix
A for the proof).
Proposition 1 Let m2 denote the beliefs at the beginning of period two. The unique period-two
equilibrium effort strategy satisfies
′
c (α2 (m2 )) = δRφ
m1 − µm2
; m2 , H .
1−µ
(8)
Thus, for any reputation m2 , the equilibrium period-two effort level α2 (m2 ) is positive.
The incumbent benefits from exerting effort because this increases the reelection probability.
The right-hand side of (8) represents the marginal benefit of exerting effort in period two. This
marginal benefit is given by the change in the probability of reelection multiplied by R (the
value of winning the election) and the discount factor. This probability change is given by the
likelihood of the realization of the minimum signal required for receiving R next period.
Understanding the relationship between the incumbent’s reputation and the strength of his
reelection incentives is crucial for comprehending why the model generates cycles. Even though
on the equilibrium path the voters’ and the incumbent’s beliefs always coincide, we can study the
way each of these beliefs affects the incumbent’s decision separately. Voters’ belief determines
the minimum signal required for reelection. For example, if at the beginning of period two voters
believe the incumbent is very talented, he may be reelected even if the period-two signal is low
11
(because voters’ period-three belief may still be good enough). The incumbent’s belief determines
the signal density function he uses for evaluating his problem. Loosely speaking, it determines
how likely he thinks it is that a certain signal is realized. In the marginal benefit of exerting
−µmv2
effort, the density reads φ m1 1−µ
; mi2 , H (but on the equilibrium path mi2 = mv2 ). The
next lemma describes the relationship between the incumbent’s belief and the marginal benefit
of exerting effort in period two (Appendix B presents the proof).
−µmv2
Lemma 1 On the equilibrium path, φ m1 1−µ
; mi2 , H is continuous, increasing in mi2 if
mv2 < m1 , and decreasing in mi2 if mv2 > m1 .
The intuition for this result is as follows. First, suppose voters expect the incumbent’s
ability to be low (mv2 < m1 ). Then, the current-period minimum signal the incumbent needs
−µmv2
for receiving R next period ( m11−µ
) is high. When the incumbent believes he is better (i.e.,
m1 −µmv2
when mi2 is higher), a high signal is more likely (i.e., φ
; mi2 , H is higher). A parallel
1−µ
argument applies when mv2 > m1 . The next lemma characterizes the relationship between the
voters’ belief and the marginal benefit of exerting effort in period two (Appendix C presents the
proof).
Lemma 2 On the equilibrium path, φ
m1 −µmv2
; mi2 , H
1−µ
mv2 < m1 , and decreasing in mv2 if mv2 > m1 .
is continuous, increasing in mv2 if
The intuition behind this result is straightforward. Recall that more extreme signal values
are less likely. In other words, if the value of a signal is more extreme (i.e., if it is further from
the mean), the density function evaluated at the signal is lower. Suppose voters expect the
incumbent’s ability to be low (mv2 < m1 ). Then, the minimum signal realization that would
−µmv2
> m1 ). In particular, the minimum signal
allow the incumbent to be reelected is high ( m11−µ
is higher than mi2 (recall that on the equilibrium path
m1 −µmv2
1−µ
> m1 > mv2 = mi2 ). If voters
believe the incumbent is better, the minimum signal is lower and, therefore, is closer to mi2 and,
consequently, is more likely. A parallel argument applies when mv2 > m1 .
The previous lemmas imply that the marginal benefit of exerting effort is a hump-shaped
function of the incumbent’s reputation. Thus, the equilibrium effort strategy in (8) is a humpshaped function of the incumbent’s reputation. The following proposition states this result:
12
Proposition 2 The period-two equilibrium effort strategy α2 (m2 ) is continuous, increasing in
the incumbent’s reputation m2 if m2 < m1 , and decreasing in m2 if m2 > m1 .
As the intuition behind the previous lemmas suggests, the equilibrium effort would be humpshaped in reputation under more general assumptions. Equilibrium effort is hump-shaped in
the incumbent’s belief if better incumbents are less (more) likely to produce bad (good) signals
(Lemma 1). Equilibrium effort is hump-shaped in the voters’ belief if extreme signals are less
likely than average signals (Lemma 2).
3.2.3
Period one
Let a∗1 denote the period-one equilibrium effort level. Let M(m, s) ≡ µm + (1 − µ)s denote the
mean in the posterior belief when m is the mean in the prior belief and s is the signal used to
update the prior. The following proposition presents the incumbent’s period-one effort-smoothing
decision (Appendix D presents the proof).
Proposition 3 There exists a unique and positive period-one equilibrium effort level a∗1 that
satisfies
c
′
(a∗1 )
= δµ
Z
∞
c′ (α2 (M(m1 , s1 ))) φ (s1 ; m1 , H) ds1 .
(9)
−∞
Eq. (9) represents the typical intertemporal tradeoff in dynamic models: having less utility
today allows the incumbent to have more utility next period. In this case, a lower expected
marginal cost of exerting effort in period two compensates a higher marginal cost of effort in
period one. The incumbent knows that he can affect the reelection probability by exerting effort
in periods one and two. He could exert more effort in period one and less effort in period two
(or vice versa) and still have the same reelection probability. Eq. (9) shows that the optimal
effort-smoothing decision is such that the marginal cost of exerting effort in period one equals
the expected marginal cost of exerting effort in period two discounted by δ and µ.
13
3.2.4
The cycle
Recall that differences in the incumbent’s behavior during his term in office could be the result
of changes in his reputation and may not imply that he is deciding differently because the
election time is closer. We want to focus on differences in the incumbent’s behavior that are due
to the proximity of the next election. Therefore, we refer to differences in behavior across the
incumbent’s term for a given reputation level as political cycles. Proposition 4 shows that our
model generates such cycles (Appendix E presents the proof).
Proposition 4 For the same reputation level (m1 ), the period-two equilibrium effort level is
higher than the period-one equilibrium effort level.
For the same reputation level, why is the equilibrium effort level higher closer to the election?
There are three forces that may account for political cycles in our model.
The first force is discounting. As explained above, when exerting effort in period one, the
incumbent substitutes period-two effort. When deciding how much effort to exert in period one,
the incumbent discounts the cost of exerting effort in period two by δ. Thus, if δ is lower, incentives to exert effort in period one are relatively weaker and political cycles are more important.
Note that proposition 4 shows that our model generates cycles for all possible values of δ and,
in particular, it would generate cycles for δ = 1 (see Appendix E). That is, discounting is not
necessary for generating cycles in our environment.
The second force that generates cycles arises because the incumbent’s period-two performance
provides more information about his post-election (period-three) ability than his period-one
performance. This is the case because his ability evolves over time and, therefore, his periodthree ability is more correlated with his period-two ability than with his period-one ability. Recall
that the incumbent exerts effort because he wants to be reelected, and the reelection decision
depends on mv3 . Using (6), it is easy to show that
mv3 = µ2 m1 + (1 − µ)sv2 + µ(1 − µ)sv1 ,
(10)
which shows that the weight in mv3 of the signal voters compute in period one is lower than
the weight of the signal they compute in period two. This implies that period-one effort—which
14
affects sv1 —is less effective in increasing the reelection probability than period-two effort. The
relative effectiveness of period-one effort is represented in (9) by µ, which indicates the relative
weight of sv1 (compared with sv2 ) in mv3 —see (10). Proposition 4 shows that our model generates
cycles for all possible values of µ. In particular, it generates cycles if the effectiveness of periodone effort is arbitrarily close to the effectiveness of period-two effort (µ is arbitrary close to 1).
That is, the lower effectiveness of period-one effort is not crucial for generating cycles in our
environment.
Why would political cycles arise in an economy in which there is no discounting and manipulating policy is equally effective in every period? Our model indicates that cycles could still arise
in such an economy because of the incumbent’s effort-smoothing decision. Cycles arise because
at the beginning of his term, the incumbent knows that his reputation is likely to change, and
he anticipates that this change will lead him to choose an effort level lower than the one he
would choose at the end of his term for his beginning-of-term reputation level. In period one,
the incumbent anticipates (using his next-period optimal strategy) that if his reputation does
not change, he will choose α2 (m1 ) in period two. He also anticipates that, for example, if his
period-one performance turns out to be either very good or very bad (and, therefore, his periodtwo reputation is either very good or very bad), he will exert a lower effort level in period two. In
particular, the expected period-two effort level is lower than α2 (m1 ), and the expected marginal
cost of exerting effort in period two is lower than c′ (α2 (m1 )). Therefore, the effort-smoothing
rule in (9) implies that c′ (a∗1 ) < c′ (α2 (m1 )), and the incumbent chooses a∗1 < α2 (m1 ).
3.2.5
Comparison with previous theories of political cycles
Previous studies of political cycles succeeded in showing that cycles can arise with forward-looking
and rational voters. These studies gained expositional simplicity by making three assumptions
that imply that the incumbent can only affect his reelection probability with his last action before
the election (see, for example, Rogoff [29], and Shi and Svensson [31]). The first assumption is
that at the time of the election, only the end-of-term ability is not observable. If beginning-ofterm ability is observable, the incumbent cannot influence the voters’ belief with his beginning-
15
of-term actions and, therefore, cycles arise.
The second assumption is that only end-of-term ability is correlated with post-election ability. Consequently, only the voters’ inference about end-of-term ability directly influences their
reelection decision.
The third assumption is that output is a perfect signal of ability. This implies that voters
can learn the incumbent’s end-of-term ability (which is correlated with his post-election ability)
perfectly from his end-of-term performance, without considering his performance in previous
periods. Therefore, beginning-of-term actions are not effective in changing the reelection probability. The implications of this assumption can be analyzed by looking at the limit of our results
when hε goes to infinity. As (4) shows, this implies that µ goes to zero. Proposition 3 shows
that this in turn implies that the period-one equilibrium effort level is zero. Recall that, in (9),
the relative effectiveness of period-one effort is represented by µ.
The three assumptions described above imply strong asymmetries across periods. Political
cycles in previous studies are a direct result of these asymmetries. Our paper explains how cycles
can arise without these asymmetries.
3.2.6
Comparative statics
Comparative-statics exercises have been used to identify under which circumstances political
cycles would be of higher magnitude. For example, in a model in which only the end-of-term
action can affect the reelection probability, Shi and Svensson [31] show that if the per-period office
value is higher, political budget cycles are amplified. They find empirical evidence that supports
this prediction. The intuition behind this result is simple. A higher office value implies that
incentives to increase reelection probabilities are stronger. In their model, since the incumbent
can increase his reelection probability only with his end-of-term action, an increase in the office
value enhances the importance of cycles.
In contrast, in our model, a higher office value (R) implies a higher effort level in every period.
Eq. (8) shows that a higher R implies a higher period-two effort level for any reputation. Eq.
(9) shows that if the incumbent expects a higher period-two effort level, he chooses a higher
16
period-one effort level. In particular, the next proposition shows that if the marginal cost of
effort is a homogeneous function, the office value only has a scale effect on political cycles: The
difference between the equilibrium effort levels in periods one and two as a percentage of the
period-one effort level is independent of R (Appendix F presents the proof).6
Proposition 5 Assume that the marginal cost of effort is a homogeneous function of order j.
Then, for any period-two reputation m,
3.3
α2 (m)−a∗1
a∗1
does not depend on R.
Multiple elections
In this section, we analyze the multiple-election version of the model, which allows us to study
situations that do not arise in the one-election version: with multiple elections, the beginningof-term reputation may be better than the average reputation, and the end-of-term effort is
not maximized at the beginning-of-term reputation. Recall that in the one-election version of
the model, at the beginning of the term, there is a new incumbent with an average reputation.
Moreover, the proof of proposition 4 (which shows that a political cycle arises in the one-election
version of the model) is based on the end-of-term equilibrium effort strategy being such that it is
optimal to exert the maximum effort level for the beginning-of-term reputation (see Appendix E).
We show that the insight described in the one-election version of the model helps us understand
political cycles with multiple elections: for the same reputation, end-of-term effort is higher if,
at the beginning of the term, the incumbent anticipates that changes in his reputation will on
average lead him to choose an end-of-term effort level lower than the one he would choose for his
beginning-of-term reputation.
3.3.1
Definition of equilibrium
If t is a period without elections, for any incumbent’s belief mi and voters’ belief mv , the expected
lifetime utility of the incumbent (who will face an election at t + 1) is given by
6
Note that our result is not inconsistent with the empirical findings in Shi and Svensson [31]. Our result
does not refer to differences between the effort levels in periods one and two but rather to these differences as a
percentage of the period-one effort level.
17
Z
Wt (mi , mv ) = max R − c(a) + δ
a≧0
∞
−∞
Wt+1 (M (mi , s), M (mv , s + a − αt (mv ))) ιt+1 (M (mv , s + a − αt (mv ))) φ (s; mi , H) ds ,
(11)
where
Z
Wt+1 (mi , mv ) = max R − c(a) + δ
a≧0
∞
Wt+2 (M(mi , s), M(mv , s + a − αt+1 (mv ))) φ (s; mi , H) ds .
−∞
(12)
The voters’ expected lifetime utility at the beginning of period t is denoted by
Z ∞
Vt (mv ) ≡ αt (mv ) +
[s + δVt+1 (M (mv , s))] φ (s; mv , H) ds,
(13)
−∞
where their expected lifetime utility at the beginning of election-period t + 1 satisfies
h
i
R∞
I αt+1 (mv ) + −∞ [s + δVt+2 (M (mv , s))] φ (s; mv , H) ds
h
i
Vt+1 (mv ) = max
.
R∞
I∈{0,1}
+ (1 − I) αt+1 (m1 ) + −∞ [s + δVt+2 (M (m1 , s))] φ (s; m1 , H) ds
(14)
Definition 1 An equilibrium consists of the functions Wt (mi , mv ) and Vt (mv ), an effort strategy
α̂t (mi , mv ), and a voting strategy ιt (mv ), such that if t is a period without elections, Wt (mi , mv )
satisfies (11) and Wt+1 (mi , mv ) satisfies (12); Vt (mv ) satisfies (13) and Vt+1 (mv ) satisfies (14);
and α̂t (mi , mv ) solves (11), α̂t+1 (mi , mv ) solves (12), and ιt+1 (mv ) solves (14).
3.3.2
Beginning-of-term effort strategy
First, we will discuss incentives to exert effort at the beginning of a term. These incentives are
represented in the following Euler equation (the derivation of this equation follows the one for
the single-election case presented in Appendix D):
Z ∞
′
c (αt (mt )) = δ
[µ − (1 − µ) α′t+1 (M(mt , st ))]c′ (αt+1 (M(mt , st ))) φ (st ; mt , H) dst ,
(15)
−∞
where t is a beginning-of-term period.7
7
As in the one-election version of the model, assuming enough convexity in the cost function can guarantee
the global concavity of the incumbent’s problem. Moreover, for our numerical example, the concavity of the
incumbent’s problems is easily checked numerically.
18
As in the one-election version of the model, when deciding how much effort to exert at the
beginning of a term, the incumbent considers the marginal cost and the relative effectiveness of
exerting effort across his term. Eq. (9), which describes incentives to exert effort in period-one
in the single-election version of the model, is a special case of (15).
The main difference between (9) and (15) is in the relative effectiveness of beginning-of-term
effort. The incumbent’s effort in period t affects mvt+1 directly, and affects mvt+2 through mvt+1 .
His period-t + 1 effort affects mvt+2 directly. Thus, the relative effectiveness of period-t effort
is given by the derivative of mvt+2 = M (mvt+1 , s + at+1 − αt+1 (mvt+1 )) with respect to mvt+1 ,
which is equal to µ−(1 − µ) α′t+1 (mvt+1 ). The second term in the relative effectiveness represents
the impact of a higher mvt+1 on the signal computed by voters in t + 1. This effect is weighted
by 1 − µ, the weight of the signal in the posterior belief. As explained in Appendix D, because
of the symmetry of the period-two effort strategy around m1 , the expected effect of a higher m2
on the signal computed by voters in period two is zero.
Note that the relative effectiveness of beginning-of-term effort could be higher than one. That
is, beginning-of-term effort could be more effective in changing the reelection probability than
end-of-term effort. This is the case because even though we assume that the incumbent’s more
recent performance provides more information about the quality of his future performance (see
Section 3.2.4), beginning-of-term effort affects both the signal computed by voters at the beginning and the end of the term. This is not the case in previous studies: If one assumes that only
the incumbent’s end-of-term performance provides information about his future performance,
beginning-of-term actions are not effective (see Section 3.2.5). Thus, the force driving cycles in
previous studies (i.e., the lower effectiveness of beginning-of-term actions) may not survive when
one assumes weaker asymmetries across periods.
3.3.3
End-of-term effort strategy
The characterization of the end-of-term equilibrium effort strategy depends directly on the voters’
reelection strategy. For expositional simplicity, we present the end-of-term equilibrium effort
19
strategy only for the case in which
1 if m > m
vt
1
ιt (mvt ) =
0, otherwise.
(16)
That is, we restrict attention to equilibria in which the optimal reelection rule is to reelect the
incumbent if and only if his expected ability is higher than the one of a policymaker who was
not in office before.8 In this case, end-of-term incentives are represented in the following Euler
equation (the derivation of this equation follows closely the one for the single-election Euler
equation presented in Appendix D):
c′ (αt (mt )) = δWt+1 (m1 , m1 ) φ
Z ∞
m1 − µmt
; mt , H +δ m −µm µ − (1 − µ) α′t+1 (M (mt , st )) c′ (αt+1 (M (mt , st ))) φ (st ; mt , H) dst ,
t
1
1−µ
1−µ
(17)
where t is an end-of-term period.
The first term in the right-hand side of (17) represents the gain from increasing the nextperiod-election reelection probability. This gain is similar to the one the incumbent considers at
the end of his term (period two) in the one-election version of the model. The difference is that
with multiple elections, the value of winning the next election is endogenous (Wt+1 (m1 , m1 ))
while in the one-election version of the model, the value of winning the election is equal to R
(which is equal to WT (m1 , m1 )).
The second term in the right-hand side of (17) represents the gain from increasing the reelection probability in future elections. This gain is similar to the beginning-of-term gain. In order
to increase future reelection probabilities, the incumbent can exert effort now or in the future.
Therefore, he evaluates the marginal cost and the relative effectiveness of exerting effort in each
8
If for reputations better than average the incumbent’s equilibrium effort is decreasing with respect to his
reputation (as it is the case in equilibrium), voters could be worse off with an incumbent with higher expected
ability (who chooses to exert lower effort). Therefore, voters could choose not to reelect an incumbent with a
reputation better than average. For instance, consider an augmented one-election version of the model in which
in period one, voters can choose whether to appoint a new policymaker with an average reputation m = 0 or
to reelect an initial incumbent with a better reputation m = 0.6. Suppose that c(a) = a5 , δ = 0.9, R = 200,
m1 = 0, and hη = hε = 7.5. It is easy to show that in this example, voters would not reelect the incumbent.
One can rule out cases as the one presented in the previous example by limiting the responsiveness of effort to
changes in reputation. For example, one can assume low-enough values of R and/or of the precision in the signal
distributions H (see the example presented in Section 3.3.4).
20
period. Note that at the end of a term, the incumbent knows that he may lose the next election
and, therefore, he may not enjoy the benefits of increasing the reelection probability in future
elections. In (17), this is represented by the lower bound in the integral.
3.3.4
The cycles
To illustrate how cycles arise in the multiple-election version of the model, we will focus on
the limit of the finite-horizon solutions when voters and the incumbent are far enough from the
termination of the game. The limit case has particular interest because of its stationarity: the
incumbent’s incentives (the number of future elections, the value of winning these elections, and
his future actions) do not depend on time.
Given the complexity of the problem we study, a numerical approach is necessary. The
expected lifetime utility of an incumbent at the beginning of a term evaluated in equilibrium
(when the incumbent’s and the voters’ beliefs coincide), and Euler equations (15) and (17)
constitute a system of three functional equations with three unknowns. One can easily obtain a
numerical solution.9 We discuss the following example: c(a) = a5 , δ = 0.9, R = 20, m1 = 0, and
hη = hε = 0.75.
Recall that for deriving the incumbent’s effort strategy in (17), we assumed that voters follow
the reelection strategy in (16). We need to confirm that the strategies in (16) and (17) are in fact
equilibrium strategies. That is, we need to check that when we assume that the effort strategy is
given by (17), voters’ optimal reelection strategy is given by (16). This is the case in our example
and, therefore, (16) and (17) give us the equilibrium strategies.
Figure 1 presents the equilibrium effort strategies and the expected end-of-term effort level as
a function of the beginning-of-term reputation. The figure only presents beginning-of-term and
expected end-of-term effort levels for better-than-average beginning-of-term reputations, because
9
Given the next-period equilibrium strategy, it is easy to obtain the current-period equilibrium strategy using
(15) and (17). Therefore, it is easy to find the limit of the T -period solutions when voters and the incumbent
are far enough from the termination of the game by computing first the period-T − 1 solution using (8), then
computing the period-T − 2 solution using (15), then computing the period-T − 3 solution using (17), and so
on. Moreover, recall that (8) gives us a unique period-T − 1 equilibrium strategy and, therefore, (15) gives us a
unique period-T − 2 equilibrium strategy, (17) gives us a unique period-T − 3 equilibrium strategy, and so on.
This procedure converges monotonically to the solution presented in the paper.
21
in equilibrium, at the beginning of a term, only policymakers with reputations that are better
than average may be in office.
Figure 1: Equilibrium effort levels.
Figure 1 illustrates how the numerical approach used for computing the equilibrium effort
strategies produces results that are consistent with the closed-form solutions in the one-election
version of the model: The equilibrium effort strategies are hump-shaped functions of reputation,
and effort levels tend to be higher at the end of a term. Figure 1 also shows that in our example,
the expected end-of-term effort level tends to be higher than the beginning-of-term effort level.
Note also that beginning-of-term effort is not maximized at m1 .
For reputations close to the average, why is effort higher at the end of a term? For any
beginning-of-term period t and for any reputation m, let us compare the equilibrium marginal
cost of exerting effort at the beginning and at the end of a term, c′ (αt (m)), and c′ (αt+1 (m)).
Eq. (15) shows that c′ (αt (m)) is equal to the expected marginal cost of exerting effort at
the end of the term (weighted by the relative effectiveness and discounted by δ). In order to
understand the way in which this expected marginal cost (and, therefore, c′ (αt (m))) compares
with c′ (αt+1 (m)) (the marginal cost of αt+1 (M(m, st )) evaluated at the expected M(m, st )),
the shape of the equilibrium effort strategy must be considered. If c′ (αt+1 (M(m, st ))) were a
concave (convex) function, Jensen’s inequality would imply a positive (inverted) political cycle,
i.e., it would predict that the incumbent’s effort level is higher (lower) at the end of the term.
However, as Figure 1 illustrates, the end-of-term equilibrium strategy is neither globally concave
22
nor globally convex. But for reputations close to the average, effort is higher closer to the election
because the equilibrium effort strategy is hump-shaped in reputation (and concave). In contrast,
for better reputations, the end-of-term equilibrium effort strategy is less concave (and convex).
Therefore, political cycles are less important.
Why are the equilibrium effort strategies in Figure 1 hump-shaped in reputation? First,
note that in general, the gain from increasing the reelection probability in future elections—
represented in the right-hand side of (15) and in the second term of the right-hand side of (17)—
preserves the shape of the next-period effort strategy (i.e., this gain is higher for reputations for
which next-period effort is higher). Recall that this gain is given by the expected marginal cost of
exerting effort next period (discounted and weighted by the relative effectiveness). Suppose the
incumbent’s current-period reputation is represented by m. Then, he knows that his next-period
reputation is likely to be close to m. If, for reputations close to m, next-period equilibrium effort
is high (low), then the expected marginal cost of exerting effort next-period is high (low). That
is, typically, the gain from increasing the reelection probability in future elections is higher for
reputations for which next-period effort is higher. Proposition 2 shows that the period-T − 1
equilibrium effort strategy is a hump-shaped function of reputation. The previous discussion implies that in general, the period-T − 2 equilibrium effort strategy is hump-shaped. Consequently,
in general, the period-T − 3 equilibrium effort strategy is hump-shaped and so on.10
It is also worth mentioning that the incumbency advantage generated by the model because
of selection may be significant. Table 1 presents the average reelection probability (computed
using simulations) as a function of the number of terms the incumbent has been in office, for the
example discussed in this section.
4
Conclusions and extensions
The paper shows that a standard political-agency model generates political cycles without assuming strong asymmetries across periods (other than the proximity of the next election). Because
10
This general intuition is complicated by the relationship between the incumbent’s reputation and the expected
relative effectiveness. For a more extensive discussion of the relationship between the relative effectiveness and
the strength of incentives to exert effort, see Martinez [20].
23
Number of previous terms in office
Reelection probability
1
50%
2
3
75% 83%
4
5
88% 90%
6
7
92% 93%
8
9
94% 94% 95%
Table 1: Average reelection probability as a function of the number of terms the incumbent has been
in office.
the incumbent’s equilibrium effort strategy is a hump-shaped function of his reputation, his
effort-smoothing decision implies that he chooses a higher effort level closer to the election. We
also find that political cycles are typically less important when the incumbent’s reputation is
very good. Furthermore, we show that when policymakers can affect reelection probabilities
with their decisions in every period, results from comparative-statics exercises are different from
the findings in previous work.
More generally, our analysis helps us understand agency relationships in which an important
part of the compensation is decided upon infrequently. With the exception of previous studies on
political cycles (which assume that the agent would only exert a costly action in the last period
before his compensation is decided), previous studies of career concerns assume that the agent’s
compensation is decided after every output observation.11 Since considering incentives from
career concerns is necessary for designing optimal contracts that complement these incentives
(see Gibbons and Murphy [13]), analyzing the way in which career-concern incentives depend
on the proximity of the renegotiation decision could be important for understanding the way in
which contracts should depend on the proximity of this decision.12
Analyzing the way in which our framework could help us account for differences in the frequency of elections (or the length of contracts) could be an interesting extension of this paper.
Moreover, our dynamic model may help us account for changes in the frequency of elections over
time. Additional natural extensions are analyzing cases with asymmetries in the learning pro11
See, for example, Dewatripont, Jewitt, and Tirole [8, 9], Gibbons and Murphy [13], Holmstrom [17], and
Prendergast and Stole [27].
12
In a model that does not include learning about ability and assumes the principal uses long-term contracts for
providing incentives to the incumbent, Spear and Wang [32] present an alternative reason for which the principal
may want to replace the incumbent: it may be more costly to induce the incumbent to exert effort than to induce
a new agent to exert effort. If the career-concern incentives we discussed were complemented with incentives
contracts, the firing motives considered by Spear and Wang [32] could appear.
24
10
cesses, term limits and/or retirement for the policymakers, and a finite number of policymakers
(political parties) participating in elections. Situations in which the incumbent’s action affects
the voters’ capacity to learn could also be studied.
A
Proof of proposition 1
Let a∗2 (α2 (m2 )) denote the optimal period-two effort level when voters compute the signal using
α2 (m2 ). This optimal effort level satisfies
m1 − µm2
′ ∗
∗
c (a2 (α2 (m2 ))) = δRφ
+ α2 (m2 ) − a2 (α2 (m2 )) .
1−µ
(18)
Plugging α2 (m2 ) = a∗2 (α2 (m2 )) into (18), we obtain (8), which for each reputation m2 , gives
us the unique period-two equilibrium effort level α2 (m2 )—the right-hand side of (8) does not
depend on the effort level and is positive.
B
Proof of lemma 1
Recall that φ (s; m, H) decreases with respect to m if and only if s < m. On the equilibrium
−µmv2
−µm1
path, mi2 = mv2 . Consequently, if mi2 = mv2 > m1 , then m1 1−µ
< m11−µ
= m1 < mi2 .
−µmv2
Therefore, φ m1 1−µ
; mi2 , H is decreasing with respect to mi2 . If mi2 = mv2 < m1 , then
m1 −µm1
m1 −µmv2
m1 −µmv2
> 1−µ = m1 > mi2 . Therefore, φ
; mi2 , H is increasing with respect to
1−µ
1−µ
mi2 .
C
Proof of lemma 2
Recall that φ (s; m, H) increases with respect to s if and only if s < m. On the equilibrium path,
−µmv2
−µmv2
mi2 = mv2 . If mi2 = mv2 > m1 , then m1 1−µ
< m1 < mi2 , and φ m1 1−µ
; mi2 , H is decreasing
−µmv2
−µmv2
with respect to mv2 . If mi2 = mv2 < m1 , then m1 1−µ
> m1 > mi2 , and φ m1 1−µ
; mi2 , H is
increasing with respect to mv2 .
25
D
Proof of proposition 3
The period-one incumbent’s maximization problem is given by
Z
∗
max δ W2 (M(m1 , s1 ), M(m1 , s1 − a1 + a1 )) φ (s1 ; m1 , H) ds1 − c (a1 )
a1 ≥0
where
W2 (mi2 , mv2 ) =
m1 − µmv2
R − c (α̂2 (mi2 , mv2 )) + δR 1 − Φ
+ α2 (mv2 ) − α̂2 (mi2 , mv2 ); mi2 , H
1−µ
denotes the incumbent’s expected utility at the beginning of period two when his belief and
the voters’ belief are characterized by mi2 and mv2 , respectively. Let W2,2 (mi2 , mv2 ) denote the
derivative of W2 (mi2 , mv2 ) with respect to the mean voters’ belief. Using the envelope theorem
we obtain that
µ
m1 − µmv2
′
W2,2 (mi2 , mv2 ) = δR
− α2 (mv2 ) φ
+ α2 (mv2 ) − α̂2 (mi2 , mv2 ); mi2 , H ,
1−µ
1−µ
(19)
where α′t (m) denotes the derivative of the incumbent period-t equilibrium effort strategy αt (m)
with respect to his reputation. When evaluated at mi2 = mv2 = m2 ,
µ
m1 − µm2
µ
′
′
− α2 (m2 ) φ
; m2 , H =
− α2 (m2 ) c′ (α2 (m2 )) .
W2,2 (m2 , m2 ) = δR
1−µ
1−µ
1−µ
(20)
Plugging (20) evaluated at M(m1 , s1 ) into the first-order condition of the incumbent’s period-one
problem evaluated at a1 = a∗1 gives
c
′
(a∗1 )
=δ
Z
∞
−∞
[µ − (1 − µ) α′2 (M(m1 , s1 ))]c′ (α2 (M(m1 , s1 ))) φ (s1 ; m1 , H) ds1 .
It remains to show that
Z ∞
(1 − µ) α′2 (M(m1 , s1 )) c′ (α2 (M(m1 , s1 ))) φ (s1 ; m1 , H) ds1 = 0.
(21)
−∞
Recall that φ (s1 ; m1 , H) is symmetric with respect to s1 with maximum at s1 = m1 . Therefore,
(8) shows that c′ (α2 (M(m1 , s1 ))) is a symmetric function with maximum at s1 = m1 . Moreover, α′2 (M(m1 , m1 )) = 0, and, for any A ∈ ℜ, α′2 (M(m1 , m1 + A)) = −α′2 (M(m1 , m1 − A)).
Consequently, (21) holds.
26
Since the right-hand side of (9) is positive, a∗1 is positive. Since there is a unique period-two
equilibrium strategy α2 (m2 ) defined by (8), there is a unique period-one equilibrium effort level
a∗1 that can easily be obtained from (9) (the right-hand side of this equation does not depend on
the period-one effort level).
E
Proof of proposition 4
Since δµ < 1,
c
′
(a∗1 )
<
Z
∞
c′ (α2 (M(m1 , s1 ))) φ (s1 ; m1 , H) ds1 .
−∞
Since α2 (m1 ) > α2 (m) for all m 6= m1 (see proposition 2), c′ (α2 (m1 )) > c′ (α2 (m)) for all
m 6= m1 . Therefore,
′
c (α2 (m1 )) >
Z
∞
c′ (α2 (M(m1 , s1 ))) φ (s1 ; m1 , H) ds1 .
−∞
Consequently, c′ (α2 (m1 )) > c′ (a∗1 ), and α2 (m1 ) > a∗1 (by c′′ > 0).
F
Proof of proposition 5
Consider any office values R0 and R1 = λR0 , with λ ∈ ℜ. Let αt (m; R) denote the equilibrium
period-t effort level when m represents the beliefs and the office value is R. Eq. (8) implies that
for any m,
c′ (α2 (m; R1 )) = λc′ (α2 (m; R0 )) .
(22)
1
Since c′ is homogenous of order j, λc′ (α2 (m; R0 )) = c′ λ j α2 (m; R0 ) . Therefore, α2 (m; R1 ) =
1
λ j α2 (m; R0 ).
Equations (9) and (22) imply that c′ (α1 (m1 ; R1 )) = λc′ (α1 (m1 ; R0 )) and, therefore, α1 (m1 ; R1 ) =
1
λ j α1 (m1 ; R0 ). Thus,
α2 (m; R0 ) − α1 (m1 ; R0 )
α2 (m; R1 ) − α1 (m1 ; R1 )
=
.
α1 (m1 ; R0 )
α1 (m1 ; R1 )
That is,
α2 (m)−a∗1
a∗1
does not depend on R.
27
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