17

the case where W'(0) > 0, so the solution to (18) intersects the a axis from
above instead of asymptoting to it. To understand the figure, it is helpful to
consider two closely neighboring realizations of the competency shock, a, and
Gyr Gy > ay. Suppose both a types are thought to cheat by the same amount,
a*(a,) = a*(a,). Then when the incumbent party draws ay it would have to set
a= a*(a,) + a - a, in order to convince the public it had drawn a. In
deciding whether to take this action, the incumbent would compare the increased
expectation of winning with the marginal social welfare cost of distortions. If
U' is high, and if & and hence W' is low, then the temptation to cheat is great.
To discourage an a, type incumbent from defecting, it is necessary to force it
to cheat more to gain any given increase in votes. This implies aS(a,) > aS(a,)
Good types must "run away" from bad types. As A rises, W' rises and the temp-
tation to cheat falls. Since U' begins to fall at a, A° must eventually begin
to fall. The dotted line in figure 1 is a graph of as for the case where type

ax

zero must use seignorage since g - ca > 0. In this case, cheating may be
9 ig may

strictly decreasing ina. It is simple to prove that aS(a) iS nondecreasing
ing - max 15

We have shown that a unique separating equilibrium exists. '6 We now discuss
pooling equilibria. First, note that a pooling equilibrium would have to
involve a range of a types setting the same level of t (taxes), not the
same level of A (seignorage). Recall that voters observe tT = g - a - A(a),
not A(a@). Second, there cannot be a pooling equilibrium where voters ignore
taxes completely, even though politicians would then have no incentive to
cheat. In this case, the incumbent party's tax bill would fully reveal its
type, and it would not be rational for voters to ignore this information. So

a pooling equilibrium must involve having all a on some interval [a,,a5] set

the same level of taxes. But this cannot be an equilibrium if voters' mapping


38

Swhen R is the opposition party, then uP e1- uv. The winner receives all the

votes only because we have not allowed for different voters to have different
values of q (party-specific preferences related to exogenous non-economic
factors). Our later analysis requires only that a party be more likely to win
the more competent it is perceived to be.

10the analysis would have to be modified slightly if parties also cared directly
about the competency of the government. The main qualitative effect would be
that when the incumbent knows it has a high level of competency relative to the
mean value of the opposition's competency shock, then it will place a greater
weight on being elected. If on the other hand, the incumbent knows its com-
petency to be very low, then there is the (perverse) possibility that it might
prefer to see the opposition win.

"the main effect of relaxing the assumption that both parties place the same
weight (x) on social welfare would be that expected inflation in t+2 is no longer
the same for both parties. See proposition 4 in section III below. Because we
allow for fairly general U functions, the analysis can be generalized to where
parties care about their plurality, and not just their Probability of winning.

1 * : : 5 «a5
2e discuss the possible existence of pooling equilibria later on.

1 soe . : : . .

tn the finite horizon case, (12) is the only possible form for rationa expec-
tations. Expectations conditioned on past behavior "unravel backwards". Other
equilibria are possible in the infinite horizon case, however; see section IV
below.

416 g - 1% = a*(0), then the type zero agent will not (cannot) defect by
setting A lower. (He is so incompetent that he cannot make ends meet without
relying on seignorage.) It is straightforward to show that no type a > 0 will
defect if a* = 4° follows (18).

157, is also straightforward to examine the case where there is some maximum
level of cheating, amex a may have a maximum if, for example, there

is a limit_to how much seignorage can be extracted from money holders.
Suppose A exists, and the solution to (18) reaches ATMTM at some a

Then there is an equilibrium where A pgmains at aTMTM* for a>a untif the
lowest a such that xU'(a) < (1-x)W'(A ). Denote this a asa’. Fora2ra
the equilibrium path is again governed by (18), and A begins tS decline. TKe
proof is analogous to the proof of proposition 3.

8the analysis of this section bears a resemblance to the information theory
models surveyed in Riley (1979). The question arises as to why our model is not
Plagued by the existence problems familiar from that literature. "Unraveling in
the middle" does not occur here because the voters face a single large agent.
The issue of having pooling equilibria which bid away subcoalitions of types
(because everyone in the coalition gets a better vote/social welfare package
than in the separating equilibrium) does not arise.


34

Appendix B

Here we provide an example for which eq.(18) has a closed-form solution.
The example has U" < 0, which requires that q(t) be asymmetrically distributed
and/or that the incumbent's current popularity v be high. (We can have q(t)
being asymmetrically distributed and yet retain eq.(6) if q is viewed as an

incumbent-specific shock rather than a party-specific shock.)

Example. UE (alt)] = Infv + EP(alt)], and W(A) = InB - 1n(B - |A|). Note
that W" > 0, so the example satisfies the second-order conditions derived in the

proof of proposition 1. For 4 > 0, substituting into (25) gives

oy — XA
(81) are (1-x) (v+a) ~ (1-x) (v+a) 7

a linear first-order differential equation with solution
(B2) A(a@) = B - (1-x)(v+a) + C(vta) */(1-X) |

First we observe that if the arbitrary constant C = 0, then cheating is

linear ina. This may seem like a natural equilibrium, but A(0) = 0 requires

Cs (tex) p17) poX/(1-X) ore ig easy to confirm that A = 0 Va is an

equilibrium if i > x, If x > ix , then

B v v Aso * 0 and there is a range

2a
aa
of @ over which the candidate cheats. It is also Straightforward to confirm

proposition 4 with this example.


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1

Another variant of our model involves interpreting G + € as the effective
level of government services. The public only observes expenditures, G. This
seems quite realistic if one views G + € as national defense, given the secrecy
of military documents. (The country would not necessarily want to make these
documents public just to mitigate the political business cycle.) During an
election year, the incumbent might have an incentive to shave G and Tr, claiming
that military preparedness is adequate because the funds are being used
efficiently (high €). More generally, our model will suggest why the incumbent
is likely to cut back on expenditures which have low short-run visibility, and

focus on expenditures (and tax reductions) which have high immediate visibility.


21

However, proposition 5 does not allow us to unambiguously state whether expected
t+2 inflation is higher or lower for the high € party. Also, we have only ana-
lyzed the case where the "R" party is the incumbent, but obviously the analysis
is the same when the "0" Party is the incumbent. The fact that the parties tend
to move in and out of power (depending on competency shocks and voter preference
shifts) does not complicate our analysis because we assume that the public
directly observes a, in period t+1. If they were only able to observe
ea, + a4" then there would be a "start-up" problem whenever a new party
moves into office. (The public doesn't know @,_, and therefore can't sort out
G1 and a,.) This case is much more complicated.

It should be emphasized that elections are not necessarily a bad thing,
just because they result in excessive inflation or a suboptimal distribution
of tax distortions over time. By holding elections, the public gets a more
competent government, on average. It is possible in Principle to explicitly
analyze the tradeoff in a model such as the present one. But the exercise would

not be interesting without allowing for a richer stochastic structure.


39

T5ee, for example, Frey and Schneider (1978). There are several other empiri-
cal studies which also attempt to relate Presidential popularity to the severity
of the political business cycle. Golden and Poterba (1980) find no evidence of
a relationship.

18that the infinite horizon case may yield outcomes with higher social welfare
js well-known. In the macroeconomics literature, see Barro and Gordon (1983),
or Canzoneri (1985), for example. The “election game" may end with some posi-
tive probability each period; this effectively raises the incumbent's discount
rate.

"9atesina (1985a,b), in the context of a Barro-Gordon (1983) /Kydland-Prescott
(1977) model, has shown how cooperation between two political parties can serve
to dampen post-election volatility in prices and employment. Wage setters
aren't sure which party is going to win, and hence cannot predict prices. The
two parties may implicitly "cooperate" and always choose the same post-election
monetary policy despite their different preferences vis-a-vis inflation and
unemployment. The winner of the election will follow the cooperative strategy
if the long-term gain in reduced volatility outweighs the short-run costs. The
"third" channel we refer to in the text is similar to Alesina's, except that our

analysis pertains to pre-election volatility.

20the trigger-strategy equilibria are analogous to those discussed in Friedman
(1971),

This is a fairly general result in infinitely repeated games. The class of
punishment strategies considered here involve a reversion to the finite horizon
equilibrium. It is possible that there exist more severe punishment strategies
in our model, analogous to those described by Abreu (1985).

22 canzoneri (1985) models private information in the framework of Barro and
Gordon (1983). His analysis is an application of the trigger strategy
equilibrium proposed by Green and Porter (1984). The mode? here is different in
part because there are a continuum of types (instead of one known type), each of
whom must be presented with the appropriate incentives.


In section II we present the basic model. Governments in this setup are
differentiated in part by their level of “competency". Although the analysis
could be extended to encompass other aspects of a government's performance,
the notion of competency that we use is as follows: The more competent that
a government is, the less revenue it needs to provide a given level of govern-
ment services. This particular measure of competency stresses the administra-
tive abilities of the policymaker. Naturally, other things being equal,
voters prefer more competent governments. The government obtains information
about its (serially correlated) competency more quickly than voters can.
However, at the beginning of each period voters receive a signal; they learn
the level at which the government is going to set income or poll taxes.
Because taxes are set at the beginning of the period, any (intentional or
unintentional) error the government makes must be made up through issuing
bonds or drawing seignorage. (In another variant of the model, government
spending can also be adjusted.) Thus the public opserves the government's
competency directly, but with a lag. During an election year, a vote-
conscious incumbent party has incentive to make its most recent “competency
shock" appear large. The incumbent party's incentive to "cheat" (rely on
excessive seignorage or bond financing) is tempered only by the fact that it
places some weight on social welfare.

There is only one really important difference between the incumbent party
and the opposition party in our model. The opposition party has no credible
way to "signal" the effectiveness of its policies. Obviously, there are many
factors other than “competency” which influence the outcome of an election, but
our model treats these as exogenous. Nevertheless, the framework developed here

is sufficient to capture the essential elements of a political business cycle.



32

Lemma If a solution to (A.2) exists, then there exists a 4, <a such that

4.(@) <4, Wa eA, ife <A).

Proof. Suppose aia) > 0. Then

Wala] < Fy via) <@,

W! + as A ~ A; hence a, (a) must have an upper bound which is strictly less

than A.

f is a continuous function on the domain D # R x }-4,aC and (0,c) is an
interior point of D. Thus by theorem 10.1 in Ross (1965), (A.2) has a
solution ala), on [-h, h], where h = (A - A) /max| ¢ | > 0. Likewise, (A.2)
with initial condition A(h) = Beh), has a solution on [0,2h]. Proceeding
in this manner we can show (A.2) has a solution on R. Then clearly (A.1)
has a solution on A, denoted by 4, (a), force 10,4]. Denote the right-
hand side of (A.1) by g(a,4). By construction of W and the assumption that
W"/W' is uniformly bounded, ag(a,d)/ad < M for some M < ~, Wae A,

VA € [c, aL. Thus, by the mean-value theorem, g satisfies a Lipschitz
condition on A. By theorem 10.1 in Ross (1965), the solution 4, (a) is unique

It remains to show that dig 4,(@) exists and satisfies (A.1) for c = 0.

If c decreases to zero, (A(a)) is non-increasing and bounded from below by
zero. Thus g(a) = din 4, (a) exists. Ayla) is clearly unique and satisfies
(A.1). Wa' > 0, {a} is equicontinuous on a@ 2 a'. Hence, 4o is continuous on
a>o. g(a) is strictly increasing in a (for small a) and bounded from below

by zero. Thus, lim 49 = 0 and 49 is continuous on A. The first and second
an


18

from taxes to competency is monotonic. Suppose all types a € [a,,a5] set

t= T*. Then ePtalr 2 1) = a°, where a? is the mean value of a on [a,,a5].
But faced with these expectations, type a, has an incentive to set taxes
slightly lower. He thereby gains a discrete increase in votes, a, - a, at the
cost of an infinitesimal rise in seignorage distortions. The only pooling
equilibria that might exist in our model would have to involve perverse (and
nonmonotonic) expectations. For example, the following type of pooling
equilibrium might exist: Voters believe a = a if t = T, and believe a = 0
otherwise. It would seem very reasonable to rule out such equilibria.

We now turn to establishing some comparative statics properties of the
model. Define A(a,x) as an equilibrium path for a given value of x, the
weight the two parties place on votes. (For the remainder of this section,

we omit the "s" superscript.)
Proposition 4. aA(a,x)/ax 2 0 Va, with strict inequality if A > 0 anda > 0.

Proof. If a > 0, then xU" (a, ) = (1-x)W'(0), and u"(ai) > 0. Thus da, /dx <0,

with strict inequality if a> 0. Consider first the case ae 0. Then by

(18), for a ¢ ay,

(23) 3a! (asx) U' (a) _ XU (a)W"[A(a,x)] . 9A(a,x)
ax

“- x) W' [A(a,x)] (1-x)W' [A(a,x)]

Evaluating (23) at a = 0 we get
ae {oun} u'(0)
* ex) W' (0)

3A(0,x
ax

in a; hence there exists a deleted RHS-neighborhood N of zero where

because A(0,x) = 0 Vx implies = 0. The function 34/ax is continuous

a4(a,x)/3x > 0, Wa éeN. Now suppose there exists an a > 0 such that


24

expected inflation rate (level of cheating) during the punishment period:

a
(26) P = e2( f WlaS(a)]dF(a) - W(0)),
0

where dF(a) is the continuous probability density of a. The expected cost P
will be the same for any 4, > 0

In deciding whether or not to defect, the incumbent party has to compare
its expected utility from defecting with its expected utility from
cooperating. If in period t voters believe the incumbent party will not cheat,

then by (13), T = 9, - a,, and

t

(27) (alr) =g-Te+a+a,
where time subscripts are omitted. If the party does not cheat, it attains
(28) LS(a) = xU(a) + (1-x)W(0),

where Le stands for the party's utility level if it cooperates. If, on the
other hand, the party defects, it will attain
(29) (a) = max (xU(atd) - (1-x)W(A)) - PL

4

Proposition 6. 1f L°(a) - L%a) 30 va, then a’ =0 vais a

at trigger strategy equilibrium.

The particular trigger strategy described in (a)-(d) involves only a one-
election punishment period. If P is insufficient to prevent all a types from
cheating, then it still may be possible to attain the zero-cheating equilibrium

by extending the (threatened) punishment interval. In general, the threatened


20
Define A(a,v) as an equilibrium A function for a given value of v.

Proposition 5 aA(a,v)/av 2 0 foratv< @, with strict inequality if

a> .
fi

Proof. The proof is analogous to the proof of Proposition 4.

Proposition 5 contradicts the conventional notion that more popular

17 16 the

incumbents are less likely to engage in a political business cycle.
incumbent party draws a low a, then a rise in its popularity increases its
temptation to cheat. As a + v rises from zero to &, a small amount of

cheating yields larger and larger benefits in terms of increased probability

of election. Only for a+ v > @, so that U" < 0, can an increase in popularity
lead to a lower level of cheating.

Proposition 5 deals with a small change in popularity, v. We conjecture
that a large increase in v will move most types into the range where U' is
declining, and whereas some types will cheat more, most will cheat less.
Proposition 5 has a second important interpretation. The shift parameter v
may be viewed as an observable component of the incumbent's competency shock.
We illustrate proposition 5 in figure 3.

Thus far, we have only analyzed the government's behavior during even
(election) periods. But clearly, there will be no gain to cheating in off-
election years, since the public will have observed Le) by period t+2. This
supports our assumption in section II that expected inflation is the same
regardless of which party wins the election. If the e's followed a higher-
order MA process, then expected inflation in t+2 (conditional on t information)

would not necessarily be the same for both parties (by proposition 5).


33

derivatives of 4 are given by (A.1). They exist and are continuous except
at a@ = 0 (if W'(0) = 0) and Gy.

We have treated the case where a= 0, but clearly the case a> Oisa
trivial extension. The second part of Proposition 3 is a straightforward
consequence of the fact that A> obeys (18) for all a €a< Gy, and that W" > 0.
Note that the proof of Theorem 1 required, in addition to the assumptions on W

given below eq.(2) of the text, that W"/W' be uniformly bounded on (cal for

c>o.


15

Since U(0) is the lower bound on U, D(a) represents the minimum gain to a type a
who defects and sets A = 0 instead of equal to a*(a) > 0. Clearly 0(0) > 0 for
4*(0) = 6. Furthermore, since D is continuous in a, there exists a neighborhood
of zero in R, such that D(a) > 0. Hence defection is not a probability measure

zero event, since all a@ within a neighborhood of zero will defect.

Comment. It might seem that the "natural" boundary condition would be
a(a) = 0 rather than A(0) = 0. The best type cannot gain by posing as a better
type. Proposition 2 tells us, however, that it is a = 0 who gains nothing by

cheating in equilibrium. As long as other agents are cheating enough so that,

by (18), it is not worthwhile for a = 0 to raise A above A*(0), then he might as
well not cheat at all since he will be recognized as a zero in equilibrium
anyway.

The proof of proposition 2 relied on the fact that tT = g is a feasible
level of taxes. Suppose there exists some 1TM* < g such that taxes cannot

exceed 1%,

(It would be plausible to posit that taxes cannot be raised
above 14x without fundamental changes in the tax system, and that these changes
would take several periods to implement.). Then 4*(0) =zg- max is the only
consistent conjecture for voters. The proof is analogous to the proof of
proposition 2.14

The incumbent party's maximization problem (15) does not always have an
interior solution on all of A. This issue definitely arises if
W'(O) = lim W'(A) > 0. Suppose for example that

A+0+

(21) Qla) # xU'(a) - (1-x)W'(0),

is strictly negative for a = 0. Then (18) implies that 4'(0) is negative, but

this possibility is ruled out by propositions 1 and 2. Since U" > 0 fora < a,


of the more general case. The fact that the shocks vary over time may be
justified by noting that the leaders of a political party change, and that
policy prescriptions suited for one historical episode may be imappropriate in
other circumstances.

We assume that exogenous, party-specific, non-economic preference shocks

also follow a MA(1) process so that

R_ 10
(4) Me ~ ME = % + Ay

where (q,) is an i.i.d, stochastic process on R. Furthermore, the density
function of q is unimodal, continuously differentiable, and symmetrically
distributed around zero.

Any promises the two parties might make before an election have no
impact on the voters in our model. (All the voters are from Missouri.) What
does make an impression on voters is their observations on the incumbent party's
performance, from which they can infer something about its most recent
competency shock. (We will describe the information structure shortly.) The
macroeconomic policy cycle will arise because the incumbent party has an incen-
tive to try to signal that its most recent competency shock is high. The
opposition party can make promises, but it lacks an effective way to reveal
how well it would have performed if it were currently in office. In fact,
all the public knows about the opposition party is the probability distribu-
tion of its competency factor, €. Because € is an MA(1) process, the fact
that the opposition party may once have been in power is not relevant.”

It would be pointless for the incumbent party to try to deceive the public
unless it has an information advantage. Our assumptions about the information

structure and the timing of elections are as follows. The incumbent party has


30

parties. A related extension would be to allow for imperfect information
about the incumbent's preferences. But the rationale presented here for an

electoral policy cycle should carry over to these more general frameworks.


27

V. Equilibrium When Cheating is Never Directly Observed

Until now, we have assumed that the incumbent government's information set
becomes common knowledge after a one-period lag. Thus in period t+1, the
public can directly confirm its inference about how much the government was
cheating in period t. Here we modify the information structure so that

22 Our

electorate can never know for certain how much the incumbent cheated.
main purpose is to show that the results of section III generalize.
We assume as before that in period t+1, all agents observe the lagged

competency shock, ae and lagged seignorage (or bond financing), A However ,

t
seignorage and cheating are not equivalent if the incumbent party cannot
perfectly predict its revenue needs when setting Tee In particular, suppose it
only observes the noisy signal §. = aoe. where >, is an i.i.d. stochastic
Process on [0,1], and is independent of ae
The analysis can be simplified somewhat with <ne assumption that the

social-loss function takes the special form:
(33) wz a2y2,

Since the quadratic form of W implies certainty equivalence, it would be

socially optimal for the government to set
H
(34) rT [4(@)] = g - &(@),

where &(6) = E(a|@).
Since the government has incentives to cheat as before, the public

believes that taxes are actually set according to

(35) t*[@(9)] = g - (6) - x*[a(a)],


In section IV, we examine other equilibria which can arise if the two
political parties have infinite horizons. If their discount rates are low
enough, then there will exist an equilibrium in which there is no macroeconomic
policy cycle. "Enforcement" of the cooperative equilibrium can come through a
number of related channels, including both voters' expectations and the strate-
gic interactions of the two political parties. If the parties have sufficiently
high discount rates, there will always be a cycle, but it may be damped. we
illustrate some properties of sustainable equilibria for this case. Finally, in
section V, we briefly consider what happens if the incumbent party cannot per-
fectly project its revenue needs; i.e., has imperfect information about its com-
petency shock. The analysis of section III generalizes even if the forecast
error is strictly private information. In the conclusions, we discuss some

empirical implications of our model, and some possible extensions.


In section III, we consider equilibria in which voters’ expectations
depend only on known characteristics of the two parties, past observations on
the competency shocks, and the government's most recent tax bill. In particular,
expectations do not depend on how much a party might have "cheated" (set taxes
suboptimally low) in previous elections. The equilibria in section III are the
only possible equilibria when the political parties have finite horizons, given
the assumed information structure. Using only mild restrictions on the utility
functions, we are able to prove the existence of a unique separating equilibrium.
In the separating equilibrium, voters are able to exactly infer the incumbent's
information from the tax bill. Because the competency shock is directly observed
by the public with a lag, there is never any "cheating" (suboptimal use of
seignorage or bond financing) during off-election periods.

During election periods, the equilibrium has the following characteristics
If the incumbent party knows that its competency shock is the lowest possible,
then it will not cheat at all. This result obtains even if the incumbent's
temptation to improve its image is greatest when its competency is lowest.
Cheating is increasing over a range of realizations of the unobservable
compepetency shock, but is declining at higher levels of competency. Incumbents
of intermediate ability cheat the most. (This conclusion must be amended if
there is an upper bound.to the level at which taxes can be set.) The model also
yields some interesting conclusions with respect to changes in observable
factors. For example, the conventional wisdom is that the incumbent party will
inflate less if it becomes more popular for noneconomic reasons. We show,
however, that although an increase in popularity may cause the incumbent to
cheat less if its unobservable competency shock is high, it will cheat more if

its competency is low.


16

it is possible that Q(a) > 0 for some a > 0. Denote the smallest a such that

Q(a) 3 0 as a. (If Q(a) <0 Va, let a= a.) Then consider a solution
path to (18) initiating at A(a,) = 0. It is easy to show that 4 will initially
rise from zero, but eventually will decline and cross the a axis from above at
some oy <o, (See Appendix A. If W'(0) = 0, the curve may asymptote to the a
axis rather than intersecting it.) Furthermore, it must be true that Q(a) is

strictly negative for all a 3 a, (since U" <0 for a> a). This leads us to

proposition 3.
Proposition 3. A =0 for a 3 a, and for a ¢ a.
Proof. Suppose voters believe A*(a) = 0 Va > ay, SO that

(22) Ng-a-d) = g - (g-a@- A) aed va ray,

Then since Q(a) <0 Va 3 ay, 4 = 0 is optimal for a > ay. The proof of the
second half of proposition 3 is similar and is subsumed in the proof of

Theorem 1 in Appendix A.

Definition 1. A separating equilibrium is a continuous function as: a = (0,a[

such that
(i) a =0 ifas a.
(ii) a® satisfies (18) if a €a< ay.

(iii) AS 2 0 fara,
Theorem 1. A unique separating equilibrium exists.

Proof. See Appendix A.
The solid line in figure 1 is a graph of the equilibrium a® function. (To insure

a unique turning point, we require U"' <0 for a < a.) Figure 1 is drawn for


bond financing if taxes are distortionary. In this case there generally
exists an optimal distribution of tax distortions over time.>

Each of the identical voters has a time-separable indirect utility
function, which depends on the incumbent's competency, ¢€, the level of distor-
tions arising from seignorage or suboptimal deficit financing, A, and exoge-
nous noneconomic factors which depend on which party is in power, n. (It is
straightforward to extend the model to allow for different voters to have
different preferences with respect to exogenous factors.) Social welfare in

period t is given by

(2) Q = - WA) te, +n

te!
where W:)-4,4L ~R is twice-continuously differentiable, except possibly at zero.
W'>(<)O for A >(<)0, and W" > 0. Thus, W is minimized at A = 0, and

W(A) + as a= a.°

Each party's competency shock is serially correlated so that for party

(3) ej = ad + ad

where (ad) is an i.i.d. stochastic process on A # (0,a[, where a may be

p and a® are independent Ws,t, that they have the

infinite. It is assumed that a
same twice-continuously differentiable distribution function, and that

e,_,(a3) =a. It is crucial that ted) display some serial correlation, or there
would be no reason to vote for a party just because it appears more competent
today. The assumption that the competency shocks follow MA(1) processes simpli-

fies the analysis by making it possible for elections to be independent.

However, we present results in section III indicating some qualitative features


22

IV. Infinite-Horizon Equilibria
If the time horizons of the two political parties extend infinitely far
into the future, then equilibria which Pareto-dominate the "noncooperative"

18 There are three

equilibrium of section III will generally be attainable.
related mechanisms for enforcing a more "cooperative" equilibrium. First, the
public can assume that a party which has defected in the recent past will] not
cooperate during its next term(s) in office. The incumbent party must then
recognize that a defection will lower expected future social welfare and
thereby its own, since the social costs of distortions are an element of its
utility function. Second, if the public assumes there will be a reversion to
the noncooperative equilibrium next time the defector wins an election (and
hence higher expected distortions), then it will be less likely to vote for the
defector. The third mechanism for enforcing cooperation involves strategic
interactions between the two political parties. If the incumbent party cheats,
the opposition party can respond by cheating more the next time it is in office.
Since both parties care about social welfare, each may be willing to restrain
itself to keep the other party in tine. 19

When politicians have high discount rates, even the best attainable
equilibrium will involve some cheating. We illustrate why in this case, any
sustainable equilibrium will still have the characteristic that higher a types
cheat to “run away" from low a types, just as in the finite horizon case.
However, to enforce any cooperative equilibrium, it is typically not necessary
(to threaten) to punish very high a types as severely. Here punishments are

never actually meted out along an equilibrium path, but the issue is relevant in

the strictly private information setting of section V.


International Finance Discussion Papers
Number 271

December 1985

ELECTIONS AND MACROECONOMIC
POLICY CYCLES

by

Kenneth Rogoff
and

Anne Sibert

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


This assumption will turn out to be correct in equilibrium because:
(a) no party ever chooses to inflate in the off-election year t+1, and

and a® have the same distribu-

a . . . D
(b) conditional on time t information, Cree) t+2

tion. Thus, despite the fact that 442 Will turn out to be a function of
Geer voters have no information at time t to help Predict which party will set
Bee higher. Assumption (6) must be relaxed when ¢ and q follow more general

stochastic processes.

As the opposition party D has no way to signal its most recent competency

shock, then by (3),

PLR R D D ,) __POR) _
(7) Ele eaat eben” Sted 42) = E (ae) a.
By (4),

PR R D _ oD _
(8) Ee (Mear* Nea2” ter” tea) = %

Thus, by (2) and (5) - (8), when party R is the incumbent it will win 38
PR -
(9) E, (ay) -a+q, 20.

The incumbent party, R, does not observe the disturbance to voters’

preferences, Ge: at the time it sets taxes. Therefore its estimate of the

probability it will win the election is?

Ro Pe Ry) = oe PR, _ +
(10) U, = ULE (a,)] = ¢ ProblE,(a.) - a + 4,701, ¢> 0.

Given our assumptions about the distribution of q (see eq. (4)), we can infer
that U is twice-continuously differentiable, strictly increasing in EF (ak),

; ee opeP) oR PR .
ne! (E, (@,)] = 0, and that UTE (a.)] >(<) 0 as Ela.) <(>) a.


contemporaneous information about its most recent competency shock, a, (In
sections III and IV, the information is perfect; in section V it is not.)
Citizens observe a, only with a one-period lag. However, at the beginning of
each period, voters receive a signal from the incumbent in the form of a
period-t tax bill. (When setting taxes, the incumbent party does not yet know
A: the most recent exogenous shock to voters' preferences.) After receiving
their tax bill and observing Qe citizens vote if it is an election period. At
the end of the period, markets meet and the price level is determined. If the
government set taxes too low, it will be forced to use the inflation tax to
balance its budget. (Again, our results extend to bond financing if taxes are
distortionary.) Citizens can infer 4, and €, at this point, though we make
the stronger assumption that citizens learn enough by t+#1 to observe
a, directly. The above scenario is consistent with the lag between conception
and implementation of fiscal policy.

At time t, voters will prefer party R to party D if their expected utility
from having party R in office during periods t+1 and t+2 is greater than that

from having party D in office. Thus, party R will win if

Prat + a - (a?

(5) E t+] t+2 t+1

0)
+ MM y2)] 20,

where e is the expectations operator conditioned on time-t public information,

which includes a G

tar Te See Igy and %: We will temporarily assume that

voters' expectations about bead and Sea (suboptimal use of seignorage) do not

depend on which party wins. Thus

R R D fy) .
+ Weae) - (Wear? Wey] = 0.

P
(6) ER E(WE,


4l

ce 3



29

Conclusions

Our analysis illustrates the essential role of temporary information
asymmetries in explaining electoral cycles in macroeconomic policy variables.
Much of the extant empirical evidence on political business cycles deals with
national elections in the United States and Germany. Our model is broadly
consistent with this evidence. However, the general framework we develop can
also be used to examine other types of electoral policy cycles, such as those
associated with state and local elections. (In this case, of course,
seignorage will not be a factor, but bond financing still is.) A limitation
in applying our model to some other countries is that it does not allow for
endogenous timing of elections. Nevertheless, if there is a sufficient lag
between the time when elections are called and the time when they are held,
the general notion that the incumbent will try to look good remains relevant

The analysis of the text focused on an electoral cycle in taxes, inflation
and deficits. But it is straightforward to extend the model to allow for
variable government spending. Consider again the highly plausible example where
the government has private information about the effective level of national
defense. The government might claim that it is efficiently obtaining a lot of
defense for any given input of expenditures. Therefore, it might signal its
competency by cutting defense appropriations. More generally, one would also
expect election-year shifts in the composition of government spending, towards
services with high immediate visibility (potholes), and away from expenditures
which the average voter only observes with a lag (contributions to government
pension funds).

It would be of interest to extend this framework to address other aspects of

political economy, and to allow for more differences between the two political


28

where we now refer to k rather than A as the level of cheating. gt denotes
voters' expectations of Kee given their observation on The Ex-post, seignorage
may be high in part because the incumbent party made a mistake in predicting a,

and in part because it was cheating:
(36) A= @(0) +K- a.

Despite the fact that the public never directly observes x, there still exists
a separating equilibrium analogous to that of section III. The steps involved
in deriving the equilibrium are as before. Equation (37) is the same as the

first-order condition (18), except that a and A are replaced everywhere by @(@)

and x:

(37) x'[8(8)] = EEaTICICIBI <4.

Again, the separating equilibrium reveals the incumbent's information €(6). As
long as Gea is observed in period t+2, the incumbent still has no incentive to
cheat in off-election periods. As in section III, this is the only equilibrium
in the finite-horizon case, and is an equilibrium in the infinite-horizon case.
The infinite-horizon trigger-strategy equilibria of section IV do not

extend directly to the case of imperfect monitoring. The observation that the
incumbent party needed to rely on seignorage or suboptimal bond financing no
longer implies that it has cheated. An interesting topic for future research
would be to find the optimal equilibria for this case, drawing on the literature

on repeated principal agent problems [see, for example, Radner (1985)].


II. The Model

Every other period, atomistic voters choose between the two political
parties, "R" and "D". A major factor in the election is voters’ perception of
the relative "competencies" of the two parties. A party's competency is
defined as follows: All governments are required to provide a fixed level of
government services, G. (The operative assumption is that G is observable,
not that it is fixed.) The more competent the government, the less revenues

it requires to deliver G:
(1) Gzettt+a,

where € is the government's competency, Tt is direct taxes (or transfers if
negative), and A represents seignorage or bond-financing. "Competency" is
a broad index which captures the administrative abilities of the incumbent
party, and the success of its policies in providing necessary government services
efficiently. For example, a highly competent government will use well-designed
bidding procedures on government contracts, and will make good choices about
which weapons systems to purchase. (Although the analysis below focuses on the
definition of competency embodied in (1), it would be possible to apply our
framework to other measures of a government's performance).

The underlying macroeconomic model is rudimentary. In each period t,
every voter receives a fixed, known amount of the perishable good. Agents pay
a total of tT, in real taxes. Money is required to conduct transactions, and

t

the transactions technology is such that the equilibrium price level, Pee is

increasing in the money supply, My We assume that there is no bond financing

and that seignorage revenues, A, = (M, : Meiy)/Pee are strictly increasing in

M, over the relevant range. Our analysis applies equally well to the case of


19

aA(a,x)/ax <0. Then it must be the case that there exists an a* > 0 such that
aA(a*,x)/ax = 0, and aA'(a,x)/ax < 0, within a RHS-neighborhood of a*. But by

(23),

aa! (ak,x) _ u" (a*) >0.
ax (1-x) W! [A (ae, x) J

' ial
Hence aa" (er x) > O within a RHS-neighborhood of a*. And this is a
contradiction.

If a =@> 0, then clearly dA (a) /dx > 0 and otherwise the proof is the

same as when a = 0.

As x rises, the incentive to cheat rises. Equilibrium requires that
each type must cut taxes by more to pose as the next highest type. Thus
“good" types must cheat more relative to "bad" types. Since by proposition 2,
4(0) = 0, then A must rise for all a > 0. The same result obtains if

max

(0) g-T . In figure 2, we illustrate proposition 4.

Now suppose that voters like the incumbent party for observable "noneconomic"
reasons, indexed by the parameter v. In particular, suppose we modify equation

(9) so that the incumbent party wins if
Pp -
(24) vet E,(a,) -a+q>30.

In order to maintain our assumption that EP (Ae go) is the same for both parties,
we will assume that the popularity disturbance is transitory, affecting only
the current election. Then the first-order condition, (18), can be rewritten

as

1 = -xUL(aty) |
(25) A'(a) (i-x)w'(ay 7 Te


36

Jonung, Lars, "Business Cycles and Political Changes in Sweden,” Skandinaviska
Enskilda Banken Quarterly Review 2 (1985), 26-39.

Kimbrough, Kent P., "The Optimum Quantity of Money Rule in the Theory of Public
Finance," mimeo, Duke University, (1985).

Kirchgaéssner, Gebhard, "Rationality, Causality and the Relation Between Economic
Conditions and the Popularity of Parties: An Empirical Investigation for

the Federal Republic of Germany, 1971-1982," European Economic Review 28,
(June-July 1985), 243-268.

Kreps, David M., and Robert Wilson, “Reputation and Imperfect Information,"
Journal of Economic Theory 27 (August 1982).

Kydland, Finn E., and Edward C. Prescott, "Rules Rather than Discretion: The
Inconsistency of Optimal Plans," Journal of Political Economy 85 (June
1977), 473-92.

McCallum, Bennett T., "The Political Business Cycle: An Empirical Test,"
Southern Economic Journal 44 (1978), 504-515.

McRae, Duncan, "A Political Model of the Business Cycle," Journal of Political
Economy 85 (1977), 239-263.

Nordhaus, William 0., "The Political Business Cycle," Review of Economic Studies
42 (1975), 169-190.

Radner, Roy, "Repeated Principal-Agent Games with Discounting," Econometrica 53
(September 1985), 1173-1198.

Riley, John G., "Informational Equilibrium," Econometrica 47 (March 1979),
331-359.

Ross, Shepley L., Differential Equations. (New York: Blaisdel1 Publishing Co.,
1965).

Stigler, George, "General Economic Conditions and National Elections," American
Economic Review 62 (1973), 160-167.

Tabellini, Guido, “Accommodative Monetary Policy and Central Banking
Reputation," mimeo, UCLA (October 1983).

Tufte, Edward R., Political Control of the Economy. (Princeton: Princeton
University Press, 1978).



37
FOOTNOTES

‘There continues to be a significant amount of empirical work on the topic,
especially in the political science literature. See, for example, Kirchgassner
(1985), Hibbs (1985), or Jonung (1985).

eMccal lum (1978) makes this point. The conventional rationale for political
business cycles is also questioned by Stigler (1973). Nordhaus (1975) notes
that the cycle will disappear in his model, once voters understand the process.

3cukierman and Meltzer (1985) have independently developed a related line of
research, though their approach is very different from ours. Backus and Oriffil
(1985) adopt Kreps and Wilson's (1982) model of reputation to provide another
rationale for political business cycles. [Barro (1985) and Tabellini (1983)
present similar models.] In the Backus-Driffill model, the Probability the
incumbent will inflate increases towards the end of his final term in office.
He inflates precisely because he is not worried about the future, and has no
incentive to maintain his reputation as an inflation fighter. By inflating
(more than anticipated), the benevolent and rational elected offical is able to
temporarily reduce the effect of distortions which keep employment below its
socially optimal value. This repeated game "reputational" model of political
business cycles seem at odds with Tuffe's (1978) evidence that the political
business cycle is more pronounced when the incumbent is up for re-election.

‘McCallum (1978), and Golden and Poterba (1980), find little evidence of a poli-
tical business cycle in employment. The evidence cf a political business cycle
in variables such as transfers and money supply growth is stronger (see Tufte
(1978), or the Hibbs and Fassbender volume (1981).)

Sse Barro (1979).

Sour main results do not depend on the assumption that W is minimized at zero,
or that W is independent of t. There is however a general argument, due to
Kimbrough (1985), that the seignorage tax should always be set at zero even if
taxes on labor and consumption are distorting. Kimbrough argues that for any
reasonable transactions technology, money should be viewed as an intermediate
good in the production function, and not as a final consumption good. By a
standard theorem in public finance, it is suboptimal to tax intermediate goods
(since it causes a distortion on two margins), unless there are increasing
marginal tax collection costs.

"1¢ € follows a higher-order stochastic process, then the public may take into
account the opposition's performance when last in office. The main effect this
will have on the analysis is to alter the probability the incumbent will win;
the resulting political business cycle is qualitatively similar.

Sin (2), the representative voter is risk-neutral with respect to e. If the
public were risk averse, then the fact that it knows more about the incumbent's
competency than the opposition's would make the incumbent's re-election more
likely. Allowing for this possibility should not alter the general nature of
the results.


25

Punishment interval may be shorter when the incumbent draws a high a, since then
the potential gains from cheating are very small. (Since a is observed with a
one-period lag, conditional punishment strategies are feasible.)

If the incumbent's discount rate is very high, there May not exist any
punishment interval, or more general punishment strategy, sufficient to
discourage all a types from defecting. However, it may still be possible to
sustain an equilibrium which Pareto dominates the "noncooperative"

equilibrium.2?

Although we do not provide a general analysis of this inter-
mediate case, we do provide an example which illustrates some of the special
features of our model. We construct our example in a way such that the maximum
punishment, P, is given. In equation (11) of section II, the two political
parties are assumed to have a constant discount rate 8. Here we assume that B
is time varying such that B(t) is much Jess than B(s) for all s > t. We assume
that after period t, B is large enough so that a zero-cheating equilibrium can
be achieved in all future election periods. But Bit) is so small that the
zero-cheating equilibrium cannot be sustained in election period t. (It is
always possible to choose (8) to satisfy these assumptions.) To establish an
enforceable period-t trigger-strategy equilibrium, it is useful to ask what
would happen if voters think that no one cheats, so A* = 0.

Consider an agent of type a. Let
(30) G(a,A) = xU(atd) - (1-x)W(A) - P= [xU(a) - (1-x)W(0)]

be the gain from cheating by A rather than cooperating when A* 20. By hypothe-
sis, there is some a € A such that there is a A > 0 where G(a,4) 20. Let
a, be the minimum such a and let 45 be the minimum A such that G(a,.4,) 20.

Then a+ 4g is the minimum a type any other type a would ever be willing to


I. Introduction

At one time, research on political business cycles received a great deal of
attention. Nordhaus (1975) and McRae (1977) provide important examples.! Their
models suggest that politicians will inflate during election years in order to
exploit a Phillips curve tradeoff which is more favorable in the short-run than
in the long-run. Interest in these adaptive expectations models waned, however,
after the rational expectations revolution of the seventies. As long as private
agents (such as wage setters) understand the government's incentives, one would
not expect to observe any systematic rise in employment prior to elections.*
But the objections to conventional political business cycle models go beyond
their Phillips curve formulation, and apply to any model in which the government
takes artificial measures to make itself look good. Suppose the government
tries to please the public before elections by raising transfers or by lowering
taxes; according to Tufte (1978), this is the most robust empirical charac-
teristic of the electoral cycle. Why should voters prefer a candidate who is
suboptimally distributing tax distortions over time? Moreover, why should such
actions suggest that the incumbent will do a good job over the course of the
coming term?

Here we argue that electoral cycles in certain macroeconomic policy
variables -- such as taxes, government spending, deficits and money growth
-- derive from temporal information asymmetries.° We assume that the government
observes an indicator of its performance (e.g., in providing national defense
efficiently) before the representative voter does. Administrative performance
is correlated over time; hence prior to election periods the incumbent has an
incentive to try to "signal" that it is doing well. This gives rise to an elec-
toral cycle in macroeconomic policy. (It is important to stress that our model

does not directly provide a rationale for an electoral cycle in unemployment.) 4


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III. Finite-Horizon Equilibrium

In this section, we analyze the case where the two pclitical parties have
finite time horizons. Because the information asymmetries are temporary, and
because the random disturbances are MA(1) processes, each election cycie turns
out to be independent of previous election cycles. We are able to show tnat
there exists a unique (perfect) separating equilibrium, in which the incumbent
party's action (tax bill) fully reveals its information (competency snock).'@

We also demonstrate the qualitative features of the model described in the
introduction.

If there were full information, so that voters knew the competency shock
a, at election time, then the incumbent party would have no incentive to cheat.
For then it could not possibly influence voters' perceptions of its competency.
Moreover, by cheating, it would only lower social welfare by increasing
seignorage distortions. With asymmetric informat:3n, however, the incumbent
party may have an incentive to lower taxes in elec-ion years to try to
exaggerate its competency. We will temporarily posit that voters recognize this
incentive and believe that the level of inflation (cheating) depends on the
government's competency shock, a Later, we will verify that if voters have

rational expectations, then this supposition is correct. Oenote the voters

conjecture of inflation as a function of a by

* *
(12) a =A (a),

where the superscript referring to the political party has been dropped for
notational convenience. '? Then by equation (1), voters believe that the tax

bill as a function of a, is

* *
(13) T (a,) =9 a-A (a),

t 7 Me


14

“TFOe] =

-1' -
where we have made use of the fact that f 1 (f(x)] = 1/f' (x), and f
: F(X) /LF OX) PP. Inspection of (18) confirms our assumption that T* is
continuous and strictly monotonic, since T' = - (1+A').

The second-order condition (19) allows us to rule out equilibria involving

negative A:

Proposition 1. If A maximizes (15) and if voters' conjectures are consistent,

then A(a) cannot be strictly negative for any a.

Proof. By (13), T' = - (1 + 4"). By (18), T' > (<) 0 for every a if and only

if A < (>) 0 for every a. Oifferentiating both sides of the equilibrium

condition (18) with respect to a gives

2 ye
(20) an 2 CBE yma y,

Substituting (20) into (19) gives the result that the second-order condition

holds as long as W"/(1+4') > 0.

Note that the proof of proposition 1 also established that (19) holds for
every 4 > 0 which solves (18). The proof required W" > 0, but did not need
any restrictions on the sign of U". The intuition behind this is that in
equilibrium, U is not a function of the choice variable.

Equation (18) is a first-order differential equation with no apparent

initial condition. The next proposition provides a boundary condition.

Proposition 2. A perfect equilibrium requires A(0) = 0.

Proof. By proposition 1, A(0) > 0. Assume that voters' expectations A* (a) are

governed by (18) with initial condition a*(0) = 6 > 0. Define

O(a) @ xU(0) - (1-x)W(O) - xU(a) * (1-x)W[A*(a)).


Elections and Macroeconomic Policy Cycles

ABSTRACT

There is an extensive empirical literature on political business cycles, but
its theoretical foundations are grounded in pre-rational expectations macroeco-
nomic theory. Here we show that electoral cycles in taxes, government spending
and money growth can be modeled as an equilibrium signaling process. The cycle
is driven by temporary information asymmetries which can arise if, for example,
the government has more current information on its performance in providing
for national defense. Incumbents cheat least when their private information
is either extremely favorable or extremely unfavorable. An exogenous increase
in the incumbent party's popularity does not necessarily imply a damped policy

cycle.

Kenneth Rogoff

Economics Department

1180 Observatory Or., Rm. 6428
University of Wisconsin
Madison, WI 53706

(608) 262-3906

Anne Sibert

International Finance Division

Board of. Governors of the
Federal Reserve System

Washington, 0.C. 20551

(202) 452-3715


35
REFERENCES
Abreu, Dilip, “Extremal Equilibria of Oligopolistic Supergames," Harvard

Institute of Economic Research Discussion Paper Number 1167 (July 1985).

Alesina, Alberto, "Rules, Discretion and Reputation in a Two-Party System,"
mimeo, Harvard University (1985a).

Alesina, Alberto, “Credibility and Policy Convergence in a Two-Party System
with Rational Voters," mimeo, Harvard University (1985b).

Backus, David, and John Driffill, “Inflation and Reputation," American Economic
Review 75 (June 1985), 530-538.

Barro, Robert J., “On the Determination of Public Debt," Journal of Political
Economy 87 (October 1979), 940-971.

Barro, Robert J., "Reputation in a Model of Monetary Policy with Incomplete
Information," mimeo, University of Rochester (August 1985).

Barro, Robert J., and David 8. Gordon, "Rules, Discretion and Reputation in a
Model of Monetary Policy," Journal of Monetary Economics 12 (July 1983),
101-121.

Canzoneri, Matthew B., “Monetary Policy Games and the Role of Private
Information," American Economic Review 75 (December 1985), 1056-1070.

Cukierman, Alex, and Allan H. Meltzer, "A Positive Theory of Discretionary
Policy, the Cost of Democratic Government, and the Benefits of a
Constitution, mimeo, Carnegie-Mellon University (September 1985).

Frey, Bruno, and Frederich Schneider, “An Empirical Study of Politico-Economic
Interaction," Review of Economics and Statistics 60 (1978), 174-183.

Friedman, James W., "A Non-cooperative Equilibrium for Supergames," Review of
Economic Studies 28 (1971), 1-12.

Golden, David G., and James M. Poterba, "The Price of Popularity: The Political
Business Cycle Reexamined," American Journal of Political Science 24,
(November 1980), 696-714,

Green, Edward J., and Robert H. Porter, "Noncooperative Collusion Under
Imperfect Price Information," Econometrica 52 (January 1984), 87-100.

Hibbs, Douglas A., "Inflation, Political Support, and Macroeconomic Policy,"
in L. N. Lindberg and C. S. Maier, eds., The Politics of Inflation and
Economic Stagnation (Washington: The Brookings Institution, 1985).

Hibbs, Douglas A. and Heino Fassbender (eds.), Contemporary Political Economy
(New York: North Holland Publishing Co., 1981).



23

There are a rich variety of supergame equilibria. However, it is
Possible to exposit some of the general features of the infinite horizon
case if we consider a special class of equilibria in which only the first and
third enforcement mechanisms discussed above are relevant. Let al (a) 2 0 Va
be a candidate cooperative equilibrium, and suppose the public adopts the

following trigger-strategy expectations ; 22
Definition 2. Voters have a l(a) trigger strategy for A*(a) if:
(a) When t is odd, A¥(a) = 0 Va.

When t is even, then

* T . T . * s

(b) ay (a) = 4, (a) Va if 44_9(@) € 44-2 (@), or if 44-2 (@) 2 44-2 (a).
* s ag ak at T

(¢) 4, (a) 2 4, (a) if Ag (@) = 44-2(@) and 4, _2(@) > 44-2 (a).

(d) ag (a) = aa. where zero is the initial period.

The trigger strategy expectations described above have the property
that whenever the incumbent party defects and sets 4, too high, then there
will be a temporary reversion to the noncooperative equilibrium, 45, at the
next election in t+#2. This reversion takes place regardless of which party is
in office, so expected inflation is the same for both parties and assumption
(6) is intact. Note that there is no punishment for cheating too little;
ine., if A (a) ¢ al(a).

Let us first consider conditions under which al (a) = 0 Va is a trigger-
strategy equilibrium. for the trigger strategies described in definition 2

the expected cost to the incumbent of defecting, P, is solely due to a higher


December 1985

ELECTIONS AND MACROECONOMIC

POLICY CYCLES

Kenneth Rogoff*
and

Anne Sibert**

* University of Wisconsin
** Board of Governors of the Federal Reserve System

The authors have benefited from discussions with Rao Aiyagari as
well as from seminars at Harvard and Indiana Universities. The views
expressed in this paper are those of the authors and do not necessarily
represent the views of the Board of Governors of the Federal Reserve
System or any of its staff. Financial Support from the Alfred P. Sloan
Foundation is gratefully acknowledged.


31

Appendix A

Proof of Theorem 1: Suppose xU'(0) - (1-x)W'(0) > 0. Then an
equilibrium is given by

xU' (a

(-x)W'(a(a)] 7 1 if A(a) >corifas=o,

4'(a) =
(A.1) 0 otherwise.

A(0) =c,

where c = 0. Note that we have not ruled out W'(0) = 0 and A is not a domain;
hence we cannot immediately apply the standard existence and uniqueness theorem

for differential equations. Force }0,aC, it is useful to construct

WA) if Ae (c,AL,

W(A) =
) Wie) - (1 - e4Sywi(c) if Ae J-d,cf.

( U'(a) if a2o,

Y(a) =
U'(-a) ifa<o.
xY(a) : .
Let f(a,A) = ~ - 1. The strategy is to show that there exists a
(1-x)W' (4)

unique solution, a,(e), to

(a2) $2 os flan); (0) = © © 10,80,

and hence to (A.1) for c > 0. We then show that as c + 0, the limit of the
solutions to (A.1) exists and satisfies (A.1) for c = 0. In order to prove

the existence of a solution to (A.2), we need the following lemma.




26

"impersonate". Let Ay be the subset of A where the differential equation given
by (18) with initial condition A(a,+4,) = 0 has a strictly positive solution,

a, Let

(31) aia) =<

0 ae A/A,.
Clearly ab ¢ ae Va e€ A. See figure 4. We now show

Proposition 7. a = ae is the time-t part of a trigger-strategy equilibrium.

Proof. Clearly a party with a > a+ 45 will not defect. If a party with

a< a, + 4 wants to defect, then by construction it must want to pose as

a, >a, + 4,. But, since W" > 0, then Va < a, +4, and Va, € Ay:
'(q! "al
(az) = (aay (a8 (a, 404-0) < © yn a8 lay] = 0
144 (a,) +h (a,)

by (18); hence this is impossible.

a8

is not necessarily the best sustainable trigger-strategy equilibrium.
However, it seems clear that the optimal A function must still begin rising

from zero at a, + Ao: though possibly more slowly than by (18). Thus the optimal
equilibrium should have similar qualitative features to figure 4. Although we
have structured our example so that the punishment P is constant, the

equilibrium will have the general shape of figure 4 even if P is determined

jointly with a,+ bor the "start-up" point.


10

We now specify the objective functions of the two political parties. fach
party aims to maximize a present-discounted-value functional which depends on
(a) their probability of being in office, and (b) the social welfare losses aue
to suboptimal use of seignorage or bond-financing. '° The R party's odjective

function is

T T
k-t OR k-t
(11) Peed at uk - (oy ak * wa,
tt kes k kst k

where S is the set of even-numbered (election) periods, T is the (possibly)

infinite time horizon, and x is the weight the party places on being elected;

x is contained in the open interval ]0,1[. Party O's utility function is

identical."!
Because the assumption that the policymaker has a temporary information

advantage over the public (at least, off the equi'ibrium path) drives the

results below, we need to discuss the reasonableness of our specification:

The representative voter understands the model. However, it is not worth it

for him as an individual to monitor the government closely enough to have

complete contemporaneous information on how effectively the government is

spending his tax dollars. It is certainly reasonable to assume that a voter

does not engage in costly information-gathering activities solely to decide

his own vote, which has infinitesimal weight. Implicitly, we are assuming

that other information which the voter does gather (because it is worthwhile

in his production or consumption activities) does not allow him to directly

observe a, (until t+1). We are, of course, also assuming that statements from

the opposition party cannot be trusted, and that there is no public watch group

which can provide free, complete and unbiased information.


13

where 8 G- Gy. We temporarily assume that t* is continuous and
strictly decreasing. Then t* has an inverse function, T*-1 and the public's

time-t expectation of a, is

P te
(14) EY (a) = r#N(g,- a - a).

Substituting (14) into (10), and the result into (11) yields the incumbent
party's maximization problem
(15) max (xU[T*"1(g-a-4)] - (1-x)W(A)), Va,
A
where the subscript t has been dropped. Given the public's beliefs (12), the
incumbent party's choice of A affects only current-period elements of its
objective function, (11).

The first- and second-order conditions for an interior solution to (15) are

(16) MUN [TT (g-a-a) Jr" 1" (g-a-a) - (1-x)W'(A) = 0,

(17) xu" [r*1 (g-a-a)) [2*=1" (9-8) ]? + XU! [1-1 (g-a-a) }e*-1" (gaa)
= (1-x)WH(A) << 0.

In a separating equilibrium, voters' conjectures must be consistent, and
hence a* = A Va. Thus, equations (16) and (17) can be rewritten as the interior

equilibrium conditions:

xU' (a

. ——xui{a)
(18) 4'(a) = (1-x)W'[A(a)]

1,

(19) xU"(a) XU (a) A" (a)

2 307 (1-x)W"[A(a@)] < 0,
[1+A'(a)] [1+4'(a)]