88 M
. Bordignon, E. Minelli Journal of Public Economics 80 2001 73 –98
L
of efficiency: at best, the voter can now obtain only W in each period. Is the
higher transparency of the simple rule enough to compensate for this efficiency loss?
4. Comparison
In this section we compare the relative merits of simple and complex rules from an ex ante point of view. To do so, it is useful to introduce some new notation. Let
c
v R,T indicate the expected discounted utility of the consumer over periods 1
and 2 under rule R, conditional upon government being of type T at the beginning of period 1. R [ hS,Cj can either be ‘simple’ or ‘complex’, and T [ hB,Gj can
c
either be ‘bad’ or ‘good’. Thus, for example, v S,G is the expected discounted utility of the voter over periods 1 and 2 under the simple rule when the incumbent
is good in period 1. In the following, we assume that the initial reputation of the incumbent government is large enough to satisfy
u u , so that the partial pooling
1
equilibrium described in the previous section can be supported under the complex rule. Building upon the results of the previous section, the conditional expected
utility of the consumer in the four possible cases is: ]
c L
v S,B 5
du W
c L
L
v S,G 5 W 1
d W
c H
v C,B 5
pg 1 1 2 pg 2 c
1 2
c H
H
v C,G 5 W 1
d W
c
To understand these formulas, take for example v S,B . Under the simple rule and bad government, period 1 expected utility of the consumer is zero. However,
in period 2, the incumbent is not elected and the opponent takes its place. At the ]
beginning of period 1, the consumer assigns probability u to this candidate being
L
of the good type, in which case her period 2 discounted expected utility is d W ,
] and probability 1 2
u to this candidate being of the bad type, in which case her ]
c L
period 2 expected utility is zero. Hence, v S,B 5 du W . By the same token, if
H
the rule is complex and the incumbent is good, the consumer gains W in both
H H
periods, so that her expected utility at the beginning of the period is W 1 d W .
On the other hand, if the rule is complex and the government is bad, the consumer
H
only earns pg 1 1 2 pg 2 c in the first period, and then since the bad type is
1 2
re-elected for sure, she earns zero in the second period. Finally, under the simple rule and a good government in period 1, the consumer does not run the risk of not
L
re-electing it, so that she gains with certainty 1 1 d W .
It is then easy to check that the citizen is certainly better off under the complex
M . Bordignon, E. Minelli Journal of Public Economics 80 2001 73 –98
89
rule if the incumbent turns out to be good, and she is certainly better off under the
H
simple rule in the other case, provided that pg 1 1 2 pg 2 c is small, that is,
1 2
provided that g is ‘sufficiently close’ to x. However, to enforce a partial pooling
equilibrium, the reputation of the government in period 1 must be high, so that the second case appears to be more likely. It is possible, nevertheless, that the simple
rule dominates the complex rule from the point of view of the citizen? To answer this question, we define:
]
C C
C C
H u,u,d,g 5 uv S,G 1 1 2 u v S,B 2 uv C,G 2 1 2 u v C,B
] In the range of
u which supports the partial pooling equilibrium, Hu,u,d,g measures the difference in expected utility for the consumer between the simple
and the complex rule, where the ex ante probability of the two types to occur is determined by the citizen’s a priori. Clearly, the simple rule dominates the
complex rule, from the point of view of the citizen, if H ? . 0.
Proposition 3. Let g 5 x. Then, under A1 and A2, there exists a g , 1, such that
] for
g . g , there exists u . u such that Hu,u,d,g . 0 for all u [ u ,u .
g 1
1 g
Proof. See Appendix.
Hence, if the difference in efficiency between the two rules is sufficiently small, there is an interval of
u ’s in the support of the partial pooling equilibrium where the simple rule dominates the complex rule, and this interval is the larger, the
smaller is this difference. Indeed, for g
→ 1, the simple rule dominates the
complex rule for all u ’s that supports the partial pooling equilibrium. Quite
intuitively, the lower is the cost in terms of efficiency to get rid of the bad type in the second period by enforcing the simple rule, the more the voter would be
willing to pay this cost, even for high level of reputation of the incumbent. This then raises the question on who would choose the simple rule. One could
wonder, for instance, if the choice of the rule might be used by the policymaker itself as a signalling device, with, say, a good government choosing the simple rule
to signal its type to the voter. Interestingly, this turns out not to be possible in our case. As an inspection of the formulas above clearly show, both types of
government are indeed better off under the complex rule. The bad type, because the complex rule allows it to pool, and so to extract more resources from the
citizen. The good type, because the complex rule allows it to maximize the welfare of the citizen. Thus, if asked, each government, upon observing its type, would
vote for the complex rule, although for opposite reasons.
One could then ask if a Constituent could choose the simple rule, interpreting here the Constituent as is in the tradition of public choice theory e.g. Buchanan
and Tullock, 1963, that is, as a higher level agent who, behind a ‘veil of ignorance’ but understanding the results of the game being played by the agents in
90 M
. Bordignon, E. Minelli Journal of Public Economics 80 2001 73 –98
the economy, sets up the rules so as to maximize consumer’s welfare. The answer, perhaps unsurprisingly, is yes, provided that again the difference in efficiency
12
between the two rules be not too large. While this result follows easily from the above proposition, it is worthwhile
pointing out that we obtain it in a society where expectations on the ‘honesty’ of incumbent policymakers have reached a very high standard, high enough to
support a partial pooling equilibrium for all realizations of u. Thus, it is certainly
not necessary to have very bad expectations on politicians in order to want to enforce ‘flat’ rules. Indeed, in a sense the issue of the transparency of the rules is
important when governments are expected to be good, rather than the other way round. It is only when the reputation of the average politician is high, in fact, that a
badly minded policymaker can use the complexity of the contingent rules to ‘pool’ and exploit citizens without being discovered.
5. Related literature
