THE CONSERVATIVE CENTRAL BANKER PROPOSITION

The traditional model for analyzing the effectiveness of monetary discipline and policy credibility is due to Barro and Gordon (1983) and Rogoff (1985). They take a “surprise” supply function

where y denotes output as a deviation from its natural rate, π is inflation (and

is the private sector's expectation of that inflation rate), and

is a random supply shock with mean zero and variance

. This relationship has been justified variously as a new Keynesian Phillips curve

In this type of model, the central bank chooses monetary policy to minimize the following loss or social welfare function:

where b denotes the relative preference for output stabilization, or aversion to output instability, compared to inflation control.

denotes the level around which output should be stabilized.

Substituting (1) into (2) implies that the optimal choice for inflation, under rational expectations for the private sector, is

with

as the inflation bias, and

as the deviation of output from its natural rate. Hence, E(y)=0 and

.

The Conservative Central Bank proposal comes from the observation that social welfare can be increased for everyone if monetary policy can be delegated to an independent central banker whose preferences are more conservative than society as a whole—that is, to a bank that is more inflation averse and has a value of

in (2). However, the bank has to be independent. Its policymakers cannot be removed if their policies conflict with those of the government. In such a case the Bank would follow (3) as the optimal policy rule, but using its own value of

for b. Hence, to maximize social welfare, we need to choose a value

that solves

The first-order conditions for this are

Since the first term of (5) is positive, the solution has to be

as claimed. That solution implies policies that are more inflation averse than society would have chosen for itself. However, they are also welfare improving because the bank's operating parameters become subject to choice ex ante. The institutional environment has therefore become an additional choice variable.

POLITICAL UNCERTAINTY

To compare these “conservative” policies with the outcomes that could be expected in their absence, we must recognize that alternative regimes will produce outcomes that are subject to the preferences of whichever government is in power at the time. Alesina and Gatti (1995) provide a model that analyzes that case.

Suppose there are two competing political parties, D and R, with preferences

and

where

. So, party R is more inflation averse (conservative) and D is more unemployment averse (liberal). Suppose wages and costs are set first, and then there are elections. Suppose party D is expected to win those elections with probability p. Then the party that gains power gets to set monetary policy. That means the private sector must expect, ex ante, an inflation rate of

where, if left to themselves, the two parties would choose

and

by minimizing (6) or (7), subject to (1) but constrained, in each case, by the market's expectations

. This implies

and

We would get (

) with probability p; and (

) with probability 1−p. Hence,

but

are the long-run outcomes. The first term of (11) represents the volatility induced by the electoral cycle. It increases with the differences between the two parties, but vanishes with political certainty: that is, if p=0, or p=1, or k=0. The second term represents the volatility caused by supply shocks, but dampened by the degree of stabilization and altered by the different political preferences for output stability.

These outcomes must be compared to those from our independent central bank with

for b:

and

where

, society's preferred value. So, expected inflation is lower, but output volatility is higher, the smaller is

.

LOWER INFLATION AT NO COST?

Can we get lower inflation on average, and also a lower variability of output, by appointing an independent central bank? Lower average inflation requires

which is always positive since the right of (12) is strictly positive. So

, but small enough, will do it.

Similarly, we can always choose

to provide lower output variability. That happens if we choose

to satisfy

where

is the political “signal to noise” ratio.

value above a certain limit; and implies that the variability in output due to changing political preferences has been eliminated, and that the impact of stochastic shocks has been reduced—at least to the extent that the term on the left of (13) is smaller than the second and third terms on the right. However, there is no guarantee that this last condition will hold. It is also entirely possible that the inequality in (13) will hold even though the second and third terms are smaller than

—in which case the stochastic element in the output variability has risen on the appointment of the independent central bank, even though the political influences have been eliminated.

The key question, therefore, is: can we choose

to give both benefits—lower inflation and lower output volatility—at the same time? The problem is that lower inflation requires a smaller

, but lower volatility requires a larger

. However, we can always achieve both outcomes together, for any given value of p, if there is a nonempty interval for

between the upper bound implied by (12) and the lower bound implied by (13). Starting from (12), we can show that this interval exists if

holds, whereas, from (13),

also holds, where A=[1+b^D−p(b^D−b^R)]². In other words, the term on the right of (14) must be strictly larger than that on the right of (15). Clearly, that cannot be true if p=0, p=1, or if b^D=b^R. In those cases, there is no

that will achieve both lower inflation and lower output variability. However, in all other cases such an interval exists, and gets larger as g increases, so long as

Hence, a nonzero interval exists for all values of 0<p<1, and for any finite values of k,

, or

.

can always be found to give us both outcomes. However, that does not mean that the same value of

will remain valid if the parameters—and p in particular—change around the cycle or with the electoral calender.

It is worth noting that this result only holds because both the political cycle and the supply-side volatility enter the problem in the politically influenced case. Repeating the same analysis when the upper bound on

comes from (12), but the lower bound comes from just the last two terms of (13), shows that the independent and conservative central bank will only generate lower output volatility in the face of stochastic shocks if p<1/2. Conversely, it will only generate less volatility than that induced by the electoral calender if

that is, if k is large and

is small, but

,

,

. These results show that it is the independence of electoral outcomes, not the conservative nature or credibility of the policies per se, that is providing the lower inflation at no cost result. However, it is not a general result. It requires k large enough, or

small enough, and b^D−b^R large enough for the current value of p. If those quantities were to change over time, in particular if b^D−b^R or p were to change, then the permitted interval for

and the lower inflation at no cost property would both be lost. That is a problem because the whole point of this analysis is that the chosen value of

should not be changed until the current Banker's term of office expires—and especially not each time a different government appears likely to take office.

ONCE THE BANK IS CHOSEN, WILL IT CONTINUE TO DELIVER LOWER INFLATION AT NO COST?

The problem with the results in Section 4 is that the appropriate value of

is specific to the particular value of p (or, equivalently, to the specific values of b^D and b^R) which held at the moment the central bank was chosen. If the choice interval for

at that time was relatively narrow (perhaps because b^D−b^R was small, or p was small), then the chosen

could easily become invalid as the electoral fortunes of each party change. And that means the economy would no longer be getting lower inflation at lower output volatility, compared to what it could be getting with policies appropriate for the new circumstances.

From (12), we can see that we will get lower inflation from the chosen

only if the value of p is such that

For this to make sense, we will require

, so that

. However, (18) also implies that p_L>0 only if

. In other words, we can only get lower inflation for all possible values of p if

is chosen. Otherwise we will only get lower inflation when the probability of party D forming a government is greater than p_L.

Conversely, we will get lower output variability only if p satisfies (13) for the given

: that is, only for the values of p that satisfy

where

and

if

. This sign pattern means that we are faced with two possible situations, depending on whether

or

. These two cases

Case A:

. In this case, a₀<0 and the results follow from Figure 1 in which f(p)=0 has one root p_F, in the interval [0, 1]. This leads to Proposition 1.

PROPOSITION 1. If

, then we will always have one property (lower inflation), or both properties (lower inflation and lower output variability), depending on whether p is currently below p_F or above it. However, we can never have both properties for all values of p.

Proof. See Appendix A.

Comment 1. It is possible to get both properties together if

exactly, if k² exceeds

sufficiently, and if b^D is sufficiently different from b^R. However, that is a very special case: Insufficient differences between the political parties, say

, or a nonpartisan central bank (

), or a moderate target for the level of output (k) compared to its natural variance (

), would prevent it. Appendix A contains the details.

Case B:

. In this case, a₀>0 and f(p)=0 will have either one or three positive roots. Figures 2A and 2B apply accordingly. There are various possible positions for h(p) depending on the value of P_L, and hence on the choice of

.

PROPOSITION 2. If f(p)=0 has a single positive root, Figure 2A, then either one or the other of the desired properties will always hold—lower inflation if p>p_L, or lower output variability if p<p_F. It is also possible that both properties will hold simultaneously if

.

Proof. See Appendix A.

Comment 2. Again we can never get both properties for all values of p. However, there is one, possibly small, interval of values for p where both properties apply, and there is no interval where neither property applies.

PROPOSITION 3. If f(p)=0 has three positive roots, Figure 2B, then there will always be some interval where both properties (lower inflation and lower output volatility) hold. However, there will also be intervals for p where only one of the two properties holds (lower output volatility when

or p_L3>p_F2, but lower inflation when

or p_L2<p_F2). These different values of p_L1, p_L2, and p_L3 reflect three possible choices of

[and hence of h(p)]. Moreover, there is also always one interval, (p_F1, p_L2), if

is chosen to yield p_L2, or (p_F1, p_F2) if

is chosen to yield p_L3, in which neither property holds, although

has been correctly chosen.

Proof. See Appendix A.

Comment 3. The upshot of our results so far is that there will always be intervals for p where both lower inflation and lower output volatility hold, but no case where both properties hold for all values of p. Conversely, unless we can find some mechanism by which the central bank can be made both independent and to rule out Proposition 3 altogether by imposing a suitable choice for

, there will also be intervals for p where neither property is achieved. In between these extremes, there will always be times when only one or the other of the two properties holds.

PROPOSITION 4. It is only possible to rule out Proposition 3 in the case where all possible governments are very liberal, and where the central bank is permitted to be no more liberal than the most conservative of the possible governments. That occurs when governments are either “target liberal” (g² is large) or “weight liberal” such that

is large enough and inequality (A.4) applies. However, Proposition 3 may also be ruled out if the central bank is very conservative in itself

, so that Proposition 1 applies.

Proof. See Appendix A, Case C.

Comment 4. All the results so far assume

has been verified to lie within the nonzero interval implied by the lower bound in (15) and the upper bound in (14). If it does,

can be said to be “correctly chosen” for current value of p. However, if it does not, perhaps because it has been evaluated for some past value of p, or because it has proved too hard to obtain accurate measures of b^D or b^R from the political parties, then it will also be possible to get intervals p¹_L−p_F where neither property holds in Figure 2A; and also no interval where both properties apply (Appendix A). So, to be able to rely on the results derived in Propositions 1 and 2, we not only have to impose sufficient conservatism on the central bank, but also ensure that the

in question has been correctly chosen. Section 6 shows what will be risked if this is not done.

Postscript. Appendix B computes the social welfare outcomes for each of the policy strategies available here: for full commitment when the true value of b is known (or using b=0, if it is not); for the conservative central banker solution with either

or an approximately optimal

to satisfy (5); and for the political economy solution where the central bank's policies are chosen by compromise to reproduce (9)–(11) for the value of p at the moment of the central bank's appointment. These results show that:

1. An optimally conservative central bank will choose
   bounded away from zero, and
   from below if
   is small, b is small, or k is large.
2. The conservative central banker will always delivers smaller welfare losses than full commitment using b=0, unless
   is large and k² is small. However, the differences are small if the true value of b is small, or if
   is set small.
3. The optimally conservative central bank will deliver smaller losses than full commitment with the true value of b, unless
   is very small. Again, however, the differences are small if the true b is small.
4. The political economy compromise is always worse than any conservative central bank.

These results are included to provide a benchmark for the conservative-central-banker and political-economy solutions. They make the point that a conservative central banker may be a convenient way of achieving optimal outcomes when full commitment is not available. However, it may not always be possible to achieve such outcomes, for example, if

is small, and it may not matter much if the true b is small. Thus, these results also make the point that getting “lower inflation at no cost” may not be a particularly important property. Such a property may not be optimal and may not be attained when the electoral probabilities can shift significantly from those that held when the current central bank was appointed. Better to stick with a solution with much lower inflation across all likely electoral outcomes, or one with lower output volatility, even if the other property is made a little worse.

THE PROBABILITY OF GETTING LOWER INFLATION AT NO COST

Finally, if we are able to make specific assumptions about the probability function that governs the chance that party D wins each election, we can compute the probability or proportion of time spent with lower inflation, or with lower output variability, or with both properties, when

is correctly chosen. Conversely, if

takes an inappropriate value, we can compute the proportion of time spent without one or the other property, or with neither of them.

Uniformly Distributed Probabilities

If we take probability that party D wins the next election to be distributed uniformly over the unit interval, as Alesina and Gatti (1995) do, then we will get lower inflation only 1−p_L of the time in the long run (assuming

). Similarly, if f(p)=0 is constructed to have a single positive root, then we will have lower output volatility p_F of the time (where p_F is that root). If

, then we get lower volatility all the time.

Tables 1 and 2 summarize the results of three constructed examples, to show the proportions of time spent with lower inflation, lower output variability, or with both properties. Examples 1 and 3 use a correctly chosen value of

, and Example 3 is the numerical example originally used by Alesina and Gatti to demonstrate the advantage of delegating monetary policy to an independent central bank. Example 2 shows what can happen when

is incorrectly chosen¹¹

Following (18),

would be necessary to get lower inflation at no cost in this case.

—even though the remaining parameters have been chosen to ensure that a value of

could exist such that at least one of the two desired properties holds all the time. However, as we pointed out in Section 5, some of those parameters (b^D, b^R, and k) are under the policymakers' control. So this situation could be avoided with sufficiently astute policymakers. It is interesting to speculate whether the political parties would also learn to adjust b^D or b^R in order not to be seen to be offering policy preferences that could predictably lead to periods where neither lower inflation nor lower volatility would be achieved.

Table 2, which is derived from the p_L and p_F figures reported in Table 1, shows that the conservative central bank—under reasonable values for the preference and output variance parameters—would do a pretty good job of providing lower inflation than a dependent or politically influenced central bank. It would also do a pretty good job of providing lower variability in output. In most cases, it can provide one or the other property alone for 50–75% of the time.

The striking result, however, is the fact that the conservative central bank seldom achieves both properties together—that is, low inflation at no cost. In fact, this occurs just 6% of the time in Examples 1 and 3, and never in Example 2. On the other hand, it achieves neither property rather more often if

is poorly chosen: 26% of the time in Example 2.

These results are significant because they show that the advantages of having a conservative and independent central bank are sensitive to small shifts in the underlying political or economic conditions. For example, a small shift of 6 percentage points in the probability of party D getting elected can destroy those advantages altogether, and put us back into a position where we have to trade off lower inflation for greater output stability in the traditional manner (Examples 1 and 3). In addition, similar-sized shifts in the public's preference for more conservative policies (i.e., toward party R) from a position of neutrality (p=0.5), can cost us both properties for significant periods of time: 26% of the time in Example 2.

Hence the disadvantage of an independent and conservative central bank is that, once chosen, it is hard to change and may go on delivering unfavorable policies long after the economic and political circumstances of its birth have changed. It may therefore turn out to have been a bank for particular circumstances, rather than a “bank for all seasons.”

Normally Distributed Probabilities

Table 3 recomputes the proportions reported in Table 2, for the same four examples, when the probability of party D being elected is normally distributed, N(0.5, 0.2). The interval

then contains 95% of the potential realizations of p. The results, however, are not very different from those in Table 2. The probability of getting lower inflation at no cost in output stability increases somewhat, but still reaches only 10–12%. However, the proportion of time spent with neither property when

is poorly chosen also increases, to around 33%. So, our conclusions about the low-inflation-at-no-cost result being very sensitive to small shifts in the political and economic circumstances remain. In fact, the lower-inflation-at-no-cost property will hold only for a small minority of the time.