Voting in Central Banks: Theory versus Stylized Facts

A1 Model Technical Details

Appendix A1 includes more technical details for the models from the main body of the paper so that it becomes apparent how to generate C’s proposals and Ps’ voting behaviour. We further explain finer details of our simulation exercise and the methods used.

First note that for all the models equilibrium exists by simple (within-period) backward induction argument and it is unique. The Markovian restriction (Maskin and Tirole 2001) then means all the committee members condition their actions only on the state given by the status-quo policy and their signals, not on payoff irrelevant histories.

Throughout the explanation we will often work with a vector of random variables. All those variables form a random vector r={iˉ∗,iP1,…,iPN,iC}′ that has a multivariate normal distribution with, conditional on i∗, mean ρi∗ and variance-covariance matrix equal to a matrix with the vector of {σu2,σu2+σP2,…,σu2+σP2,σu2+σC2}′ on the main diagonal and all the off-diagonal elements equal to σu2. ^[37] Often we will need to compute the conditional expectation of r given some specific value of one or more of its elements. For this we use the well known result for the multivariate normal distribution that states that for a vector of (possibly more than two) random variables {x1,x2}′ distributed according to N(μ,Σ) with μ={μ1,μ2}′ and Σ=σ11σ12σ21σ22 where the partitioning of μ and Σ conforms to the partition of {x1,x2}′, the conditional distribution of x1 given a specific value of x2 is N(μ1′,σ11′), where μ1′=μ1+σ12σ22−1(x2−μ2) and σ11′=σ11−σ12σ22−1σ21.

In the autarkical model, at the beginning of each period we have status-quo x, last-period optimal interest rate i∗ and r={iˉ∗,iP1,…,iPN,iC}′. First we need to derive C’s proposal y. This will be a solution to

where p(x,y) is the policy adopted given proposal y and status-quo x. The optimization problem can be rewritten as

where a is the event of y being accepted, aˆ is the event of y being rejected and pz is the probability of event z.

We need to calculate the probability of y being accepted against status-quo x, pa. Chairman C knows, and we show below, that the remaining players will vote for y if and only if their signal is above (or below, but this case is symmetric) a certain cut-off that we denote here by k. The other relevant information C has is iC and i∗. Hence we need to calculate the probability of at least N2P members voting for y given iC and i∗.

The probability of the first N′ committee members voting for y is equal to P(#|iP≥k|=N′,#|iP<k|=N−N′|i∗,iC) and is straightforward to calculate. We know the distribution of {iP1,…,iPN}′ and can transform the probability into P(#|iP≤k|=N|i∗,iC) by multiplying the whole problem (that is the mean and variance-covariance matrix) by {−1,…,−1,1,…,1}′, where there are N′ negative ones and N−N′ positive ones. Denoting the probability of the first N′ members accepting by PN′, the probability of y being accepted becomes ∑i=N/2NNiPi.

The key computational problem in simulating the autarkical model is computing the expected value of iˉ∗ given iC, i∗ and the event of y being accepted. Acceptance means that the signals iP of N2 or more P members must have been above (or below) a certain threshold k, which carries information about the unknown iˉ∗. We use two simple results to simplify the computation. For random variable X and two mutually exclusive and exhaustive events A and B we have

and the similar result for variance states that

However, the key problem remains. We need to calculate an expectation of the form E[iˉ∗|i∗,iC,#|iP≥k|=N′,#|iP<k|=N−N′]. The first step is simple and amounts to calculating the distribution of {iˉ∗,iP1,…,iPN}′ given i∗ and iC. It is N(μ,Σ), with each element of μ being equal to ρi∗σC2+iCσu2σu2+σC2 and Σ being a matrix with the vector σ′,σ′+σP2,…,σ′+σP2′ on the main diagonal and σ′=σu2σC2σu2+σC2 off the main diagonal. We then convert the problem into one of finding E[iˉ∗|i∗,iC,#|iP≥k|=N] using the same multiplication by a vector of positive and negative ones as when calculating pa. This leaves us with a multivariate truncated normal random vector with known mean and variance. To calculate the expectation we used the results in Tallis (1961) and Lee (1979) and wrote our own MATLAB function which calculates the expectation. We checked its correctness using the “tmvtnorm” R-software package (see Wilhelm and Manjunath 2012).

With these results, we can expand the maximand in [6] and use the rules for conditional expectations and variance given above to determine the value of C’s objective function for each y∈Y. This gives us the solution to C’s optimization problem and hence her proposal.

With C’s proposal y, we can determine the voting behaviour of the remaining P committee members. For each member j we use the voting rule [2] from the text adapted to the autarkical model

with E[iˉ∗|i∗,ij]=ρi∗σj2+ijσu2σu2+σj2. This also proves that each P member votes for y if and only if his signal is above (or below, depending on the position of the status-quo) a certain cut-off.

In the consensual model, again given x, i∗ and r={iˉ∗,iP1,…,iPN,iC}′, C’s proposal y will be the policy in Y closest to C’s expectation of iˉ∗, E[iˉ∗|i∗,iC]=ρi∗σC2+iCσu2σu2+σC2.

Because the P committee members extract information from C’s proposal in the consensual model, each P member j will vote based on the voting rule [2] given i∗, ij and the information embedded in C’s proposal y. Adapting [2] to the consensual model gives

It is easy to confirm that the information embedded in C’s proposal is equal to an event of iC∈[ilC,iuC], with the bounds given by

Using the law of iterated expectations, E​[i¯*|i*,ij,iC∈[ilC,iuC]] rewrites as E​[E[i¯*|i*,ij,iC]|i*,ij,iC∈[ilC,iuC]]. The inner expectations are equal to ρi∗σj2σC2+ijσu2σC2+iCσu2σj2σu2σj2+σu2σC2+σC2σj2 and we know that the distribution of iC given i∗ and ij is normal with mean ρi∗σj2+ijσu2σu2+σj2 and variance σu2σj2σu2+σj2+σC2.

Last thing we need is to be able to calculate mean of doubly truncated normal variable. For N(μ,σ2) distributed x1 conditional expectation of x1 given x1∈[al,au] is Ex1|x1∈[al,au]=μ+σϕ(al−μσ)−ϕ(au−μσ)Φ(au−μσ)−Φ(al−μσ), where ϕ(⋅) and Φ(⋅) are, respectively, the probability density and cumulative distribution functions of the univariate standard normal distribution.

In the supermajoritarian model C knows the most preferred policies of all the committee members. For each player j this policy, given x, i∗ and r={iˉ∗,iP1,…,iPN,iC}′, will be the policy in Y closest to j’s expectation of iˉ∗, i. e. closest to E[iˉ∗|i∗,ij]=ρi∗σj2+ijσu2σu2+σj2. Denote vector of the most preferred policies by {p1∗,…,pN+1∗}, ordered such that pj∗≤pj+1∗ for j∈{1,…,N}. Policy most preferred by the median member is denoted by pm∗=pN/2+1∗.

C’s proposal will be the policy that receives a supermajority of at least N2+2 members. Naturally this policy depends on the status-quo and it is easy to show that y as a function of x satisfies

For voting we again use [2] along with the assumption that all the committee members extract no information from proposal y. This makes the voting stage in the supermajoritarian model equivalent to the voting stage in the autarkical model. However, by construction, C’s proposal is always accepted and receives at least N2+2 votes.

To simulate each of the models, we start in the first period of a given path, with the previous optimal interest rate and monetary policy rate being zero. In all the simulations we restrict the policy space to be in the [−10,10] interval so that with our choice of sˉ the policy space is equal to Y={−10,−9.75,…,9.75,10}′. We do not need to look at a larger policy space, as the optimal interest rate and players’ signals stay well away from its border. As explained in the text, it is inconsequential that we allow the optimal interest rate and the monetary policy rate to attain negative values, as all the results and estimates are invariant to adding a constant to the optimal interest rate.

The values of the random variables used in the simulations are kept constant across the different models. That is, when we simulate, say, the first path for the baseline scenario of the autarkical model, the random variables used are the same as when simulating the first path of any other scenario for the same model or of any other model for the same scenario. This holds even across the N=4 and N=6 simulations, where we naturally have to add two more random variables for the two extra players, but the remaining random variables are kept the same.

A2 Further Simulation Results