Optimal Conservatism and Collective Monetary Policymaking under Uncertainty

Abstract

We study how the optimal degree of conservatism relates to decision-making procedures in a Monetary Policy Committee (MPC). In our framework, central bank conservatism is required to attenuate the volatility of monetary decisions generated by the presence of uncertainty about the committee members’ output objective. We show how this need for conservatism varies according to the number of MPC members, the MPC’s composition as well as its decision rule. Moreover, we find that extra central bank conservatism is required when there is ambiguity about the MPC’s true decision rule.

Appendix

Proof of Result 1

From expression (15), it is easy to see that $\frac {\partial f}{ \partial V\left ({\epsilon _{t}^{j}}\right )}>0$. Hence, a rise in $V\left ({\epsilon _{t}^{j}}\right )$ causes an upward shift of the function f and thereby a shift to the right of the intersection point between the 45^∘ line and the function f curve, implying an increase in λ _{C B∗}.

As $\epsilon _{t}^{AR}={{\sum }_{i}^{n}}{\epsilon _{t}^{i}}/n$, the aggregation process implies: $E\left (\epsilon _{t}^{AR}\right )=0$ and $V\left (\epsilon _{t}^{AR}\right )=\sigma _{\epsilon }^{2}/n$. Further, since $\epsilon _{t}^{MR}=\text {median} \left [{\epsilon _{t}^{1}},...,{\epsilon _{t}^{n}}\right ]$, we have $E\left (\epsilon _{t}^{MR}\right )=0$ and $V\left (\epsilon _{t}^{MR}\right )=\frac {\Pi }{2n}\sigma _{\epsilon }^{2}$.¹⁸

• i) It is obvious from Eq. 15 that λ _{C B∗} > λ _G even if ρ = 0.

• ii) Since $V\left (\epsilon _{t}^{CBi}\right )=\sigma _{\epsilon }^{2}$ , it follows that $V\left (\epsilon _{t}^{CBi}\right )>V\left (\epsilon _{t}^{MR}\right )$ and $V\left (\epsilon _{t}^{CBi}\right )>V\left (\epsilon _{t}^{AR}\right )$. Consequently, $\lambda _{CB\ast }^{CBi}>\lambda _{CB\ast }^{MR}$ and $\lambda _{CB\ast }^{CBi}>\lambda _{CB\ast }^{AR}$.

• iii) Since $\frac {\partial V\left (\epsilon _{t}^{AR}\right )}{ \partial n}<0$ and $\frac {\partial V\left (\epsilon _{t}^{MR}\right )}{\partial n}<0$, $\lambda _{CB\ast }^{AR}$ and $\lambda _{CB\ast }^{MR}$ depend negatively on n.

• iv) Finally, as $V\left (\epsilon _{t}^{AR}\right )<V\left (\epsilon _{t}^{MR}\right )$ we have $\lambda _{CB\ast }^{AR}<\lambda _{CB\ast }^{MR}$, according to the graphical analysis.

□

Proof of Result 2

To demonstrate result 2, we begin by deriving $V\left (\epsilon _{t}^{GEN}\right )$:

$$\begin{array}{@{}rcl@{}} V\left(\epsilon_{t}^{GEN}\right) &=&E\left\{p\ \epsilon_{t}^{chair}+(1-p)\left[q\epsilon_{t}^{ARc}+(1-q)\epsilon_{t}^{MRc}\right]\right\}^{2} \\ &=&p^{2}\sigma_{chair}^{2}+(1-p)^{2}\left[ q^{2}\frac{{\sigma_{b}^{2}}}{n_{b} }+(1-q)^{2}\frac{\Pi \sigma_{ext}^{2}}{2n_{ext}}\right] \end{array} $$

(19)

Differentiating this expression with respect to p yields:

$$ \frac{\partial V\left(\epsilon_{t}^{GEN}\right)}{\partial p}\,=\,2p\left[\sigma_{chair}^{2}\,+\,q^{2}\frac{{\sigma_{b}^{2}}}{n_{b}}\,+\,(1-q)^{2}\frac{\Pi \sigma_{ext}^{2}}{2n_{ext}}\right]-2\left[q^{2}\frac{{\sigma_{b}^{2}}}{n_{b}}+(1-q)^{2} \frac{\Pi \sigma_{ext}^{2}}{2n_{ext}}\right] $$

(20)

This derivative is negative if $p<\frac {q^{2}\frac {{\sigma _{b}^{2}}}{n_{b}} +(1-q)^{2}\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}{\sigma _{chair}^{2}+q^{2} \frac {{\sigma _{b}^{2}}}{n_{b}}+(1-q)^{2}\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}} }=p_{\min }$ and becomes positive otherwise. Hence, $p_{\min }$ minimises $V\left (\epsilon _{t}^{GEN}\right )$ and the optimal degree of conservatism $\lambda _{CB\ast }^{GEN}$ as well.

We then turn to the council and differentiate $V\left (\epsilon _{t}^{GEN}\right )$ with respect to q. In doing this, we obtain:

$$ \frac{\partial V\left(\epsilon_{t}^{GEN}\right)}{\partial q}=2q\frac{{\sigma_{b}^{2}}}{ n_{b}}-2(1-q)\frac{\Pi \sigma_{ext}^{2}}{2n_{ext}} $$

(21)

This derivative is negative for $q<q_{\min }=\frac {\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}{\frac {{\sigma _{b}^{2}}}{n_{b}}+\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}$ and positive otherwise. As a consequence, $q_{\min }$ minimises $V\left (\epsilon _{t}^{GEN}\right )$ and thereby the optimal degree of conservatism $\lambda _{CB\ast }^{GEN}$. □

Proof of Result 3

To solve problem (16), the first stage is to identify the realizations of (p ^UN,q ^UN) that maximise the expected social loss:

$$\begin{array}{@{}rcl@{}} \max\limits_{p^{UN},q^{UN}}E\left[{L_{t}^{G}}\left(\epsilon_{t}^{UN}\right)\right]&=&\max\limits_{p^{UN},q^{UN}}\left\{ \frac{\lambda_{G}\alpha^{2}+1}{ \left( \alpha^{2}\lambda_{CB}+1\right)^{2}}\ E\left(\epsilon_{t}^{UN}\right)^{2}\right.\\ &&\qquad\qquad\left.+\frac{\lambda_{G}+\alpha^{2}\left( \lambda^{CB}\right)^{2}}{(1-\rho^{2})\left( \alpha^{2}\lambda_{CB}+1-\beta \rho \right)^{2}}\right\} \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} \text{where } E\left(\epsilon^{UN}_{t}\right)^{2}&\,=\,&E\left\{p^{UN}\ \epsilon^{chair}_{t}\,+\,(1-p^{UN})\left[q^{UN}\epsilon^{ARc}_{t}+(1-q^{UN}) \epsilon^{MRc}_{t}\right]\right\}^{2} \\ &=& E\left(p^{UN}\right)^{2} \sigma^{2}_{chair}+ E(1-p^{UN})^{2}\\ &&\times\left[ \left(q^{UN}\right)^{2}\frac{{\sigma^{2}_{b}}}{n_{b}}+(1-q^{UN})^{2}\frac{ {\Pi}\sigma^{2}_{ext}}{2n_{ext}}\right]. \end{array} $$

Note that $E\left [{L_{t}^{G}}\left (\epsilon _{t}^{UN}\right )\right ]$ only depends on p ^UN and q ^UN via $E\left (\epsilon _{t}^{UN}\right )^{2}$. The social planner first determines the allocation of decision power within the council that maximises the expected social loss. Differentiating $E\left (\epsilon _{t}^{UN}\right )^{2}$ with respect to q ^UN yields

$$\frac{\partial E\left(\epsilon_{t}^{UN}\right)^{2}}{\partial q^{UN}}=2q^{UN}\frac{ {\sigma_{b}^{2}}}{n_{b}}-2(1-q^{UN})\frac{\Pi \sigma_{ext}^{2}}{2n_{ext}}. $$

As has already been demonstrated, for a given p, $E\left (\epsilon _{t}^{UN}\right )^{2}$ attains its minimum for $q_{min}=\frac {\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}{\frac {{\sigma _{b}^{2}}}{n_{b}}+\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}$ and thus its maximum for extreme values of q in the interval [0,1]. We finally compare $E\left (\epsilon _{t}^{UN}|_{q=0}\right )^{2}= \frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}$ with $E\left (\epsilon _{t}^{UN}|_{q=1}\right )^{2}=\frac {{\sigma _{b}^{2}}}{n_{b}}$ to show that if $\frac { {\Pi } \sigma _{ext}^{2}}{2n_{ext}}>(<)\frac {{\sigma _{b}^{2}}}{n_{b}}$, q _{m a x} – i.e. the value of q that maximises $E\left (\epsilon _{t}^{UN}\right )^{2}$ and thus $E\left [{L_{t}^{G}}\left (\epsilon _{t}^{UN}\right )\right ]$ – is equal to 0 (1).

Once q _{m a x} has been determined, the social planner turns to p _{m a x} , the value of p that maximises $E\left (\epsilon _{t}^{UN}\right )^{2}$ and thus $E\left [{L_{t}^{G}}\left (\epsilon _{t}^{UN}\right )\right ]$.

Taking the derivative of $E\left (\epsilon _{t}^{UN}\right )^{2}$ with respect to p ^UN yields

$$\begin{array}{@{}rcl@{}} \frac{\partial E\left(\epsilon_{t}^{UN}\right)^{2}}{\partial p^{UN}}&=&2p^{UN}\left[\sigma_{chair}^{2}+q_{max}^{2}\frac{{\sigma_{b}^{2}}}{n_{b}}+(1-q_{max})^{2}\frac{ {\Pi} \sigma_{ext}^{2}}{2n_{ext}}\right]\\ &&-2\left[q_{max}^{2}\frac{{\sigma_{b}^{2}}}{n_{b}} +(1-q_{max})^{2}\frac{\Pi \sigma_{ext}^{2}}{2n_{ext}}\right] \end{array} $$

(23)

For a given q=q _{m a x} , $E\left (\epsilon _{t}^{UN}\right )^{2}$ has its minimum at $p_{min}=\frac {q^{2}\frac {{\sigma _{b}^{2}}}{n_{b}}+(1-q)^{2}\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}{\sigma _{chair}^{2}+q^{2}\frac {{\sigma _{b}^{2}}}{n_{b}} +(1-q)^{2}\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}}$ and its maximum for extreme values of p in the interval [0,1]. We thus compare $E\left (\epsilon _{t}^{UN}|_{p=0}\right )^{2}=\left (q_{max}\right )^{2}\frac {{\sigma _{b}^{2}}}{n_{b}} +(1-q_{max})^{2}\frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}$ with $E\left (\epsilon _{t}^{UN}|_{p=1}\right )^{2}=\sigma _{chair}^{2}$.

If $\left (q_{max}\right )^{2}\frac {{\sigma _{b}^{2}}}{n_{b}}+(1-q_{max})^{2} \frac {\Pi \sigma _{ext}^{2}}{2n_{ext}}>\sigma _{chair}^{2}$, then p _{m a x} =0 , otherwise p _{m a x} =1. In both cases, $E\left (\epsilon _{t}^{UN}\right )^{2}=V\left (\epsilon _{t}^{UN}\right )>V\left (\epsilon _{t}^{GEN}\right )$ – the latter being defined by Eq. 19 – which means that the optimal degree of conservatism, $\lambda _{CB\ast }^{UN}$, when the MPC’s decision procedure is unknown is higher than $\lambda _{CB\ast }^{GEN}$, the one obtained under transparency about the decision procedure. □