Central bank policy in a monetary union with heterogeneous member countries

Abstract

We analyse the policy of an independent central bank in a monetary union. The monetary policy equilibrium, prevailing under either discretion or commitment, is analogous to the one country case, although the stabilization policy is less than optimal for each single country in the monetary union. The extent of optimality of the monetary rule changes with the cross-country heterogeneity in economic shocks. Heterogeneity of preferences implies, that in a dynamic setting there is variation in the incentives of each member country. A country with a low target level of output or output cost weight might not reap any benefit from a deviation from the commitment equilibrium. The commitment policy can be enforced with a proper definition of the inflation expectations rule. With homogeneous preferences the advantages and disadvantages of the monetary union commitment policy relatively to the own discretionary one, for any new candidate or existing member country, are a function of its relative size and degree of asymmetry.

Appendix: Proofs of statements

We provide the basis for the monetary policy equilibrium in the monetary union model, with alternative assumptions about the institutional constraints on the behaviour of the monetary authority.

Conditional on the institutional constraints, the monetary policy equilibrium is determined under the assumption of rational expectations. In each model the monetary policy rule is defined as a function of the realized values of the demand or supply shocks.

In the discretionary equilibrium the monetary authority chooses in each time period the average inflation rate $\pi _{t}$, in order to minimize the loss function (4) subject to (1)–(3) and to the rational expectations constraint that the expected inflation rate in each country is equal to the average expected inflation rate: $E_{t} \left( \pi _{i t}\right) =E_{t} \left( \pi _{t}\right)$ for all $i =1 ,\ldots ,n$. In addition, under discretion the optimal monetary rule is determined in each period conditional on inflation expectations.

The first order condition for the central bank optimization problem is the following:

$$\begin{aligned} E_{t} {\displaystyle \sum _{i =1}^{n}}w_{i} \{b_{i} [\pi _{t} -E_{t} \left( \pi _{t}\right) +\varepsilon _{i t} +\upsilon _{i t} -k_{i}] +\pi _{t} +\upsilon _{i t}\} =0 \end{aligned}$$

(29)

Since the expected value of the purchasing power parity shock $\upsilon _{i t}$ for $i =1 ,\ldots ,n$, conditional on information available in period t, is by assumption equal to zero, solving Eq. (29) for the average inflation rate $\pi _{t}$ yields Eq. (5). The assumption of a quadratic loss function implies moreover, that the second order condition is satisfied.

In the commitment equilibrium the central bank selects a monetary policy rule of the form $\pi _{t} =\pi \left( \varepsilon _{1 t} ,\ldots ,\varepsilon _{n t}\right)$, to minimize the loss function (4) subject to (1)–(3) and to the rational expectations constraint $E_{t} \left( \pi _{t}\right) =E_{t} \left[ \pi \left( \varepsilon _{1 t} ,\ldots ,\varepsilon _{n t}\right) \right]$. In order to derive the optimal commitment rule, we use the linear properties of the conditional expectation operator. Denote respectively with $F \left( \varepsilon _{1 t} ,\ldots ,\varepsilon _{n t}\right)$ and $G \left( \upsilon _{1 t} ,\ldots ,\upsilon _{n t}\right)$ the period t distributions of the demand or supply shocks and of the purchasing power parity shocks. Since the demand or supply shocks and the purchasing power parity shocks are assumed to be stochastically independent, the central bank loss function resulting from Eqs. (1)–(4) takes the form:

$$\begin{aligned} \int {\displaystyle \sum _{i =1}^{n}}w_{i} \left\{ \frac{b_{i}}{2} \left[ \pi _{t} -E_{t} \left( \pi _{t}\right) +\varepsilon _{i t} +\upsilon _{i t} -k_{i}\right] ^{2} +\frac{1}{2} \left( \pi _{t} +\upsilon _{i t}\right) ^{2}\right\} d F d G \end{aligned}$$

(30)

and the rational expectations constraint is:

$$\begin{aligned} E_{t} \left( \pi _{t}\right) =\int \pi \left( \varepsilon _{1 t} ,\ldots ,\varepsilon _{n t}\right) d F \end{aligned}$$

(31)

Substituting the rational expectations constraint (31) in the loss function (30), the first order condition for the central bank optimization problem can be represented as follows:

$$\begin{aligned}&\int {\displaystyle \sum _{i =1}^{n}}w_{i} \left\{ b_{i} [\pi _{t} -E_{t} \left( \pi _{t}\right) +\varepsilon _{i t} +\upsilon _{i t} -k_{i}] +\pi _{t} +\upsilon _{i t}\right\} d G \nonumber \\&\qquad -\int {\displaystyle \sum _{i =1}^{n}}w_{i}b_{i} [\pi _{t} -E_{t} \left( \pi _{t}\right) +\varepsilon _{i t} +\upsilon _{i t} -k_{i}]\ d F d G =0 \end{aligned}$$

(32)

Computing the integrals in Eq. (32) leads to the condition:

$$\begin{aligned} \left( 1 +b\right) \pi _{t} =b \left[ E_{t} \left( \pi _{t}\right) -\varepsilon _{t}\right] \end{aligned}$$

(33)

where as before $b =\sum _{i =1}^{n}w_{i}b_{i}$ is the weighted average of the output cost weights, $\varepsilon _{t} ={\displaystyle \sum\nolimits _{i =1}^{n}}\omega _{i} \varepsilon _{i t}$ is the weighted average of the demand or supply shocks in period t, compiled using the equivalent set of weights, and the equivalent weights are defined as $0 \le \omega _{i} =w_{i}b_{i}/b \le 1$, $\sum _{i =1}^{n}\omega _{i} =1$.

The rational expectations constraint and Eq. (33) in turn imply, that in equilibrium $E_{t} \left( \pi _{t}\right) =0$ in each period t. Substituting this result in Eq. (33) yields the optimal commitment monetary policy rule (10). We note again, that the assumption of a quadratic loss function implies, that the second order condition for the central bank optimization problem is fulfilled.