Cooperation between Central Bank and Government

The model of unemployment, inflation, and the structural deficit can be characterized by a system of three equations:

$${\rm u} = {\rm A} - {\rm M} - {\rm G}$$

((1))

$${\rm \pi } = {\rm B} + {\rm M} + {\rm G}$$

((2))

$${\rm s} = {\rm G} - {\rm T}$$

((3))

The policy makers are the European central bank and the European government. The targets of policy cooperation are zero inflation, zero unemployment, and a zero structural deficit. The instruments of policy cooperation are European money supply and European government purchases. There are three targets but only two instruments, so what is needed is a loss function:

$${\rm L} = {\rm \pi }^{\rm 2} + {\rm u}^{\rm 2} + {\rm s}^{\rm 2}$$

((4))

L is the loss caused by inflation, unemployment, and the structural deficit. We assume equal weights in the loss function. The specific target of policy cooperation is to minimize the loss, given the inflation function, the unemployment function, and the structural deficit function. Taking account of equations (1), (2) and (3), the loss function under policy cooperation can be written as follows:

$${\rm L} = ({\rm B} + {\rm M} + {\rm G})^2 ({\rm A} - {\rm M} - {\rm G})^2 + ({\rm G} - {\rm T})^2 $$

((5))

Then the first-order conditions for a minimum loss are:

$$2{\rm M} = {\rm A} - {\rm B} - 2{\rm G}$$

((6))

$$3{\rm G} = {\rm A + T} - {\rm B} - 2{\rm M}$$

((7))

Equation (6) shows the first-order condition with respect to European money supply. And equation (7) shows the first-order condition with respect to European government purchases.