Monetary Policy B

The model of unemployment and inflation can be characterized by a system of two equations:

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

((1))

$${\rm \pi } = {\rm B} - \alpha \in {\rm M}$$

((2))

The targets of the European central bank are zero inflation and zero unemployment in Europe. The instrument of the European central bank is European money supply. There are two targets but only one instrument, so what is needed is a loss function. We assume that the European central bank has a quadratic loss function:

$${\rm L}_1 = {\rm \pi }^2 + {\rm u}^2 $$

((3))

L1 is the loss to the European central bank caused by inflation and unemployment. For ease of exposition we assume equal weights in the loss function. The specific target of the European central bank is to minimize the loss, given the inflation function and the unemployment function. Taking account of equations (1) and (2), the loss function of the European central bank can be written as follows:

$${\rm L}_1 = {\rm (B + \alpha \varepsilon M)}^2 + {\rm (A} - {\rm \alpha M)}^2$$

((4))

Then the first-order condition for a minimum loss is:

$${\rm M} = \frac{{{\rm A} - \in {\rm B}}}{{{\rm \alpha + \alpha } \in ^2 }}$$

((5))

Here M is the optimum level of European money supply. An increase in A requires an increase in European money supply. And an increase in B requires a cut in European money supply. From equations (1) and (5) follows the optimum rate of unemployment in Europe:

$${\rm u} = \frac{{ \in ^2 {\rm A} - \in {\rm B}}}{{{\rm 1 + } \in ^2 }}$$

((6))

And from equations (2) and (5) follows the optimum rate of inflation in Europe:

$${\rm \pi } = \frac{{ \in {\rm A} - {\rm B}}}{{{\rm 1 + } \in ^2 }}$$

((7))

The comparison of equations (6) and (7) gives:

$${\rm u = } \in {\rm \pi }$$

((8))

Unemployment in Europe is not zero, nor is inflation there.