Fiscal Policy B

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

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

((1))

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

((2))

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

((3))

The targets of the European government are zero unemployment, zero inflation, and a zero structural deficit. The instrument of the European government is European government purchases. There are three targets but only one instrument, so what is needed is a loss function. We assume that the European government has a quadratic loss function:

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

((4))

L₂ is the loss to the European government caused by inflation, unemployment, and the structural deficit. We assume equal weights in the loss function. The specific target of the European government 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 of the European government can be written as follows:

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

((5))

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

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

((6))

Here G is the optimum level of European government purchases. An increase in A requires an increase in European government purchases. And an increase in B requires a cut in European government purchases. From equations (1) and (6) follows the optimum rate of unemployment in Europe:

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

((7))

From equations (2) and (6) follows the optimum rate of inflation in Europe:

$${\rm 3\pi } = 2{\rm A} + {\rm B} + {\rm T}$$

((8))

And from equations (3) and (6) follows the optimum structural deficit ratio:

$${\rm 3s} = {\rm A} - {\rm B} - 2{\rm T}$$

((9))

Unemployment in Europe is not zero. And the same holds for inflation and the structural deficit there.