Interaction between Central Bank and Government

An increase in European money supply lowers unemployment in Europe. On the other hand, it raises inflation there. Correspondingly, an increase in European government purchases lowers unemployment in Europe. On the other hand, it raises inflation there. The primary target of the European central bank is zero inflation in Europe. By contrast, the primary target of the European government is zero unemployment there. The model of unemployment and inflation can be represented by a system of two equations: (1) $${\rm u} = {\rm A} - {\rm \alpha M} - {\rm \beta G}$$ (2) $${\rm \pi } = {\rm B} + {\rm \alpha } \in {\rm M} - {\rm \beta } \in {\rm G}$$ Of course this is a reduced form. Here u denotes the rate of unemployment in Europe, π is the rate of inflation in Europe, M is European money supply, G is European government purchases, α is the monetary policy multiplier with respect to unemployment, αε is the monetary policy multiplier with respect to inflation, β is the fiscal policy multiplier with respect to unemployment, βε is the fiscal policy multiplier with respect to inflation, A is some other factors bearing on the rate of unemployment in Europe, and B is some other factors bearing on the rate of inflation in Europe. The endogenous variables are the rate of unemployment and the rate of inflation in Europe.