Austerity versus Expansionary policies

When an economy is in recession, its government is the mostly likely to use expansionary policies to stimulate its economy. However, when the government is also in deep deficit, austerity may be introduced in the hope of lowering the government debt level. Government’s deficit is equal to government expenditure minus tax incomes. If we assume that tax received is positively correlated to the GDP and government expenditure is negatively proportional to the GDP as governments tend to expand fiscal and monetary policies in recession, then government balance is in the same direction as the economy growth.

Now we assume:

GDP=f(G), dGDP/dG>0, T=g(GDP), dT/dGDP>0 (GDP:GDP, G: government expenditure, T: tax income);

then: dT/dG=(dT/dGDP)*(dGDP/dG)>0.

However, dT>dG has to be greater than 1 in order to use expansionary policies to improve government budget.

dGDP/dG could be considered as the government expenditure multiplier effect ratio, dT/dGDP is the rate of taxes returning from the economy. The product of these two figures which could change with GDP figures has to be greater one in order to change expansionary policies to improve both GDP and government budget. Otherwise, austerity is a wiser choice to prevent worsening government deficit.