# Why does decrease in fixed taxes accompanied by a decrease in Government Spending of the same amount will decrease GDP?

I personally don't agree because change in GDP caused by decreased in fixed taxes should be

$\triangle$ fixed taxes $\times \frac{-b}{1-MPE}$(fix tax multiplier), which turns out to be a positive number while change caused by decrease of Government Spending is

$\triangle$ Government Spending $\times \frac{1}{1-MPE}$(autonomous multiplier) which is a negative number. Therefore I don't think we can make conclusion. Does anyone know what did I do wrong?

• b is the marginal propensity to consume (which is less than one) so the government multiplier is going to be greater than the tax multiplier which implies that the negative change in GDP will be greater than the positive change, so GDP decreases Mar 25, 2016 at 22:19

The question basically boils down to whether the inequality below is true or not. $$\bigg|\Delta G\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\Delta T\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$ If $\Delta G=\Delta T$, we can divide by either and get $$\bigg|\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$ If we multiply each side by $1-MPC$ and we evaluate the absolute value, we get $$1>MPC$$ This statement is true, so that implies that the first inequality is true. By this inequality, we know that $\Delta GDP_G>\Delta GDP_T$. Since $\Delta GDP_G<0$ and $\Delta GDP_T>0\implies \Delta GDP<0$