91Ó°ÊÓ

The given is the annual budget deficit of the government $\$40$billion

The objective is to determine how much should the government increase taxes

Every ten-billion-dollar reduction in the amount the government borrows boosts private investment by five-billion-dollars. In other words, for every $x$ reduction in government borrowing, $0.5x$ dollars is lost in private investment.

$0.75$ is the marginal propensity to consume.

To achieve a net leftward shift of $\$40$ billion in the aggregate demand curve, the government must raise taxes. To put it another way, the government needs to raise taxes to produce a net reduction in aggregate demand of $\$40$ billion.

Assume that an increase in taxes of $x$reduces aggregate demand by $\$40$billion.

Changes in AD are solely related to a tax increase of $x$dollars.

$=\frac{-MPC}{1-MPC}×x$

$$\begin{gathered} =\frac{-0.75}{1-0.75}×x \\ =-3x \end{gathered}$$

An increase in tax of $x$dollars will result in a decrease in government borrowing of $x$dollars, which will result in a decrease in private investment of $0.5x$dollars.

Change in $mathrmAD$ as a result of a drop in private investment of $0.5x$dollars.

$=\frac{1}{1-MPC}×0.5x$

$$\begin{gathered} =\frac{1}{1-0.75}×0.5x \\ =2x \end{gathered}$$

Calculate the value of $x$as follows:

Total change in $\mathrm{AD}=-\$40$billion$(-3x)+\left(2x\right)=-\$40$billion

$$\begin{gathered} -x=-40billion \\ x=40billion \end{gathered}$$

As a result, a tax increase of $\$40$ billion reduces aggregate demand by $\$40$ billion.