Consider an economy described by the following data:

$$\begin{gathered} \overline{C}=\backslash (4trillion \\ \overline{I}=\backslash )1.5trillion \\ \overline{G}=\backslash (3.0trillion \\ \overline{T}=\backslash )3.0trillion \\ \overline{NX}=\backslash (1.0trillion \\ \overline{f}=0 \end{gathered}$$

mpc = 0.8

d = 0.35

x = 0.15

a. Derive an expression for the IS curve.

b. Assume that the Federal Reserve controls the interest rate and sets the interest rate at r = 4. What is the equilibrium level of output?

c. Suppose that a financial crisis begins and f increases to f = 3. What will happen to equilibrium output? If the Federal Reserve can set the interest rate, then at what level should the interest rate be set to keep output from changing?

d. Suppose the financial crisis causes f to increase as indicated in part (c) and also causes planned autonomous investment to decrease to I = \)1.1 trillion. Will the change in the interest rate implemented by the Federal Reserve in part (c) be effective in stabilizing output? If not, what additional monetary or fiscal policy changes could be implemented to stabilize output at the original equilibrium output level given in part (b)?

$$\begin{gathered} \overline{C}=\$4trillion \\ \overline{I}=\$1.5trillion \\ \overline{G}=\$3.0trillion \\ \overline{T}=\$3.0trillion \\ \overline{NX}=\$1.0trillion \\ \overline{f}=0 \end{gathered}$$

mpc = 0.8

d = 0.35

x = 0.15

The IS curve can be derived from the output equation

$$\begin{gathered} Y=(C+I-df+G+NX-mpc\times T)\times \frac{1}{mpc}-\frac{d+x}{1-mpc}\times r \\ Y=(4+1.5-0.35(0)+3.0+1.0+-0.8\times 3)\times \frac{1}{1-0.8}-\frac{0.35+0.15}{1-0.8}\times r \\ Y=7.1\times \frac{1}{0.2}-\frac{0.5}{0.2}r \\ Y=35.5-2.5r \end{gathered}$$

When r=4,

$$\begin{gathered} Y=35.5-25r \\ Y=35.5-2.5\left(4\right) \\ Y=25.5 \end{gathered}$$

When f=3,

$$\begin{gathered} Y=(C+I-df+G+NX-mpc\times T)\times \frac{1}{mpc}-\frac{d+x}{1-mpc}\times r \\ Y=(4+1.5-0.35(3)+3.0+1.0+-0.8\times 3)\times \frac{1}{1-0.8}-\frac{0.35+0.15}{1-0.8}\times r \\ Y=6.05\times \frac{1}{0.2}-\frac{0.5}{0.2}r \\ Y=30.25-2.5r \end{gathered}$$

The output level at r=4,

$$\begin{gathered} Y=35.5-25r \\ Y=30.5-2.5\left(4\right) \\ Y=20.5 \end{gathered}$$

Now, suppose the federal reserve wants to keep the output constant at $25.5 trillion.

$$\begin{gathered} 25.5=30.25-2.5r \\ -4.75=-2.5r \\ r=1.9 \end{gathered}$$

Now, if the financial crisis causes investment to decrease to $1.1 trillion then, the total output level would be

$$\begin{gathered} Y=(C+I-df+G+NX-mpc\times T)\times \frac{1}{mpc}-\frac{d+x}{1-mpc}\times r \\ Y=(4+1.1-0.35(3)+3.0+1.0+-0.8\times 3)\times \frac{1}{1-0.8}-\frac{0.35+0.15}{1-0.8}\times r \\ Y=5.65\times \frac{1}{0.2}-\frac{0.5}{0.2}r \\ Y=28.25-2.5r \end{gathered}$$

Now, if the interest rate is kept at 1.9% then,

$$\begin{gathered} Y=28.25-2.5(1.9) \\ Y=23.5 \end{gathered}$$

So, the equilibrium level of output would be reduced to $23.5 trillion.

The reduction in the interest rate has not been able to bring back output to the previous equilibrium level. Other policy tools such as reduction in taxes, increase in government spending, or reducing interest rates even further can be adopted.