Question

Consider an economy described by the following:

$$\begin{gathered} \overline{C}=\backslash (4trillion \\ \overline{I}=\backslash )1.5trillion \\ \overline{G}=\backslash (3.0trillion \\ \overline{T}=\backslash )3.0trillion \\ \overline{NX}=\$1.0trillion \\ \overline{f}=0 \\ mpc=0.8 \\ d=0.35 \\ x=0.15 \\ \lambda =0.5 \\ \overline{r}=2 \end{gathered}$$

(a) Derive expressions for the MP curve and the AD curve.

(b) Calculate the real interest rate and aggregate output when$\pi =2$and$\pi =4$

(c) Draw a graph of the MP curve and the AD curve, labeling the points given in part (b).

Short answer

(a) MP curve - $r=2+0.5\mathrm{\pi }$, AD curve -$Y=30.5-1.25\mathrm{\pi }$

(b) At$\mathrm{\pi }=2$, r = 3%, Y= 28; At$\mathrm{\pi }=4$, r=4%, Y=25.5

(c)

Step-by-step solution

Step 1. Given information

$$\begin{gathered} \overline{C}=\$4trillion \\ \overline{I}=\$1.5trillion \\ \overline{G}=\$3.0trillion \\ \overline{T}=\$3.0trillion \\ \overline{NX}=\$1.0trillion \\ \overline{f}=0 \\ mpc=0.8 \\ d=0.35 \\ x=0.15 \\ \lambda =0.5 \\ \overline{r}=2 \end{gathered}$$

Step 2. Explanation

(a) The IS curve shows the different points where the goods market is in equilbrium.

$$\begin{gathered} Y=\overline{C}+\overline{I}-d\overline{f}+\overline{G}+\overline{NX}-mpc\times \overline{T}\times \frac{1}{1-mpc}-\frac{d+x}{1-mpc}\times r \\ Y=[4+1.5-0.35(0)+3+1-0.8(3\left)\right]\times \frac{1}{1-0.8}-\frac{0.35+0.15}{1-0.8}\times r \\ Y=35.5-2.5r \end{gathered}$$

The expression for MP curve would be,

$$\begin{gathered} r=\overline{r}+\lambda \pi \\ r=2+0.5\mathrm{\pi } \end{gathered}$$

Combining the two expression can give the expression for AD curve,

role="math" localid="1648125126675" $$\begin{gathered} Y=35.5-2.5\mathrm{r} \\ \mathrm{Y}=35.5-2.5(2+0.5\mathrm{\pi }) \\ \mathrm{Y}=35.5-5-1.25\mathrm{\pi } \\ \mathrm{Y}=30.5-1.25\mathrm{\pi } \end{gathered}$$

Step 3. Explanation Part (b)

Now, the output level at different rate of inflation can eb obtained by putting the value in the AD curve expression.

$$\begin{gathered} \pi =2\% \\ Y=30.5-1.25\left(2\right) \\ Y=28 \\ \pi =4\% \\ Y=30.5-1.25\left(4\right) \\ Y=25.5 \end{gathered}$$

The interest rates can be obtained similarly,

$$\begin{gathered} \pi =2\% \\ r=2+0.5\left(2\right) \\ r=3 \\ \pi =4\% \\ r=2+0.5\left(4\right) \\ r=4 \end{gathered}$$

Step 4. Explanation part (c)