Question

Consider an economy described by the following data:

$$\begin{gathered} \overline{C}=\backslash (3.25trillion \\ \overline{I}=\backslash )1.3trillion \\ \overline{G}=\backslash (3.5trillion \\ \overline{T}=\backslash )3.0trillion \\ \overline{NX}=-\backslash (1.0trillion \\ \overline{f}=1 \end{gathered}$$

mpc = 0.75

d = 0.3

x = 0.1

a. Derive simplified expressions for the consumption function, the investment function, and the net export function.

b. Derive an expression for the IS curve.

c. If the real interest rate is r = 2, what is equilibrium output? If r = 5, what is equilibrium output?

d. Draw a graph of the IS curve showing the answers from part (c) above.

e. If government purchases increase to \)4.2 trillion, what will happen to equilibrium output at r = 2? What will happen to equilibrium output at r = 5? Show the effect of the increase in government purchases in your graph from part (d).

Short answer

a. C = 1+0.5Y; NX = -1+0.1Y; I = 1.3-0.3r

b. Y = 32-2r

c. r=28; r=22

d.

e.

Step-by-step solution

Step 1. Given information

$$\begin{gathered} \overline{C}=\$3.25trillion \\ \overline{I}=\$1.3trillion \\ \overline{G}=\$3.5trillion \\ \overline{T}=\$3.0trillion \\ \overline{NX}=-\$1.0trillion \\ \overline{f}=1 \end{gathered}$$

mpc = 0.75

d = 0.3

x = 0.1

Step 2. Explanation Part (a)

The consumption can be expressed as,

$$\begin{gathered} C=3.25+0.75(Y-T) \\ =3.25+0.75(Y-3) \\ =3.25+0.75Y-2.25 \\ =1+0.75Y \end{gathered}$$

Similarly, the net exports can be expressed as,

$NX=-1+0.1Y$

And investment can be expressed as,

$I=1.3-0.3r$

Step 3. Explanation Part (b)

The IS curve can be derived from the output equation

$$\begin{gathered} Y=C+I+G+NX \\ Y=1+0.75Y+1.3-0.3r+3.50-1+0.1Y \\ Y=4.8+0.85Y-0.3r \\ Y-0.85Y=4.8-0.3r \\ 0.15Y=4.8-0..3r \\ Y=(4.8/0.15)-(0.3/0.15)r \\ Y=32-2r \end{gathered}$$

Step 4. Explanation Part (c)

$$\begin{gathered} Whenr=2, \\ Y=32-2\left(2\right)=28 \\ Whenr=5, \\ Y=32-2\left(5\right)=22 \end{gathered}$$

Step 5. Explanation Part (d)

The graph below shows the IS curve.

Step 6. Explanation Part (e)

If the government expenditure increases to $4.2 trillion, the new IS curve would be

$$\begin{gathered} Y=1+0.75Y+1.3-0.3r+4.2-1+0.1Y \\ Y=5.5+0.85Y-0.3r \\ Y-0.85Y=5.5-0.3r \\ 0.15Y=5.5-0..3r \\ Y=(5.5/0.15)-(0.3/0.15)r \\ Y=36.67-2r \end{gathered}$$

Now, when r=2,

$Y=36.67-2\left(2\right)=32.67$

and, when r=5

$Y=36.67-2\left(5\right)=26.67$

The relationship can be expressed as follow: