Answer to Question #193601 in Macroeconomics for Jack

Question

Consider the following IS-LM model in a closed economy with prices fixed at  (we are in the short run):

Md/P = Y - r

C = 1 + Y/2

I = 1 - r/2

G = G

Ms/P = M/P

Md/P ≤ Ms/P with equality if r>0

Assume that expected inflation is zero.

Explain the minimum value that the real interest rate, r, can take. Derive the IS curve. Write down the LM curve.                                                                             

Answer 1

Solution:

The minimum value that the real interest rate, r, can take is the value at equilibrium, where the IS and LM curves intersect each other. The crossing of these two curves is the combination of the real interest rate and real GDP, denoted (r*, Y*), such that both the money market and the goods market are in equilibrium.

Derive the IS curve:

In the derivation of the IS curve, we will find out the equilibrium level of national income as established by the equilibrium in the goods market by a level of investment determined by a given interest rate. Therefore, the IS curve is associated with different equilibrium levels of national income with multiple interest rates.

Y = AD

Y = C + I + G

Y =$I + \frac{Y}{2} + 1 - \frac{r}{2} + G$

$Y - \frac{Y}{2} = 1 + 1+G - \frac{r}{2}$

$\frac{Y}{2} = 2+G - \frac{r}{2}$

Y = 4 + 2G – r

IS Curve: Y = 4 + 2G – r

 

Derive LM curve:

To derive the LM curve, we set Money demand (Md/P) equal to the money supply (Ms/P).

LM: Md/P = Ms/P

Y – r = $\frac{M}{P}$

Y = $\frac{M}{P} +r$

LM Curve: Y = $\frac{M}{P} +r$

At equilibrium: IS = LM

4 + 2G – r = $\frac{M}{P} +r$

After simplification the value of r (real interest rate) at equilibrium will be:

r = $-\frac{M}{2P} + G + 2$