ECON 222Macroeconomic Theory I

Fall Term 2012/13

Assignment 5SOLUTIONS

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Question 2: Open Economy IS-LM-FE

(a) The IS curve is derived using the equilibrium equation Sd − Id = NX or Y = Cd + Id + G + NX.Using the second:

Y = 20 + 0.7(Y − T ) − 130rw + 30 − 170rw +G+ 40 − 0.1Y − 6e

We want to rearrange this equation to get an expression for r in terms of Y:

Y = 20 + 0.7(Y − T ) − 130rw + 30 − 180rw +G+ 40 − 0.1Y − 6e

170rw + 130rw = 20 + 0.7(Y − T ) + 30 +G+ 40 − 0.1Y − 6e− Y

rw =90 +G− 0.7T − 6e− 0.4Y

300

The LM curve is derived directly from the money demand equation:

Md

P= 15 + 0.5Y − 300(rw + πe)

rw =15 + 0.5Y − 300πe − Md

P

300

The AD curve is derived by equating the IS and LM curves, and solving the resulting equation for Y:

15 + 0.5Y − 300πe − Md

P

300=

90 +G− 0.7T − 6e− 0.4Y

300

15 + 0.5Y − 300πe − Md

P= 90 +G− 0.7T − 6e− 0.4Y

0.9Y = 90 +G− 0.7T − 6e− 15 + 300πe +Md

P

Y =105 +G− 0.7T − 6e+ 300πe + Md

P

0.9

(b)First we can use the LM curve to solve for Y:

rw =15 + 0.5Y − 300πe − Md

P

300

0.1 =15 + 0.5Y − 50

30030 = 15 + 0.5Y − 50

0.5Y = 30 + 50 − 15

Y = 65/.5

Y = 130

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To solve for the real exchange rate (e), use Y = 130 in the IS curve:

rw =90 +G− 0.7T − 6e− 0.4Y

300

0.1 =90 + 8 − 0.7 ∗ 10 − 6e− 0.4 ∗ 130

30030 = 39 − 6e

6e = 9

e = 1.5

To find the price level we can use the definition of the real exchange rate: e = enom(P/PFor): e =1.5, enom = 1.2, Pfor = 2

P = e ∗ PFOR/enom = 2.5

With P = 2.5 and M/P = 50, we can find M :

M = 50 ∗ P = 125

(c)In the short run, the price level (P ) is unchanged. The shift in the LM curve will put downward pressure

on domestic interest rates and the exchange rate, causing the IS curve to shift out as well. The process willstop when domestic interest rates are equal to rw and enom stops changing. We can solve for the short-runlevel of output by using the LM curve, which has shifted out because of the 5% rise in M but with theinterest rate unchanged at rw = 0.1

rw =15 + 0.5Y − 300πe − Md

P

300

0.1 =15 + 0.5Y − 125∗1.05

2.5

3000.5Y = 30 − 15 + 52.5

Y = 135

We now use the IS curve, along with the higher level of Y(= 135) to get the new value of e:

rw =90 +G− 0.7T − 6e− 0.4Y

300

0.1 =90 + 8 − 0.7 ∗ 10 − 6e− 0.4 ∗ 135

30030 = 91 − 6e− 54

e = 1.6667

With P and PF unchaged at 2.5 and 2, respectively, we can use the definition of the real exchange rateto solve for the nominal exchange rate:

enom = e ∗ PF /P = 1.6667 ∗ 2/2.5 = 0.9333

The short-run equilibrium position of the economy is one of excess demand, which will now cause the pricelevel to start to rise. The process continues until the economy has shifted back to its long-run equilibrium

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level of output and world rate of interest but now the price level will be higher. Nothing “real” has changed,only the price level and the nominal exchange rate. We can use the Lm curve to solve for the long run valueof P , given the equilibrium levels of Y (= 130), rw = 0.1 and the now higher level of M(= 131.25):

rw =15 + 0.5Y − Md

P

3000.1 ∗ 300 = 15 + 0.5(130) − 131.25/P

P =131.25

15 + 0.5 ∗ 130 − 30

P = 2.625

(2.625 − 2.5)/2.5 = 0.05

The price level rose by 5%, and the money supply rose by 5%, which means that money is neutral.The nominal exchange rate can be derived from the definition of the real exchange rate, which is back at

its equilibrium level (e), given the new price level (P = 2.625):

enom = e ∗ PF /P = 1.5 ∗ 2/2.625 = 1.1428

(1.1426 − 1.2)1.2 = −5%

The nominal exchange rate depreciated by 5%. Relative PPP holds.

(d)Over the short run, when the level (P) is assumed to be unchanged, the increase in the foreign price level

to 3 will lower e to:e = enom ∗ P/PF = 1.2 ∗ 2.5/3 = 1

which will shift the IS curve out. The outward shift, in turn, will put upward pressure on the domesticinterest rate and the nominal exchange rate. To offset this pressure, the central bank increases the moneysupply, shifted out by enough to equate the domestic interest rate with rw. To find the short-run level of Ywe need to use the IS curve, with the new value of e and the world interest rate of 10%:

rw =90 +G− 0.7T − 6e− 0.4Y

3000.1 ∗ 300 = 90 + 8 − 7 − 6 ∗ 1 − 0.4Y

0.4Y = 91 − 6 − 30

Y = 137.5

At the short-run equilibrium position, the economy is experiencing excess demand and this will putupward pressure on the price level, which in turn will cause both the IS and the LM curves to start shiftingback. The process continues until Y has returned ot its long-run equilibrium level (Y = 130) at a point wherer = rw = 10%. Since nothing real has changed, e must as well be back at e = 1.5, its original equilibriumlevel. With enom = 1.2, we can solve for the new domestic price level:

P = PF ∗ e/enom = 1.5 ∗ 3/1.2 = 3.75

Prices have also risen by 5 percent.

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Question 3: Exchange Rates and Interest Rates Define and explain interest rate parity, purchasingpower parity and relative purchasing power parity. Which do you think is more likely to hold in the realworld?

See Abel & Bernanke pages 337-339 and 345-351

Question 4: Open IS-LMThe nation of Scooby-Doo has the following

Cd = .62(Y − T ) − 200, 000r

T = 75, 000

Ms = 934, 500

G = 85, 000

Md/P = .65(Y ) − 90000(r + πe)

Id = 215, 500 − 300, 000r

NX = 36, 000 − .05Y − 50, 000r

πe = 0

Y = 500, 000

Where Y is output at the full level of employment

(a) Find equations for the IS and LM curves. What is the long-run real rate of interest and price level?

(b) Suppose the price level is fixed in the short-run and firms are willing to produce any level of outputat the price level. Find the AD curve for Scooby-Doo with output isolated on the left hand side (LHS) ofyour equation.

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Solution

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(c) Continuing from part b, suppose the Scooby-Doo central bank increases the money supply by 3%.What is the short-run level of output and short-run real rate of interest?

(d) Suppose the nation of Scooby-Doo is now a small-open economy where the real rate of interest isequal to the world real rate of interest (rw). If rw = 12% and the money supply is equal to 934,500 what isthe new level of savings, investment, Net Exports and price level.

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Solution

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(e) Explain in words what would happen to the nation of Scooby-Doo if it increased the money supplyas a small open economy. Describe the impact on the nominal and real exchange rate and the short-run andlong-run effects using the IS-LM diagram.

Solution

See Abel & Bernanke pages 360-363

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