14.02 Principles of Macroeconomics

Problem Set 5 Solutions

1)
*(3 points) *True.
The *eventual* effect of a depreciation should be an increase in
net exports. However, after the
depreciation people and firms may not be able to immediately adjust their
transactions to the new prices. Because
imports will now be more expensive due to the weaker currency, net exports will
fall. (That is, X and Q stay about the
same, but e rises. So, NX = X - eQ will
fall.) Over time people will adjust to
the new exchange rate (change X and Q) and net exports will rise. This is the J-curve – prices change
immediately, but quantities move more slowly.

2)
*(3 points) *True. When the government reduces the deficit (by
either raising T or lowering G), it decreases aggregate demand and lowers Y. This means that, for a given exchange rate,
NX rises: NX(Y, Y*, E) = X(Y*, E) –
EQ(Y, E). Y only directly enters this
expression through its positive effect on imports, so when Y falls imports
fall, and NX rises (reducing the trade deficit or increasing the trade
surplus.) Additionally, if you allow
the exchange rate to move, this enhances the effect – the reduction in the
deficit leads to lower interest rates, and thus through the interest parity
equation, implies that the currency depreciates (E rises). This increases the positive effect on net
exports. NOTE: One of the basic macro
accounting identities tells us: NX = (T – G) + (S – I). If you answer this question in terms of this
identity, and assume (S-I) changes, the change in (S-I) may offset the change
in (T-G). (You must explicitly state
that (S-I) changes in order for this to question to be false.)

3)
*(3 points) *False. Bondholders require equal expected rates of
return in each country. As the interest
rate parity condition shows, this consideration includes not just the interest
rates but expected changes in the exchange rate. People may hold a country’s bonds at a zero interest rate if they
expect currency appreciation in the future.
(Although it is possible that they would just hold the currency rather
than the bonds because they hold equivalent returns.)

4)
*(Does not count – everyone gets 3 free points here.) *The answer to this question is a bit
confusing, which is why we are not going to take off any points for either a
true or false answer. The key point
that you should realize is that under fixed exchange rates, the central bank
does not have freedom to adjust monetary policy as it might see fit to affect
the economy. So it will not conduct
OMOs in order to use monetary policy to change interest rates and output. However, the central bank still has the
freedom to engage in transactions in the bond market, and may do so as part of
its policy to sustain the exchange rate and to keep the interest rate parity
equation in balance. With E fixed, the
bank must agree to buy and sell domestic and foreign currency at the fixed
exchange rate in the open currency market to keep interest parity, and may also
buy and sell bonds in the open market as part of its more complicated decisions
about portfolio structure.

5)
*(3 points) *False.
The direct effect of increased foreign income is greater demand for
exports. This has an expansionary
effect on the domestic economy. In
general, statements regarding economic ‘competition’ between countries are
questionable. Larger foreign economies
provide larger market for domestic goods.
Cheaper foreign goods provide benefits to domestic consumers.

(a) *(5
points) *Closed Economy:

Y = Z = 20 + .6(Y – T) + I + G

Y = (5/2) (20 -.6T + I + G) à multiplier is 5/2

Open Economy:

Y = Z = 20 + .6(Y – T) + I + G +
.1Y^{*} - .1Y

Y = 2(20
-.6T + I + G + .1Y^{*}) à multiplier is 2

The multiplier is lower in the open economy because .1 of every additional dollar of income is spent on foreign goods and is therefore not cycled back through the domestic economy.

(b) *(5
points) *From part 1:

Y = 2(20
-.6T + I + G + .1Y^{*})

Since the two economies are the same:

Y^{*}
= 2(20 -.6T^{*} + I^{*}
+ G^{*} + .1Y)

Plugging the second into the first and simplifying:

Y =
(25/24)*2*[24 – .6T + I + G + (1/5)(-.6T^{*} + I^{*} +G^{*})]

Plugging in for taxes and investment (doing this for later):

Y =
(25/12)[36 + G + (1/5)G^{*}]

à Y = Y^{*} = 100

The multiplier is 25/12 > 2. It is greater now because increasing domestic output through
increased demand (say, a fiscal expansion) will result in an increase in
foreign output (Y^{*}) because of increased demand for foreign goods in
the form of exports (from the foreign country’s perspective). The increase in foreign output will likewise
result in an increase in domestic demand in the form of exports to the foreign
country. Of course, this process
repeats. This feedback effect raises
the multiplier.

If the domestic economy is ‘small’, changes in it’s income (and demand for imports) will have little significant effect on the foreign country. In other words, foreign output/income will not change in response, and the feedback effect just described will be negligible. As a result, the domestic multiplier will not be noticeably different in open economy than in the close economy.

(c) *(5 points) *From
above, recall that:

Y = C + I + G – Q + X = 20 + .6Y – 6 + 16 + G - .1Y +.1Y*, and

Y* = 20 + .6Y* – 6 + 16 + 10 - .1Y* +.1Y

If the home country changes G such that Y=125, and the foreign country does not change G* (G* = 10), you can calculate Y* -- just plug Y=125 into the second equation above. This gives Y*= 105. Then take Y=125, Y*=105 and plug them into the first equation, solving for G. This means G=22, so the budget deficit is (G-T) = 12.

The other method is to use the multiplier – you found it in (b). The change in Y (25) divided by the multiplier gives you the change in G – this also gives you 12 as the budget deficit, with total G =22.

Imports will be .1Y = 12.5.
Exports will be .1Y^{*} = .1(105) = 10.5. This makes the net exports equal to -2.

(d) *(5
points) *If G = G^{*}, then
we can use the equation from (b) to find what they must be equal to for Y =
125.

Y = (25/12)[36 + G + (1/5)G] = 125

à 36 + (6/5)G = 60 à G = 20

Both countries will have a budget deficit of G – T = 10. Since everything is symmetric, imports will equal exports, and net exports will be 0 for both countries.

(e) *(5
points) *Coordinating a joint fiscal
expansion as in part (d) is difficult because each country will want to shirk
on the agreement and benefit from the other’s expansion while not increasing
its budget deficit as much as agreed.
This is hard to police in practice because often fiscal variables are
affected by thing not directly in the government’s control (such as taxes,
which depend on incomes and other taxed variables). Furthermore, fiscal variables are largely in the control of
legislatures, who can’t necessarily coordinate their activities well.

(a) *(6
points) *IS:

Y = Z = 10 + .6(Y – 10) +10 – 20i + .2Y + 10

à (1/5)Y = 24 – 20i

à Y = 120 – 100i (IS)

LM:

Y/10i = 50

à Y = 500i (LM)

Solving for the equilibrium:

120 – 100i = 500i

à i = 1/5

à Y = 100

(b) *(6
points) *IS:

Y = Z = 10
+ .6(Y – 10) +10 – 20i + .2Y + 10 + .2Y^{*} - .2Y – 20/E

à (2/5)Y = 44 – 20i – 20/E

Using the interest rate parity condition:

(2/5)Y = 44 – 20i – 20(.6 + i)

à Y = (5/2)(32 – 40i)

à Y = 80 – 100i (IS)

LM is the same as in part a.

So, in equilibrium, 500i = 80 – 100i. This gives i = 2/15 = .13. Then, Y = 200/3 = 66.7. The exchange rate is E = 1/(1 + .13 - .4) = 1.37.

(c) *(6
points) *Adjusting the IS:

Y = Z = 10
+ .6(Y – 10) +10 – 20i + .2Y + 20 + .2Y^{*} - .2Y – 20/E

à (2/5)Y = 54 – 20i – 20(.6 + i)

à Y = (5/2)(42 – 40i)

à Y = 105 – 100i

So, the IS curve shifts outward. LM is still the same.

In equilibrium, 500i = 105 – 100i. This gives i = 7/40 = .175. Then, Y = 87.5. The exchange rate is E = 1/(1 + .175 - .4) = 1.29.

In words: the fiscal expansion has increased demand, therefore output is higher. But with fixed money supply, this led to higher interest rates, and, through interest parity, we had an appreciation. Note (not required in answer): NX has unambiguously decreased - higher Y, lower E.

(d) *(6
points) *New LM:

Y/10i = 100

à Y = 1000i à i = (1/1000)Y

So, the LM shifts down. IS is the same as part (b).

In equilibrium, 1000i = 80 – 100i. This gives i = 4/55 = .073. Then, Y = 73. The exchange rate is E = 1/(1 + .073 - .4) = 1.48.

In words: the monetary expansion has decreased interest rates, which stimulated investment, and, through a depreciation, boosted net exports. These led to higher output. Note (not required in answer): the total effect on net exports is ambiguous with a monetary expansion -- higher Y but higher E. The effect here was an overall decrease, if you check the numbers.)

(e) *(6
points) *The fiscal expansion
increases output, but it also raises the interest rate. Everything else equal, this will cause a
currency appreciation. Under a system
of fixed exchange rates, this isn’t allowed.
So, the central bank must respond with a monetary expansion in order to
lower the interest rate to the original level.
(What might also happen is that people will start giving foreign
exchange for domestic currency, which the CB must be giving them - with this,
they automatically increase the money supply.)
This will, of course, result in a further expansion of output.

Under a system of fixed exchange rates, fiscal policy changes have a magnified effect on output because of the necessary monetary adjustment that follows them.