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
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)
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.