Answer for Homework 7: Modern Macroeconomics I

1. Suppose that s fraction of employed workers are separated. Assume that m proportionof unemployment workers meets such a job. Let E and U denote the number ofemployed workers and unemployed workers, respectively.
(a) Derive the dynamics of the number of unemployed workers.
Answer Note that the number of unemployed workers at period t + 1 who areemployed workers at period t is sEt and that the number of unemployedworkers at period t + 1 who are unemployed workers at period t is (1 m) Ut .
Then, the total number of unemployed workers at period t + 1 is
Ut+1 = sEt + (1m) Ut :Therefore, the dynamics of unemployment is
Ut+1 = Ut + sEt

mUt :(b) Suppose that Ut+1 = Ut on the steady state. The natural rate of unemployment
is de…ned as the unemployment on the steady state. Derive the natural rate ofunemployment as a function of s and m.
Answer In the steady state, we havesEt = mUt :Since unemployed workers and employed workers consists of labor force:
Nt = Et + Ut
where Nt denotes labor force. Then, we have
Et = Nt

Ut :Therefore, we haves (Nts 1Ut ) = mUt
UtUt= mNt
Nt

I thank to Wataru Tamura who made these sample answers.

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By solving this equation with respect to Ut =Nt , we have natural rate of unempolyment in the steady state:
Uts=
Nts+m
1:
=
1 + ms(c) What in‡uences the natural rate of unemployment through changes in s and m.
Discuss it.
Answer People may change their taste and it may reduce demand for an industry.
Emergence of new technology may destroy job opportunities for old skill.
Thus, a change in economic environment may cause mismatch in the labormarket and may increase s. However, when a new business demands morepeople, it will increase m and reduce frictional unemployment. Hence, thenet e¤ect is ambiguous.
2. Suppose that Bank of Japan wants to minimize the following loss function by choosing
the in‡ation rate at date t, gpt ,2
L (u; ) = ut + gpt;
2subject to the following Phillips curve,
ut = ungptegptwhere ut is the unemployment rate at date t, un is the natural rate of unemploymente
gpt is the in‡ation rate, gpt
is the expected in‡ation rate held by Japanese people, andand are parameters.
(a) Suppose that the Bank of Japan announces that it will choose 0 in‡ation rate,
but they can not commit their announcement. What would be the equilibrium
unemployment rate and in‡ation rate?
Answer Timing of this model is as follows. First, Bank of Japan announces thein‡ation rate. Second, given this announcement, Japanese people expectedethe in‡ation rate, gpt. Third, given Japanese people’s expectation, Bankof Japan determines the in‡ation rate, gpt . Finally, equilibrium outcome isobserved. We …rst examine Bank of Japan’s optimal choice of in‡ation rate,given the expected in‡ation rate held by Japanese people. The minimization
eisproblem for Bank of Japan given gpt
min = ungpt

gpt

FOC is
= gpt
22e
gpt
+ gpt
2

Therefore, the optimal choice of in‡ation rate is
gpt =>0This implies that at the decision stage of interest rate, Bank of Japan has
an incentive to choose the positive in‡ation rate, = ; regardless of Japanesepeople’s expectation. Therefore, if Bank of Japan can not commit it willchoose 0, then Japanese people expect that the in‡ation rate becomes = .
Therefore, the equilibrium unemployment rate and the in‡ation rate become
ut = u n
gpt =This implies that unemployment does not change but the in‡ation occurs.
The social loss is un + 2 =2 .
(b) Suppose that the Bank of Japan announces that it will choose 0 in‡ation rate
and builds up a commitment device (for example, the Bank of Japan writes thechosen in‡ation rate in Japanese constitution). What would be the equilibrium
unemployment rate and in‡ation rate?
Answer Since Bank of Japan builds up a commitment device, Japanese people
believe and expect that the in‡ation rate is zero. Then, Bank of Japan chooses
zero in‡ation rate. The equilibrium unemployment rate and the in‡ation rate
becomeuct = unc
gpt=0
In this case, the social loss is un :(c) What can you learn from your answers in question (a) to (b)?
Answer By comparing the results, social loss when Bank of Japan builds up commitment device is smaller. This implies that establishment of commitmentdevice leads to socially better equilibrium outcome. This result follows frometime inconsistency problem. Note that social loss is minimized1 when gpt=0and gpt = = . Before announces the in‡ation rate, Bank of Japan wants to
set zero in‡ation rate. However, after announces the in‡ation rate, it has anincentive to break the announcement and to choose positive in‡ation rate inorder to reduce unemployment. Because of time inconsistency problem, public can not fully trust government announcement unless they actually commit
the policy. In this case, more government’s discretionary power makes worse
situation.
3. Answer following questions.
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Assume that theet0;t0:3

(a) The permanent income hypothesis predicts that a temporal change in current
income should not in‡uence a change in consumption. However, data shows that achange in consumption is partially in‡uenced by the movement of current income.
What is two possible explanations?
Answer There are two possible reasons.
Serially correlated income When today’s income is highly correlated withthe future income, consumers can easily predict the steam of the future
income based on the current income. In the labor market, when consumers are promoted, they typically expect not only a high income today,
but also a high income in the future. Hence, a change in current income
can be seen as the change in the permanent income.
Liquidity constraint The permanent income hypothesis implicitly assumes
that every consumer can borrow money as long as they can return it in
the future. But if some consumers can not borrow enough money, they
can not buy consumption goods more than their income. Therefore,
current disposable income limits consumption:
Ct

Yt :

If many consumers are constrained by current income, and if current
income increases, then obviously consumers will increase their consumption. In this way, a change in current income may a¤ect a change inconsumption.
(b) What is Ricardian equivalence? Explain its logic.
Answer The general principle is that government debt is equivalent to futures
taxes, and if consumers are su¢ ciently forward looking, future taxes are equivalent to current taxes. Hence, …nancing the government by debt is equivalent
to current taxes. The implication of Ricardian equivalence is that a debt…nanced tax cut leaves consumption una¤ected. When consumers care about
their permanent income, they are worried not only today’s income but also
tomorrow’s income. Since today’s government debt can be seen as the future tax burden, it does not change their permanent income. Therefore,consumers do not change their consumption decision.
(c) Data shows that Ricardian equivalence does not literally hold. What are threepossible reasons?
Answer There are three possible reasons.
Transfer among generations: If parents do not care about their children,parents can enjoy low tax today and enforce their children to pay for them.
Hence, the issue is how much parents care about their children. If theycare their children like themselves, Ricardian Equivalence can still hold.
However, if they do not care their children, an increase in government
debt induce their demand.
Liquidity constraint: If there is liquidity constraints, the reduction of tax
can increase disposable income and raise consumption.
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Distortional tax: If a tax change the marginal bene…t or cost of consumption, it a¤ects consumers’decision and a¤ect consumption. In particular,
if government raises capital income tax, consumers are discouraged to saveand increase consumption.
4. Suppose that a consumer lives two periods, dates t and t + 1. In each period, he earns
the same labor income w and initially does not have any asset. When he receive his
income at date t, he can consume ct or leave it for tomorrow, ct+1 . Assume that heconsume every income at date t + 1 and does not leave asset for future. When he saves
for t + 1, he can earn the interest rate at date t + 1. Assume that the consumer has
an instantaneous utility function U (c) and discount factor, . Suppose that there isno tax at date t. Government will levy a lump sum tax, at date t + 1.
(a) Formulate the consumer’s decision problem.
Answer The consumer’s utility function takes the following form
U (ct ) + U (ct+1 )The budget constraint in period t is
st + c t = w
where st is saving for t. Similarly, the budget constraint in period t + 1 is
ct+1 = w + (1 + ) st:

Combining these budget constraints, we have the lifetime budget constraint:
ct +ct+1w=w+:
1+1+Thus, the consumer’s decision problem is to choose the consumption in periodt as follows:
max U (ct ) + U (1 + ) xP

ctw ct 0

where xP = w + w1+ denotes the permanent income.
(b) Assume that the consumer has a quadratic instantaneous utility function: U (ct ) =act 2b c2t . Derive the …rst order condition of the problem above.
Answer When the consumer has a quadratic instantaneous utility function, theconsumer’s decision problem above can be rewritten asmax
ct 0actb 2
c2 t+a (1 + )(xPct )Using the Chain rule. For U (ct+1 ) = act+15b(1 + )(xP2b 2
c2 t+1ct )2and ct+1 = g(ct ) =(1 + )(xPct ), the chain rule givesdU (g(ct )) dU (ct+1 ) dg(ct )=dct
dct+1dct0P=Uct )) f (1 + )g=b(1 + )(xP

(1 + ) act ) :Then the …rst-order condition is given by
U 0 (ct )fa(1 + )U 0 (ct+1 ) = 0; orbct g(1 + ) ab(1 + )(xPct ) = 0:

Without the chain rule. Since the instantaneous utility function is speci…ed as a quadratic function, we can directly obtain the …rst-order condition. The objective function for the consumer isactb 2
c2 t+a (1 + )(xPct )b(1 + )2 [(xP )222xP ct + c2t ] :Then the …rst-order condition isaa(1 + ) + b(1 + )2 (xPbctct ) = 0:(c) Assume that the consumer has a quadratic instantaneous utility function: U (ct ) =act 2b c2t . Derive the optimal ct . How does a change in a¤ect ct ? Why?
Answer Recall that the …rst-order condition implies
U 0 (ct ) = (1 + )U 0 (ct+1 ); orfa

bct g = (1 + ) ab(1 + )(xPct ) :(Euler eq.)

This is called Euler equation. By solving this equation, we havect =(1 + )]a + (1 + )2 bxP[1 + (1 + )2 ]b[1where xP = w + (w)=(1 + ). An increase in tax decreases the lifetimeP
income x , and hence the consumer reduces the current consumption ct (i.e.," =) xP # =) ct #).
Euler equation represents the key trade-o¤ between consumption and saving. The left-hand side of the equation is the marginal loss from an decrease in current consumption while the right-hand side is the marginal gainfrom an increase in saving. Therefore the optimal consumption in period
t is determined as a cross point of the left- and right-hand sides. Figure
1 illustrates how the optimal consumption in period t is determined. To
understand how a change ina¤ects ct , recall that the lifetime income
xP = w + (w)=(1 + ) is decreasing in and that U 0Pct )) is de6

Figure 1: HW7-3: Euler equation.
creasing in xP . That is, " =) xP # =) RHS ". Intuitively, an increasein decreases the consumer’s lifetime income. Since the marginal bene…tfrom consumption is decreasing, U 00 (ct ) = b < 0, consumers prefer stableconsumption to unstable consumption. In order to smooth consumption, heshould decrease the current consumption.

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