Topics in Macroeconomics, Goethe University Frankfurt, WS 2013-14
Problem set 7
Instructor: Ctirad Slavı́k, email: slavik@econ.uni-frankfurt.de
Teaching Assistant: Zixi Liu, email: zixifrankfurt@hotmail.com
Website: http://www.wiwi.uni-frankfurt.de/professoren/slavik/teach.html
see the syllabus for instructions on how to work on the problem sets
1. Government policies and productivity in general equilibrium. (30 points)
Consider the following economy. There is a representative consumer, a representative firm and a government.
There are 2 goods, a general consumption good and labor (time). We normalize the price of the consumption
good to 1 and denote the wage rate as w.
The representative firm hires labor at the labor market paying the wage w and produces the general consumption
good using the following technology: y = z · nd , where nd is the labor demand and z the level of technology.
The consumer is endowed with 1 unit of time and he decides how much to work, and how much to consume.
We denote as c his consumption of the general consumption good and as ` his leisure. Therefore ns = (1 − `) is
his labor supply. The consumer’s utility is u(c, `) = log c + log `. The consumer owns the firm, so he gets any
profits that the firm creates.
The government is using lump sum taxes T to finance its expenditure G, i.e. it must be G = T .
(a) (5 points) Write the profit maximization problem of the firm. Argue that in equilibrium w = z. What are
equilibrium profits? Use that information in the next question.
Solution:
• (1p) maxnd π = y − wnd s.t. y = znd
• (3p) arguing that in equilibrium w = z
i. if z > w, then π > 0, try to hire as many workers as possible.
ii. if z < w, then π < 0, no labor hired.
iii. if z = w, then π = 0, nd can be any number in [0, 1].
• (1p) zero equilibrium profits.
(b) (5 points) Write the consumer’s maximization problem and the Lagrangean. Solve for the optimal choices
of c and ` as functions of T and w.
Solution: (3p) maximization setup, (2p) the optimal choices
max u(c, `) = log c + log `
s.t.
c,`
c ≤
ns
≤
wns − T
1−`
The Lagrangian read:
L = u(c, `) + λ(w(1 − `) − T − c)
Solution for the optimal choices: c =
w−T
2
,`=
w−T
2w
(c) (5 points) Define the competitive equilibrium.
Solution: A competitive equilibrium is a price w, an allocation {c, `, ns , nd , y} and a government policy
{T, G} such that given price and government policy,
i. an allocation solves the utility maximization problem and the profit maximization problem.
ii. markets clear: c + g = y, nd = ns
iii. a government’s budget constraint is balanced: G = T
(d) (5 points) Determine the equilibrium output, consumption and labor supply ns as a function of government
expenditure G (recall that in equilibrium G = T ) and the level of technology z.
Solution: y =
z+G
2 ,
c=
z−G
2 ,
ns =
1
2
+
G
2z
(e) (5 points) Comparative statics. What happens with output, consumption and labor supply if G increases
by ∆? What does it say about the efficiency of government expenditure?
Solution:
• (3p) y increases by ∆/2, c decreases by ∆/2, ns increases by ∆/(2z).
• (2p) inefficient in the sense that it crowds out the private consumption, even though it contributes
to the increase of output.
(f) (5 points) Comparative statics. What happens with output, consumption and labor supply if z increases
by ∆? Is this result in line with the data? Can we use it to argue that shocks to technology z are sources
of business cycle fluctuations?
Solution:
• (3p) y and c increase by ∆/2, ns decreases by
G
2z 2 ∆.
• (2p) Yes, especially in western Europe.
2. Ricardian equivalence. (30 points)
Consider the following economy. There are 2 agents, one consumer and one government. Both the consumer
and the government live for 2 periods.
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The consumer has endowments y1 , y2 and chooses how much to consume c1 , c2 in period 1 and period 2, respectively. The utility function is u(c1 , c2 ) = log(c1 ) + β log(c2 ) with β > 0. In addition the consumer can issue or
buy bonds. We will denote a positive bond issuance as b > 0. Issuing one unit of bond means that the consumer
gets 1 unit of the consumption good in the first period (from the trading partner) and agrees to pay back 1 + r
units of the consumption good in the second period, where r is the endogenous net interest rate.
The government needs to finance the expenditure G1 and G2 in the first and in the second period, respectively.
It does so by taxing the consumer and issuing/byuing bonds. The lump-sum tax is T1 in the first and T2 in the
second period. Government bond issuance will be denoted as B and we shall assume that B > 0 when bond
issuance is positive, i.e. the government is borrowing the consumption good in the first period (agreeing to repay
in the second).
(a) (6 points) Write the consumer’s maximization problem as one with 2 budget constraints, one for the first
and one for the second period. Write the government’s budget constraints for the first and second period.
Solution: (3p) The consumer’s problem:
max u(c1 , c2 ) = log c1 + β log c2
s.t.
c1 ,c2 ,b
c1
≤
c2 + (1 + r)b ≤
y1 − T1 + b
y2 − T2
(3p) The government’s BCs:
G1
=
T1 + B
G2 + (1 + r)B
=
T2
(b) (6 points) Define competitive equilibrium. Hint: there are no international markets, so the bond market
clears if B + b = 0. This means that if the government issues bonds the consumer is buying them and the
other way around.
Solution: A competitive equilibrium is a price r, an allocation {c1 , c2 , b, G1 , G2 , T1 , T2 , B} such that
given price and government policy,
i. an allocation solves the utility maximization problem.
ii. goods markets and a bond market clear: c1 + G1 = y1 , c2 + G2 = y2 , b + B = 0
iii. a inter-temporal government’s budget constraint is balanced: G1 +
G2
1+r
= T1 +
T2
1+r
T2
(c) (8 points) Suppose that T1 increases by ∆, but the net present value of taxes, i.e. T1 + 1+r
does not change.
How does T2 change? Show that in this case the consumer decisions c1 and c2 are not affected if the interest
rate r remains the same. Does b and B change? Denote the new bond issuances bnew and B new and express
them as functions of the parameters, b, B and the tax change ∆. Discuss why the old equilibrium interest
rate r is also the equilibrium interest rate after the tax change.
Solution:
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i. (2p) T2 decreases by (1 + r)∆.
y2
1+r , P VT
β(1+r)
1+β (P Vy −
ii. (2p) define P Vy = y1 +
1
1+β (P Vy
= T1 +
T2
1+r .
Optimal choices of consumption are c1 =
− P VT ), c2 =
P VT ). If P VT does not change, the consumption decisions
are unaffected if the interest rate remains the same.
iii. (2p) As optimal private bond issuance is b = c1 − y1 + T1 , bnew = b + ∆. From the 1st period
Gov’t BC, B new = B − ∆.
iv. (2p) When the new bond market clears, bnew + B new = 0. ⇒ b + ∆ + B − ∆ = 0 ⇒ b + B = 0.
Therefore, the new bond market clears under the old equilibrium interest rate.
(d) (4 points) The previous result stating that the timing of taxes is irrelevant is know as Ricardian equivalence.
Discuss briefly under which conditions and why it will be violated.
Solution: (Examples)
i. A consumer is borrowing constrained.
ii. The government and consumer face a different interest rate.
(e) (6 points) Suppose that G1 increases by ∆, but the net present value of government expenditure under the
G2
old interest rate, i.e. G1 + 1+r
does not change. How does G2 change? Is it true that the old allocations c1
and c2 and the interest rate r are parts of the equilibrium after the tax change? Justify your claim carefully.
Solution: (2p) The following condition holds: G1 +
Gnew
− G2 =
2
G2
1+r
= G1 + ∆ +
Gnew
2
1+r new .
Therefore,
(rnew − r)
G2 − (1 + rnew )∆
1+r
(1)
(4p) B increases by ∆ in order to meet the Gov’t BC of the first period. Then, we have a new clearing
condition for a bond market:
1
y2
T2
(y1 +
− T1 −
) − y1 + T1 + B + ∆ = 0
new
1+β
1+r
1 + rnew
As an old clearing condition for a bond market is
the following relationship between rnew and r.
1
1+β (y1
(2)
y2
T2
+ 1+r
− T1 − 1+r
) − y1 + T1 + B = 0, we find
1
1
(1 + β)∆
=
−
new
1+r
1+r
y2 − T2
(3)
Therefore, rnew increases when ∆ is positive. The old allocations are not parts of the equilibrium any
longer.
3. The AK Model. (30 points)
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Note: this is probably the hardest problem on this exam. You might want to leave it for the end.
Consider a version of the growth model discussed in class (and in PS4).
∞
X
max ∞
{ct ,kt+1 }t=0
βt
t=0
∀t : ct + kt+1
≤
c1−σ
t
1−σ
s.t.
Akt
In this model, A is the constant level of technology (plus possibly the undepreciated capital). Note that we have
an infinite sequence of budget constraints, one for each time period t.
(a) (6 points) Denote the Lagrange multiplier for the budget constraint at time t as λt . Write the associated
Lagrangean.
Solution:
L ({ct , kt+1 , λt }∞
t=0 )
=
∞ X
t=0
βt
c1−σ
t
+ λt (Akt − ct − kt+1 )
1−σ
(b) (4 points) Take first order conditions with respect to an arbitrary ct and an arbitrary ct+1 . Note: if
1−σ
f (x) = x1−σ , then f 0 (x) = x−σ .
Solution: β t c−σ
= λt , β t+1 c−σ
t
t+1 = λt+1
(c) (4 points) Take first order conditions with respect to kt+1 . Do not forget that kt+1 shows up in the budget
constraint at time t as well as at time t + 1.
Solution: λt = Aλt+1
(d) (2 points) Plug in from the FOC wrt ct and ct+1 into the FOC wrt kt+1 for λt and λt+1 . This is the Euler
equation.
1
Solution: ct+1 = (βA) σ ct
(e) (2 points) We will guess that the solution in terms of kt+1 as a function of kt at any point in time t is:
1
kt+1 = (βA) σ kt . What is ct as a function of kt ? Hint: Use the fact that the budget constraint must clear
at the solution.
1
Solution: ct = Akt − kt+1 = (A − (βA) σ )kt
(f) (6 points) Express ct+1 as a function of kt+1 and then as a function of kt using the guess above. Verify that
the Euler equation is satisfied for this ct+1 and the ct you derived above.
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Solution: We start with the equation for ct+1 as a function of kt+1 .
ct+1
=
1
(A − (βA) σ )kt+1
1
1
⇒ =
(A − (βA) σ )(βA) σ kt
⇒ =
(βA) σ (A − (βA) σ )kt
⇒ =
(βA) σ ct
1
1
1
The last equation is the Euler equation we derived in (d). Q.E.D.
(g) (6 points) Define output in this economy as yt = Akt . What is the the growth rate output in this model?
Which countries grow faster, those with more or those with less patient consumers (again in the context of
this model)?
Solution:
yt+1
yt
1
= (βA) σ
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