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Macroeconomía II – Problem Set
TA Session 4
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Some of those exercises will be solved on the ayudantia of September 28, 2018.
Monetary equilibrium
Exercise 1
[Optional] Consider the basic Cagan’s model in which the log of the price level (pt) and
the log of the money supply (mt) satisfy the following difference equation:
mt − pt = −η (pt+1 − pt) (1)
where η is a constant larger than zero. For simplicity, suppose that mt is constant and
equal to m̃, for every t. We know that a solution to (1) is given by
pt =
1
1 + η
∞∑
i=0
(
η
1 + η
)i
mt+i = m̃
The solution above is called the fundamental solution, but we know that there are other
solutions to (1). Answer the questions below:
1. Propose another (non-fundamental) solution and show that it also satisfies (1).
2. Discuss the economic intuition behind the non-fundamental solutions in which prices
increase over time, even though the money supply is constant. (Tip: you may want to
start with “Suppose everyone expects prices to increase a lot in the future. Then...”)
Seignioriage
Macroeconomía II, 2018/2 1
Exercise 2
Consider an economy with a single final good. The seigniorage revenue in real terms in a
given period is given by
S = ∆M
P
,
where ∆M is the increase in the money supply and P is the price of the good. The demand
for money is given by M/P = Y e−αi, where i is the nominal interest rate, Y is output
and α > 0. Using the Fisher equation (and assuming expected inflation is equal to actual
inflation), we can write M/P = Y e−α(r+π), where r is the real interest rate and π is the
inflation rate. Suppose that in the long run the interest rate and output are constant (i.e.,
∆i = 0 and ∆Y = 0). Find the inflation rate that maximizes S in the long run.
Exercise 3
Consider the following money demand:
Md
P
= y [1 − (r + πe)]
where y is the real output, r is the real interest rate and πe is expected inflation.
1. Derive an expression for the seigniorage revenue assuming y = 250, r = 0.05 and
πe = 0.1 (and all those are independent of money growth). Compute the seigniorage
revenue for the following growth rates of the money supply: 25%, 50% and 75%.
What is the relationship between seigniorage and money growth?
2. Assume now that expected inflation is equal to actual inflation (π = πe), while the
other variables are the same. Compute the seigniorage revenue for the following
growth rates of the money supply: 25%, 50% and 75%. Why are your answers
different from the previous item? Explain.
3. Assuming π = πe, find the rate of monetary expansion that maximizes the seignior-
age revenue.
Exercise 4
[Adapted from de Gregorio] Suppose an economy in which agents keep money as currency
and as deposits. The money multiplier is denoted by µ̃. The demand for real balances in
Macroeconomía II, 2018/2 2
this economy is given by:
L(i, y) = ay (b− i) ,
where y is real output.
1. Compute the seigniorage if the inflation rate is 10%. What assumptions do you need
to make to be able to compute the seigniorage?
2. Suppose b > r, where r is the real interest rate. Compute the inflation rate that
maximizes the government revenue. What happens with the inflation rate that you
found if the real interest rate increases?
3. Suppose now that in reality the multiplier increases. How does this change your
answer in item 1?
Exercise 5
[Adapted from de Gregorio] The money demand is given by:
M
P
= α− βi+ γy.
The notation is the usual one.
1. Find the seigniorage (S), assuming π = πe, and discuss how π relates to S. In
order for a revenue maximizing seigniorage to exist, do we need some restriction on
parameters?
2. If it exists, compute the inflation rate that maximizes seigniorage.
Suppose now that in this economies output grows at a rate g.
3. Write the seigniorage as a function of the parameters α, β, γ, the real output y, the
growth rate g, the inflation rate π and the nominal interest rate i. Use the Fisher
equation.
4. Find the inflation rate that maximizes seigniorage. Hows does it compare to item
2?
Macroeconomía II, 2018/2 3
Proposed exercises
Exercise 6
Consider the basic Cagan’s model in which the log of the price level (pt) and the log of
the money supply (mt) satisfy the following difference equation:
mt − pt = −η (pt+1 − pt) (1)
where η is a constant larger than zero. You may assume that mt is bounded.
1. Write pt as a function of the current and future money supply. Interpret the results.
2. [Optional] Are there solutions to (1) that depend on variables other than the money
supply and the parameter η (i.e., non-fundamental solutions, bubbles)? If your
answer is yes, provide one example and show that it satisfies (1). If your answer is
no, show that there is no bubbles.
3. Suppose the money supply is constant and equal to m̃ and everyone is certain it will
remain at that level forever. Using your solution in item 1, characterize the price
level.
4. Suppose now that at date 0 the centrak bank announces that the money supply will
be equal to m̃ from date 0 to date T − 1, and that at date T the money supply will
increase to m̃′ > m̃ (and remain at the new level forever). In a graph and using
the fundamental solution, show the evolution of the price level as a function of time.
Intepret the results.
Macroeconomía II, 2018/2 4
Macroeconomía II – Solutions
TA Session 4
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
Item 1
Guess a solution of the form (as seen in class, we know that it is a good guess):
pt = m̃+ b0gt.
Plugging into (1):
−b0gt = −η
(
b0g
t+1 − b0gt
)
Solving for g:
g = 1 + η
η
Thus,
pt = m̃+ b0
(
1 + η
η
)t
is a solution to (1).
Item 2
The idea is for this kind of equilibrium is the following: suppose people expect pt to rise
in the future (b0 > 0). Then, the Fisher equation implies that the nominal interest rate
will be high, reducing agents incentives to hold money. But then, there is too little money
demand for too much money supply, and prices have to increase to equate the demand and
the supply for real balances. But this confirm agents initial expectations (a self-fullfiling
prophecy).
Macroeconomía II, 2018/2 1
Exercise 2
Multiplying and dividing the right-hand side of S = ∆M
P
by M we get:
S = ∆M
M
M
P
(1)
We know that M = Pye−αi. Since we assumed that Y and i are constant, changes in M
come only from changes in P and therefore, applying differentials:
∆M = ye−αi∆P
Dividing the right-hand side by M and the left-hand side by PY e−αi (since both are
equal):
∆M
M
= ∆P
P
= π (2)
Plugging (2) into (1):
S = M
P
π
Using M/P = ye−α(r+π) we get
S = ye−α(r+π)π (3)
Taking the first order condition and solving for π:
dS
dπ
= Y e−α(r+π) − αye−α(r+π)π = ye−α(r+π)(1− απ) = 0 (4)
π = 1
α
From (3) it is easy to see that π ≤ 0 does not maximize seignoriage, since it yields a
zero or negative S (and any π > 0 yields a positive S). From (4), one can easily see
that dS/dπ > 0 for π ∈ [0, 1/α) and dS/dπ < 0 for π > 1/α. Therefore, the first order
condition characterizes the maximum and the revenue maximizing seigniorage is 1/α.
Exercise 3
Item 1
Multiplying and dividing the right-hand side of S = ∆M
P
by M we get:
S = ∆M
M
M
P
= ∆M
M
y [1− (r + πe)] (5)
Macroeconomía II, 2018/2 2
Now, we are going to compute the seigniorage revenue assuming exogenous values for r,
πe, y and ∆M/M . We assume y = 250, r = 0.05 and πe = 0.1. Then, the seigniorage
revenue for different ∆M/M is given by the table below:
∆M/M Seigniorage
25% 53.125
50% 106.25
75% 159.375
Therefore, as we increase the money growth, the higher the seigniorage. Notice that it is
implicit here that the increase in the inflation rate is not affecting πe, and that is why we
get this monotonic effect (instead of the usual Laffer curve).
Item 2
Following the same steps we did in Exercise 1 (do it yourself again to make sure you
understand, assuming a SS with constant i and y),you can show that:
∆M
M
= ∆P
P
= π
Hence, using πe = π the seigniorage revenue is given by
S = ∆M
M
y
[
1−
(
r + ∆M
M
)]
= πy [1− (r + π)]
We assume y = 250, r = 0.05. Then, the seigniorage revenue for different ∆M/M is given
by the table below:
∆M/M Seigniorage
25% 43.75
50% 56.25
75% 37.5
Therefore, we get the usual hump shaped thing. That is because now, as inflation increases,
people adjust their inflation expectations, demanding less real balances. Thus, while the
“tax rate” is increasing, the “tax base” is decreasing.
Item 3
We need to maximize:
S = πy [1− (r + π)]
Macroeconomía II, 2018/2 3
One can easily see that this function is strictly concave. The FOC is:
dS
dπ
= y [1− (r + π)]− πy = 0
Hence, the revenue maximizing seigniorage is:
π∗ = 1− r2
Exercise 4
Item 1
We have to be very careful here. So far, for simplicity we have implicitly assumed that
the government had the monopoly of creating money. In other words, we have assumed a
money multiplier of 1 (M = monetary base). But when people use deposits, we know it is
not true (banks also create money). Let M = C +D = µ̃H (the notation is the standard
one). Then, we have:
St =
∆Ct
Pt
Note that the the increase in money is ∆Mt = µ̃∆Ct, but only a fraction of this increase
becomes government revenue. Therefore, we have:
St =
∆Mt
µ̃Pt
= 1
µ̃
∆Mt
Mt
Mt
Pt
Using i = r + πe and the money demand function (and dropping the time subscripts):
S = 1
µ̃
∆M
M
ay (b− r − πe) (6)
But, applying differentials to the money demand M = Pay (b− i) we have (assuming we
are in a SS with y and i constant):
∆M = ay (b− i) ∆P
Dividing it by M = Pay (b− i)
∆M
M
= ∆P
P
= π
Macroeconomía II, 2018/2 4
Hence, (6) becomes:
S = 1
µ̃
πay (b− r − πe)
Finally, assuming that π = πe we get and π = 10%:
S = 1
µ̃
πay (b− r − 0.1)
To get to this expression we had to assume π = πe. We also make the assumption that
we were in a SS with constant i, π and y, but we could do it assuming a constant output
growth, for instance (try it).
Item 2
Using the results and assumptions from the previous item, we need to maximize:
S = 1
µ̃
πay (b− r − π)
This is clearly strictly concave. The FOC is:
dS
dπ
= 1
µ̃
{ay (b− r − π)− πay} = 0
Which implies:
π∗ = b− r2
Item 3
When the money multiplier increases, S decreases, everything else constant. That is
because now, for a given inflation rate, a smaller fraction of the increase in the money
supply is taken by the government (since banks “contribute” to part of that increase in
the money supply).
Exercise 5
Item 1
Following the same steps and making the same assumptions we did in Exercise 2 (do it
again) we get:
S = πM
P
Macroeconomía II, 2018/2 5
Hereafter we assume that inflation equals expected inflation. Thus:
S = [α− β (r + π) + γy] π
If β > 0, S is a concave parabola, and so a revenue maximizing seigniorage obviously
exists. If β < 0 it is a convex convex parabola, thus a maximum will not exist. If β = 0,
it is a linear an increasing function, thus a maximum will not exist either. Therefore we
need β > 0.
Item 2
The first order condition is:
−βπ + [α− β (r + π) + γy] = 0
Which yields:
π∗ = 12β [α− βr + γy]
Item 3
We will still assume that ∆i = 0, but now ∆y/y = g. Then applying differentials to
M = P [α− βi+ γy]:
∆M = [α− βi+ γy] ∆P + Pγ∆y
Dividing it by M = P [α− βi+ γy]:
∆M
M
= ∆P
P
+ γ∆y
α− βi+ γy = π + γ
∆y
M/P
Hence:
S = ∆M
M
M
P
=
[
π + γ ∆y
M/P
]
M
P
= πM
P
+ γ∆y
= π (α− βi+ γy) + γ∆y
y
y
= π (α− βi+ γy) + γgy
Macroeconomía II, 2018/2 6
Using the Fisher equation i = r + πe, with π = πe:
S = π [α− β (r + π) + γy] + γgy
Item 4
It is easy to see that the inflation that maximizes seigniorage is the same, since we have
just summed a constant (γgy) on the seigniorage expression. The only difference, is that
now for a given inflation rate, the seigniorage revenue will grow in time, since y is growing.
Macroeconomía II, 2018/2 7

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