Ayudantia 2

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Macroeconomía II – Problem Set
TA Session 2
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 August 31, 2018.
Money demand
Exercise 1
In the money in the utility function model (without labor) seen in class, we have shown
under the optimal choice of money and consumption, the following equation holds:
um (ct,mt)
uc (ct,mt)
= it1 + it
, (1)
where um(ct,mt) and uc(ct,mt) denote the marginal utility of real money balances (mt)
and consumption (ct), respectively, and it denotes the nominal interest rate. We also know
that the household choice satisfies a standard Euler equation:
uc(ct,mt) = β (1 + rt)uc(ct+1,mt+1). (2)
1. Show that the household choice satisfies:
um (ct,mt) =
it
1 + πt+1
βuc(ct+1,mt+1).
Interpret this equation.
2. Assume that u(c,m) = ln c−(m−5)2. What is the nominal interest rate and inflation
rate that maximize the representative household utility in the steady state? If the
nominal interest rate is chosen to maximize the household’s utility at the steady
state, how much money is demanded at the steady state? Provide an intuition for
the optimal nominal interest rate. (Tip: remember that money is superneutral in
this model.)
Macroeconomía II, 2018/2 1
Exercise 2
[Adapted from Walsh] Consider the basic MIU model in Section 2.2 of Walsh’s book (also
seen in class), with one modification: now the real money balances that the agent decides
to hold at date t yield utility only at date t+1. More precisely, the instantaneous utility at
date t depends on ct and the real money balances the agent choose at t− 1 (instead of the
real money balances he choose at t). Define Mt as the amount of money the agent choose
to hold at date t − 1 (instead of the amount he chooses to held at date t, as in Section
2.2.). Defining Mt that way, the agent utility can be written as before:
∑∞
t=0 β
tu
(
ct,
Mt
Pt
)
.
1. Write the budget constraint in nominal terms (i.e., with the interest rate, money
holdings and bond holdings expressed in nominal units).
2. Write the budget constraint in real terms (i.e., with the interest rate, money holdings
and bond holdings expressed in real units).
3. Show that in the optimal, the equation below holds for every t:
um (ct+1,mt+1)
uc (ct+1,mt+1)
= it.
Interpret it.
Exercise 3
Consider the basic MIU model in Section 2.2 of Walsh’s book (also seen in class), with some
modifications: now agents can work to increase their production, but working causes them
some disutility. More precisely, the production function is now given by yt = f(kt−1, nt),
where nt is the number of hours worked. Normalizing the total number of hours the worker
has available to 1, we have nt = 1 − lt, where lt denotes the number of hours spent on
leisure. The instantaneous utility function of the agent is given by u(ct,mt, lt), where the
usual assumptions on u(·) and f(·) to guarantee an interior solution are satisfied.
1. Write the budget constraint in nominal terms (i.e., with the interest rate, money
holdings and bond holdings expressed in nominal units).
2. Write the budget constraint in real terms (i.e., with the interest rate, money holdings
and bond holdings expressed in real units).
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3. Derive the equations that determine the steady state level of consumption, leisure
and capital. Under which conditions we have superneutrality? Compare the results
obtained here to the ones obtained in the model without labor.
Exercise 4
Consider a centralized economy where households produce and consume. The production
function Y (Kt,nt) is homogeneous of degree one in both capital and labor. Here, household
incomes are production Yt, monetary transfers from the government Tt and money balances
Mt. On the other hand, the outcomes are consumption ct, investment in capital It and
money balance held for the next period Mt+1. Capital evolves according to the following
expression: Kt+1 = Kt(1 − δ) + It.
Now let’s think of a decentralized economy, where households own the resources capital
Kt and labor nt, which is then rented every period by the firms receiving as payment
interest rate Ptrt for capital and wages Ptwt for labor (hence, the real rental rate of capital
is rt and the real wage is wt). In this economy, there are transfers Tt and money Mt+1
that households accumulate in t to use in t+ 1.
1. Present the budget constraint for both centralized and decentralized economies in
nominal terms.
2. Present the budget constraint for both centralized and decentralized economies in
real terms.
3. Show that we can recover decentralized budget constraint from centralized one in
this setting assuming perfect competition.
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Proposed exercises
Exercise 5
Consider the basic MIU model in Section 2.2 of Walsh’s book (also seen in class) with one
modification: assume that there is no capital and the output yt is endogenously given.
Assume that:
u(c,m) = v(c) + ω lnm,
where v(·) satisfies the usual assumptions that guarantee an interior solution.
1. Compute the current price as a function of current and future money supplies.
2. Compute the equilibrium nominal interest rate as function of current and future
money supplies.
Exercise 6
Consider an economy with the following state at some date t: β = 0.95, it = 0.07,
πet = 0.02. Expected inflation remains constant while nominal interest rate increases by
1pp each period. Households preferences are logarithmic on consumption, having then
u(ct) = ln(ct) in the CIA model and u(ct,mt) = ln(ct) + ln(mt) in the MIU model. (The
notation is the standard one and the CIA and MIU model referred to are the ones in
Section 2.2 and Section 3.3.1 in Walsh’s book).
1. For each of the models (CIA and MIU) present the household problem, estimate the
first order conditions and find the Euler condition by relating current and future
consumption.
2. How do both results relate? Explain each difference you find.
Macroeconomía II, 2018/2 4
Macroeconomía II – Solutions
TA Session 2
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
Item 1
Using the Fisher equation, (1) becomes:
um (ct,mt) =
it
(1 + rt) (1 + πt+1)
uc (ct,mt)
Using (2) we get:
um (ct,mt) =
it
���
��(1 + rt) (1 + πt+1)
��
��(1 + rt)uc(ct+1,mt+1) =
it
1 + πt+1
βuc(ct+1,mt+1)
The intuition is the following. In equilibrium, the household must be indifferent between
having more money today or consuming tomorrow. If she gives up holding one unit of
money at t, she gets it extra dollars at t+ 1, which implies a real increase of it1+πt+1 in the
amount of goods she can buy. Thus, her utility increases by it1+πt+1βu
′(ct+1,mt+1). For her
to be indifferent, this must be equal to the marginal utility of holding money today.
Item 2
Since consumption is independent of the monetary side of the economy, we maximize the
household utility by maximizing his utility of holding money. Since u(·) is concave in m,
this happens when um(c,m) = 0. Thus, (1) implies that interest rate that maximizes the
household utility in the steady state is:
0 = i
ss
1 + iss ⇒ i
ss = 0
The Fisher equation implies that
1 + iss = (1 + rss) (1 + πss)
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And thus, the steady state welfare maximizing inflation is 1+πss = 11+rss ⇒ π
ss = − rss1+rss .
The money demand when iss = 0 is given by um(css,mss) = 0, implying mss = 5.
The intuition for it = 0 being optimal is the following. The opportunity cost of
holding money is the nominal interest rate (it is the “price” of money). The marginal cost
of producing money is zero. Efficiency implies that price equals the marginal cost, thus
it = 0.
Exercise 2
Item 1
Given that now the money accumulated at date t only yields utility at date t + 1, it is
useful to introduce a small change in notation. We denote the money the agent decides to
carry from date t to date t+ 1 by Mt+1 (and not by Mt as we did in class). The remaining
notation is the same usedin class: Bt denotes the amount invested in bonds at date t; kt
is the amount of capital carried from t to t+ 1; ct is the amount of consumption at date t
and Tt are the cash transfers received from the central bank at the beginning of date t; it
is the nominal interest rate paid at t+ 1 for each dollar invested in bonds at t.
The household budget constraint in nominal terms can then be written as:
ctPt + ktPt +Mt+1 +Bt = f(kt−1)Pt + kt−1Pt(1− δ) + (1 + it−1)Bt−1 +Mt + Tt
Item 2
First, we define the real variables as we did in class: bt ≡ Bt/Pt, mt ≡ Mt/Pt and
τt ≡ Tt/Pt. Moreover, rt is the real interest rate.
To write the budget constraint in real terms we divide it by Pt as usual:
ct + kt +
Mt+1
Pt
+ Bt
Pt
= f(kt−1) + kt−1(1− δ) + (1 + it−1)
Bt−1
Pt
+ Mt
Pt
+ Tt
Pt
But since:
Mt+1
Pt
= Mt+1
Pt+1
Pt+1
Pt
= mt+1 (1 + πt+1)
and
Bt−1
Pt
= Bt−1
Pt−1
Pt−1
Pt
= bt−11 + πt
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We can write the budget constraint as:
ct + kt +mt+1 (1 + πt+1) + bt = f(kt−1) + kt−1(1− δ) + (1 + it−1)
bt−1
1 + πt
+mt + τt
But by the Fisher equation 1 + rt−1 = (1 + it−1) / (1 + πt). Therefore:
ct + kt +mt+1 (1 + πt+1) + bt = f(kt−1) + kt−1(1− δ) + (1 + rt−1) bt−1 +mt + τt
Item 3
First, we need to write the agents problem. Note that given our change in notation forMt,
the agent utility can be written exactly as before: ∑∞t=0 u(ct,mt) (if we had not changed
the notation, the budget constraint would look the same, but mt−1 would show up in the
instantaneous utility at t).
max
{ct,kt,mt+1,bt}t≥0
∞∑
t=0
βtu (ct,mt)
s.t. ct + kt +mt+1 (1 + πt+1) + bt = f(kt−1) + kt−1(1− δ) + (1 + rt−1) bt−1 +mt + τt, ∀t ≥ 0
ct, kt,mt+1, bt ≥ 0, ∀t ≥ 0
m0 = m > 0, b−1 = b > 0, k−1 = k > 0
Here we will use the fact that the solution is interior, and hence we will ignore the non-
negativity constraints. The Lagrangian of this problem is:
L =
∞∑
t=0
βt [u (ct,mt)− λt (ct + kt +mt+1 (1 + πt+1) + bt − f(kt−1)− kt−1(1− δ)− (1 + rt−1) bt−1 −mt − τt)]
We will take the FOCs with respect to ct,mt+1, bt:
βtuc(ct,mt)− βtλt = 0 (FOC1)
−βtλt(1 + πt+1) + βt+1 [um (ct+1,mt+1) + λt+1] = 0 (FOC2)
−βtλt + βt+1λt+1(1 + rt) = 0 (FOC3)
Which simplifying becomes:
uc(ct,mt) = λt (FOC1’)
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um (ct+1,mt+1) =
λt(1 + πt+1)
β
− λt+1 (FOC2’)
1 + rt =
λt
βλt+1
(FOC3’)
Dividing (FOC2’) by (FOC1’) forward:
um (ct+1,mt+1)
uc(ct+1,mt+1)
= λt(1 + πt+1)
βλt+1
− 1
Using (FOC3’) it becomes:
um (ct+1,mt+1)
uc(ct+1,mt+1)
= (1 + rt) (1 + πt+1)− 1
Finally, by the Fisher equation (1 + rt) (1 + πt+1) = 1 + it. Therefore, it becomes:
um (ct+1,mt+1)
uc(ct+1,mt+1)
= it (*)
Remember from your micro class, that if an is choosing between two goods, say z and y,
then the marginal rate of substitution between z and y must be equal to the price of z in
units of y (that is, using the usual notation, Umgz/Umgy = Pz/Py). Here, the LHS of (*)
is the marginal rate of substitution between mt+1 and ct+1. To understand why we can
interpret it as the relative price of mt+1 in terms of ct+1, consider an agent who at date
t decides to consume one extra unit of the consumption good at t + 1. To do that, she
must increase her wealth at the beginning of t+ 1 in Pt+1 units. She can achieve that by
purchasing Pt+1/ (1 + it) dollars in bonds at date t and then using all the bond payment
at t + 1 to buy the consumption good (Pt+1/ (1 + it) is the “price” of consumption at
t+ 1 in date t dollars). Now suppose the agent decides to enter date t+ 1 with one extra
unit of real balances. To achieve that, she must hold Pt+1 units of cash at date t. But
since that cash remains with her at date t + 1, she can reduce her bond holdings at t
in Pt+1/ (1 + it) dollars and still get the same wealth at the beginning of t + 1. Hence
Pt+1 − Pt+1/ (1 + it) is the “price” of holding one unit of real money balances at t+ 1 (in
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date t dollars). Therefore, the “relative price of money and consumption” is:
"Price" of mt+1︷ ︸︸ ︷
Pt+1 −
Pt+1
1 + it
Pt+1
1 + it︸ ︷︷ ︸
"Price" of ct+1
= it
Exercise 3
Item 1
The budget constraint in nominal terms is similar to the standard MIU model without
labor, we only use nt = 1− lt in the production function:
ctPt + ktPt +Mt +Bt = f(kt−1, 1− lt)Pt + kt−1Pt(1− δ) + (1 + it−1)Bt−1 +Mt−1 + Tt
Item 2
Dividing the nominal budget constraint by Pt (the notation is the same as in Walsh’s
book)
ct + kt +mt + bt = f(kt−1, 1− lt) + kt−1(1− δ) + (1 + it−1)
Bt−1
Pt
+ Mt−1
Pt
+ τt
But:
Mt−1
Pt
= Mt−1
Pt−1
Pt−1
Pt
= mt−11 + πt
(1 + it−1)
Bt−1
Pt
= (1 + it−1)
Bt−1
Pt−1
Pt−1
Pt
= 1 + it−11 + πt
bt−1 = (1 + rt−1) bt−1
Note that we used the Fisher equation above. Hence, we can write:
ct + kt +mt + bt = f(kt−1, 1− lt) + kt−1(1− δ) + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ τt
Macroeconomía II, 2018/2 5
Item 3
The
max
{ct,kt,mt,bt,lt}t≥0
∞∑
t=0
βtu (ct,mt, lt)
s.t. ct + kt +mt + bt = f(kt−1, 1− lt) + kt−1(1− δ) + (1 + rt−1) bt−1 +
mt−1
1 + πt
+ τt, ∀t ≥ 0
ct, kt,mt+1, bt ≥ 0, lt ∈ [0, 1] ∀t ≥ 0
m0 = m > 0, b−1 = b > 0, k−1 = k > 0
Since we are assuming the solution is interior, the first order conditions must be satisfied
at the optimal. The Lagrangian of this problem is:
L =
∞∑
t=0
βt
{
u(ct,mt, lt)− λt
[
ct + kt +mt + bt − f(kt−1, 1− lt)− kt−1(1− δ)− (1 + rt−1) bt−1 −
mt−1
1 + πt
− τt
]}
The first order conditions are:
βtuc(ct,mt, lt)− βtλt = 0 (FOC1)
−βtλt + βt+1λt+1 [fk(kt, 1− lt+1) + 1− δ] = 0 (FOC2)
−βtλt + βt+1λt+1 (1 + rt) = 0 (FOC3)
βtum(ct,mt, lt)− βtλt +
βt+1λt+1
1 + πt+1
= 0 (FOC4)
βtul(ct,mt, lt)− βtλtfn(kt−1, 1− lt) = 0 (FOC5)
Simplifying a bit:
uc(ct,mt, lt) = λ′t (FOC1)
fk(kt, 1− lt+1) + 1− δ =
λt
βλt+1
(FOC2’)
1 + rt =
λt
βλt+1
(FOC3’)
um(ct,mt, lt) = λt −
βλt+1
1 + πt+1
(FOC4’)
ul(ct,mt, lt)
fn(kt−1, 1− lt)
= λt (FOC5’)
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Combining (FOC1’) and (FOC5’) we get:
ul(ct,mt, lt)
uc(ct,mt, lt)
= fn(kt−1, 1− lt) (1)
Dividing (FOC1’) by (FOC1’) forward, multiplying both sides by 1/β and using (FOC2’)
we get:
uc(ct,mt, lt)
βuc(ct+1,mt+1, lt+1)
= λt
βλt+1
uc(ct,mt, lt) = β (1 + rt)uc(ct+1,mt+1, lt+1) (2)
Combining (FOC2’) and (FOC3’):
fk(kt, 1− lt+1) = rt + δ (3)
Finally, dividing (FOC4’) by (FOC1’) we get:
um(ct,mt, lt)
uc(ct,mt, lt)
= 1− 11 + πt+1
βλt+1
λt
Using (FOC2’) we have βλt+1
λt
= 11+rt . Hence:
um(ct,mt, lt)
uc(ct,mt, lt)
= 1− 1(1 + πt+1) (1 + rt)
= 1− 11 + it
(4)
Note the last equality follows from the Fisher equation.
Denote the steady state variables with superscripts “ss”. Equation (2) implies that:
1 + rss = 1
β
Hence, (4) in the steady state becomes:
um(css,mss, lss)
uc(css,mss, lss)
= 1− β1 + πss (5)
Equation (3) in the steady state becomes:
fk(kss, 1− lss) =
1
β
− (1− δ) (6)
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Using the budget constraint we get
css = f(kss, 1− lss) + τ ss + m
ss
1 + πss − δk
ss −mss (7)
But since Mt −Mt−1 = Tt, we have that
Mt −Mt−1
Pt
= τt
τt = mt −
mt−1
1 + πt
Hence, τ ss = mss − mss1+πss . Therefore, (5) becomes:
css = f(kss, 1− lss)− δkss (8)
Finally, using (1):
ul(css,mss, lss)
uc(css,mss, lss)
= fn(kss, 1− lss) (9)
Equation (5), (6), (8) and (9) characterize the steady state in this economy. Note that
inflation will affect the real money holdings in steady state by equation (5). Moreover, for
a given mss, (6), (8) and (9) characterize consumption, capital and leisure in the steady
state, and mss only affect these variables through equation (9). Hence, if ul(c
ss,mss,lss)
uc(css,mss,lss)
does not depend on mss,css, kss and lss will not depend on mss, and therefore, will not
depend on inflation πss. In that case, we have superneutrality. If ul(c
ss,mss,lss)
uc(css,mss,lss) depends on
mss then superneutrality fails: inflation affects mss, which affects the steady state level of
consumption, capital and leisure. Intuitively, inflation may distort agent decisions of how
much to work if indirectly their money holdings affect the marginal utility of labor and
consumption. In practice, it is hard to tell how money holdings affect this marginalutility,
and that is why a more micro-founded model of money (such as the CIA) is desirable.
Exercise 4
Item 1
Note the notation in this exercise is a little different from the one we used in class: We
denote the money the agent decides to carry from date t to date t + 1 by Mt+1 (and not
by Mt as we did in class). Similar change of notation holds for capital (Kt+1 is the capital
decides to accumulate at date t). Also, note that we did not introduce bonds. The budget
Macroeconomía II, 2018/2 8
constraint of the centralized economy is:
Ptct + PtIt +Mt+1 = PtY (Kt, nt) +Mt + Tt (1)
(You could replace It by Kt+1− (1− δ)Kt in the equation above). The budget constraint
of decentralized economy is:
Ptct + PtKt+1 +Mt+1 = Pt(1− δ)Kt + Ptwtnt + PtrtKt +Mt + Tt (2)
Item 2
Dividing by Pt the budget constraint of the centralized economy:
ct + It +
Mt+1
Pt
= Yt +
Mt
Pt
+ Tt
Pt
Defining mt ≡Mt/Pt and τt ≡ Tt/Pt and using that
Mt+1
Pt
= Mt+1
Pt+1
Pt
Pt+1
= mt+11 + πt+1
We can write it as:
ct + It +
mt+1
1 + πt+1
= Yt +mt + τt
Similarly, for the decentralized economy we have:
ct +Kt+1 +
Mt+1
Pt
= (1− δ)Kt + wtnt + rtKt +
Mt
Pt
+ Tt
Pt
ct +Kt+1 +
mt+1
1 + πt+1
= (1− δ)Kt + wtnt + rtKt +mt + τt
Item 3
The homogeneity of degree one property of Y (K,N) allows us to write:
Y (Kt, Nt) = Yn(Kt, Nt)nt + Yk(Kt, Nt)Kt
In perfect competition, the maximizing firm would pay each factor according to its marginal
contribution to production:
wt = Yn(Kt, Nt) rt = Yk(Kt, Nt) (3)
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So the nominal budget constraint (1) in the centralized economy becomes:
Ptct + PtIt +Mt+1 = Pt (Yn(Kt, Nt)nt + Yk(Kt, Nt)Kt)︸ ︷︷ ︸
Y (Kt,nt)
+Mt + Tt
Replacing It = Kt+1 − (1− δ)Kt and (3) above we get:
Ptct + PtKt+1 +Mt+1 = Pt(1− δ)Kt + Ptwtnt + PtrtKt +Mt + Tt
Which is identical to (2).
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