Solutions Problem Set 2 Macro II (14.452)
Francisco A. Gallego
04/22
We encourage you to work together, as long as you write your own solutions.
1
Intertemporal Labor Supply
Consider the following problem. The consumer problem is:
M ax E0
fCt g;fNt g
t=T
X
t
t=o
f
s:t:
At+1 = R(At + Nt Wt
$1
1 Ct
+
!
$2
2 Nt g
Ct )
Where C is consumption, N is labor supply, A0 is initial wealth, R = 1 + r,
and the greek letters are parameters. Only W is stochastic
1). Derive and interpret the …rst-order conditions for Ct and Nt .
What are the intertemporal and intratemporal optimal conditions
There are at least two (equivalent ways of setting up and solving this problem). I will implement both of them
1. Lagrangian method I:
In this case the Lagrangian is:
M ax
fCt g;fNt g;fAt+1 g
$
= E0
t=T
X
t
t=o
f
$1
1 Ct
+
$2
2 Nt
t (At+1
R(At + Nt Wt
s.t. A0
F.O.C.
fCt g :
@$
=0,
@Ct
1
$1
1 $ 1 Ct
1
=
tR
(1)
!
Ct )g
fNt g :
@$
=0,
@Nt
$2 1
2 $ 2 Nt
=
t Wt R
The interpretation is more or less natural. The …rst condition tells you
that the marginal utility of Ct has to be equal to the marginal utility of
wealth (do not worry too much about R–it is an implication of the fact
that C in this case is realized at the beginning of the period t–so the value
of saving one unit of consumptions is 1 times R). The second condition
tells you that marginal disutility of work has to be equal to the marginal
utility of wealth times how much consumption you can get from one our
of work (the wage rate). Why no expectation? This is important, at time
t, the only source of uncertainty is realized (Wt ) and, roughly speaking,
t is a function of expected future wages (see below).
Combining both equations we get the intratemporal condition:
$2 1
2 $ 2 Nt
=W
$1
1 $ 1 Ct
1
To get the intertemporal condition, we need another FOC:
@$
=0,
@At+1
t
t
t+1
=
t+1 R
Using the FOC for consumption, we get our intertemporal condition:
t
$1
1 $ 1 Ct
1
=
"
1 = RE
t+1
RE(
Ct+1
Ct
$1 1
1 $ 1 Ct+1 )
$1
1
#
2. Lagrangian method II:
Notice that you can iterate the constraint and write the intertemporal
budget constraint.
T
X
R t Ct = A0 +
t=0
T
X
R t Nt Wt
t=0
Two comments: (1) Why no expectation? This has to hold exactly. (2)
what about AT +1 ?
M ax $ = E0
fCt g;fNt g
t=T
X
t=o
t
f
$1
1 Ct
+
F.O.C.
2
$2
2 Nt g
T
X
(
R t Ct
t=0
A0
T
X
t=0
!
R t Nt Wt )
fCt g :
fNt g :
@$
=0,
@Ct
@$
=0,
@Nt
t
$1
1 $ 1 Ct
1
t
$2 1
2 $ 2 Nt
= R
t
= WR
(2)
t
(3)
If you combine both equations, you get the same intratemporal condition
as before.
To get the intertemporal condition write the FOC for t + 1 taking expectations at t.
h
i
@$
$1 1
fCt+1 g :
= 0 , Et t+1 1 $1 Ct+1
R t 1 =0
@Ct+1
Using the FOC for Ct :
h
$1
Et t+1 1 $1 Ct+1
1
t
1
$1
1
$1 Ct
R
t 1
i
=0
and you get the same as before:
1 = RE
"
Ct+1
Ct
$1
1
#
What is more interesting about this method is that can be interpreted
as the su¢ cient statistic you need to solve the problem.
Notice that (1) and (2) imply that:
tR
=
( R)
t
How can you interpret this condition?
We can plug (1) and (3) into the budget constraint and we will get a
decreasing implicit function of as a function of fWi g with i t and, of
course, some constant terms (among them R). Thus, the only thing you
need to compute is to know something about fWi g. Therefore, you will
keep your constant in the future if the realizations of fWi g were more
or less what you expected in the past.
2). What is the link between Ct and Nt in this model? What
assumption(s) is (are) producing this result
The only link between both variables is (or t ). This implies the intratemporal condition and the negative correlation between both variables if W is kept
constant. And, the lack of correlation if W increases without a¤ecting .
Assumption: (1) both terms are additively separable in the utility function
and (2) perfect capital markets (what may happen if we introduce frictions as
At > 0)
3
3). How can you analyze changes to wages that do not a¤ect the
expected wealth of the consumers? (Hint: what is –the Lagrange
multiplier–in this model?). Let’s de…ne the wealth-constant elasticity
ln Nt
of labor supply as = @@ ln
in
Wt when wealth is constant. What is
this model? Is it positive or negative? Why?
Given our previous discussion, these are changes that do not a¤ect , therefore we can take logs in (3) and get:
log(
2 $2
( R)
t)
+ ($2
1) log(Nt ) = log Wt
Then
=
log(Nt )
1
=
log Wt
($2 1)
Notice that > 0. There are several ways of understanding this. Here is
one:
First, notice that obviously 1 > 0; 2 < 0; $1 < 1. Why? People like C,
dislike N , and marginal utility of consumption is decreasing in general. Therefore if the instantaneous utility function is to be concave (and, therefore, the
solution we just proposed has sense), it has to be true that:
@2U
$
2
= 2 $2 ($2 1) Nt 2 < 0 ) ($2 1) > 0
@N 2
4). Take the FOCs and discuss the e¤ect of the following situations
on fCt g and fNt g :
1.
Cross-sectional di¤erences associated with permanent differences in human capital (assume W varies in the crosssection)
Higher human capital will probably imply higher W . Therefore, #,
C ".
What about N ? Assume all other parameters do not vary in the
cross-section and take two individuals i and j
log(
Nj
1
)=
Ni
($2 1)
log
Wj
+ log
Wi
j
i
So e¤ect is ambiguous. One of them is richer, but at the same time
the alternative cost of leisure is higher.
Life-cycle changes in wages (i.e. if Mincerian equations are
correct, wages follow an inverted-U behavior over the lifecycle)
Notice that these evolutionary changes were predictable at the beginning of the life-cycle and, therefore, as W changes, is constant.
Thus C is also constant–see (2). But given that
> 0 and
is
constant, N moves together with W .
4
Unexpected temporary shocks to wages
If change is temporary, it shouldn’t a¤ect
by much, then C is
constant, and N moves together with W .
Unexpected permanent shocks to wages
In this case should be a¤ected. Therefore, C moves in the same
direction as W and the e¤ect on N is unclear.
2
Log-Linear RBC
Consider the following problem. Consumers maximize
2
3
1
X
et+j ; Nt+j )5
max Et 4
bj U (C
j=0
Subject to
e t+1 = Rt K
et + W
ft Nt
K
et
C
et ; Nt ) = log C
et + log(1
U (C
while …rms solve
n
o
e t = arg max Zt F (At N; K)
e
Nt ; K
e
1) K
(Rt +
e
N;K
and we assume
Nt )
e = Y = ZK
e1
F (At N; K)
(At N )
ft N
W
At+1 = At :
This is the standard RBC model. trending variables are those with a hat.
1) De-trend the model
e
De…ne V = VA .
1. Utility function:
U (Ct ; Nt ) = log (Ct At )+ log(1 Nt ) =
log (At )
| {z }
+log (Ct )+ log(1 Nt )
Does not m atter
2. Consumers B.C.Kt+1 At+1 = Rt Kt At + Wt At Nt
Ct At
Kt+1 At+1 = Rt Kt At + Wt At Nt
Ct At
,
,
Kt+1
At+1
= Rt Kt + Wt Nt
At
| {z }
Kt+1 = Rt Kt + Wt Nt
5
Ct
Ct
3. Pro…ts
1
fNt ; Kt g = arg max Zt (Kt At )
(At N )
N;K
1
, fNt ; Kt g = arg max Zt (Kt )
(Rt +
(N )
N;K
1) At Kt
(Rt +
1) Kt
At Wt Nt
Wt Nt
2)Solve for the F:O:C of the consumer problem, either using a Lagrangian or a Bellman Equation. You should …nd two …nal equations,
the Euler Equation and the labor supply.
1. Bellman equation
V (K)
= M ax flog (C) + log(1
C;N
s:t:
K0 = RK + W N
N ) + bE [V (K 0 )]g
C
FOC:
@V
1
@ [V (K 0 )] @K 0
1
=
+ bE
=
0
@C
C
@K
@C
C
@V
=
@N
1
N
+bE
@ [V (K 0 )] @K 0
=
@K 0
@N
1
b
N
E
+
@ [V (K 0 )]
=0
@K 0
Wb
E
(4)
@ [V (K 0 )]
=0
@K 0
(5)
Envelopment theorem:
@V
@ [V (K 0 )] @K 0
Rb
@ [V (K 0 )]
= bE
=
E
@K
@K 0
@C
@K 0
(6)
(a) Euler Equation:
Using (15) and (4), we get:
@V
R
=
@K
C
Then, we get:
1
b
R0
= E
=0
C
C0
(a) Labor supply
(5) and (4) imply:
1
6
N=
C
W
(7)
2. Lagrangean:
t=T
X
M ax $ = E
fCt g;fNt g
t=o
t
b flog (C) + log(1
N)
t(
Kt+1
(Rt Kt + Wt Nt
FOC
fCt g :
fCt g :
1
=
C
1
N
(8)
t
=
t Wt
(9)
From (5) and (4) we get:
1
N=
C
W
(10)
Take additional FOC:
fKt+1 g :
t
t
t+1
=E
t+1 Rt+1
(11)
Then, combining (16) and (11), we get our Euler equation:
t
1
=E
Ct
t+1
1
Ct+1
1
=E
Ct
t
Rt+1 , 1 =
Ct
Rt+1
Ct+1
3)Solve the …rm problem to …nd the labor and capital demands.
e t+1 = Rt K
et + W
ft Nt C
et
Give the law of motion for capital (re-write K
to get the usual equation)
This should be easy.
, fNt ; Kt g = arg max
N;K
1
= arg max Zt (Kt )
N;K
(N )
(Rt +
1) Kt
Wt Nt
FOC:
fNt g :=
fKt g :=
@ t
= 0 , Zt
@Nt
@ t
= 0 , (1
@Kt
) Zt
Kt
Nt
Nt
Kt
1
= Wt
= (Rt +
1)
4) We have a total of 5 equations. Discuss how would you proceed
to log-linearize our system (you don’t need to do all the math, just
apply the method to one or two equations).
There is no unique way of log-linearize things. I will use here my way (which
I think is simpler and does not use any math trick you didn’t know). Let go
one by one:
7
Ct ))g
!
1. Consumers BC:
Kt+1 = Rt Kt + Wt Nt
, Kt+1 = (1
Ct
)Kt + Yt
Ct
It is also true that (where X denotes the steady state of X):
K = (1
)K + Y
C
Then:
Kt+1
Kt+1
K = (1
K
K
= (1
De…ne xt = log (Xt )
) Kt
)
Kt
K + (Yt
K
+
K
(Xt
log X
(Yt
Y)
Ct
Y) Y
Y
K
Ct
C
C C
C
K
X)
X
Y
C
ct
kt + yt
K
K
But, take the production function (I’ll skip some steps):
, kt+1 = (1
)
, yt = zt + nt + (1
(12)
) kt
Then (12) becomes:
kt+1 + 1
+ (1
Y
K
)
C
Y
Y
ct +
n t + zt = 0
K
K
K
kt
2. The wage equation:
Zt
Kt
Nt
Z
K
N
1
= Wt
It is also true that
Then
Zt
Z
Kt N
K Nt
1
=W
1
=
Wt
W
Take logs and use small letters to denote log deviations from the steady
state and you get:
zt + (1
)(kt nt ) = wt
8
3. Labor supply
1
Nt =
Ct
Wt
1
N=
C
W
Then,
N
CWt
=
Nt
Ct W
1
1
Take logs
log(
Let’s de…ne L = 1
1
1
N
) = wt
Nt
N and l = log(L)
ct
(13)
log(L):
Then, the LHS can be written as:
log(
1
1
N
1
)=
Nt
l
1
N
1
N
N
n 1 N
N
=
nN
N
Then, (13) becomes:
n 1
N
N
= wt
ct
So there is a mistake in the log-linearization that appears in the program.
4. Price of capital
As with the wage equation:
(1
) Zt
Nt
Kt
= (Rt +
1)
Then (after some steps)
Zt
Z
Nt K
N Kt
=
(Rt +
R+
1)
1
Take logs:
zt + n t
kt
(Rt +
1)
R+
R+
1
9
1
=
Rt
R+
R
1
R
R+
1
rt
5. Euler equation
b Ct
Rt+!
Ct+1
1=E
bC
R
C
1=
Then,
1=E
Ct C Rt+1
= E [(1 + ct ) (1
C Ct+1 R
ct+1 ) (1 + rt+1 )]
E [1 + ct
ct+1 + rt+1 ]
5) Our log-linearized system is the following (small variables denote logs):
The law of motion and the market clearing conditions:
kt+1 + 1
+ (1
)
Y
K
C
Y
Y
ct +
n t + zt
K
K
K
kt
1
1
(kt
N
N
kt +
nt )
1
ct + nt
R
R+
1
wt + zt
N
N
rt
nt
= 0
=
0
wt
= 0
zt
=
0
The Euler Equation (or any other asset pricing equation):
Et [rt+1
ct+1 ] + ct = 0:
We assume
zt = z t
1
+ "t :
We want to write the system of six equations in order to solve it numerically
using the process in Uhlig(1997). Let st be the vector of endogenous state variables (capital stock), xt = [ct ; rt ; nt ; wt ]0 the endogenous control variables and
zt the stochastic processes. To avoid confusion, matrices are named using two
capitalized letters. There are two conceptually di¤erent sets of equations. The
…rst set describes the law of motion of the economy, together with the market
equilibrium conditions: I use M _ for these matrices (Motion and Market). In
the second set, the equations are forward looking and they involve the expectations of the agents: I use F _ for these equations (Forward). Finally, we have
the equation for the stochastic process zt . Write the system in the following
way:
M S1 st+1 + M S st + M X xt + M Z zt = 0
Et [F S2 st+2 + F S1 st+1 + F Sst + F X1 xt+1 + F Xxt + F Z1 zt+1 + F Zzt ] = 0
zt+1 ZZ zt "t+1 = 0:
10
In other words, what does each matrix contain?
Just by inspecting the log-linearize system we get:
M S1
MX
2
2
3
Y
+ (1
)K
(1
)
0
1
6 0 7
6
7
6
= 6
4 0 5 ; MS = 4
0
2
C
C
0
K
K
6 0
0
(1
)
= 6
4 1 N
0
1
N
R
0
R+ 1
F S2
;FX
3
7
7;
5
3
2 Y
0
K
6 1
1 7
7
6
;
M
Z
=
1 N 5
4 0
N
0
1
3
7
7
5
2
30
1
6 1 7
7
= 0;F S1 = 0;F S = 0;F X1 = 6
4 0 5
0
2 3
1
6 0 7
0
7
= 6
4 0 5 ; F Z1 = 0;F Z = 0
0
6) We want to …nd numerically a solution of the type
st+1
xt
c zt
= Sb st + SZ
b st + XZ
d zt ;
= X
because we are looking at small deviations from steady state. You have to solve
that using matlab. I give you a program, linear, containing the function linear
which does the solution of the system, so basically what you have to do is create
a program containing the following:
a) Values of the parameters of the model. We will assume
2
; = 1; = :979;
3
= 1:0163; Y =K = 1=2=4; K = 1; C = Y 68=100;
= :2; e = :0072
=
R
N
0:025;
= 1:004,
=
Before solving the problem, notice that we need a few clari…cations here:
There are some conditions that the steady state values have to satisfy:
(1
)Z
N
K
=
)
(1
Y
K
Y
= R+
1
K
R+
1
=
= 3 (0:0163 + 0:025)
(1
)
)
11
1
8:0701
So
K
Y
K
is not a parameter that you can change.
=
(1
)K + Y
)
C
= (1
Y
C,
)
C
= (1
K
K
+1
Y
)+
Y
K
8:0701( 0:025
0:004) + 1
0:7629
So the value that is in the PS is inconsistent with the budget constraint.
Sorry about that.
Moreover,
1
C
=
N=
W
C
Y
N
=
C
Y
N
,N =
1
1+
C
Y
:3180
So, if we want to have N = 0:2, we need
6= 1! Fortunately, this does
not a¤ect the simulations Olivier showed in class because it is the sum of two
mistakes that cancel out each other. In any case, keep in mind that if you change
N you have to change : This is completely intuitive because is a measure of
how much you value leisure. Notice that if = 0, then N = 1:
b) The matrices derived in 5).
c)Then you have to call the function linear.
7) Now we want to do impulse-response functions. For that, attach
to your program (copy and paste after part c)) the program linear 2.
This programs calls two other functions, impulse and impulse-plot.
The …rst …nds the reduced form model and the second hits it with a
shock. That should be the end of your program. Explain the results.
How do they depend on the values assumed in a)? (run the program
for di¤erent values of at least two parameters and compare results).
Interpret.
Several exercises can be implemented:
Base case (Figure 1)–Minor di¤erences with the …gure Olivier showed in
class, same intuition.
12
Baseline case
Change
to 0.999 and 0.300 (Figures 2 and 3)
= 0:999
13
= 0:9
Change N = 0:8 (Figure 4).
N = 0:8
R = 1:1 (Figure 5)
14
R = 1:1
= 1:01 (Figure 6)
= 1:01
3
(Optional) Discrete Time Dynamic Program15
ming1
Use the discrete time setup of Philippon and Segura-Cayuela (2003) (up to section 5) and the MATLAB programs contained in the folder DynamicP rogramming.
Read tutorial:doc and the comments in DP:m.
1) Derive the Euler equation in discrete time in the model with
exogenous labor, …rst using dynamic programming and then using
the Lagrange multiplier method. Compare it to the continuous time
Euler equation.
The problem is:
M ax$
fCt g
=
t=1
X
1
bt
1
t=o
e t+1
K
At+1
et
C
s:t:
et + W
ft
= Rt K
=
At
et
C
Divide all the trending variables by At :De…ne V =
A0
t
e
V
A
and notice that At =
Utility function:
M axV
= M ax
t=1
X
bt
1
t=o
1
= M ax A0
t=1
X
t=o
= M ax
t=T
X
t
t
1
| {z }
1
t=o
1
b
1
t
Ct A0
(Ct )
(Ct )
1
1
The budget constraint (same as before):
Kt+1 = Rt Kt + Wt
Thus, the problem becomes M ax
t=T
X
t
1
Ct
(Ct )
1
s.t.
Kt+1 = Rt Kt +
t=o
Wt
Ct
Bellman equation
1 This exercise is optional in the sense that you could skip it if you feel comfortable with
stochastic DP, and do it if you want to get more training.
16
V (K)
t
= M ax
1
C
s:t:
= RK + W N
K0
1
(C)
+ E [V (K 0 )]
C
FOC:
@V
=C
@C
1
e
C
+ E
@ [V (K 0 )] @K 0
=C
@K 0
@C
1
e
C
E
@ [V (K 0 )]
=0
@K 0
(14)
Envelopment theorem:
@ [V (K 0 )] @K 0
R
@V
= E
=
0
@C
@K
@C
@E [V (K 0 )]
@K 0
(15)
Using (15) and (4), we get:
@V
= RC
@K
Then, we get:
2
0
C
C
E 4R0
1=
But notice that:
1
e
C
1
e
C
3
1
e
C
3
5
1
e
C
=b
Then
2
1 = bE 4R0
Lagrangean:
M ax$ =
fCt g
t=1
X
t=o
t
1
C0
C
1
(Ct )
t(
5
Kt+1
(Rt Kt + Wt Nt
Ct ))
FOC
1
fKt+1 g :
fCt g : Ct
t
t
=E
=
(16)
t
t+1
t+1 Rt+1
Then, combining (16) and (11), we get our Euler equation:
"
#
1
h
i
1
1
Ct
t
t+1
Ct = E
Ct+1 Rt+1 , 1 = E
Rt+1
Ct+1
17
(17)
Continuous time
Z1
M ax
fCt g
1
et
C
1
0
s:t:
et
K
et + W
ft
1)K
= (Rt
=
At
At+1
exp (
t) dt
et
C
Detrending:
M ax
Z1
1
1
0
1
= M axA0
Z1
et
C
exp (
t) dt = M ax
Z1
1
(Ct At )
1
exp (
t) dt
0
1
1
(Ct )
1
t
( )
exp (
t) dt
0
1
Notice that ( )
1+
:
1
= exp log (
) = exp
1
log ( )
1
exp
Then
1
M axA0
Z1
1
1
(Ct )
t
( )
1
exp (
t) dt
0
= M ax
= M ax
Z1
0
Z1
1
1
1
(Ct )
1
(Ct )
1
exp
exp (
t dt
t) dt
0
1
with
The budget constraint becomes
Kt = (Rt
1)Kt + Wt
Ct
Then the Euler equation will be:
Ct
Ct
=
(Rt
1) = (Rt
=
(Rt
1)
18
1
1)
,
2) Numerical analysis using DP:m. Describe the dynamics of the economy
when it starts far away from its steady state (say 30% below). How does the
recovery depend on the evolution of productivity? What does it tell you about
the rapid growth in Europe after the second world war? (Hint: for the question
on the evolution of productivity you may want to replicate the model, say 100
times, and see what happens with the recovery path)
See …gures
Evolution of output after the recovery (Figure 7):
Recovery
Mean and median and max and min in 100 simulations (Figures 8 and 9)
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Mean and median (100 recoveries)
Fastest and Lowest Recovery (100 simulations)
3)DP:m produces 5 plots. Describe brie‡y each of them (what they represent
and how they are computed. I am not asking for the details of the computations.
20
Simply show me that you understand the architecture of the program).
Figure 1: 3D graph showing optimal consumption as a function of realizations of Z and initial capital stock. By-product of the search for the
policy rule.
Figure 2: Policy rule. Kt+1 as a function of Kt . Capital in the steady
state goes from 4.6 to around 5.4
Figure 3: Evolution of consumption in a stochastic simulation after starting from some initial capital that is below the steady state value.
Figure 4: Evolution of consumption in the same simulation.
Figure 5: stationary distribution of capital in the steady state.
5) Numerical analysis using DP:m. We have seen in the continuous time
setup that the reaction of consumption to a positive technology shock depends
on the elasticity . Use the policy function estimated by DP:m to illustrate this
…nding. (note: in the program, they de…ne gamma, the CRRA instead of .
How are they related? Why?) Hint: DP:m produces 5 plots. The …rst one is
called consumption, and it is 3 dimensional. Consumption is measured on the
vertical axis. The two horizontal axis are for K and Z. If you do not see well,
you can rotate the picture with the mouse.
1
Notice that CRRA =
Di¤erent
= f0:1; 1; 10g Figures 10 to 12.
21
= 0:1
=1
22
= 10
6) Show numerically how the behavior of consumption is a¤ected by the
persistence of the shocks. Interpret.
Di¤erent
= f0:05; 0:99g
= 0:05
23
= 0:99
24