1. Consider the following nonlinear dierence equation:
1
(1 + n) kt+1 = s (1
a) A akt
+1
a
where A > 0, a 2 (0; 1), s 2 (0; 1), and
1. To give economic
content to this equation, it is taken from a variant of the standard Solow
growth model. Savings is a c
Homework #5
Econ 7020
1. Consider a two-state Markov chain with realizations fz1 ; z2 g (where z2 > z1 ) and transition
matrix
2
=4
p
1
1
q
p
q
3
5:
(a) Set p = q. Prove that the autocorrelation is 2p
1. Remember that the correlation
coe cient is given by
Homework #8
Econ 7020
1. Consider a growth model economy with preferences given by
1
X
t
log (ct )
t=0
and a resource constraint
ct + it
at kt :
Capital takes multiple periods to build, as in Kydland and Prescott (1982):
kt+1 = (1
) kt + s1;t
s1;t+1 = s2;
Homework #9
Econ 7020
1. Consider the dynamic program
v (x) = max Qx2 + Ra2 + 2W ax + v x0
a
with law of motion
x0 = Ax + Ba:
Assume that (Q; R; W ) are such that the objective function is strictly increasing and strictly
concave over the relevant ranges
Homework #7
Econ 7020
1. Consider the following deterministic growth model:
(1
)
X
t
max 1
Nt u (Ct )
fCt ;Kt ;It gt=0
t=0
subject to
Ct + It = Yt
Yt = Kt (Zt Nt )1
Kt+1 = (1
) Kt + Vt It :
Let population Nt , labor-augmenting technical progress Xt , and
Homework #1
Econ 7020
1. Prove the continuous-compounding result:
lim
n!1
n
1+
r
n
no
= er :
Take natural logs to get
r
n
ln 1 +
r
n ln 1 +
=r
n
r
n
!
:
Now use lHopitals rule to evaluate the limit, since the expression at n = 1 yields
(
!)
n
ln 1 + nr
r
Homework #6
Econ 7020
1. Consider a deterministic economy populated by two types of households (each with population weight 12 ) with dierent discount factors:
identical preferences given by
1
X
1
<
2.
Each household has otherwise
t
i u (ci;t )
t=0
where
Homework #4
Econ 7020
1. Consider a Markov chain with realizations fz1 ; z2 g and transition matrix
3
2
p
1 p
5:
=4
1 q
q
(a) Find the invariant distribution using both the eigenvector method and the iterative
method.
We want to solve for the eigenvector
Homework #3
Econ 7020
1. Suppose we have a quadratic version of the permanent income model, with consumer preferences represented by
E0
"1
X
t
#
u (ct ) ;
t=0
where
u2 2
c ;
2 t
u (ct ) = u0 + u1 ct
and a ow budget constraint
1
at+1 = yt + at
1+r
ct :
Sup
Homework #10
Econ 7020
1. Consider an economy with a continuum of identical households with preferences given by
1
X
t
t=0
ct1
1
and a budget constraint of the form
ct + kt+1
(1 + rt
) kt + wt :
Factor prices rt and wt are competitively determined as a fu
Homework #2
Econ 7020
1. Consider the following decision rule for consumption:
ct = Et (yt+1 )
where
2 (0; 1). Aggregate output is determined by
yt = ct + it + xt
where it is investment and xt is net exports. Investment is determined by
it = it
and (et ;
Homework #7
Econ 7020
1. Consider a growth model economy with preferences given by
1
X
t
u (ct )
t=0
and a resource constraint
ct + it
f (kt ) :
Capital takes multiple periods to build, as in Kydland and Prescott (1982):
kt+1 = (1
) kt + s1;t
s1;t+1 = s2;
Homework #9
Econ 7020
1. Consider the following problem.
A vintner has one unit of labor to use each day.
He
can allocate that labor between the making of bread and the pressing of grapes for grape
juice.
The bread he makes today he can consume today.
tod
Homework #5
Econ 7020
1. Consider a standard Lucas tree economy with preferences given by
"1
#
X
t
u (ct )
E0
t=0
and dividends generated by some Markov process with cdf G (d; d0 ).
Dividends are not
storable.
(a) Obtain the price of a two-period European
Homework #3
Econ 7020
1. Take the linear stochastic dierence equation
Et yt+1
yt = x t
where
xt = a (L) "t
= a0 "t + a1 "t
1
+ a2 "t
2
+ :
and a (L) is "square-summable":
1
X
j=0
a2j < 1:
Let j j > 1. Look for a bounded solution that takes the form
yt = c
Homework #2
Econ 7020
1. A stock price equation that is implied by the e cient markets hypothesis is
1
(pt+1 + dt+1 )
rm
pt = Et
where rm > 1 is the constant return to the market portfolio.
(a) Solve this equation for fpt g assuming nothing about the path
1. A currently-popular model of price-setting behavior can be expressed as
t
= Et [
t+1 ]
+ yt + ut
where
> 0, t is the in
ation rate ( t = pt ), yt is output relative
to capacity for a typical producer, and the shock ut is rst-order moving
average:
ut =
1. Actual data give not consumption at a point in time but average consumption over an extended period, such as a quarter. Suppose that
consumption follows a random walk:
Ct = Ct
1
+ et ;
where et is white noise. The data provide average consumption over
1. Suppose that we have a symmetric Markov chain with transition matrix
1
=
with
2 (0; 1).
1
Find the invariant distribution
prove that limn!1 f
n
1
1
g=
2
=
1
and
2
2
.
The invariant distribution solves the eigenvector equation
1
1
2
=
1
1
:
2
Multiplyin
1. Take the standard growth model with preferences given by
t u (Ct )
t=0
and resource constraints
Ct + Kt+1 (1 ) Kt AF (Kt , 1) .
A is a parameter.
(a) Find the equations that determine the steady state.
The steady state is given by
1 = (AD1 F (K, 1) +
1. Consider the problem of choosing a consumption sequence fct g1 to maximize
t=0
1
X
t
[log (ct ) + log (ct
1 )]
t=0
subject to the constraint
ct + kt+1 = Akt :
Here, ct is current consumption, kt+1 is tomorrow capital, kt is today capital, and ct
s
s
is
1. Take the growth model and add a scal authority. The government budget constraint is
Gt + Tt =
and the household problem is
(
max E0
"
1
X
K
t
(rt
) Kt +
N wt Nt
(log (ct ) + log (1
#)
nt )
t=0
subject to
ct + kt+1
(1 + (1
K ) (rt
) kt + (1
N ) wt n t
+
1. Consider an economy in which converting old capital and investment into new capital is
costly. Households have preferences given by
1
X
t
u (ct )
t=0
and face a sequence of budget constraints given by
ct + pt kt+1
(rt + qt ) kt + wt :
qt is the market
1. Consider a consumer who seeks to solve the following optimization problem:
(1
)
t
X
1
max
log (ct )
1+
fct g1
t=0
t=0
subject to
At+1 = (1 + r) st
st + ct = At
A0 > 0 given.
At is the consumer stock of nancial assets at the beginning of period t, ct is
1
Supplemental Notes on Mathematics
These notes should be used to supplement the main material in class. Here I provide proofs for a
large number of relevant mathematical theorems, as well as cover some more general points than
we do in class (for example
Homework #4
Econ 7020
1. Consider the quadratic version of the permanent income model with preferences represented
by
E0
"1
X
t
#
u (ct ) ;
t=0
where
u2 2
c ;
2 t
u (ct ) = u0 + u1 ct
and a ow budget constraint
1
at+1 = yt + at
1+r
ct :
Let labor income b
Homework #6
Econ 7020
1. Take the standard growth model with preferences given by
1
X
t
u (Ct )
t=0
and resource constraints
Ct + Kt+1
A is a parameter.
(1
) Kt
AF (Kt ; 1) :
Both u and F are continuously-dierentiable at least twice, strictly
increasing,
Homework #7
Econ 7020
1. Consider the growth model with preferences represented by
"1
#
X
t
(log (ct ) + log (1 ht )
E0
t=0
and resource constraint
ct + qt kt+1 = kt ht1
:
qt is a random shock to embodied technological progress that follows an AR(2) proce
Homework #1
Econ 7020
1. Prove the continuous-compounding result:
lim
n!1
n
1+
r
n
no
= er :
2. Use a Taylor expansion to prove that, for z small enough,
(1 + z)
1 + z:
Using Matlab, plot the actual function on the interval z 2 [ 0:1; 0:1] and your approx
Homework #1
Econ 7020
1. A common approximation used in macroeconomics is
log (1 + x) = x
for small x.
(a) Use a Taylor expansion argument to prove this approximation is locally valid.
The Taylor expansion of log (1 + x) at x = x0 is
log (1 + x) = log (1