Problem
Set
4
Received
on
Wednesday
(10/8),
due
on
Monday
(10/13)
in
class.
1.
An
international
summit
has
been
held
to
discuss
what
do
to
improve
the
living
conditions
of
the
people
that
live
in
the
country
of
Poorville.
You
are
their
economic
advisor.
This
country
has
a
Cobb-­?Douglas
production
function
Y=AK?L1-­??,
where
K
is
the
capital
stock,
and
L
the
number
of
workers,
saves
a
constant
fraction
of
its
income
s,
has
a
constant
population
growth
rate
of
n,
a
constant
depreciation
of
its
capital
stock
of
?,
and
can
be
approximated
as
a
closed
economy
with
no
government.
a) Derive
the
fundamental
equation
of
the
Solow
model
for
?k,
the
change
in
capital
per
worker,
and
solve
for
the
steady-­?state
level
of
k
and
the
steady-­?state
level
of
output
per
worker
y.
b) We
know
that
Poorville
has
the
same
growth
rate
of
population
and
depreciation
rate
as
the
rest
of
the
world.
We
know
that
it
has
the
same
capital
share
?=0.5.
But
we
also
know
that
it
saves
10%
of
its
income
relative
to
a
savings
rate
20%
in
the
rest
of
the
world.
Knowing
that
Poorville
is
4
times
poorer
than
the
rest
of
the
world
in
terms
of
output
per
capita,
would
you
infer
that
its
total
factor
productivity
is
the
same
as
the
rest
of
the
world
or
lower?
c) The
rest
of
the
world
considers
sending
a
foreign
aid
gift
in
the
form
of
extra
capital
to
Poorville.
Discuss
the
consequences
of
this
policy
for
the
evolution
of
output
per
worker
over
time
in
Poorville.
d) Another
proposal
on
the
table
is
to
implement
policies
that
raise
the
savings
rate
of
Poorville
to
the
world
average
of
20%.
Discuss
the
consequences
of
this
policy
for
the
evolution
of
output
per
worker
over
time
in
Poorville.
e) Assume
that
the
golden
rule
rate
of
savings
is
30%.
Plot
what
happens
to
consumption
over
time
after
the
gift
of
capital.
Would
the
citizens
of
Poorville
like
to
receive
the
free
capital
gift?
How
does
your
answer
depend
on
their
patience?