OLG
M ax U (c0 (t) ; c1 (t + 1))
S:T: (e0 c0 (t)) t = (c1 (t + 1)
FOC:
U0 (c0 (t) ; c1 (t + 1))
= t
U1 (c0 (t) ; c1 (t + 1))
U0 (c0 (t) ; c1 (t + 1))
U1 (c0 (t) ; c1 (t + 1))
Market Clearing:
(e0
c0 (t))
e1 )
= (c1 (t + 1)
c0 (t) + c1 (t) = e0 + e1
e0 c0 (t) = c1 (t) e1
Equilibrium:
(e0
= (e0
U0 (c0 (t) ; e0 + (e1
U1 (c0 (t) ; e0 + (e1
c0 (t + 1))
c0 (t))
1
c0 (t + 1)))
c0 (t + 1)))
e1 )
Under some assumptions, this is a …rst
order di¤erence equation:
Note: if
U (c0 (t) ; c1 (t + 1)) = V (c0 (t)) +W (c1 (t + 1)) ;
then
(c0 (t) e0 ) V 0 (c0 (t)) = W 0 (c1 (t + 1)) (e1 c1 (t + 1))
If (c0 (t) e0 ) > 0 (implying (e1 c1 (t + 1)) > 0) RHS is large for
small positive c1 (t + 1) and decreasing, so that by the implicit function theorem, c1 (t + 1) = F (c0 (t)). Similar argument if utility is homothetic, that is U0 =U 1 is a decreasing
function of (c0 =c1 ) :
2
Let
= 1;
that is Population growth is zero.
3
4
Steady States:
1. Autarkic, (e0
2. Trading =
rule.
TRY
c0) = 0;
U0 (c0 ;e0 +e1 c0 )
U1 (c0 ;e0 +e1 c0 )
b (c0)2 + c1
1: U = ac0
a; b >
T ry a =
2: U =
Let c0
a >
e0 >
= 1; golden
a
0; (e0; e1) = (0; w) ; w > :
b
4; 2; 1:5
(c0 + b)1 a
+ c1;
1 a
0; a 6= 1; ; b > 0;
0: T ry = 51; a = 5:1
OLG WITH GOVERNMENT
M ax V (c0 (t)) + c1 (t + 1)
S.T.
e0 + e1 g = c0 (t) + c1 (t)
c1 (t) e1 = e0 c0 (t) g
5
(e0
c0 (t + 1) g) = V0 (c0 (t)) (e0 c0 (t))
x (t + 1) = x (t) V0 (e0 x (t)) + g
Does an increase in governement expenditures and taxes increase or decrease savings?
6
Try interpretation with money:
mt = pt (e0
c0 (t))
rtmt + pt+1e1 = pt+1 (c1 (t + 1))
rtpt
(e0 c0 (t)) = (c1 (t + 1) e1)
pt+1
rtpt
(e0 c0 (t))
M ax U c0 (t) ; e1 +
pt+1
e0
U0 (c0 (t) ; c1 (t + 1))
rtpt
=
pt+1
U1 (c0 (t) ; c1 (t + 1))
U0 (c0 (t) ; c1 (t + 1))
(e0
c0 (t + 1) =
U1 (c0 (t) ; c1 (t + 1))
U0 e0
mt+1
=
pt+1
U1 e0
mt
pt ; e1
t+1
+m
pt +1
mt
pt ; e1
t+1
+m
pt +1
c0 (t))
mt
pt
Trading
Autarkic
Samuelsonian Unstable,PO Stable, Not PO
Classical
Stable,Not PO Unstable, Not PO
= 1; e0 6= c0
6= 1; e0 = c0
7
DIAMOND
M ax (1
c0) (1+R)
) `n c0+ `n (w
FOC
1
(1
=
c0
w c0
) (w c0) = c0; (w
f (k) = k + (1
w
(f
kt+1 =
=
1+n 1+n
SS (Why only one?)
c0 ) = w
)k
f 0k) =
k =
(1
)
k
1+n
k =
(1
)
1+n
1
1+n
(1
1
Golden Rule:
(1 + n) kt+1 = (kt) + (1
) kt ct
ct = (kt) + (1
) kt (1 + n) kt+1
c0 = k 1 ( + n) = 0
k =
1
8
1
( + n)
1
)k
Note possible ine¢ ciency
k
1
=
(1 + n)
(1
)
(1
)
k =
1+n
Stability of SS
(1
)
dkt+1
=
k
dkt
1+n
(1
)
=
1+n
k
1
does not contain
? ( + n)
1
1
?
1
1
( + n)
(1
)
1+n
1
kt+1 = S (r (kt) ; w (kt))
1 sr r 0
0
0
=
>
0
s
;
w
>
0;
r
<0
r
w0
9
=k
1
Simple OLG:
dkt+1
dkt
1
=
<1
:
OLG with endogenous labor, consumption in second period:
M ax U (wtLtRt+1)
V (Lt)
FOC and Dynamics:
U 0 (w (kt; Lt) ; LtR (kt+1; Lt+1)) w (kt; Lt) R (kt+1; Lt+1)
= V 0 (Lt)
kt+1 = w (kt; Lt) Lt
These are two di¤erence equations in (kt; Lt) :
10
Simplify by assuming log utility:
M ax (1
)`n(1
Lt) + `n (LtwtRt)
FOC
1
= ; Lt =
1 Lt Lt
Production (note change in exponents of
Prod. fn):Golden Rule
M axk c (k) = k 1
= (1
k =
+ (1
)k
1
11
(1
)
1
)k
k
Dynamics
(kt)1
kt+1 = wtLt =
Steady State
k=(
1
) =
1
Stability
dkt+1
= (1
) k
=1
<1
dkt
Note: With log utility, dimension is reduced by 1.
12
OLG with Money
M ax U (ct+1)
V (Lt)
S.T.
kt+1 + Mt+1Qt = wtLt
ct+1 = Rt+1kt+1 + Mt+1Qt+1
FOC and dynamics
Rt+1wtU 0 (Rt+1wtLt) = V 0 (Lt)
kt+1 + Mt+1Qt = wtLt
Qt+1 = Rt+1Qt
These are three di¤erence equations in
(L; k; Q)
Note that there are two steady states,
with Q = 0 and with Q 6= 0; R = 1
(golden rule):
13
Simplify with Log Utility and
F = k1
+ (1
) k: Then L = :
Dynamics:
kt+1 = (kt)1
Qt+1 = Rt+1Qt
Mt+1Qt
At golden rule steady state:
^
R k;
) k^
= (1
k^ = (1
)
=1
1
Note that M Q ? 0 since:
k^ =
k^
k^
1
MQ
1
(1
=
=
)
1
(1
1
14
1
)
1
? (1
)
1
= k^
0.1
Stationary Sunspot Eq. In a Finance Constrained
Economy, JET 1986 (Hand to Mouth Workers?)
(Hand to Mouth?)Workers:
M axfct g
1
X
t 1
u (cw
t )
t
v (nt )
1
u0 + cu”(c) > 0; v (0) = 0:
w
pt cw
t + kt
pt cw
t
u (cw
t )
=
+
uw
t
ktw
dktw 1
dktw
Mtw + rt ktw
+
Mtw
1
w
Mt+1
Mtw + rt ktw 1 + wt nt
u0 (cw
t )
+
w
Mt+1
pt
1
w
rt kt 1
pt ktw
1
1
+ Eu0 cw
t+1
pt
pt+1
+ w t nt
dktw
1
!
0
First constraint is a …nance constraint stating that a given period’s expenditures must be …nanced either out of money held at the beginning of the
period, or by borrowing against the value of capital held then. (Both the rents
rt ktw 1 and wages wt nt , are received only at the end of period t but the former
can be borrowed against by pledging the capital as collateral, while future wage
payments cannot be borrowed against, because of moral hazard problems.) The
depreciation factor is d; where a fraction 1 d of capital depreciates. Second
constraint is the usual budget constraint.
FOC with respect to kt+1 :Look at equilibria such that:
u0 (cw
t )>
tE
u0 cw
t+1
rt+1
+d
pt+1
pt
is less than return capital
Also return to holding money pt+1
rt+1 + dpt+1 > pt ;
rt+1
pt+1
+d
rt+1
pt
+d>
pt+1
pt+1
so workers set ktw = 0 since current marginal utility of forgoing a unit of current
consumption exceeds discounted expected utility obtained by investing it (they
can’t borrow to consume more today), even though the net return on investing
t+1
capital is positive: rt+1 +dp
> 1: Then they maximize a static labor market
pt
problem
M axnt E [u (ct+1 ) v (nt )] S:T: pt+1 ct+1 = wt nt
to get (since workers set ktw = 0; ct+1 =
nt v 0 (nt ) = V (nt ) = E u0 (ct+1 )
w t nt
pt+1
15
wt nt
pt+1
=
Mt+1
pt+1 ;
t
u0 (ct+1 ) pw
= v 0 (nt )):
t+1
= E [ct+1 u0 (ct+1 )] = E U
wt nt
pt+1
where (bad notation!) U (c) = cu0 (c) : (Like olg with a month; workers consume
Mt+1
t nt
their wage bill w
pt+1 = Pt+1 !)
Capitalists
1
X
t 1
ln cct
1
ST
pt cct + ktc
dktc
c
+ Mt+1
1
Mtc + rt ktc
1
Solution:
cct
=
kt
=
(1
rt
+ d kt
pt
)
rt
+ d kt
pt
1
1
with
rt
+d
pt
>
so kt
=
bd;
1
b=
rt
+ d kt
pt
;
1
1
>
d kt
1
= dkt
1
so that kt > dkt 1 ; all capital from last period is held and then added to–no
decumulation. Thus Capitalists hold no money if prtt + d > bd:
Fixed Coe¤. production: Must use m units of labor with each unit of capital
to produce a > b d units of new output plus d units of old undepreciated
capital. (Needed so steady state capital >0).
EQ:
wt nt
=
w
Mt+1
=M :
nt
=
mkt
w
Since pt cw
t = Mt = wt
(b
1 nt 1
M oney M arket
Labor M arket
1
and cct = (1
)
rt
pt
+ d kt
1
= (1
1) kt ;
c
cw
t + ct + kt
=
M
+ bkt = (a + d) kt
pt
M
pt
=
(a + d) kt
h
Using these in V (nt ) = E U
w t nt
pt+1
i
1
1
= output
bkt = cw
t
we obtain
w t nt
= E [U (ct+1 )]
pt+1
V (mkt 1 ) = Et (U ((a + d) kt bkt+1 ))
V (nt )
= E U
16
)
rt
pt
+d
kt
( prtt +d)
=
V (mkt
1)
= Et (U ((a + d) kt
bkt+1 ))
Di¤erence eq. that can produce indeterminacy and sunspots. See Reichlin
(JET 1986, set b = 1)) , Benhabib and Laroque (JET 1988) for more general
i 1
h
(n)V 0
@n
setups. Let the elasticity of labor supply " = w
1
=
: The
0
n
@w
cU
unique steady state k is stable if:
">
a+d
2b a
b
d
1
giving indeterminacy: for initial k0 locally all trajectories of fkgt=1 converge to
k :
If we assign the pure rates the values = 0:04=year, g = 0:33=year (reciprocal of the U.S. capital/output ratio), and = 0:10=year, then for = 3months;
indeterminacy requires " > 0:05. For smaller values of , the required elasticity
is even smaller.
17
YAARI
Utility
M axc( )
Z
T
(t) g (c (t)) dt
0
Wealth is accumulated savings, on which
interest is earned:
Z t R
t
S (t) =
e j(x)dx (m ( ) c ( )) d
0
where m ( ) is earnings, c ( ) is consumption, and j ( ) is the interest rate and
S (T ) 0:
Z t R
t
_
S = j (t)
e j(x)dx (m ( ) c ( )) d + m (t)
0
= j (t) S (t) + m (t)
c (t) = j (t) S (t) + m (t)
18
c (t)
S_ (t)
c (t)
So
M axS( )
Z
T
(t) g j (t) S (t) + m (t)
S_ (t) dt
0
ST
Z
T
e
0
M axS( )
Rt
Z
j(x)dx
(m ( )
c ( )) d
0
T
(t) g j (t) S (t) + m (t)
S_ (t)
0
Use calculus of variations:
_ S; t = (t) g j (t) S (t) + m (t) S_ (t)
F S;
FS = g 0 j = _ g 0
g 00c_ = F_ _
c_ =
c_
=
c
Note: if
=e
0
0
_g
g 00
g0
g 00c
t
;
_
0
g
g
j
=
g 00
g 00
_
j
=
19
:
S
j
_
Random Death
Z
T 2 0; T ; Density :
T
(t)
(t) dt = 1
0
De…ne
(t) =
Z
T
(t) dt ;
t
t(
t(
)=
( )
(t)
) is the conditional density of ( )
given that death must occur in t; T : So
(t) is the probability that death occurs
after t;or the probaility of being alive at
t:
20
0
Utility:
Z
EV =
T
(t)
0
Z
t
(t) g (c (t)) d dt
0
Utility mass up to t multiplied by probability of being alive until t (dying at
t)): Let (t) g (c (t)) = f (t) : Let
Z t
f (s) = F (t) F (0)
0
Then
EV =
Z
T
(t) (F (t)
F (0)) dt
0
Now integrate by parts (u = (F (t) F (0)) ; dv = (t))
Z T
(1
(t)) (F (t) F (0)) jT0
(1
(t)) f (t) dt
0
=F T
F (0)
Z
T
f (t) dt+
0
=F T
Z
T
(t) f (t) dt
0
F (0) F T +F (0)+
Z
0
21
T
(t) f (t) dt
So
Z
EV =
T
(t) (t) g (c (t)) dt
0
So
_ S; t =
F S;
FS =
(t) (t) g (c (t))
(t) g 0j = _ (t) (t) g 0
!
_ (t)
_
g0
c_ = 00 j + +
g
(t)
But
(t) =
Z
T
_ (t) =
(t) dt;
(t)
t
_ (t)
=
(t)
So
c_ =
(t)
(t)
g0
_
t (t)
j+
t (t)
g 00
So the discount rate can be interpreted
_
as t (t)
: the standard discount
_
rate
plus an adjusment for the possibility of death (hazard rate) at t.
22
(t) _ g 0