1
The Maximum Principle:
 yt+1 - yt = Q  yt , zt ,t 
s.t. 
 G  yt , zt ,t   0
T
max  F  yt , zt ,t 
yt ,zt
t =0
1. For each t : zt maximizes
the Hamiltonian H  yt , zt ,πt+1 ,t  s.t. G  yt , zt ,t   0
2.
πt+1 - πt = -H *y  yt ,πt+1 ,t 
3.
yt+1 - yt = H π*  yt , πt+1 ,t 
H *  yt ,πt+1 ,t  = max H  yt ,zt ,πt+1 ,t  s.t. G  yt ,zt ,t   0
zt
H  yt ,zt ,πt+1 ,t  = F  yt ,zt ,t   πt+1Q  yt ,zt ,t  2
The Maximum Principle:
 yt+1 - yt = Q  yt , zt ,t 
s.t. 
 G  yt , zt ,t   0
T
max  F  yt , zt ,t 
yt ,zt
t =0
1. For each t : zt maximizes
the Hamiltonian H  yt , zt ,πt+1 ,t  s.t. G  yt , zt ,t   0
πt+1 - πt = -H *y  yt ,πt+1 ,t 
2.
3.
yt+1 - yt = H
*
π
 yt , πt+1 ,t 
H *  yt ,πt+1 ,t  = max H  yt ,zt ,πt+1 ,t  s.t. G  yt ,zt ,t   0
H  yt ,zt ,πt+1 ,t  = F  yt ,zt ,t   πt+1Q  yt ,zt ,t 
zt
H  yt ,zt ,πt+1 ,t  = F  yt ,zt ,t   πt+1Q  yt ,zt ,t  3
The Maximum Principle:
 yt+1 - yt = Q  yt , zt ,t 
s.t. 
 G  yt , zt ,t   0
T
max  F  yt , zt ,t 
yt ,zt
t =0
1. For each t : zt maximizes
the Hamiltonian H  yt , zt ,πt+1 ,t  s.t. G  yt , zt ,t   0
2.
3.
πt+1 - πt = -H *y  yt ,πt+1 ,t 
yt+1 - yt = H
*
π
 yt , πt+1 ,t 
H  yt ,zt ,πt+1 ,t  = F  yt ,zt ,t  ππt+1
t+1 Q  yt ,zt ,t 
4
Continuous Time
t = 0,1,2,3,....T
t  0
t = 0, t,2 t,3t, ....,nt, ......
T
n = 0,1,2,3......,
t
5
Continuous Time
y  t + t  - y  t  = Q  y  t  , z  t  ,t  t
y  t + t  - y  t 
t
= Q  y  t  , z  t  ,t 
y  t  = Q  y  t  , z  t  ,t 
G  yyt t,z
 ,t z,tt ,t0  0
G
T/Δt
max  F  y  i t  , z  i t  ,i t t
i=0
6
Continuous Time
T/Δt
 F  y  i t  , z  i t  ,i t t
i=0

Τ
0
F  y  t  ,z  t  ,t  dt
t  0
defining the Hamiltonian
H  y,z,π,t  = F  y,z,t   πQ  y,z,t 
The conditions:
2.
3.
πt+1 - πt = -H *y  yt ,πt+1 ,t 
yt+1 - yt = Hπ*  yt , zt ,t 
become ……7
Continuous Time
π  t  = -H
*
y
y t  = H
*
π
 y  t  ,π  t  ,t 
 y  t  , z  t  ,t 
Thus……
8
The Maximum Principle:
T
max 0
y t  ,z  t 

 y = Q  y  t  , z  t  ,t 
F  y  t  ,z  t  ,t dt s.t. 

 G  y  t  , z  t  ,t   0
1. For each t : z  t  maximizes
the Hamiltonian H  y  t  , z  t  ,π  t  ,t 
s.t. G  y  t  , z  t  ,t   0
 y  t  ,π  t  ,t 
2.
π = -H
3.
y t  = H
*
y
*
π
 y t  , z t  ,t 
H *  y  t  ,π  t  ,t  = max H  y  t  , z,π  t  ,t  s.t. G  y  t  , z,t 9  0
z
Example:
capital
(stock variable)
k = w + rk - c
wage rate
interest rate
k  0  = k T  = 0
0 
T
max

consumption
(control variable)
ln c  t  e dt
-ρt
utility discount rate
(individual time discount rate
ln  c 
c
10
Example:
H = ln  c  e + π  w + rk - c 
-ρt
First Order Condition:
-1 -ρt
c e
-ρt
e
c=
π
-π =0
substitute the optimal value of c in H to find H*
H* = -  ln  π  + ρt  e
-ρt
H *
k=
= w + rk - π -1e -ρt
π
H *
π== -rπ
k
+ π  w + rk  - e
-ρt
11
Example:
H *
-1 -ρt
k=
= w + rk - π e
π
H *
π== -rπ
k
π  t  = π0 e
-rt
k = w + rk - π e
-1
0
 ke t 
-rt
t =T
 π0
'
 r - ρ t

-rt
-rt
-1 -ρt
k
rk
e
=
we
π
=
0 e
-rt
-ρt
1
e
1
e
ke -rt - k  0  = w
- π0-1
r
ρ
12
Example:
-ρt
-ρt
e
e
=
c=
-rt
π0 e
π
= π0 e
-1
 r - ρ t
if r > ρ
then c < w for small t's and later c > w
if r < ρ
then c > w for small t's and later c < w
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