Lecture XIV
Review of Dynamic Mathematics I
I.
Introduction to the Dynamic Optimality
A.
To introduce the problem of dynamic optimization, I would like to start with a
discrete time problem.
Specifically, we will use the example from Howitt
“Some Microeconomics of Agricultural Resource Use.”
1.
The basic problem is that y
t
, the firm’s output at time t, is a function of
variable inputs, x
t
, and the level of capital stock, k
t
.
Further, capital
stock can be augmented by investment, I
t
.
The firm’s overall problem
can then be written as:
(29
(
29
max
,
,
,
,,
xkI
tt
t
t
t
T
TT
T
ttt
t
xIk
xk
kkk
I
kk
βπ
β π
δ
=

+
∑
+
= +
=
0
1
1
0
Within this formulation, the first equation embodies the objective
function, the second equation is referred to as the equation of motion
depicting how the capital changes over time, and the third equation is
the initial condition on capital.
2.
Rewriting this equation in Lagrangian form yields
(
[]
Lx
I
k
k
k
k
I
x
k
t
t
t
t
T
t
=



+
++
=

∑
λ
δ
,
11
0
1
This form of the problem yields three types of conditions.
a.
The first type of conditions are with respect to the control
variables.
This problem has two control variables, x
t
and I
t
which the decision maker can directly control.
Specifically, the
capital variable, k
t
, is a state variable which can only be
controlled through investment, I
t
.
The first of the control
variables, x
t
, has no long term effect.
Thus, setting the level of
x
t
relies on similar rules as a static optimization problem:
∂
∂
β
∂π
∂
∂
L
xx
x
t
t
t
t
==
=
(.)
(.)
0
0
However, the derivative with respect to the investment variable,
I
t
, yields conditions dependent on
λ
t+1
.
λ
t+1
is the Lagrange
multiplier on the equation of motion.
As such, it depicts the
marginal value of a change in the capital stock.
In the jargon of
dynamic optimization, this variable is referred to as the costate
or adjoint variable.
The second condition with respect to I
t
is
then: