A Simple Version of a Heuristic Derivation of Hamiltonian Danyang

A Simple Version of a Heuristic Derivation of Hamiltonian
Danyang Xie and CHINHUA
Let us examine a simple problem:
max
Z
∞
u(c)e−ρt dt
0
subject to : k̇ = g(c, k)
f (c, k) ≥ 0
k0 given
Think of the integral sign as if it is a summation, and apply Lagrangian
blindly:
Z ∞
Z ∞ h
Z ∞
i
−ρt
−ρt
L=
u(c)e dt +
μ g(c, k) − k̇ e dt +
λf (c, k)e−ρt dt
0
0
0
Use integration by parts:
Z ∞
Z
μk̇e−ρt dt = kμe−ρt |∞
−
0
0
∞
0
k(μ̇ − ρμ)e−ρt dt
Thus,
L =
Z
∞
u(c)e
−ρt
dt +
0
Z
0
∞
[μg(c, k) + k(μ̇ − ρμ)] e−ρt dt
Z ∞
λf (c, k)e−ρt dt − kμe−ρt |∞
+
0
Z ∞0
=
[u(c) + μg(c, k) + λf (c, k)] e−ρt dt
0
Z ∞
(μ̇ − ρμ) ke−ρt dt − lim μt kt e−ρt + k0 μ0
+
t→∞
0
Z ∞
Z ∞
=
H(c, k, μ, λ)e−ρt dt +
(μ̇ − ρμ) ke−ρt dt − lim μke−ρt + k0 μ0
0
FOCs:
0
∂L
∂H
=
=0
∂c
∂c
∂H
∂L
=
+ (μ̇ − ρμ) = 0
∂k
∂k
which can be rewritten as:
μ̇ = ρμ −
1
∂H
∂k