Current Value Hamiltonian VS Present Value Hamiltonian in Economics

There is no clear right and wrong about this, it's just a matter of convenience. The current-value Hamiltonian is likely to be more convenient when the objective function includes a discount factor. Following Chiang (1), suppose the problem is:

$qquad$Maximise $V = int_0^T G(t,y,u)e^{-rho t}$

$qquad$subject to $dot y=f(t,y,u)$

$qquad$and boundary conditions

The standard (present value) Hamiltonian is:

$qquad H=G(t,y,u)e^{-rho t} + lambda f(t,y,u)$

If we proceed from this Hamiltonian, the co-state equation (one of the first-order conditions) is:

$qquad dot lambda = -frac{partial H}{partial y}= -frac{partial [G(.)e^{-rho t}]}{partial y}-lambdafrac{partial f}{partial y}$

While it is possible to obtain a solution this way, the discount factor complicates the derivatives and can make interpretation more challenging.

Suppose instead we use the current-value Hamiltonian:

$qquad H_c = G(t,y,u) + mf(t,y,u)$

where $m$ is a current-value Lagrange multiplier defined by $m=lambda e^{rho t}$. The co-state equation is then:

$qquad dot m -rho m = -frac{partial H_c}{partial y} = -frac{partial G}{partial y} - mfrac{partial f}{partial y}$

This is simpler because it contains no discount term.

Reference

1. Chiang A C (1992) Elements of Dynamic Optimization pp 210 ff