Dynamic optimization with assets as state variable: interpreting capital gains and losses Given a hamiltonian of the form:
\begin{equation}
H_{t} = ln(c_{t}) \dot{} e ^{-\rho t} + \lambda_{t}(w+ra_{t}-c_{t}),
\end{equation}
with $c_{t}$ consumption at time t (the control variable), $\rho &gt; 0$ time preference, $w$ a constant (for example wage), $a$ assets held in period t (this is the state variable for which the dynamics are described by: $\frac{da_{t}}{dt}=\dot{a_{t}}= w+ra_{t}-c_{t})$ and $r$ the return on assets. 

The first order conditions are:

$\frac{\partial H_{t}}{\partial c_{t}}=0$

$\frac{\partial H_{t}}{\partial a_{t}}=-\dot{\lambda_{t}}$, or equivalently, $\frac{\partial H_{t}}{\partial a_{t}}+\dot{\lambda_{t}}=0$.

We were taught that $\frac{\partial H_{t}}{\partial a_{t}}$ can be interpreted as the marginal return on assets in t, and $\dot{\lambda}$ as a capital gain. Together they are the overal return. An accompanying sketch with $a_{t}$ on the horizontal and $H_{t}$ on the vertical axis shows $H_{t}$ is a concave function with a maximum. 

The growthpath for consumption is given by $\frac{\dot{c_{t}}}{c_{t}}=-\frac{\dot{\lambda_{t}}}{\lambda_{t}}-\rho$, so that consumption grows over time if $\rho+\frac{\dot{\lambda_{t}}}{\lambda_{t}}&lt;0$, that is: consumption grows if the relative capital losses exceed time preference. As such, the optimizing agent should let his consumption grow when there are sufficiently big capital losses.

My questions are the following:


Why can $\dot{\lambda_{t}}$ be interpreted as a capital gain?
If there are capital gains or losses ($\dot{\lambda_{t}} \neq 0$), how does this intuitively and/or graphically (in the $(a_{t},H_{t})$-diagram) effect optimization?
Why would consumption grow only in the case of capital losses? Is there an intuitive explanation?
Firstly, the $\lambda$ is by definition the shadow price of the wealth/capital. It is the price when you increase one unit of additional capital. $\lambda$ measures the rate of increase of the maximum value of the utility function as the constraint is relaxed. So, the dynamic equation of $\lambda$ is the change on this variable, which intuitively is the capital gain.

What do you mean by $(a_{t},H_{t})$ diagram ? Do you mean a phase diagram on $(a_{t},c_{t})$ ? 

For your last question, you can realize on your constraint that the consumption increases at the expense of capital gain (c.f $\dot{a}$.) So, this means that every unit of consumption instead of accumulation is a somewhat loss in capital accumulation.