# 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), $\rho > 0$ 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 $\rho+\frac{\dot{\lambda_{t}}}{\lambda_{t}}<0$, 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.
• [Part 1/2] Thanks for your reply. With $(a_{t},H_{t})$-diagram, I refer to a sketch of H in function of a. I would think that the optimal a is where H is maximized ($\frac{\partial H_{t}}{\partial a_{t}}=0$), but the second foc requires this partial derivative to equal $\dot{\lambda}$, as it seems to me, optimality therefore does not occur at the max. of H. W.r.t. $\dot{\lambda}$, as I realize now, this should reflect changes in shadow price of a. Why can the relative change in shadow price ($\frac{\dot{\lambda}}{\lambda}$) be interpreted as capital gain? Dec 21 '15 at 11:13
• [Part 2/2] W.r.t. to your last remark about the growth path of consumption and its relation to capital gain: does this mean that the growth path of consumption effects capital gains (not vice versa) through the fact that higher consumption implies lower capitalstock, and therefore lower capital gains, should they occur? Should I interpret capital gain as accumulation of capital ($\dot{a}$) and not as, for example, increasing stock prices? Dec 21 '15 at 11:21
• $H_{t}$ is simply the Hamiltonian and not really a function. For the control variable, when you find the partial derivatives and equalize it to zero, you find the maximum because when you look at the the second derivative of Hamiltonian with respect to to $a$, you will see that it is negative, so for optimality, the concavity of Hamiltonian is essential. $\dot{\lambda}$ shows the variation of $\lambda$ with respect to time, which means that the change in capital price, so if the price of the capital is higher, the return from this capital will be higher. Dec 21 '15 at 19:13
• There is a trade-off between the fact that you provide utility from consumption, for which there exists a opportunity cost, which is the price of the wealth. In your optimality condition, you will have $u_{c}(c)=\lambda$, if $\lambda$ is higher, this means the optimal consumption level will be less because it becomes more and more costly to consume instead of investing. Briefly, yes, higher consumption implies a lower capital stock and yes, in this case, as there exists less capital in economy, probably the marginal return of capital will be higher, so there will be an incentive to invest. Dec 21 '15 at 19:18