# How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as:

$$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$

Note that this is, for the most part, a typical calculus of variations set up. The integrand could be defined in the more familiar notation $L(x,\dot{x},t;r)=\dot{x} p(x,\dot{x},t) e^{-rt}$. In physics, $P$ would be the "action."

(I won't go into the economic significance Hotelling attaches to the variables because I am interested in this only as a calculus of variations problem.)

In Hotelling's mine problem, the point is to solve for the $\dot{x}$ that maximizes $P$. To achieve this one of course solves the Euler-Lagrange equation for $\dot{x}$ (and then checks for sufficiency via the Legendre and Jacobi conditions).

Before solving the Euler-Lagrange equation, however, Hotelling reduces it from its usual form ($\frac{d}{dt}\frac{\partial L}{\partial{\dot{x}}}-\frac{\partial L}{\partial x}=0$) to this:

$$L-\dot{x}\frac{\partial L}{\partial{\dot{x}}}=0$$

This is my question. On what basis can you justify this reduction? Hotelling's justification is that he invokes the transversality condition that $P$ is a maximum. Another way he states this is that $x$ is a limited quantity (the amount of gold or whatever in the mine) and will approach an asymptote.

I don't get how this means you can reduce the E-L equation as he does. If someone understands, please help.

Hotelling's paper is: "The Economics of Exhaustible Resources"

Imagine a small perturbation in which one increases $\dot{x}_t$ (the rate of mining at time $t$) by 1 at time $t$; at all other times, the plan stays the same. This causes an extra $dt$ units to be mined at time $t$, and thus $x_t$ (the amount of resources mined so far) rises by $dt$ at time $t$ and for all future times.

We know that the cost of increasing $x_t$ by 1 unit is exactly the (present value of) the price of the good at time $t$, $L/\dot{x}$. For example, if it were less costly to increase $x_t$ than the price of the good, one should just increase $x_t$ by 1 to raise profits.

Thus, the cost of a perturbation of size $dt$ is $\frac{L}{\dot{x}} dt$.

The benefit is $\frac{\partial L}{\partial \dot{x}} dt$ as one directly increases flow profits ($Ldt$) at time $t$.

Because $P$ is already maximized, this perturbation must lead to no change in profits. Hence we have the result

$\frac{L}{\dot{x}} = \frac{\partial L}{\partial \dot{x}}$

Which is Hotelling's equation.

• So the "transversality condition" is just the usual marginal cost = marginal benefit, except dynamically? – ben Nov 26 '13 at 22:58
• As I understand it (and it's always risky to interpret these old papers), he calls it a transversality condition because it allows him to pin down the long-run behavior of the asset stock, $x$. Specifically, one possibility (that is wrong) is that $x$ hits its maximum after a finite amount of time. The above argument shows that this isn't optimal; if $\dot{x}=0$, the left-hand side would be infinite, while the right-hand side would be finite. – Tom P Nov 27 '13 at 0:40
• Do you mean $P$ hits its maximum after a finite time? The resource $x$ is presumably at its maximum at the start of the extraction period. Finally, when Hotelling tinkers with the E-L equation like this, doesn't he risk making the action non-stationary? – ben Nov 27 '13 at 3:51