What are directional derivatives used for in economics?

Question

In a basic mathematics for economists course one is exposed to the concept of directional derviative. Recall that a directional derivative is defined as:

$$\nabla f \frac{\mathbb{v}}{\|\mathbb{v}\|}$$

I know that it mathematically tells you the direction the function is increasing on a two or three dimensional graph, however I am yet to see an application of such a formula in any economics problems.

What are some applications of the directional derivative in Economics?

Answer 1

Recall, that the directional derivative of a function is defined as the the rate at which the function $f(x,y,z)$ at a given point $(x_0, y_0, z_0)$ changes in direction of the vector $v$, where $||v||$ is the norm of $v$¹.

What comes to mind

Applications of the directional derivative can be used in determining the rate of switching inputs in production functions, which can be very helpful in determining/forecasting switching costs for a given bundle of inputs. Though it does not tell you where the optimum points are it does tell you the rate of change (i.e. the switching costs) of moving to a different bundle.

Hope this is helpful

_{¹ http://mathworld.wolfram.com/DirectionalDerivative.html}