# What are directional derivatives used for in economics?

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?

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.

1 http://mathworld.wolfram.com/DirectionalDerivative.html

• By "the rate of switching inputs" do you mean MRTS? – EconJohn Nov 19 '17 at 19:58
• To summarize what you said, you use the directional derivative to identify the direction of the optimal bundle? – EconJohn Nov 19 '17 at 20:00
• @EconJohn The directional derivative can be used for any bundle be that optimal or not, – FreakconFrank Nov 19 '17 at 20:15