Matthew Olckers Economist by day; graffiti nerd by night.

How to remember the Karush-Kuhn-Tucker first order conditions

I teach the maths refresher and microeconomics tutorial for the PSME program. Students need to maximize utility functions or profit functions and minimize expenditure functions or cost functions—but subject to constraints. I have trouble remembering the signs and direction of inequalities in the Karush-Kuhn-Tucker first order conditions. This blog post proposes a method to remember the first order conditions that should suffice for all the problems my students will tackle.

Put a ceiling on max, and a floor on min. Constraints are negative (because we would rather be free). And Lambda always does the opposite.

Maximize

Let’s start with maximization.

Max $f(x,y)$ subject to $g(x,y) \leq 0, x \geq 0, y \geq0 $.

The first order conditions are:

Four points to remember:

  • Since we are maximizing, I wrote the constraint $g(x,y) \leq 0$ as a ceiling (with a less than or equal inequality).
  • $\frac{\partial L}{\partial x}$ and $\frac{\partial L}{\partial y}$ is less than or equal to zero (which is the same inequality as the ceiling).
  • $\frac{\partial L}{\partial \lambda}$ has the opposite sign inequality to $\frac{\partial L}{\partial x}$ and $\frac{\partial L}{\partial y}$
  • $\lambda \geq 0$

Minimize

Now for the case I always get mixed up… MINIMIZATION!

Min $f(x,y)$ subject to $g(x,y) \geq 0 , x \geq 0, y \geq0 $.

The first order conditions are:

Five points to remember:

  • The Langrangian stays the same with the constraint entering as a negative.
  • Since we are minimizing, I wrote the constraint $g(x,y) \geq 0$ as a floor (with a greater than or equal to inequality).
  • $\frac{\partial L}{\partial x}$ and $\frac{\partial L}{\partial y}$ is greater than or equal to zero (which is the same inequality as the floor).
  • Again, $\frac{\partial L}{\partial \lambda}$ has the opposite sign inequality to $\frac{\partial L}{\partial x}$ and $\frac{\partial L}{\partial y}$
  • And again, $\lambda \geq 0$

How do I remember this?

Put a ceiling on max, and a floor on min. Constraints are negative (because we would rather be free). And Lambda always does the opposite.

More details

If you would like more details, read chapter 13 of Chiang and Wainwright.