# 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

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.