what does $\min()$ and $\max()$ mean in a function?

Question

Could someone please explain to me what is the difference between these equations how it would affect the result and the way they are solved?

$$Q2(K,L) = A\min(BK,CL)\text{ s.t. }A,B,C > 0$$ $$Q2(K,L) = A\max(BK,CL)\text{ s.t. }A,B,C > 0$$ $$Q2(K,L) = BK,CL\text{ s.t. }A,B,C > 0$$

(Do note that invented the last two equations so I'm not sure if they make any sense mathematically)

I'm encountering these equations quite often in economics and never really understood how to solve them.

Thank you very much for your help!

Answer 1

Usually $\min(x,y)$ denotes the smallest of $x$ and $y$, and $\max(x,y)$ the greatest.

Answer 2

Here is a possible interpretation of such a function.

$$Q(B,S,N) = \min(\frac1{2}B,\frac1{6}S, \frac1{12}N)\text{ s.t. }B,S,N, Q \in \mathbb N_0$$

This function can be seen as production function of a chair. You probably need $2$ boards ($B$) for the back rest and the seat surface. $6$ wooden sticks ($S$): 4 for the chair legs and 2 to fix the back rest. And probably $12$ nails (N) to fix all the parts

Now with Q you can evaluate how many chairs you can produce if some quantities of boards, wooden sticks and nails are available. Let´s say $B=30, S=100, N=190$ Then the output of the production is

$$\min\left(\frac1{2}\cdot 30,\frac1{6}\cdot 100, \frac1{12}\cdot 190\right)=\min\left(15,16\frac2{3}, 15\frac5{6}\right)=15 \ \ \color{grey}{\textrm{chairs}}$$