"Off you go".
Those were the Queen's final words. No more pleas; no buts nor ifs. It's a do or die situation. You are to climb to the very summit of Fateful Mountain to plant there HM's Royal Banner. Success will be handsomely rewarded; you'll pay failure with your life.
However high the stakes, the task is simple … or so you think. That's what I haven't told you: you are a termite. You are blind.
How will you find the summit?
You use your imagination and you think logically. That's how.
If the task is doable at all, the mountain must look like depicted above either in Red or Blue: it can be higher or lower, nearer or farther away, that doesn't really matter. What matters is that it must be "smooth" [*] and it must have a single peak [#].
The general shape of the mountain settled, you think that as you move from west to east, the mountain's slope varies: sometimes it's steeper, sometimes less so. But, no matter how more or less steep, you always struggle against gravity You are, after all, climbing up a mountain. Sometimes it's harder, sometimes it's easier. As you move eastwards, you also move upwards: two positive variations. The quotient of two positive numbers is also positive: that's the slope.
On the east face similar happens, only that now you are descending: you struggle against gravity as well, this time to avoid sliding down out of control; an eastward movement now implies a movement downwards: a positive and a negative variations. The slope is negative.
Two questions for you to ponder:
- Where should your journey be easiest and what is the mountain slope there?
- What if, instead of starting west of the mountain, you had started east of it? Would that change this analysis?
Hopefully, it didn't hurt a bit.
If you didn't answer correctly, be thankful you aren't a termite.
Optimisation is part of applied mathematics/operations research and includes many more complex cases. Other fields (including statistics, engineering, physics, biology, and yes, economics) often employ optimisation techniques. In the next post in this "It's a Long Way to the Top" series we'll provide some explicitly math flesh to the bare bones presented here.
(TO BE CONTINUED)
[*] Why? Hint: Think of accidents and obstacles and what they would mean for you. Think of multiple peaks.
[#] Indeed, Fateful Mountain cannot be like Yellow. Why? Readers interested in economics should give this question some thought: this new assumption is particularly important to theoretical economics. Hint: read the post to the end.