Types of Elasticity

OK so I’m going to skip past shifting demand and supply curves.

Because if you’ve actually internalized the endogenous/exogenous variable stuff from before, this kind of shifting is honestly a piece of cake. Moving on.

(…wait, no. If you’re the type who’s like “I HAVE to understand this or I will literally die,” drop a comment. If enough people show up, I’ll write it up properly.)

Anyway — onward to elasticity.

Elasticity is just an indicator that tells us how sensitively (or insensitively) some ‘something’ reacts when some ‘something’ else changes — and by how much.^^

Depending on what those two ‘somethings’ are, you get different flavors: price elasticity of demand, income elasticity of demand, cross-price elasticity of demand, price elasticity of supply, and so on, and so on, and so on.

Now — easy mistake incoming — you might confuse this with the derivative (the slope).

But the elasticity concept is not the same as a derivative in math.

Why isn’t it!!!!!~~

Because of how elasticity is defined.

Let me throw the definition down first.

And the definition of a derivative (slope) is:

You see the difference now, right?!!!????

OK so, first elasticity on deck:

Let’s look at “price elasticity of demand.”

It’s the indicator for how much the % change in ‘quantity demanded’ responds to a % change in ‘price’.

Writing the definition out:

Even just eyeballing this, you can spot the difference from a derivative.

The red part — that’s the ‘derivative’ bit.

But there’s a blue thing multiplied onto it (which, if you squint, is the slope from the origin).

So this is a different value from the derivative.

And because of that fundamental difference, this kind of difference shows up too.

What kind of difference, you ask?????

In math, a first-degree straight line has the same derivative at every point.

(In plain English: the slope is constant everywhere.)

But with elasticity, the elasticity value can be different at every single point.

(I didn’t say it’s always different. Just can be.)

Why?!?!??!

Because even though the red value is constant, the blue value is different at each point!!!

So — let me use this little math observation to introduce a few concepts.

Choke price

By the definition:

(Quick aside — one thing that makes economics weird compared to math: in math, the independent variable goes on the x-axis and the dependent variable on the y-axis. In econ, it’s flipped. Independent variable on y, dependent variable on x….. Every single other field puts the cause-side independent variable on x and the result-side dependent variable on y,,, only econ. ONLY ECON.

It’s because whichever economist first dragged math into econ did it backwards, and after that everyone just kept rolling with the convention, and here we are…. ugh, it’s so annoying. -- ;; ;; ;; ;_;

OK. Q is technically not an independent variable, but when things are linear, we can treat Q as if it were one (inverse function exists, no harm done), so I’m going to rewrite the price-elasticity-of-demand formula like this:

And

If we follow this figure,

That’s the deal.

So:

We can write it like this.

Since the figure above is a straight line, this’ll hold at every point, so the only thing we actually need to look at point-to-point is

In other words, the only thing we need to check is “ah, this kind of slope, got it.”

So — where’s the choke price?

Right here.

And because of this, the elasticity value blows up to ∞.

Price elasticity of demand being infinite means:

“if you nudge the % change in price by even a tiiiiiny~~~ bit, the % change in quantity demanded is ∞.”

Which actually makes total sense — past that price, nobody is buying. Zero customers left.

Make sense?

OK so we’ve got price elasticity of demand down.

And we’ve got that ’elasticity can be different at every point’ down too.

Now — as the elasticities at each point shift around, they’re not going to jump around discontinuously for no reason.

Because:

Meaning it won’t change in a way that breaks differentiability. (It changes smoothly.)

In our straight-line case, since it’s linear,

becomes a constant, and the actual elasticity is

(I’ll write it as alpha to keep this general heh heh heh heh)

What I’m hammering on right now is that ε changes smoothly.

Why????????????????????????

So that I can casually say:

“Find the point where

!!!”

Why do we find this point? Because once you’ve got it, things get sorted into buckets like this!!!!!

Now — why on earth do humans slice things up around an elasticity of ‘-1’ specifically???!!!??

Because by reading off the elasticity at each point, we get to interpret it like this:

Quantity demanded doesn’t respond to price at all.

Quantity demanded is relatively insensitive to price.

The % rise in quantity demanded equals the % drop in price.

Quantity demanded is relatively sensitive to price.

If price rises, quantity demanded drops to 0; if price falls,

quantity demanded shoots up to infinity.

So at any given price level — if we roughly know what kind of elasticity zone we’re in, that’s actually pretty useful for a firm trying to make pricing decisions heh heh heh heh

※Side note※

Constant elasticity demand curve

This shape is called the constant elasticity demand curve. Let’s just double-check that the elasticity really is the same everywhere before we move on.

Turns out to be ridiculously simple heh heh heh heh heh heh

OK let’s hit the other elasticities too!!!@@@@@@

I’m going to touch on income elasticity of demand and cross-price elasticity of demand. But first, one more time, the meaning of the price elasticity of demand we just did:

it’s the indicator that tells us “if price changes by this~~~much %, by how many % does demand change????”

The trick now is — define a few more elasticities by swapping out the ‘cause.’

1. Income elasticity of demand

The indicator that says “if income changes by this~~~much %, by how many % does demand change~~?!”

That definition doesn’t seem unreasonable, right?

Let me model a formula straight off the definition:

Let me throw down one more.

2. Cross-price elasticity of demand

“If the price of another good changes by this~~~much %, by how many % does the demand for our good change??!?”

Same deal — write the formula straight from that English meaning:

Let me look at the economic meaning a bit before we move on.

In the case where

when the other good’s price moves in the (+) direction, our good’s demand also moves in the (+) direction!!!

“Ughhh, I hate how expensive that thing’s getting!!! I’ll just sub in this one!!!!”

Something like that. Yep — for goods in a “substitute” relationship, cross-price elasticity is greater than 0.

On the other hand, when

the other good’s price moves (+) and our good’s demand moves (-) instead — that means:

“Ugh… ;; ;; ;; if ’the other good’ got more expensive, then I don’t need this one either ;; ;; ;; ;; ;; I’m just not buying either of them!!!! >_<@@”

Yep, that’s complementary goods.

It’s maybe a little weird that I framed it as (+), so let me flip it — describe what happens when ’the other good’ moves (-) and our good’s demand moves (+):

“Yesss~~~~ the other one got cheaper!!! Then I’m grabbing this one with it!!!!!!!”

Beer and peanuts, basically.

If beer gets cheaper, peanut demand will rise too, to some degree.

Soju and jokbal.

Soju and bossam.

Beer and chicken.

Soju and samgyeopsal, soju and galbi.

Soju and jjukumi-bokkeum.

Soju and hoe.

Soju and bossam.

…OK fine. I’m just listing things I want to eat right now. ;; ;; ;; ;; ;_;

Complementary goods. ;; ;;

Originally written in Korean on my Naver blog (2016-07). Translated to English for gdpark.blog.