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### Course: AP®︎/College Microeconomics > Unit 2

Lesson 3: Price elasticity of demand- Introduction to price elasticity of demand
- Determinants of price elasticity of demand
- Determinants of elasticity example
- Price Elasticity of Demand and its Determinants
- Perfect inelasticity and perfect elasticity of demand
- Constant unit elasticity
- Total revenue and elasticity
- More on total revenue and elasticity
- Determinants of price elasticity and the total revenue rule

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# More on total revenue and elasticity

In this video, take a deeper dive into the total revenue rule and the relationship between total revenue and elasticity. Created by Sal Khan.

## Want to join the conversation?

- In this example the Total Revenue is maximum when the elasticity of demand is unity, is this example specific or does it happen in any scenario? I mean, does the total revenue peak whenever the elasticity of demand is one and vice versa?(57 votes)
- Good question!

Short answer: Yes! When the elasticity of demand equals 1, the Total Revenue is ALWAYS at a maximum.

Long answer: If you're familiar with Differential Calculus, this fact is easy to prove because Total Revenue = Price x Quantity Demanded (which is the same as saying Price x Amount sold) and the maximum amount of Revenue occurs at the point where the derivative of Total Revenue with respect to Price is zero. Then, rearranging the terms gives us a definition of elasticity of demand.

Here is a proof:

d/dP(TR) = d/dP(QP)

= P*(dQ/dP) + Q

Setting equal to zero:

0 = P*(dQ/dP) + Q

Subtracting Q from both sides and dividing by P:

-Q/P = dQ/dP

then dividing both sides by Q/P gives us:

-1 = (P/Q)(dQ/dP)

Now, taking the absolute value of the left side (as we normally do for Elasticity of Demand) we have:

(P/Q)(dQ/dP) = abs(-1) = 1

where the left side is the Point-Price Elasticity definition of the Elasticity of Demand.

If this sounds confusing (either because you are unfamiliar with Calculus or with Point-Price Elasticity or both), don’t worry: the important thing is that Total Revenue is ALWAYS at a maximum when the elasticity of Demand equals 1 (It’s just that sometimes I find a proof more satisfying than a simple “yes” or “no” answer).

Hope this helps!(135 votes)

- You are wrong about the change in TR being not equal to one at unity elasticity.

If you were allowed go from P3 to P3'=0.99*P3 AND go from Q3 to Q3'=1.01*Q3

Then the Elasticity would not be unity:

[0.01/1.005]/[0.01/0.995] ~ 0.99

Truth is, if your price drops by 1%=1/100, then your QD will go up by 1/99, in which case you will get elasticity one and TR the same as before.(8 votes)- No he's right because a 1% increase in quantity & a 1% decrease in price is by definition an absolute value elasticity of 1. The 1.01 & 0.99 is the result of having this unit elasticity. If it came out to be 1 & 1 it was never unit elasticity.(6 votes)

- In this example the Total Revenue is maximum when the elasticity of demand is unity. But how about the cost of revenue (salaries, etc.)? The firms real objective is to maximise profit, not revenue. But higher quantity at a lower price increases Total Revenue but will also increase Total expense. Surely that would be taken into account when setting up prices?(11 votes)
- You're correct, it is extremely unlikely that the point of elasticity of unity is going to match the point of maximized profits, however, it is still quite useful in determining a minimum price point, or as one factor to consider in relation to variable overhead, fixed overhead and direct costs.(11 votes)

- Awesome videos I am really enjoying them. I noticed all the examples given have a negative slope or a decrease in price. My question is what if the price increased (positive slope), is the area of elasticity still furthest to the left on the graph and in-elasticity toward the right (highest ascending point) or vice-versa. Thanks for any and all advise.(7 votes)
- The calculations would be exactly the same. However, the scenario itself would be bizarre.

If there was a positive slope, that would would mean that the more expensive a commodity is, the more people are willing to buy it! This weird situation might happen a bit in some circumstances (e.g. someone might believe that an expensive item must be of higher quality), but overall, not really. Eventually it would be too expensive for anyone at all to buy.

Suppose that it happens anyway, and we have a demand curve that has a positive slope everywhere. Then there would be no "maximum" revenue, since whatever price you charge for the items, you could increase your revenue further by charging even more.(5 votes)

- but how does the knowledge of Price Elasticity be of any use to a producer?(3 votes)
- If a good is inelastic, then an increase in price will not result in a huge decrease in consumption from the consumers. Therefore, if a producer understands that their product is inelastic, then they can charge a higher price and have higher profits.(8 votes)

- What does role of thumb mean?(5 votes)
- It's a trick you can use to quickly and easily figure something out. However, it's not the most accurate.(4 votes)

- Is unit elasticity the maximied possible Total Revenue in the linear graph? I would assume so since a square maximizes area.(3 votes)
- Yes. Try taking a graph or table of price and quantity and multiplying each point to get corresponding total revenue for a given quantity.

Now create a graph of TR on the y-axis and Q on the x-axis. When you graph each point, you'll get a parabola. At the point where the slope of the parabola = 0 is the same point where unit elasticity occurs, it's also where you get the greatest total revenue.(4 votes)

- I know this is kind of late in the video or the chapter but, Can someone explain to me how to understand the graph that sal makes at the beginning of the video?(1 vote)
- Hello, Nightmare252! If you mean the demand curve, this is pretty simple. The demand curve basically shows how much product is demanded at a given price by consumers. It slopes downwards since the higher the price the less the consumers will demand.

You can not sell more than demanded that is why the revenue determined by the price you sell your product.

The elasticity of demand is a term related to price. And it is basically indicating whether for a given sacrificed unit of price (price reduction) you would get equal, more, or fewer units of quantity demanded.

It helps to decide whether it is worth reducing or increasing the price.

The elasticity of demand answers the question - if I reduce the price, would people buy much more of the product than the reduction of price or if I increase the price, would people still buy the same or nearly the same?

Sometimes your profits would be much higher with a price increase (inelastic), the same (unit elastic) or less (elastic).

Dear Nightmare252, please let me know if that what you meant.

Sincerely, Anatoly.(5 votes)

- I am confused with slope and elasticity of demand. So, if we have a horizontal straight line, then the slope is 0, elasticity of demand is 0 and therefore it is perfectly inelastic. Now, if we have a vertical straight line, then the slope is infinity, elasticity of demand is infinity and therefore it is perfectly elastic. Now, when the slope is constant the elasticity of demand isn't constant and the graph would have 3 regions of elasticity: inelastic, elastic and unit constant. When the graph has elasticity of demand constant, the slope wouldn't be constant. So, the slope and elasticity of demand coincide only at extreme cases. Please verify what I just said is accurate.(2 votes)
- Not quite, but you are getting there!

The slope of a horizontal line is 0, but a horizontal line is**perfectly elastic**, which means its elasticity is infinite. On the opposite end of the spectrum, a vertical line has an infinite slope, but a vertical demand curve is**perfectly inelastic**, which means it's elasticity is zero.

This is actually a great motivation for illustrating that slope and elasticity are similar, but distinct concepts.(3 votes)

- If the absolute value of price elasticity is greater than 1, this means the demand curve in that region is?(2 votes)
- Price elasticity of demand is the responsiveness of quantity demanded to a change in price.

If price elasticity of demand is greater than 1, then the demand curve is elastic in that region.(3 votes)

## Video transcript

I want to do one more video
on total revenue and price elasticity of demand. Just to make sure that you, the
relationship between the two is an intuitive one. So let's draw an
arbitrary demand curve. So this is my price axis. That is my quantity
demanded axis. Quantity axis. And let me just draw an
arbitrary demand curve right over here. So let's say that
is my demand curve. And let's pick some
price and quantities on this demand curve. So let's say that
the price is up here. Let's call that P1. And then, the quantity demanded. Let's call that Q1. And we already know
that the total revenue is the area of this
rectangle right over here. This is the total revenue. It's just the price
times the quantity. If I'm selling 2 burgers an
hour and for $9 a burger, I'm going to make $18 per hour. That's going to be this
area right over here. Now, let's assume in
this part of the curve that the price elasticity
of demand is greater than 1. So we are elastic. So let me write this. So the price, the elasticity
of demand-- actually, I should say the absolute value
of the elasticity of demand. It will be actually
be a negative number. But the absolute value of
the elasticity of demand is greater than 1 which means
for a 1% drop in price you have more, you have a greater
than 1% increase in quantity. And that comes straight
out of the expression or our formula for
what elasticity is. Remember, elasticity
is our percent change in quantity over
percent change in price. So if this, if the
absolute value of this is greater than 1-- these
move in opposite directions. That's why it would be negative. But if we say the absolute
value of this is greater than 1, that means that this
quantity is going to be larger than this quantity. So if we have a
1% drop in price, the change in our quantity is
going to be greater than 1%. And so for point right over
here, if we lower this by 1%, we're going to increase
this by more than 1%. So any drop in our any
reduction in our height will be more than made up for. And this is generally the case. Will be more than made up for
by an increase in our width. So total revenue will increase. So when price drops, so 1% drop
in price and a larger than 1% increase in quantity means
that total revenue will go up. Now, if we go down here. If we go down to this
part of the curve. And let's say that this-- let's
call this-- let's call that P2. And let's call that quantity 2. And then, this area right over
here would be total revenue 2. Let's call that total
revenue 1 over there. Price times the quantity. Now, what's happening over here? We're going to assume that our
price elasticity of demand, the absolute value of it
over here, is less than 1. So the absolute value of our
price elasticity of demand is less than 1 at this
point in the curve. And all that is a
fancy way of saying that for a 1% drop in price,
we get less than a 1% drop. Sorry, less than a 1% increase. They move in
opposite directions. 1% increase in quantity. So we're lowering the height. If we have a 1% drop,
we're lowering that by 1%. But we're not getting a
1% increase in our width. So the width isn't going
to be increasing that much. So in general, this
is going to result in a lowering of this area. This area will get smaller. We're reducing our
height more than we are expanding our width. So in this situation, total
revenue would go down. And remember, this is
an elastic situation. So when it is elastic, total
revenue tends to go up. And when it is
inelastic-- I want to say, when it's elastic
a drop in price tends to make total
revenue go up. And when it is inelastic,
a drop in price tends to make total
revenue go down. And then, you can imagine,
right when you're it unit elasticity,
someplace around there, a 1% a drop in price
will result in exactly 1% increase in quantity demanded. And so they will trade off. You won't get a noticeable
change in your revenue. And the reason why I say that
is that actually some, many econ textbooks will tell you that you
don't get a change in revenue. But if you actually will do a
detailed look at that math-- let me write it over here. So the absolute value
of the price elasticity of demand at that point is 1. Which tells us that
a 1% drop in price will, or goes along with
a 1% increase in quantity. But if you look at the math. So if the old area. So let's call this price 3. And let's call this
quantity 3 right over here. And so total revenue
3-- let me do this in a new color-- which
is this area right over there, is going to be equal to
price 3 times the quantity 3. Now, if we increase price by,
or if we decrease price by 1%, then this will become
0.99 times our price. And if we increase
our quantity by 1%, then this will become
1.01 times our quantity. Now, let's think about what
this number right over here is. And this is why I'm
saying it's not exactly, the total revenues aren't
going to be exactly unchanged. If you multiply 0.99 times 1.01,
you don't you get exactly 1. You don't get exactly 1. Another way to think
about it, 0.99 times 0.01 is going to be 1%
less than 1.01. And 1% of 1.01 is
slightly larger than 1. Or another way to
think about it, this value is going to
be 1% larger than 0.99. And 1% larger than, 1%
of a 0.99 is less than 1. So it's not going to get a 1. And you can see it
with your calculator. 0.99 times 1.01 gets
you to very close to 1. So this is going to be equal
to 0.9999 times P3 Q3, which is equal to 0.9999
times total revenue 3. But it is-- total revenue 3. But it is roughly unchanged. So we can-- that's the
general rule of thumb. So when you are at
unit elasticity, then, a decrease in
price roughly says, no change, approximately
no change in total revenue. So I just wanted to make
sure that it makes sense. It really just comes
from these areas. If you're reducing
the height by a less than you're
increasing the width, obviously, the area
is going to increase. Or most of the
cases, I should say. It depends on where you are. If you are, if
you're compensating, whatever you reduce the height,
you are compensating perfectly with the increase in
width, then you're not going to have a
change in revenue. And if you decrease
the height by more, if you're taking more
area from the top than you're adding on
the width, then you're going to have a total
decrease in total revenue.