Okay, so just a couple
of brief things. One, I made an
announcement about a bit. You can go back into
the first homework if you just want to
take a look at it. It also has the McGraw
-Hill explanations for each answer.
So take a look through all that.
If you still don't understand something,
please come see me next week during the
office hours. A lot of those concepts
will be on the exam. For the second
homework, it's going to come out after this
class on Monday. Reason being, we're
finishing most of the stuff today probably,
but there's one small section that has a
couple of questions that we'll talk about on
monday and i don't want to like confuse anyone
to put in a wrong answer or something like
that so i do apologize for not having it up
over the weekend for you but i feel like
this is um best case scenario and we'll have
some time again so we can talk a little bit
more about the exam but picking up from
where we left off um on wednesday we were
discussing what determines whether a product in
when it comes to demand if it's inelastic or
elastic and the three main factors we
discussed was how many substitutes are available
the more substitutes the more elastic it
is we saw that example where we looked at
like whole milk where there's you know two
percent skinny ton of others you can switch to
highly elastic versus milk is an entire category
which is inelastic not much to substitute
between there we looked at share of the
budget like if something's taken up 80% of
your budget and that price increases, there's
no wiggle room to change that anywhere
else. So you've got to kind of change that in
terms of consumption. Finally, we
spoke about time. The more time you have
to adjust, the more elastic it is. So
we gave the example, gas prices increase.
There's only so much you can do the next
couple of weeks, but in six months you
can change your whole routine in terms of
how you get into from school and work, etc.
You can buy a bicycle instead of driving
your car, et cetera. And I think with
this class, we might have finished
up here. There's just a couple of
other things that can also affect
the elasticity. We talked about
the example of this HIV drug or
something like insulin, which
people need to live. It's a necessity.
Things that are necessities
are inelastic. And luxuries, the
opposite of the necessity, are
highly elastic, so frequently change
when prices change. Also, I guess as we
talked about with milk, is the market broadly
or narrowly defined? If the market is
broadly defined, so if we're talking
about cars as a whole, and car prices increase, it's kind of hard
unless you really want to have another
vehicle to switch away from it. So
it's more inelastic. But if the
market's narrowly defined, it's one
single car, such as the 22 Black
Chevy be Colorado, the price is increased,
then the demand will heavily shift
away from it because there are so many
other cars you can buy. So knowing whether
it's broadly defined or narrowly defined
will help determine whether it's
inelastic or elastic. So finally, we have
this cute quote. This person says, I
want to either open a liquor store or
a funeral parlor. Why those two things?
I figure those are the two things
that everybody needs. So So understanding
elasticity when deciding on a business
can help a lot. So this lecture today
is going to be a little bit different
from the others. It's going to be like much
more of the weeds. We're going to go
through a lot of the mathematical details
of elasticity, which will help you
with your answers. But also if I glaze
over it, if I skip over it, it's just
not going to help you. So the first thing
to look at is this is our market demand
curve here. So, as we can tell, we
have this inverse relationship between
price and quantity. As price increases,
quantity decreases. The question is, is
elasticity the same at every point on this
linear demand curve? And a common
mistake that people make is, oh yeah,
the slope is always the same, elasticity
is the same. But this is not the
case. It changes at every point. and this is
because our elasticity of demand is two terms
multiplied together the first is the change
in quantity given the change in price
and the second is the price divided by
the quantity so this first term here the
change in quantity given the change in price
is just the inverse of the slope here so one
over the change in quantity divided by
the change in price is this and this is constant
this is the slope or the inverse of the
slope i should say here so this is constant
then price divided by quantity is going to
be different depending where you're on
the demand curve so let's say you're at this
point here so demand is 70 the quantity
is 70 and price is 5 let's say the slope
just for ease is minus 1 let's just say that
so this whole term is minus 1, and then we
have 5 divided by 70. 5 divided by 70, what
would that be like? I don't know, it's
like 15% or something like that. So you
get 0.15 multiplied by minus 1, which
gives you a number below 1. It's inelastic
at that point. We hold this minus 1
constant the entire time, but if we move
up here, now the quantity is 10 and
the price is 35. So 35 divided by 10
is 3.5. 3.5 times minus 1 is minus 3
.5. So now this number is greater, or the
absolute value of this number is greater
than 1, and it's elastic. So as you
go up the curve here, the quantity keeps
getting lower, and the price keeps getting
higher. So what that means is that whole term
is going to continue to get bigger and
bigger and bigger. So it's going to start
off pretty much being completely inelastic
down here, and by the time we move up
to the very top, it's going to be very elastic
so that's why there are changes happening
along this curve and we can use
those changes to calculate the
difference in total revenue and
marginal revenue for some producer
facing a certain demand curve so remember our
total revenue is simply the quantity times
the price um and the the the elasticity or
not the elasticity i should say this is the
the demand function here and we can see there
is this relationship between elasticity
and total revenue which means there's
gonna be a relationship between elasticity and
marginal revenue if you increase the price
by $1 how many extra units sorry how many
extra dollars will you get or will you lose
so as you can see when we start off you know
here the total revenue is relatively low and
as price increases it gets higher and
higher and higher until we get to 800 and then
as price increases it gets lower and lower
and lower so there is a maximum point
for total revenue here the next thing to
notice is we have our own price elasticity
how do we calculate that so remember our
elasticity function contains two things
the first is the change in quantity given the
change in price, so the change in quantity
given the change in price, that's just
this in our linear demand function. If
price increases by 1, quantity is going to
decrease by minus 2. So this whole term is
just going to be minus 2. And the next thing
is the price divided by the quantity at
any point. So minus 2 times 5 divided by
70 gives us minus 0 .14. That's the
elasticity at that point. If we go to G,
still minus 2 times the price divided by
the quantity minus 2 times 30 divided
by 20 and we get minus 3 it's elastic
at this point and the other thing
to notice is when we're inelastic so
below the absolute value of 1 so when we're
in between 0 and 1 in this case negative
when you increase prices the total revenue
is increasing or the marginal revenue is
increasing as well So, once you get to
the unitary elastic point here at E, this
is where your profit is maximized. This
is where marginal revenue will be zero.
You've squeezed every bit of value you can
out of it. And if you increase price
at any point above this, then your profits
are going to drop. And the way
to think about this is, so I'm going to, we'll see how this goes, see if we can all think. So imagine you're
currently selling 100 units of a
good at $10 each. Your total revenue
is going to be $1,000, 10
times a hundred. Let's say you up the
price by a dollar. And in the first
example, let's say this is totally
inelastic, so the demand doesn't change.
It's still $100. So 11 times
100 is $1,100. You do that. You
make more profit. But usually what happens
is when you increase price, demand
changes, it drops. The law of demand is this
inverse relationship. So let's say you
increase price by $1, which is
a 10% increase. And let's say
demand decreases by 20%. So demand
decreases by 20%. It goes from 100 people
wanted it to 80. So you're now selling
at $11, but only 80 people are buying it.
Your total revenue is going to be 11 times
80, which is 880. you're a hundred and twenty
dollars worse off this is because you have a
ten percent increase in price but a twenty
percent increase in demand so that extra
dollar you're getting by increasing is offset
by twenty percent of people no longer buying
the product so when your good is elastic when
you're on the elastic range you shouldn't
increase your prices anymore because more
people are going to move away from buying the
good, then how much you get from increasing
the price. But let's say it was only a 5%
decrease in demand, so only 95 people bought
it now, when it's $11, then your revenue
would be 95 times 11, which is $1,045, I
think. So now you're better on it. You're
getting 10% more based on the price,
but you're only losing 5% of your
customers. So when the demand curve is at
an inelastic point, If you raise your price
by 1%, you gain 1% more in price, but you
only lose less than 1% in terms of customers,
so it's a better deal. So you should
keep increasing the price when you're at
an elastic point until you reach the unitary
elastic point of minus one, and then never
increase your price when you're at an
elastic point because more people will
flock away from it. Hopefully that kind of
helps illustrate it. That's all that's going
on here. When you're at an elastic point
and you increase the price, you see total
revenues increasing, marginal revenues
increasing. Once you go above the
unitary elastic point and increase
prices, more people flock away from
it. You can see elasticity is above
one. It's elastic. Increasing price
makes you worse off. So we have two
graphs here. This is just the demand
curve at point E, quantity of
40, price of 20. That's where it's
unitary elastic. Anywhere down here,
it's inelastic. Anywhere up top, it's
elastic. so the price will continue to
increase, increase, increase until here
and if it's any point up here it will
decrease until it's at the unitary elastic
point and you can just see here with
the total revenue it's maximized when the
quantity is at 40 so the relationship
with total revenue and
marginal revenue is simply that
marginal revenue is the first
derivative of total revenue. And
there'll be a few ways which I
discuss this today. So total revenue,
once again, is just price times
quantity. But we can write this in
another form. I want to focus on the
left-hand side here. Total revenue equals
quantity times price as a function
of quantity. Remember, our inverse
demand function has price on the
left-hand side as a function of
quantity on the right -hand side. So here,
this is price as a function of quantity.
P equals 100 minus 2Q. If Q equals 20,
price equals 100 minus 2 times 20.
Price equals 60. If Q equals 40, price equals 20. So
the price changes as quantity
changes. That's why price is a function
of quantity. So if we take
this and we want to figure out
total revenue, we have price as a
function of quantity times quantity. We know
what this is. this is just our inverse demand
function so we can plug in the right
hand side for p as a function of q 100 minus
2q multiplied by q and that gives us our
total revenue when we expand the brackets in
terms of one variable here q and this is
important because we can use calculus here
once again the calculus isn't necessary
anything i talk about today with calculus
is just intuition and understanding is if
we take the marginal revenue which is the
first derivative also just looking out how
does total revenue change as quantity changes
if quantity increases by one what happens
to total revenue does it increase or decrease
that's what the marginal revenue is so
if we take the first derivative here of of
um the the the first derivative d total
revenue divided by d quantity we take the
one out the front minus one here so this
disappears we get 100 minus take the 2 out
the front, minus 2 times 2, 4, reduces by 1,
we get 100 minus 4Q. So, depending on
where we start, we show that a change
in Q of 1 unit or of 50 units will
have this effect on total revenue and
marginal revenue. So, I'm not going
to spend too much time on this,
but we spoke about Shkreli before,
and what he actually did was
profit maximizing. So with this elastic
demand curve, as we said before, you want
to increase prices as much as you want
because you know people are still
going to buy up until the point where people
can't afford it. The problem with
this is, we'll talk about it in a couple
of weeks, is this is exactly what happens
in a monopolistic situation while it's
a market failure. They have this
incentive to raise the prices,
reducing the quantity below
efficient levels. So don't worry about
this for now. I just wanted to come back to
this idea of marginal revenue, total
revenue in the market. so the way to think
about it is when demand is elastic the
percentage change in quantity is larger than
the percentage change in price so when you
lower your prices this will increase
the quantity demanded by more than the
decrease in price so if you decrease your
prices let's say by 10% quantity is going
to increase by 20% and it's kind of the
reverse of what we said about before this
is going to increase your total revenue
which means it's an increase in your
marginal revenue as well so this means yeah
total revenue is increasing and marginal
revenue is positive when demand is elastic
when speaking about lowering prices when we
speak about increasing prices then it's
um got a negative marginal revenue that's
just important to know we're now looking
at in kind of the other frame when
demand is inelastic so when we're down here
when you lower prices what that means is the
increase in quantity demanded is going
to be less than the decrease in price so
you decrease price by 10 but there's an
increase in quantity by only 2 so you're
not making up the lower price in terms of
more customers so what that means is lowering
the prices once you get into the
inelastic zone is going to make you a loss so
marginal revenue is negative when demand
is inelastic, when you consider the idea
of lowering prices. And when it's
unitelastic, this is just where marginal revenue
is equal to zero. You've essentially
squeezed every penny you can. You're not in a
situation where you're charging too much or
charging too little. The Goldilocks of
it is just right. So, as you can see
here, we have our demand curve, our
demand function, our unitary elastic
point. This part's inelastic. here it's
elastic and this is our marginal revenue curve
so there's a few things to notice the
marginal revenue curve intersects with zero
at the unitary elastic point this is where the
profits are maximized for a type of good or
for a monopolist as well the other thing
to notice is when it's in the elastic part
the marginal revenue is always positive remember
when you're lowering the prices at this
point, you're getting more customers to flock
in than the prices you're lowering by.
So it's good for you. And once we get into
the inelastic zone, you see marginal
revenue is now negative. Every time you lower
your price, less people than before are buying
the products, so you're making a loss on
each extra unit sold. The question here is,
why is the marginal revenue lower than the
demand curve? Why is it lower than the price
that's being sold at so at quantity one
the price is p but the marginal revenue is
below p what's going on here so ignore the graph
for now we can just look at this in terms
of numbers so let's say at five dollars
one unit is bought so the price is five
dollars here the total revenue is five dollars
and the marginal revenue if this is the first
unit is five as well but to entice more
purchases let's say sellers reduce the price
to four dollars when they move it from five
to four now two units are bought however
it's not five dollars for the first customer
and four dollars for the second it's four
dollars now for both customers so we're taking
one dollar away from the previous customer
which is also going to reduce like revenue
in in total so to calculate the total
revenue it's just the the price times units
bought so eight and the marginal revenue is
just the difference in total revenue so it's
five for the first unit and then we have eight
total revenue for the second so three
difference marginal revenue is three and
you can see the marginal revenue here is below
the price and it's above zero we can do the
same thing if we lower it again by a dollar
the price now we sell one extra unit total
revenue is three times three which is nine and
the marginal revenue is the difference in
total revenue from before and that is
going to be an increase of $1. So this is also
below the price because when you change the
price you're also reducing the marginal
revenue overall because you're changing the
price not just for the next customer but for
every previous customer as well. So even
though marginal revenue is increasing, when
you lower the price it can never be above what
the actual price is. Hence why it's always
at most of the same but below the price
slash demand function. okay so this is where
kind of where we're going to get into a bit
of the mathematics so the marginal revenue
is what we saw before can be derived from
the market demand curve and this measures the
additional revenue due to a change in
output this is really important because when
marginal revenue equals zero is when profits
are maximized or when revenue is maximized
so this link relates marginal revenue to the
own price elasticity of demand so as we saw
here we have our elasticity table at
unitary elastic that's when it's maximized and we
can see here with our total revenue it's
maximized here as well and marginal revenue
will be zero here so they're all related and
this is the equation that relates them marginal
revenue equals price times one plus elasticity
of demand divided by elasticity of
demand this comes out of absolutely nowhere
in the book so what i'm going to do on the
next page is show you how we actually get
there from total revenue you don't need to like
know this i'm going to ask you to derive
it but i just want you to know this just doesn't
come out of nowhere this is an actual way
of figuring out from total revenue getting
to marginal revenue and the important thing
to note as we said before is that if we're
in the the stage of it's been elastic then
lowering the price will increase marginal
revenue until you lower it all the way until
you're at the unitary elastic point where
marginal revenue equals zero and if you
continue to lower the price after that when
it's inelastic if you lower the price let's
say you're here and you lower the price from
10 to 5 where in the inelastic stage you
see that the marginal revenue is going to be
negative you lose 250 dollars when you go
from a price of 10 to 5. okay so recall as
before the total revenue is just price
as a function of quantity
times quantity. And what marginal
revenue is, is just the derivative
of the total revenue function, the
first derivative. So the key point here
is we actually have two separate functions
on the right-hand side. We have price as
a function of Q and Q. So in calculus, we
need to use something here called the
product rule. And I'll take you through the
abstract formula, but I'll give you
a really simple example as well to
show you how it works. So, these two separate
functions could be something else, like
fx, so f is a function of x, and g is a
function of x. So, if you want to take the first
derivative of it, in terms of dy, dx, what
we need to do is use this product rule, which
is simply, you take the first of the two
terms, and take the first derivative, and multiply
it by the second term, unchanged, and
then add the first term f as a function of
x unchanged multiplied by the first derivative
of the second function that's just
what the product rule is we're not doing any
real analysis going you know beyond that that's
just what it is so here's a very simple
example to show you how this works so if you
have a function y equals x squared then dy dx
is just 2x so 2 comes out the front 2x we
reduce the power of x by 1 we get 2x but
what if we had another function here we had y
equals x multiplied by x this is just x squared
it's the same thing but if we define this
first x as f is a function of x and the
second x as g is a function of x now we
can derive it using the product rule as well. So
once again dy dx equals f as a function of
the first derivative of f as a function
of x multiplied by gx unchanged plus f as a
function of x multiplied by g's as a function
of x first derivative. So let's look at this
term here the first derivative of this
function here x it's just one and we keep this
unchanged which is x then we look at the
second one we keep this unchanged the first arm
term fx is just x and we take the first
derivative dy dx of this which is just one then
we get x plus x which is 2x it's the exact
same thing so we can use the product rule
to get the exact same answers before so the
product rule is is a neat little trick that
we're using calculus and we're going to need
the product rule here because we have our
fx which is p as a function of q we have
our gx which is just q so we use the product rule
like before and here we want to take the
first derivative of this first term here so
we want to take dr dq and this is just dp dq
that's all that is so if we take um if we
want to take let me go back here the derivative
of q here this is just the change in p
divided by the change in q that's all it is
so that's how we get that point here this
part remains unchanged in the first part so
we get dp dq multiplied by q and the second part
we take the derivative of q which is just
one multiplied by this being the same which
is pq now what we do here is we just simplify
pq to p PQ is P, it's the same thing
we can mess with the terminology a little bit
and we end up with the first with the derivative
of total revenue equaling DP divided
by DQ multiplied by Q plus this is
just P times 1P what we do on the next
line is a little bit of algebra we take out
P as a common factor and put the two terms
in brackets, this is easy for this one here
take out P p divided by p is just 1 but
there's no p on its own here so if we take
out p from this term we've got to divide it
all by p so that's how we end up with this p
if you multiplied all this by p this would
disappear and you'd end up with this that's
what we're doing here does anyone notice
anything now about this equation down the
bottom yeah what's up it's not just like 1
over e yeah It's just 1 over E. When you think
about it, remember, our elasticity is the
change in Q divided by the change in P
multiplied by P over Q. That's our elasticity
of the amount. This is just the
complete inverse of it. So, we can just say
this whole term is 1 divided by the
elasticity of the amount. And if we substitute
that in, we get the change in revenue, given
the change in quantity, our marginal revenue
equals the price multiplied by 1 divided
by elasticity plus 1. and to put this 1 over
E we just multiply it by E so we get 1
plus E divided by E and that's how we get
this formula of what marginal revenue is in
terms of classes so you don't need to know
this you don't need to derive it I just feel
like I wouldn't be doing a good job
teaching but I just threw a random formula out
there and said no way so hopefully this gives
you a little bit of insight how deriving
the total revenue curve to get the marginal
revenue curve gives us this formula but
you can just remember p times 1 plus a over
e if you really want and using this we can
solve if the marginal revenue of something
is positive or negative given the elasticity
information so here in the first example we
have the inverse demand curve p equals 100
minus 2q we have the price and we know the
elasticity is minus 0.11 1. So we know this is
inelastic. It's less than 1 or the absolute
value is less than 1. So the marginal revenue
should be negative. So here we have
marginal revenue equals price times 1
plus elasticity divided by elasticity.
We can just plug in the elasticity
equals minus 0.11. We can plug in the
price equals 10 as we do here. And what I've
done from here to here is minus 0.111 divided
by minus 0.11 is just 1 so we get 1 plus
1 divided by minus 0 .11 which equals minus
9 you can chuck in the calculations if
you want so we have 10 multiplied by 1 minus
9 we know the number in here is going to be
negative this number is going to be negative
in fact it's minus 8 so we get a negative
number minus 80 but if it's a negative
number we know it's going to be inelastic because
marginal revenue is going to be negative
on the other hand let's say that i actually
need to fix this up um the price is 70
for example here once again we just plug
everything in you do the the the calculations
and now as you can see when the elasticity
is above minus one so it's minus 2.33 here
we get one minus 0.429 which is a positive
number so whenever you multiply the price
which has to be positive when you've got a
positive number is going to be positive. So
the marginal revenue here is positive, which
means it's elastic. Okay, I know
that was a lot. At the end of the
day, it's really just like a plug and play
type question, but I wanted to show you how
you can actually derive marginal revenue if you
have the elasticity. So if you go out
into the world and you have like these
details, you can calculate what you should be
doing based on the elasticities alone. Any
questions about that before we move on to
cross price? Yeah? It's a great question. There's one coming
up that if I ask a question on it, I'll
give it to you on the exam because
it's pretty crazy how you derive it
and I don't want you to work on things
for the sake of it. So with this one here, there's a bit of
intuition behind it. Yeah. But for something
as simple as say the equally room of supply
and demand, I'm not going to give you
that. like you should know that you it
equals equilibrium when the demand is a second
this or like it but for something like
this the one we just went through i reckon
i'm 70 percent likely to give you a formula
for it the one that you'll see in a second
like 100 percent because it's just silly
to get you to wrote you know memorize
something for no reason so cross price
elasticity we can do the same thing as before
but something could be a compliment or a
substitute and we want to know how quantity
over good increases or decreases when the
price of the substitute or the complement
increases and decreases. So directionally we know
that if the elasticity is above zero then x
and y are substitutes because if price
increases and that means quantity
increases it has to be a substitute and vice versa
if it's a complement. So we have this linear
demand function, where q equals minus
2 times its own price, minus 0.5 times the p
of the other good y, plus 3 times income.
And we get income equals 10, price of the other
good equals 4, and the price of the own
good equals 12. So we can just plug all
that in, and we know the quantity demanded
at those prices is 4. And the cross-price
elasticity of demand, so the elasticity of
the quantity of x given the price of y it's
the same thing as before except now we're
just interested in how the change in q
is as a result of the change in p y given p
y and q x so we already know what this is
this is just in front of our p here for our
linear demand function on the inverse you
just take the slope if it's the inverse
and you've got to take the inverse of the
slope that's so we're just taking the slope
here, minus 0.5, and we have what the
price of Y is, it's 4, and we just derived
what the quantity is, it's 4, and that gives
us an elasticity, or the cross-priced
elasticity, of minus 0.5. Which means these
two goods are complements in
consumption, because it's a
negative elasticity, and the absolute
value of this elasticity is
less than 1, so it's inelastic. So that means
a 10% increase in the price of
the complement will result in
a less than 10% decrease in demand
for the good. So there's two things
you want to know. The direction, which is
what the negative or positive sign tell
you, and whether it's elastic or inelastic,
which is what the absolute value of the
elasticity tells you. If the absolute value
is less than 1, in between 0 and 1, inelastic,
above 1, elastic. So, suppose it's
estimated that you have a cross-priced elasticity
of demand between clothing and food,
which is minus 0.18. So, we know in this
case it's complements, they have this
negative elasticity. That's what it tells
us. We also know it's inelastic,
because the absolute value of this is
less than one. If the price of
food is projected to increase by
10%, by how much will demand for
clothing change? So we know that the
cross-priced elasticity for clothing, given
the price of food, is going to be the change
in percentage of clothing divided by the
change in percentage of food equals the
cross-priced elasticity. We have two pieces of
information. We have the cross-priced
elasticity, we have the percentage change in
food, and we just got to figure out the percentage
change in clothing. pretty easy to figure
this out but you can verify this you
know it needs to be less than 10 because
it's inelastic and it needs to be in the
negative direction because there's a
positive increase here and the cross price
elasticity is negative all right this is where
we're going to get into a pretty funky
equation which i will give you okay so let's say
your revenue is based on more than one good
and those goods are related so you could
sell burgers and fries these are compliments
so if you increase the price of fries it's
going to change demand for fries but it's also
going to change the demand for burgers which
you also sell so by changing one price if
you sell two things that are related whether
they're compliments or substitutes it's going
to change your total revenue in a way
you've got to take both of these things into
account as you can see this is why i'll give
this to you we've got a pretty wacky equation
here but all this really is is the change in
revenue the marginal revenue just is the
difference in total revenue when you look at
the price of the fries multiplied by the quantity
of fries sold plus the price of the burgers
multiplied by the quantity of burgers
sold so this equation here over here is the
change in the price of x or the change in the
price of fries as we're showing the example
we're increasing the price of fries the
change in r is the change in total revenue and
this will depend on two things it's going to
depend on the revenue the total revenue you
currently get from fries multiplied by
one plus the elasticity of demand for fries
plus the revenue that you get from burgers
the total revenue from burgers multiplied
by the cross-price elasticity of demand
between burgers and fries. Because changing the
price of fries will change the demand
for burgers. And the reason why I'm
not going to have you remember this is you
have this weird quirk and you actually get
this equation from deriving the total
revenue term in the same way we did before.
It's just much more complex. You can do
it with the product rule if you want to
try it by yourself. But we get this point
where it's the total revenue of fries
multiplied by one plus the own price elasticity
whereas for burgers it's just the total revenue
of burgers multiplied by the cross price
elasticity so you have a one plus here which
you don't have here and this is just a
function of the way you derive it hence why
like i don't want you to memorize this i don't
think it serves any actual function if i
give this to you i just want you to be able
to plug the numbers in and solve it so for
example let's say you run this food truck.
And you know that the own price elasticity
for fries is minus 0.8. So an increase
in the price of fries is inelastic. The cross price
elasticity of demand for fries and
burgers is minus 2. Once again, they're
compliments here. So this is negative and
it's elastic, the relationship. They
currently sell 100 fries at $1 each and 40 burgers
at $2 each. how much will revenue
change if they increase the price of fries by
25% it can we use we can use this formula
here to solve it so once again this is our
formula for the change in revenue or our
marginal revenue of increasing the price
of fries by 25% the first thing we do here
is we're going to plug in the price of fries
is changing by 25% the next thing we're
going to do is calculate the current total
revenue of both fries and burgers for
fries we're selling a quantity of 100 times
one so that's our total revenue for burgers
we're selling um did i get this yeah i got
this wrong way around so this is two dollars
times 40 it's the same thing in the end
next thing we're going to do is we're going
to add the oat price elasticity for fries
which is minus 0.8 and the cross price
elasticity for burgers given the price of
fries which is minus 2. Once again, we're just
plugging this all in. Then, what we're
going to do here is we've taken 1 minus
0.8 which is 0.2, 100 times 1 which is
100 this is just minus 2 and this is just
80. Multiply by 25% so this is just 20
and this is minus 160 get this, times 0
.25 and the marginal revenue here is
minus 35 dollars. So, once you have
the formula, there's one key thing
to keep in mind, besides plugging
everything in, that Rx and Ry represent
the total revenue. So, you've got to do
the quantity times price of that product to
get the total revenue. That's the one way
you're going to really screw this up if
it's on the exam. I don't know if I'm
going to put it on the exam. I'll probably
avoid it because I don't think this is
actually a beneficial exam question, but
I'll probably hint whether it is or
isn't on the exam once I've finished writing
it on Saturday. So, similar to
the cross-price elasticity, we can
do the same thing for income,
income elasticity. How does quantity
for a good change when the percentage
of income changes? Similar to before,
directionally, if it's positive,
it means it's a normal good. As
income increases, if quantity is
increasing, normal good. If elasticity
is negative, as income increases, quantity is decreasing. so we have the exact
same linear demand function as before but
now all we want to check is the income elasticity
instead of the cross price so we care about
the elasticity of the quantity of x given
m so we get the change in quantity given
a change in income multiplied by income
divided by quantity so this here is just the slope
of what's on income which is plus three
multiplied by income at this point, which is
10, divided by quantity, which is 4. It's the
same thing as before. And that gives us an
elasticity of 7.5. So this is above 0,
so it's a normal good. And it's above, the absolute
value is above 1, so it's
highly elastic. It's highly elastic. Okay. Suppose that the
income elasticity of demand for organic
potatoes is 2.26. So we know this is going to
be a normal good because it's positive and
elastic because it's above one if income
is projected to decrease by 10 percent
what is the impact on demand for organic
potatoes so the percentage change in income
is a decrease by 10 we know what the income
elasticity is and we can figure out
this means the demand for organic potatoes
will decline by 22.6 percent so we know
that this is elastic because the change in
demand is greater than the change in income
in terms of percentage and we know it's a
normal good because when income decreases
the quantity demand of a normal good
would decrease as well so the whole idea
with elasticities is if you can measure it you can calculate
an elasticity so you can see the
percentage change in quantity demanded for
all sorts of things such as if someone moves
a mile away is their you know percentage
change of coming to class going to increase
or decrease probably decrease if they're
if they're moving further away but by how
much? Is it more than the distance that
they move away or not? What about how
responsive is quantity demanded for wait
times, for example? Increase wait time by
1%, does the quantity demanded decrease
by more than 1%? How responsive is
the quantity demanded of your good to
advertising? If you increase your advertising
spend by 1%, does that bring in more than
1% demand upon you? And as we can show
here, we can really put anything in our linear
demand function. So we have own advertising
elasticity and cross -advertising so if one
of your competitors advertises and their
advertisers affected it could mean people that
buy your product shift to the others we want
to know the percentage shift of that as well
but in the linear demand function we can calculate
anything so this is our elasticity of
own advertising so if the change in quantity
given the change in your advertising spend
multiplied by how much you're spending on
advertising divided by quantity we have a linear
demand function here that now has advertising
in here so own price advertising and
then just an intercept so assume the own price
is 10 and advertising is 5 we can calculate
the quantity our q is 15 we know the change
in quantity given the change in advertising
is just the slope here if advertising increases
by one dollar quantity increases by 6 so this
is plus 6 multiplied by 5 divided by 15
and that gives us 2. So a change in
advertising is positive increases demand and
it's elastic. There's going to be a high
percentage change in quantity than there
is in advertising. So from having a
simple linear demand function we can
calculate everything. So as we showed
before from the same equation you can
calculate the cross price elasticity you can
calculate the income elasticity you can do
all of this from the same equation and it's
just as follows it's just the slope because
we have q on the left hand side remember
if it's the inverse demand equation you
have to do 1 over the slope the inverse
but here it's just the slope on price so
alpha x multiplied by price divided by q
x so on and so forth yeah and this is just
the numerical version of it So
here we have a good that sells
at $25, PX is $25, a good Y sells at
$35, PY is $35, and advertising is, what does
it say advertising is? How many dollars? 50
units. So advertising, 50 units, and
income is $20,000. So once again, you
just plug in all these numbers, and it gives
you the quantity. And the own price
elasticity is just the number in front of the
term multiplied by the price, which is $25,
divided by the quantity gives you the own price
elasticity so in all of these you can see 65
at the bottom because that's the current
quantity and the numerator is whatever it
is we're trying to find the elasticity of
multiplied by the slope that's all we're doing
and as you can notice here let's say we we
change the the the income income went from 20
000 to 30 000 what we would do here is we
don't just plug in 30 000 here and calculate
because if we plug in 30 000 here it's going
to change the demand as well so that's no
longer going to be 65 to the bottom the slope
remains the same but as you remember with
anything if the price changes the quantity
is going to change if income changes or
advertising changes if it relates to quantity in
any way quantity will change as well so just
be careful there and that's actually where
we're going to finish up today what we're
going to do on Monday is there's a couple
of regression things I want to talk about,
including how you can run a really simple
regression either in Excel or by prompting Cord
or GPT. And I want to show you there's multiple
methods to do this. And the last thing I'll
talk about on Monday in this topic is how
you can transform a linear regression into
the elasticities, which you can do by doing
a log transformation. But given that will
take 20, 25 minutes, I don't want to start
today. So you can all leave a little
and if you have any questions about the
exam or the next homework I have time
so feel free to come up here hey Daniel how's
it going you don't need to yeah Did you
watch the BoilerCast? I didn't watch it.
Was it helpful? It was really
helpful. Okay, great. Anything else
that I liked, I didn't consider
the BoilerCast. No, no, no. Okay, sounds good. Yeah,
BoilerCast was good. Thank you. No worries. Also, I might request
you and me for office hours before the exam.
No worries. Yeah, I'll have extra office
hours before the exam. So, on Friday and
then on Monday, I'll have extra office hours.
Oh, that's awesome. Yeah, no worries.
Hey, how's it going? Sorry? Can I
waste traffic? no as long as like
that can deal with like you know one plus
zero point you know one to the power of
three is more than sufficient so I'm
not gonna ask you to graph anything it's
just interpreting yeah it's scientific
to calculate this. I'll make an
announcement, no worries. one sentence so
that's what it'll look like my ta has
some past like exam practice questions
from previous semesters and i'm
going to when i write the way i work is
when i write exams so you should
you should get an understanding
of my style yeah but also the textbook
is pretty I'm not going to go too far
away from like the textbook stuff you
can use the textbook's questions to help
you go on this day if there's not sufficient
let me know and I can try and get
more out there as well I have a question on
this so like I know you're just explaining
it but I want to like be able to Of course,
of course, yeah. But, okay, how did this
turn, like, how did they switch? Did they
have a division, or? So, this, the. Or did
I write it wrong? No, no, no, so
this, this is just, so this here
is our analysis. Yeah. Yes. And this is
upside down. Are you a stranger
thing, Span? Yeah, yeah,
yeah. This is the upside down.
Okay. So, so when, when we have this
in there, so. Okay. Let me, let me,
let me, let me. So let's say we know
this is 3, and this here is, yeah, this
whole thing, we can call the whole thing
x, let's pull out x, and then this essentially
is going to be 1 over x, so that's
the relationship between two of them,
this is 8, and that means this is just
going to be 1 over 2. So this one is X? Yep. Okay, and this one
is just one of X. Okay, and then, but this is, what is it? That is, what was the
name of this? The inverse? This has no
meaning. No name. It's just to show
how you get it. Yeah, just the relationship
is, if you flip this upside down, you get
the elasticity. Perfect. And when you have
anything that's like the wrong way around,
it's just inverse. Yeah, the
relationship here is the exact same
thing. So if you times this by one
something, sorry, but yeah, like when you
think about it, it's literally just the
opposite, when you have two of the inverse
of each other, the relationship is minus
just x and the other is one of the other. yeah
okay so it's seven and one seven yeah
yeah so that's that's how we get from this
stage to sorry from this stage from this
stage to this stage is by yeah dividing it
by one essentially got it and then why do we
have the best one again just because that's
just the derivative so from going from
here to here what we're doing is we're taking
out a common factor of So for this one here,
there is no p, so we've got to divide
it by p when we try to get out. This is the
mathematical trick that gets us the inverse
of the elasticity. So that's why we
have the plus one. Okay, so again, this
is the elasticity. Again, this is,
what is this? Okay, so, no, no,
no, don't apologize. This is the exact
reason why I did this, because I
think it's a problem. So we have our
total revenue here, so this is our total
revenue equation, and what we want
to do is take the derivative of
the total revenue, and we want to
differentiate it, because total revenue, when
we differentiate it, is when we get the
marginal revenue, which is what we care
about optimising. so to do that we use the
product rule so for p is a function of q
like for example let me go back to so
let's say we have p equals a minus q so
to find the the first derivative here it's
just going to be how does p change when q
That's all that is. That's how we get
this, this part here. So this is the derivative
of P as a function of Q, which is this
here, multiplied by the second part
of the function of Q. Then we add the
second part of the product,
which is when we differentiate
this part here.