Download Handout: Some Math Behind Elasticities

Handout: Some Math Behind Elasticities
Price Elasticity of Demand (Own and Cross)
Own Price
Own price elasticity of demand tells us how much the quantity demanded of a good is
going to change when the price of that good changes, with all other factors held constant1.
For you mathematically minded folks, that’s . . .
"Qd
ed = Qd "P
P
Where ed is the price elasticity of demand2. This makes a lot more sense if we have a
graphical example.
!
Say the graph here represents the demand for CDs. Consider moving from point A to B.
That’s a change in price of $5, and a change in quantity demanded of 10. That means the
price elasticity of demand is
Price ($)
10
" 20 + 10 %
$
'
# 2 &
!
40
A
35
= 4.33
5
" 35 + 30 %
$
'
# 2 &
i.e. a 1% change in price will bring
about a 4.33% change in quantity
demanded3. Now consider moving
from point C to point D. That’s the
same price and quantity demanded
increment change as before. But
now . . .
B
30
25
20
15
C
10
D
5
10
20
30
40
50
60
70
80
Quantity
Demanded (in
millions)
1
Because of the Law of Demand that we learned before, for all normal goods the change in quantity
demanded will move in the opposite direction of change in price. So elasticity of demand for normal goods
will be a negative number. But the negative is often dropped for the sake of simplicity.
2
For those of you unfamiliar with Greek letters, ! is called “delta”, and it means “change in”. So
"Qd Qd is the equivalent of saying “percentage change in”, as it’s the change in quantity demanded
divided by the total quantity demanded. Similar logic follows for price.
3
Note that I’ve used our textbook’s convention of using the “midpoint formula” to calculate elasticity – we
evaluate the elasticity at the average of the prior and post values for Q and P.
!
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2009
10
" 70 + 60 %
$
'
# 2 &
= 0.23
5
"10 + 5 %
$
'
# 2 &
i.e. a 1% change in price will bring about a 0.23% change in quantity demanded. That’s
quite a different result from before. What’s the deal? Let’s break the elasticity equation
!
into two parts.
ed =
!Qd P
*
!P Qd
"Qd
, is the same in both cases (10/5), since we have a
"P
P
curve with a constant slope. But the second part,
, is different as we move along the
Qd
curve, because at each
! point were dealing with different prices and quantities demanded.
In the first case, it was 32.5/15, and in the second case, it was 5/65.
The first part of the equation,
What’s the moral of the story? Depending on the shape of the demand curve and where
we are on the demand curve, price elasticity for the same good can take on different
values.
There are demand curves that can be constructed that will have a constant elasticity.
However, with at constant slope demand curve (where slope is not equal to 0), the
elasticity will vary along the curve.
Cross Price
Cross price elasticity of demand is similar to own price elasticity, but now, we’re looking
at two goods at once. Specifically, we’re asking what percentage change in quantity
demanded we’ll see for good A when the price of good B changes. Let’s go back to the
math?
eA,B
"QdA
A
= Qd
"P B
PB
!
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2009
Here, I’ve chosen the A,B subscript on the e as a way of indicating that we’re talking
about the demand of good A and the price of good B. Note that we’re holding the price of
good A constant.
The sign of the cross price elasticity of demand tells us about the relationship between the
two goods. For example, if eA,B is negative, that means as the price of good B goes up, the
demand for good A goes down, and the goods are compliments. However, if the function
is positive, as the price of good B increases, the demand for good A increases as well, and
the two goods are substitutes.
“Size” of Price Elasticity
If ed is greater than 1, a 1% change in price will have a greater than 1% change in
quantity demanded, and the good is called elastic. If ed is equal to one, the good is unit
elastic, and if it is less than 1, the good is inelastic. In our earlier constant slope demand
curve example, in the first case we had elastic demand, but in the second, it was inelastic.
If ed is infinite, then quantity demanded is so amazingly sensitive to price changes that
the tiniest change in price results in huge changes in quantity demanded, and the good is
called perfectly elastic, which corresponds to a totally horizontal demand curve. On the
other side of things, if ed is equal to 0, then no matter what happens to price, quantity
demanded won’t change at all, and the good is called perfectly inelastic, which
corresponds to a perfectly vertical demand curve4.
Own Price Elasticity and Revenue
How sensitive the quantity demanded for a good is to price changes also tells us about
how revenue for sellers will change if price changes. If the price elasticity is greater than
1, the good is elastic, and quantity demanded is sensitive to price changes. The revenue
gained by increasing the price is outweighed by the revenue lost by the decrease in
quantity sold. So raising prices when ed > 1 results in an overall decrease in revenue. On
the other side of the coin, if price elasticity is less than 1 (ed < 1), the revenue gained by
increasing the price is greater than the revenue lost by the decrease in quantity sold, and
overall revenue increases.
Who cares? For one, sellers do. Take our above example again. If I’m in the music
industry, and I know that we’re at point B on the demand curve, I know that price
elasticity is greater than 1, so an increase in the price of CDs is going to result in a loss of
revenue. But if I know I’m at point D a higher price means good news for me, because
I’ll make more revenue in the end.
4
It’s kind of hard to imagine that the demand for anything is perfectly elastic or inelastic. Is there really
some good out there that, even if the price was increased twenty-fold, quantity demanded wouldn’t change?
Probably not. They are largely theoretical concepts. But there are goods that are extreme cases in both
directions.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2009
Income Elasticity of Demand
As a concept, it’s almost identical to price elasticity, but now, we’re talking about . . . you
guessed it, income. The income elasticity of demand measures how much the quantity
demand of a good will change (at any given price) when income changes.
Note the part in parenthesis . . . at any given price. In the first case, we held income
constant, and asked what happened with quantity demanded when price changed
(movements along the demand curve). Now, we’re holding prices constant, and asking
what happens to quantity demand when income changes (shifts of the demand curve).
The math looks almost the same, with a little change of what letters we throw in. Let I
represent income. Then our equation is for income elasticity of demand (ed,w) is
!Qd
ed , w =
!I
Qd
I
Now, however, we have to watch out for negatives. Unlike price elasticity of demand, it
isn’t assumed that this is going to be a positive value. In fact, the sign of ed,w tells you a
lot about the good we’re examining. If we get a positive value, then the good is a normal
good, i.e. consumption of it increases as income increases. If we get a negative value, the
good is an inferior good, meaning as our income increases, we consume less5.
Price Elasticity of Supply
Imagine you had a price elasticity of demand equation. Now imagine replacing Qd with
Qs. If you were imagining right, you’d have:
!Qs
es =
!P
Qs
P
Nothing new here, people. Just the same basic idea, but now we’re asking by what
percentage quantity supplied changes when price changes by a certain percentage. Other
than that, it’s just about the same deal as price elasticity of demand, and the terminology
is similar (inelastic supply, elastic supply, unit elastic supply, etc. etc. etc.).
5
An example of an inferior good we've discussed is Ramen noodles. Sure, they’re cheap and you can live
off them for a little while. But if your income suddenly tripled, chances are you aren’t going to increase
your Ramen consumption. Instead, you’ll probably start eating more “real” food, and less Ramen.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2009