### Elasticity

```Elasticity
Elasticity measures the degree of one variable’s
dependence on another variable, or the “sensitivity” of
one variable to a change in another variable.
While calculating elasticity, changes are expressed in
relative (percentage) terms.
The main reason:
We don’t want the size of the market or the choice of
measurement units affect our result. Dealing with
relative changes makes our result ‘cleaner’ and less
dependent of any arbitrary conventions.
The relative change is calculated as the “nominal”
change (measured in the same units as the variable),
divided by some reference value.
% P 
P
 100 %
P
What is used as the reference value:
• Either the current value
• Or, in cases when we are discussing elasticity over
some interval of values, the average value
Which approach should be used is usually clear from
the context.
The most common use of the elasticity concept:
The own price elasticity of demand, E Q X , PX
is the percentage change in the quantity demanded, Q,
divided by the corresponding percentage change in the
price, P:
E Q X , PX 
% Q X
%  PX
(the variable causing the change is in the denominator,
the affected (“dependent”) variable – in the numerator.)
•Since QD and P always change in the opposite
directions,
E Q X , PX 
% Q X
%  PX
is always negative.
The convention:
When saying “more elastic” about own price elasticity of
demand, we will refer to a greater variation in QD given a
certain change in price, therefore a greater absolute value
of a negative number for elasticity.
For example, demand with elasticity of –3 is more elastic than
demand with elasticity –1.4 because in the first case the same
relative change in the price of the good causes a larger relative
change in quantity demanded
(stronger reaction from consumers,
consumers are more sensitive to price)
The convention:
When saying “more elastic” about own price elasticity of
demand, we will refer to a greater variation in QD given a
certain change in price, therefore a greater absolute value
of a negative number for elasticity.
For example, demand with elasticity of –3 is more elastic than
demand with elasticity –1.4 because in the first case the same
relative change in the price of the good causes a larger relative
change in quantity demanded
(stronger reaction from consumers,
consumers are more sensitive to price)
Note it is uncommon to say “more/less elastic” about any other
application of elasticity!
Demand is called
•“elastic with respect to price” or “price elastic”
or simply “elastic” if %  Q  %  P
X
X
This implies    E Q
• “inelastic” if
%  Q X  %  PX
and
•“unitary elastic” if %  Q X  %  P X
X
, PX
 1
 1  E Q X , PX  0
and
E Q X , PX   1
What factors does own price elasticity of
demand depend on?
1. Number of substitutes
More substitutes – more choices for consumers –
– the more freely they switch among products –
– more elastic demand
Price elasticity of air-travel demand over the North Atlantic is – 1.2.
The elasticity of demand for air travel inside the U.S. is – 1.98.
What factors does own price elasticity of
demand depend on?
2. How expensive is the product, in
relation to consumers’ income?
A 10% change in price matters to consumers more when the
product is expensive
What factors does own price elasticity of
demand depend on?
2. How expensive is the product, in
relation to consumers’ income?
A 10% change in price matters to consumers more when the
product is expensive, or when consumers are poor
More expensive product –
What factors does own price elasticity of
demand depend on?
2. How expensive is the product, in
relation to consumers’ income?
A 10% change in price matters to consumers more when the
product is expensive, or when consumers are poor
More expensive product – more elastic demand
Wealthier consumers –
What factors does own price elasticity of
demand depend on?
2. How expensive is the product, in
relation to consumers’ income?
A 10% change in price matters to consumers more when the
product is expensive, or when consumers are poor
More expensive product – more elastic demand
Wealthier consumers – less elastic demand
The price elasticity of white pan bread in Chicago appeared to be – 0.69,
while for premium white pan bread elasticity was measured at – 1.01.
What factors does own price elasticity of
demand depend on?
3. How much time do consumers have to
react to the change in price?
Therefore the greater the time period between the change in
price and the moment when Qd is recorded, the more elastic
the demand will look.
The price elasticity of the demand for coffee is – 0.2 in the short run
and – 0.33 in the long run.
Ceteris paribus, would you rather be producing a good
with high or with low elasticity of demand? Why?
It is better to deal with consumers who have little choice
In other words,
It is better to face less elastic demand.
That would give you greater ability to vary your price;
specifically, to set price above the marginal cost and still
remain profitable.
(Such ability is called “pricing power” or “market power”)
What can producers do to reduce the elasticity of
demand for their product?
One of the reasons to care about elasticity –
The relationship between elasticity and revenue.
(Total) Revenue, TR = P ∙ Q
Not the best measure of overall firm performance (profit is
a much better one) but is often used nevertheless.
Elasticity has a lot to do with how price affects revenue.
When demand is elastic, then Q changes more than P:
Price up, P x Q = TR
Price down, P x Q = TR
Price and revenue move in the opposite directions.
When demand is inelastic, then
Price up, P x Q = TR
Price down, P x Q = TR
Price and revenue move in the same direction.
Marginal Revenue, MR (recall that “marginal” means
“additional”) tells us what happens to the total revenue
as the quantity produced increases by one unit.
MR>0 tells us that increasing Q would increase TR.
So overall we have: Q up, P down, TR up.
We just learned that such relationship between P and
TR indicates demand is elastic.
If MR<0, then increasing Q would decrease TR.
Or, Q up, P down, TR down.
Demand is inelastic.
Consider a linear
demand curve,
QD = 6 – P
P
6
5
4
3
2
1
0
1
2
3
4
5
6
Q
We can trace what happens as we move along
the demand curve.
Q
P
TR
0
6
1
5
0
5
2
4
3
3
4
2
5
1
6
0
8
9
8
5
0
P
6
TR is not directly
proportional to the
quantity produced
because in this case
in order to sell more,
the firm needs to
lower its price.
5
4
3
2
1
0
1
2
3
4
5
6
Q
Change in total revenue due to
one more unit produced and sold
Q
P
TR
MR
0
6
1
5
0
5
--5
2
4
3
3
4
2
8
9
8
3
1
–1
5
1
6
0
5
0
–3
–5
In fact, MR represents the slope of the TR curve plotted
against output, Q.
P
6
5
4
3
2
1
0
1
2
3
4
5
MR
6
Q
Q
P
TR
MR
0
6
1
5
0
5
--5
2
4
3
3
4
2
8
9
8
3
1
–1
5
1
6
0
5
0
–3
–5
E
An alternative version of the elasticity formula may come useful:
  Q   100 %


Q
% Q


Ed 


P
% P
 100 %
P


An alternative version of the elasticity formula may come useful:
 Q 


Q
% Q


Ed 

P
% P
P


An alternative version of the elasticity formula may come useful:
 Q   P  Q


Q
% Q


Ed 

P
% P
 P Q
P


An alternative version of the elasticity formula may come useful:
Ed 
% Q
% P

Q  P
 P P  P  Q
An alternative version of the elasticity formula may come useful:
Ed 
% Q
% P

Q  P
P  Q
Recall that P and Q are the present actual price and
quantity and the ‘deltas’ are their changes in their
respective units.
Also note than, when the demand equation is QD = 6 – P,
every unit change in price causes a unit change in
quantity in the opposite direction.
Q
In other words,
P
 1
Elasticity along a linear
demand curve
P
6
P = \$2
QD = 6 – P = 4
E Q X , PX  (  1) 
ED=–5
5
P

QD
 (  1) 
4
2
  0 .5
4
3
P = \$5
QD = 6 – P = 1
ED=–0.5
2
1
E Q X , PX  (  1) 
0
1
2
3
4
5
6
Q
P
QD
 (  1) 
Along a linear demand curve, the slope is constant
but elasticity is not!

5
1
 5
We can trace what happens as we move along
the demand curve.
Q
P
TR
MR
0
6
1
5
0
5
--5
E
–∞
–5
2
4
3
3
4
2
8
9
8
3
1
–1
–2
–1
–.5
5
1
6
0
5
0
–3
–5
–.2
0
How this helps in finding the point of maximum revenue:
P
6
–5
5
–2
4
–1
3
–.5
2
–.2
1
0
1
2
3
4
5
6
Q
D is elastic,
P up – TR down,
P down – TR up
P
6
–5
5
–2
4
–1
D is inelastic,
P up – TR up
3
–.5
2
–.2
1
0
1
2
Revenue is MAX
where ED= –1
3
4
5
6
Q
Marginal revenue can also help us
find the revenue-maximization point
P
6
5
4
3
2
1
0
1
2
3
4
5
MR
6
Q
In summary,
Total revenue is maximized when
• MR changes its sign from ( + ) to ( – ) as we move
along the quantity axis, …
and
• Demand changes from elastic to inelastic
In other words, when E = –1
(You can use either one of the criteria)
Therefore, Marginal Revenue and Elasticity are related.
More specifically,
 1  E Q ,P
X
X
MR  P 
 E
Q X , PX





This formula will prove useful later.
Other applications of elasticity:
• Income elasticity of demand
E Q X ,M 
% Q X
% M
where M stands for consumer income.
What can be said about the income elasticity of demand
for normal goods?
If %  M  0 , then %  Q X  0 therefore E Q , M  0
X
… for inferior goods?
E Q X ,M  0
Cross-price elasticity of demand,
E Q X , PY 
% Q X
%  PY
measures the effect of a change in the price of
good Y on the quantity of good X demanded.
What values of E Q
X
, PY
should we expect for “complementary” or
“substitute” goods, respectively?
Complements: %  PY  0 ,
% Q X  0
E Q X , PY  0
%  PY  0 ,
% Q X  0
E Q X , PY  0
Substitutes:
Elasticity under linear demand specification:
If the demand equation is QD = 6 – 2P, then for each \$1
decrease by 2 units
incremental increase in P, QD will _________________
or  Q D
P
 2
If the demand equation is QD = A + b P, then
and
E Q X , PX 
% QD
% P

QD
QD

P
P
 b
P
QD
P
b
0
QD
where P, QD are the current price and quantity demanded.
The same holds for more complex equations such as
QD = A + b P + c Pother + d M + e T
Elasticity under the log-linear specification
Log-linear specification establishes a relationship not between
the actual values of the variables but between rates of their
change.
For example, a log-linear equation such as
lnQD,X = 6.4 – 1.3 lnPX + 0.4 lnPY – lnM
says,
“If own price increases by 10%, and everything else stays the
decrease
same, the unit sales of our product would ______________
by
13 percent
_____________.”
But be careful:
What does such an equation imply about the effect of price
variations on revenue?
Demand is elastic… the more we drop the price, the more our
revenue will grow. This doesn’t sound right.
Two problems on revenue maximization:
1. (Problem 13 on p.112 in the text)
You are a manager at the Chevrolet division of General
Motors. If your marketing department estimates that the
semiannual demand for the Chevy Tahoe is
QD = 100,000 – 1.25P, what price should you charge in
order to maximize revenues from sales of the Tahoe?
We can solve it either by using the fact we just learned
about elasticity, or by using derivatives.
DERIVATIVES
The derivative of a function y=f(x) is equal to the rate of
instantaneous change in variable y with respect to variable x.
Can be denoted in a number of different ways:
y
f ( x )
dy
df ( x )
dx
dx
Graphically, it is equal to the slope of the function at a
point.
Positive derivative – positive relationship – positive slope
Negative derivative – negative relationship –negative slope
Slope = 0
It is the maximum!
Positive
slope
Negative
slope
How derivatives help us find the maximum of a function
Four basic rules of differentiation.
(“Differentiation” is the procedure of finding the derivative of a
function.)
Rule 1. The derivative of a constant is zero.
(Easy to see if you plot it;
the derivative is the slope of the graph)
y
x
Rule 2. The derivative of a power function (a function
of the form y  x n , where n is any number).
If y =
x2,
If y =
x3,
then
dy
 2x
dx
then
dy
dx
 3x
2
, etc.
n
Overall,
d (x )
dx
nx
n 1
, or in other words…
- The power in the original function becomes the
coefficient in the derivative;
- The power of x in the derivative is one less than in
the original function.
Rule 3. A constant times a function.
If y = k f(x), then
dy
dx
k
df ( x )
dx
The derivative of a constant times a function is equal
to the constant times the derivative of the function.
Example:
Y=3
x3
dY
dx

Rule 3. A constant times a function.
If y = k f(x), then
dy
k
df ( x )
dx
dx
The derivative of a constant times a function is equal
to the constant times the derivative of the function.
Example:
Y=3
x3
dY
dx
 3 3x  9x
2
2
Finally, Rule 4.
The derivative of the sum of two functions is the sum
of their derivatives.
The derivative of the difference of two functions is the
difference of their derivatives.
Example:
Y = x3 – 2x + 3
dY
dx

Finally, Rule 4.
The derivative of the sum of two functions is the sum
of their derivatives.
The derivative of the difference of two functions is the
difference of their derivatives.
Example:
Y = x3 – 2x + 3
dY
dx
 3x  2
2
Back to the problem:
If your semiannual demand is QD = 100,000 – 1.25P,
what price should you charge in order to maximize
revenues?
- Express revenue in terms of price (and price only!)
- Find the maximum of that function (differentiate, set
derivative to zero, solve for P)
TR  Q  P 
Back to the problem:
If your semiannual demand is QD = 100,000 – 1.25P,
what price should you charge in order to maximize
revenues?
- Express revenue in terms of price (and price only!)
- Find the maximum of that function (differentiate, set
derivative to zero, solve for P)
TR  Q  P  (100 , 000  1 . 25 P ) P
Back to the problem:
If your semiannual demand is QD = 100,000 – 1.25P,
what price should you charge in order to maximize
revenues?
- Express revenue in terms of price (and price only!)
- Find the maximum of that function (differentiate, set
derivative to zero, solve for P)
TR  Q  P  (100 , 000  1 . 25 P ) P  100 , 000 P  1 . 25 P
d (TR )
dP
 100 , 000  2 . 5 P
2
Back to the problem:
If your semiannual demand is QD = 100,000 – 1.25P,
what price should you charge in order to maximize
revenues?
- Express revenue in terms of price (and price only!)
- Find the maximum of that function (differentiate, set
derivative to zero, solve for P)
TR  Q  P  (100 , 000  1 . 25 P ) P  100 , 000 P  1 . 25 P
d (TR )
dP
 100 , 000  2 . 5 P  0
2
Problem 2. The manager of a grocery store
experiments with prices. On some days a bag of
store brand bagels is priced at \$1.99. On such days,
67 bags are sold on average. On other days, the
price for the same item is reduced to \$1.49, in which
case the average quantity sold increases to 88 bags
per day.
Given this information, what price would most likely
maximize the daily revenue from selling bagels?
This problem is different from the previous one in
that we don’t have a demand equation. We do,
however, have two points on what we believe to
be the demand curve.
The most important takeaway points:
You need to be able to
• Use your knowledge of elasticity formulas to predict
changes in relevant variables;
• Derive information about elasticities from a demand
equation which is
-linear
-log-linear;
• List and elaborate on the factors that affect elasticity of
demand for a product;
• Explain how elasticity affects the relationship between price
and revenue;
• Find the revenue-maximizing price-quantity pair when given
a linear demand equation.
```