Ch. 5 Handouts on Elasticity
Elasticity: measures the responsiveness of one factor due to a change in another factor

Measures the percentage change in a dependent variable, Y, resulting from a 1% change in the
value of an independent variable, X.

General Form:
o

Two different ways to measure elasticity: point elasticity and arc elasticity
o Point elasticity: measures elasticity at one specific point and is commonly used to
measure the effect on the dependent variable, Y, of a very small change in X. (typically
less than 5% change in X)
o Arc elasticity: measures the average elasticity over a range of a function and is better
suited for larger changes in X (more than 5%).
Price Elasticity of Demand (ED or EP): the percentage change in Quantity demanded, Qd or Q, as a result
of a percentage change in the product’s own price, P.

Shows how sensitive consumers are to a change in the price of your product.

The law of demand says that as prices increases, quantity demanded will fall. Price
elasticity will tell you the magnitude of this decrease.

This will ultimately impact sellers’ decisions regarding increasing or lowering prices to
increase total revenue (TR).
% Qd
% P

Point Elasticity and Arc Elasticity general form: Ed=

The difference between the two measures is the information required to calculate each
measure and the overall applicability to specific situations.
Point Elasticity Measure of Ed:
 Measures the elasticity at one specific point on the demand curve.

Point Elasticity: Ed=
o where the second term is the inverse slope. Remember: inverse slope is the “b”
term in your demand equation or coefficient for price in regression output:
ex: Qd=100-2P (-2 is inverse slope)

Point Elasticity example:
o If the current selling price is P=5, and the current Qd=90 then we can estimate
the elasticity at that point by the following:
o Ed=
P Qd
*
Q P
o
o
o This means a 1% increase in the price of our product results in a .11% decrease
(since negative) in the quantity demanded of our product. Or that a 1%
decrease in price results in a .11% increase in Qd.
o This also means that a 3% increase in price would have a 3*.11 = 3.33%
decrease in the quantity demanded.
o Advantage of point elasticity: only need the demand equation and one P, Qd
combination to get an estimate. Easy to apply with regression output.
o Disadvantage: as we will see later elasticity changes at every point along the
demand curve. Therefore, it is only accurate for small changes in prices.
Arc Elasticity Measure of Ed:
 Measures the average elasticity between two points along the demand curve
P  P1
Qd
* 2
P
Q2  Q1

Arc formula:

Ex:

Advantage: used to analyze the average sensitivity of demand to price changes over
an extended range of prices.
Interpreting and Classifying Ed numbers
 All Ed numbers will be negative in accordance with the law of demand indicating a negative
relationship between P and Qd

Convert all to positive by taking the absolute value.

We then classify the numbers into one of 5 categories:
o Elastic demand ______________
o Inelastic demand _______________
o Unit elastic demand ________________
o Perfectly elastic demand ____________________
o Perfectly inelastic demand ___________________
Elastic Demand:
 Ed > 1

Ed=
“elastic”
% Qd
to get a number larger than one it must be that the numerator is a
% P
larger number than the denominator:

%Qd > %P meaning that consumers are price sensitive and alter the Qd
significantly in response to changes in prices

this means that for every 1% change in price there is more than a 1% change in Qd
o ex: Ed=-2.5 means a 1% change in price leads to a 2.5% change in the Qd.
Consumers have a large reaction to the price change. They are price sensitive
or price responsive.
Inelastic Demand:
 0< Ed < 1

Ed=
“inelastic”
% Qd
to get a number between zero and one (e.g. 0.5, 0.25) it must be that the
% P
numerator is a smaller number than the denominator.

%Qd < %P meaning consumers are not price sensitive; they may alter the Qd when price
changes but it is a small response

this means that for every 1% change in price there is less than a 1% change in Qd
o ex: Ed=-0.5 means a 1% change in price leads to a 0.5% change in the Qd. Consumers
have a relatively small reaction to the price change. They are NOT price sensitive or price
responsive.
Unit Elastic:

Ed = 1

%P = % Qd

a 1% change in price leads to exactly a 1% change in Qd
a 10% change in price leads to exactly a 10% change in Qd
“unit elastic”
 occurs when consumers transition from being “elastic” to “inelastic”
Perfectly Elastic Demand:
 special case: Ed =  perfectly elastic (largest positive number)
 when a small change in price results in the consumer altering quantity demanded in an extreme
way. Consumers are hypersensitive to prices.
 Realistically occurs only in rare circumstances when consumers can buy the exact same product
from different vendors and always select solely on the basis of lowest price.

occurs with a horizontal demand curve (show graph)
Perfectly Inelastic Demand:
 special case: Ed = 0 perfectly inelastic
 when any change in price results in the consumer buying the exact quantity they were already
purchasing. Consumers do NOT change their Qd at all.
 Realistically occurs only in rare circumstances when consumers must buy a product in fixed
quantities and cannot store products (so they don’t buy extra units when prices are low).
 Ex: insulin for diabetics. Use the same amount of prescription medication. You don’t buy more
when prices fall; and you can’t really buy less for health reasons.

Vertical demand curve
Factors that Influence Price Elasticity of Demand
1. degree of substitutability

o All beverages (soda, water, alcohol, juice, etc).
o With lots of choice price plays an important role in decision making

o Gasoline for your car—there is no good substitute
o With little or no choice price plays less of a role
2. Product Definition


Ex: milk (not any good substitutes for milk)

Ex: many substitutes within brands or types of milk such as 1% milk, 2% milk, ½%
milk, whole milk etc.
3. Time horizon before purchase

Ex: airlines, last minute willing to pay more
Ex: last minute gifts

Ex: vacation next year: time to search for alternatives
4. Proportion of budget spent on the good

Ex: expensive items such as cars, jewelry, computers
A 5% change in the price of an expensive good can have a large effect on your
budget. You will be more price sensitive.

Ex: inexpensive items such as pencils, pop.
A 5% change in price has little or no effect on budget. You will not alter the Qd by
very much.
Calculating Price Elasticity from Sales Data
Month
March
April
May
June
July

Sales of Sports
Watches
55000
60000
40000
50000
45000
Sports Watch
Advertising
9500
8500
8500
8500
10000
Sports Watch
Price
250
240
260
260
260
Dress Watch Price
550
510
510
570
500
To determine the price elasticity of demand for sports watches we will estimate the change in sales
of sports watches resulting from changes in its own price.



Qd
P  P1
Arc formula: Ed= P * 2
Q2  Q1

Ed=

Ed=


For every 1% of a price change sales will change by 5%.

If expect prices to decrease by 6% then sales will increase by 6*5=30%.
Elasticity changes as you move along a Linear Demand Curve:
Since the relative expense of the product influences elasticity, this means that you will not be able to
categorize demand for a good as always being elastic or inelastic it depends on where you are located on
the demand curve.
Price
$4.00
$3.00
$2.00
$1.00
Qd
0
2
4
6
Plot on graph:


Demand equation:
P=4-1/2 Qd
Inverse demand equation:
2P=8-Qd
Qd=8-2P (x-intercept is 8; inverse slope Qd/P=-2)

Calculate point elasticity Ed at various points
A: P/Q * Qd/P =
B: P/Q * Qd/P =
C: P/Q * Qd/P=
D: P/Q * Qd/P =
Hence, given regression output you should be able to determine elasticity by using the coefficient for
price as Qd/P .
 Example: Q=10,000 -200P+ ……………….




Total Revenue (TR) and Elasticity of Demand

TR= P * Q

Ed deals with percentage changes so want total revenue expression in percentage terms

If elastic, then %P < % Qd.

%TR= %P + % Qd
ex:



ED=-2
If increase price, Qd falls by more than P so TR falls
If decrease price, Qd increases by more than P so TR increases
If demand is price elastic, reduce prices to increase revenues.

Ex: discounts, coupons, specials
If inelastic, then %P >% Qd.
 %TR= %P + % Qd

ex: ED=-.25




If increase price, Qd falls by less than P so TR increases
If decrease price, Qd increases by less than P so TR decreases
If demand is price inelastic, increase prices to increase revenues.
Ex: taxes on cigarettes, alcohol, prescription drugs
If unit elastic then Ed=1
For a 10% increase in Price, Qd will decrease by 10%
%TR= %P + % Q
= 10 -10
=0
price changes have no effect on TR with unit elastic demand
Total revenue is maximized at this point.
Show Graph of Linear Demand Curve and ranges of Ed (TOP)
Graph of TR (BOTTOM)
Starting at the reservation price:
Elastic range of the curve: Ed > 1



Unit Elastic point on curve: Ed = 1



Inelastic range of the curve: 0< Ed < 1



Optimal Price Formula for Firms:
 Pricing depends on your marginal cost of production as well as demand

Specifically, on the demand side you can charge higher prices when demand is inelastic to make
more money, and charge lower prices when demand is elastic to make more money.

If you can divide up your markets into segments, the optimal price strategy would be to charge
different prices to different customers (price discrimination: review later)

If you plan to charge one price across a given population, then determining the optimal price is a
function of marginal cost and the average Ed.

Firm’s maximize profit by producing the Q where MR=MC and then set their price accordingly.
We will translate the MR equation to be a function of Ed.

MR=∆TR/∆Q and TR=P*Q. You can then algebraically substitute and simply to obtain the
following
where Ed is not in absolute value so it is a negative number

If you set MR=MC and solve you arrive at the optimal pricing formula called the “inverse
elasticity rule”

Ex: If MC=20 and Ed=-2.5
o
o

Because of the nature of TR and costs, firms will always end up charging a price in the elastic
portion of the demand curve in order to maximize profits (We will see this later in the course
when we draw out cost curves)

Thus, this rule works only with prices in the elastic portion of the demand curve

Note: This may appear that the rule then only works for “elastic goods” but this is absolutely
false. All products even if they are typically more inelastic in nature will still have prices at which
people are price elastic, unit elastic, and inelastic. Remember elasticity changes as you move
along the demand curve so all products effectively span the entire range of Ed.
Income Elasticity (EI)
 Measures how buyers change the quantity they purchase when income changes.

This is particularly useful for firms in forecasting changes in demand when the overall
economy, or local economy, is headed toward a recession or expansion.

Income elasticity will not only tell you if people will buy more or less of your product as
income changes, but how much more or less for each one percentage change in income.


There are two major classifications of income elasticity: normal goods and inferior goods.
Normal Goods:
 if EI > 0 (any positive number) then the product is considered a normal good.
o This means that there is a positive relationship between Income and Demand so
that as income increases, demand increases and as income decreases demand
decreases (I, D; ifI D)
o Ex: EI= 0.35, 1.6, 2.5, etc.
o EI=0.35 means a 1% increase in income leads to a .35% increase in demand. Or a
10% increase in income leads to a .35(10)=3.5% increase in demand.
o This means with normal goods an increase in income will shift demand to the right
o This means with normal goods a decrease in income will shift demand to the left.
There are two subcategories for normal goods: luxury goods (also called cyclical
normal goods) and necessities (also called noncyclical normal goods)


Luxury goods or cyclical goods:


Ex: automobiles, entertainment, jewelry, food items,

Ex: EI=3.5 means that for every 1% increase in income, you will
increase demand by 3.5%. This is typically true of luxury goods. The
more money you have, the more of these goods you buy.

They are considered “cyclical goods” because demand is highly
influenced by the state of the economy (recession vs. expansion:
where the economy is in terms of the business cycle). When the
economy is strong (and incomes rise), demand increases
dramatically. When the economy is weak (and incomes fall)
demand decreases dramatically.
Necessities or Non-cyclical Goods:


Ex: toiletries, movies, liquor, cigarettes

Ex: EI=.25 means that for every 1% increase in income, you will
increase demand by .25%. This is typically true of necessities. Even
when you have more income you still buy about the same amount
of those basic goods.

They are considered “non-cyclical goods” because demand is
relatively unaffected by the state of the economy. You don’t alter
consumption very much as income changes.
Inferior Goods or Countercyclical Goods
 if EI < 0 (any negative number) then the product is considered an inferior good.
o This means that there is a negative relationship between Income and Demand so that
as income increases, demand decreases and as income decreases demand increases
(I, D; ifI D)
o Examples: public transportation (buses), generic products. Goods that you will buy
less of if you have more income. And when your income really falls, goods you will
buy more of often because they are less expensive.
o Ex: EI=-0.15, -1.3,
o EI=-0.15 means a 1% increase in income leads to a .15% decrease in demand. Or a 10%
increase in income leads to a -.15(10)=-1.5% decrease in demand.
o This means with inferior goods an increase in income will shift demand to the left
(decrease demand)
o This means with inferior goods a decrease in income will shift demand to the right
(increase demand)
Determining if goods are Normal or Inferior using basic sales data:
% change in personal income
(from BEA.gov)
Nov 2008
Dec 2008
Jan 2009
-0.4
-0.2
-0.4
% change in sales (i.e., quantity purchased)
(from TWSJ)
Jan
Feb
Store
2009
2009
Abercrombie & Fitch
-20.0
-30.0
Costco
4.0
4.0
Gap
-23.0
-12.0
Hot Topic
6.0
10.8
J.C. Penney
-16.4
-8.8
Macy's
-4.5
-8.5
Neiman Marcus
-24.4
-20.9
Saks
-23.7
-26.0
Sam's Club
2.4
5.9
Target
-3.3
-4.1
Wal-Mart
2.1
5.0
Calculating EI whether goods are normal or inferior:
% Q
% I

E I=

Point Elasticity formula:

Arc Elasticity formula: Simplify to:

Typically use arc since it is applicable to wider swings in income.

If you have actual income level as data then you may plug in actual changes in income
and changes in sales to calculate income elasticity.

Otherwise, given the table of data above we only have percentage changes in income and
percentages in sales data:
I Q
*
Q I
I  I1
Q
* 2
I
Q2  Q1
o Example: from December to January income decreased by -0.4%




You may also calculate income elasticity using regression output: if Q=10,000 -200P +1.5I
o
o

Ex:
suppose current price =10; income=30,000





If median income is expected to increase to 32,000 then estimate sales:


Cross Price Elasticity (EXY or EPX):
 Tells us the relationship between goods
o Are x and y substitutes (consume one or the other)
o Are x and y complements (consume together)
o Are x and y unrelated

Measures how the quantity demanded of one good (X) changes as a result of a price change of
another good (Y).

EXY =
%Qx
%Py

EXY =
PY Q X
*
Q X PY
EXY Point Elasticity Formulation

EXY =
Q X
P  PY 1
* Y2
QX 2  QX 1
PY
EXY Arc Elasticity Formulation

Ex: Y=McDonald’s hamburger and X=Wendy’s hamburger
Py=$1.00, Qx=2000.
Py=$.80, Qx=1800.
o
EXY =
Q X
P  PY 1
* Y2
QX 2  QX 1
PY
EXY Arc Elasticity Formulation
o
o EXY =
o EXY =
o For a 1% change in the price of McDonald’s hamburgers we would expect a .47% change
in Wendy’s hamburgers. Since the number is positive: a 1% increase in price of
McDonald’s hamburgers leads to a .47% increase in the quantity or demand for Wendy’s
hamburgers.
Interpreting and Classifying EXY numbers:
 Substitute Goods:
o If EXY is positive > 0 then Goods X and Y are substitutes
o A positive number suggests a positive relationship between the Price of Y (P Y) and the
Quantity purchased of X (QX).
o This means that an increase in the price of y results in an increase the the quantity
purchased of X.


Ex: Y=coke; X=pepsi

If the price of Y, coke, increases we know that people will buy less coke. They will
increase the quantity of pepsi, X.

Ex: EXY= 2.1 means that a 1% increase in the price of coke leads to a 2.1% increase
in the quantity of pepsi. Or a 1% decrease in the price of coke leads to a 2.1%
decrease in the quantity of pepsi. Or a 4% decrease in the price of coke leads to a
4(2.1)=8.5% decrease in the demand for pepsi.

Since changes in the price of a substitute will shift the demand curve we know
that with a positive EXY an increase in the price of coke will result in an increase in
the demand for pepsi (demand shifts to the right for pepsi). Show example:
Complements:
o If EXY is negative < 0 then Goods X and Y are complements
o A negative number suggests a negative relationship between the Price of Y (PY) and the
Quantity purchased of X (QX).
o This means that an increase in the price of y results in a decrease the the quantity
purchased of X.

Ex: Y=camera; X=film

If the price of Y, camera, increases we know that people will buy less cameras.
Thus, less film will be needed so we see a decrease the quantity of film, X.

Ex: EXY= -0.15 means that a 1% increase in the price of camera leads to a 0.15%
decrease in the quantity of film. Or a 1% decrease in the price of film leads to a
0.15% increase in the quantity of film. Or a 4% decrease in the price of camera
leads to a 4(0.15)=.60% increase in the demand for film.


Since changes in the price of a complement will shift the demand curve we know
that with a negative EXY an increase in the price of a camera will result in an
decrease in the quantity of film (demand shifts to the left for film). Show example:
Unrelated Goods:
o if EXY =0 then X and Y are unrelated
o If you estimate the impact of a change in the price of pizza on the quantity of cars
purchased you would calculate a zero.
o There is no conceivably predictable relationship between pizza and cars.
Applications of EXY:
1. Understanding basic relationships between goods and predicting sales:

If Exy for McDonald’s burger (X) relative to BK burger prices (Y)=2.75 and BK plans to lower
prices by 10% then determine impact on McDonald’s sales:
%Qx
o Exy =
%Py
o
o
2. Determining your strongest competition in a market:

Stronger competition means the two goods are stronger substitutes

If you know the EXY for each related pair you can rank your competition:
Ex: McDonald’s Hamburgers
EXY with Wendy’s Hamburgers is 0.47
EXY with Burger King’s Hamburgers is 0.75
EXY with Hardees’ Hamburgers is 0.21
EXY with coke is -1.2
Your strongest competitor is the one with the largest, positive EXY number or Burger King in this
example.

For a 1% change in the price of a BK burger, you get the largest impact, of .75% change in
sales of McDonald’s hamburgers.

If you have a regression equation that includes several competitors then all variables with
a positive coefficient are competitors. Must calculate Exy for each to determine which is
your strongest competition.
3. Determining your strongest complementary good:

These are goods you may want to consider packaging together for consumers since they
consume them together

If you know the EXY for each related pair you can rank the strength of your complementary
goods:
Ex: McDonald’s Hamburgers
EXY with fries is -2.1
EXY with milkshakes is -0.45
EXY with coke is -1.2
Your strongest complement is the one with the largest, negative EXY number or fries in this
example.
For a 1% change in the price of fries, you get the largest impact, of 2.1% change in sales of
McDonald’s hamburgers.
Thus, you may want to offer something like a “value meal” including hamburger, fries, and even
coke since it’s the next strongest complement.

If you have a regression equation that includes several complementary goods then all
variables with a negative coefficient are complements. Must calculate Exy for each to
determine the strongest complementary good.

Qmcd burger=20,000 -500Pmcd burger + 300Pbk burger -600Pmcd fries -800mcd soda
o

Simplify demand equation to
o
o
o

Determine strongest complement:
o Confirm using Exy
o Exy for fries is
o Exy for soda is
o Fries are most complementary!