Utility Optimisation
Y12 Extension Class
What is Utility?
Satisfaction, happiness, benefit
Cardinal Utility vs. Ordinal Utility
• Cardinal Utility: Assigning numerical values to
the amount of satisfaction
• Ordinal Utility: Not assigning numerical
values to the amount of satisfaction but
indicating the order of preferences, that is,
what is preferred to what
Util
• A unit of measure of utility
Total Utility
•The amount of satisfaction obtained by
consuming specified amounts of a product per
period of time.
Marginal Utility
•The change in total utility (TU) resulting from a
one unit change in consumption (X).
•MU = TU/ X
Diminishing Marginal Utility
• Each additional unit of a product contributes
less extra utility than the previous unit.
Total
Utility
Graphs of Total Utility
& Marginal Utility
TU
X1
X1 is where marginal utility
reaches its maximum.
This is where we encounter
diminishing marginal utility.
The slope of TU has reached its
maximum; TU has an inflection
point here.
X
Marginal
Utility
X1 is where total utility reaches
its maximum.
MU is zero.
This is the saturation point or
satiation point.
After that point, TU falls and
MU is negative.
MU
X1
X
Imagine you are at a party and there are free cookies.
• X1 would represent the point at which you were
full. That is where you become satiated.
• After this point, you begin to feel a bit bloated
and unwell. Your utility begins to fall
The Connection between
Demand and Utility
• Instead of thinking in terms of utils, let’s
think in terms of dollars.
• Suppose the purchase of one unit of a good
gives you $10 worth of satisfaction.
• In other words, the marginal utility of that
first unit of the good is $10.
• Then you would be willing to pay up to $10
for it.
•
If a second unit of the good contributes $8
more of satisfaction, the marginal utility of
your second unit is $8 and you would be
willing to pay up to $8 for it.
• If a third unit of the good contributes $6
more of satisfaction, the marginal utility of
your third unit is $6 and you would be willing
to pay up to $6 for it.
Remember that the demand curve tells you what
people are willing to pay for various amounts of
a good or, equivalently, how many units of a
good they are willing to purchase at various
prices.
•
So, since we just found that the marginal
utility tells us what we are willing to pay for
a good, the marginal utility provides us
with information that we can use to
determine our demand curve.
For example:
Number of units
purchased (Q)
Price you are
willing to pay (P)
Marginal
Utility
0
-
-
1
10
10
2
8
8
3
6
6
If we graph that information,
we get our demand curve.
Number of units Price you are
purchased (Q)
willing to pay (P)
Marginal
Utility
price
0
-
-
1
0
1
10
10
2
8
8
3
6
6
8
6
Demand
1
2
3
quantity
Notice that by adding our MU values we can determine our
total utility at different consumption levels.
Number of units Price you are
purchased (Q)
willing to pay (P)
Marginal
Utility
Total
Utility
0
-
-
0
1
10
10
10
2
8
8
18
3
6
6
24
Indifference Curve
•A set of combinations of goods that are
viewed as equally satisfactory by the
consumer.
Indifference Map
• A collection of indifference curves
Assumptions
1. The consumer can rank all bundles of
commodities.
2. If bundle A is preferred to bundle B and B is
preferred to C, then A is preferred to C. (This
property is called transitivity.)
3. More is better.
Characteristics of indifference curves
Indifference curves slope down from left to right.
clothing
A
C
B
food
•Consider 2 points A & B on the
same indifference curve.
•At point B, you have more food
than at point A.
•If the amount of clothing you had
at point B was the same as or more
than at point A (like at point C), you
would not be indifferent between A
and B (since more is better).
•So A & B could not be on the same
indifference curve, which goes
against our initial statement that
they are.
•So you must have less clothing at
B, which means than B lies below A
and the indifference curve slopes
downward.
Indifference curves to the northeast are preferred.
clothing
E
A
IC2
IC1
food
•Point E is preferred to
point A because it has
more food & more
clothing.
•Since you are indifferent
between A & all points on
IC1, E must be preferred to
all points on IC1.
•Since you are indifferent
between E and all points
on IC2, all points on IC2
must be preferred to all
points on IC1.
Indifference curves can not intersect.
•Suppose indifference curves could intersect.
•Let the intersection of IC1 & IC2 be D.
•Then you must be indifferent between D & any
other point A on IC1.
•Similarly, you must be indifferent between D &
any other point B on IC2.
clothing
D
B
A
IC2
IC1
food
By transitivity, you must be
indifferent between A & B.
But A & B are not on the same
indifference curve, which they
should be if you are indifferent
between them.
Then, our initial supposition that
indifference curves could
intersect must be wrong.
Indifference curves are convex.
[The curves become flatter as we move from left to right.]
clothing
A
B
C
D
food
•When we have lots of
clothing & not much food
(as near A & B), we are
willing to give up a lot of
clothing to get a little more
food.
•When we have lots of
food & not much clothing
(as near C & D), we are
willing to give up very little
clothing to get a little more
food.
Odd special cases that are not consistent
with the characteristics listed previously.
Perfect Complements
• You need exactly 4 tires
with 1 car body (ignoring
the spare tire).
• Having more than 4 tires
with 1 car body doesn’t
increase utility.
• Also having more than 1
car body with only 4 tires
doesn’t increase utility
either.
tires
IC2
8
IC1
4
1
2
Car bodies
Perfect Substitutes
Mini-packs
10
5
IC2
IC1
1
2
Jumbo
packs
•Consider two packs of paper; the
mini-pack has 100 sheets & the
jumbo pack has 500 sheets.
•No matter how many mini-packs
or jumbo packs you have, you are
always willing to trade 5 mini-packs
for 1 jumbo pack.
•Since the rate at which you’re
willing to trade is the slope of the
IC, and that rate is constant, your
IC’s have a constant slope.
•That means they are straight lines.
Goods versus Bads
production
•As you get more of a
bad, you need more of
a good to compensate
you, to keep you
feeling equally happy.
•So IC of a good & a
bad slopes upward.
IC2
IC1
pollution
“Neutral” Good
Neutral good
IC1
IC2
IC3
•Your utility is unaffected
by consumption of a
neutral good.
Desired
good
Addict
Substance 1
Substance 2
•The more substance 1 the
addict has the more
he/she is willing to give up
of substance 2 to get a
little more of 1 (& vice
versa).
•So the IC’s are concave
instead of convex.
The slope of the indifference curve is
the rate at which you are willing to
trade off one good to get another
good.
•It is called the marginal rate of substitution
or MRS.
What is the MRS or slope of the IC?
Clothing C
IC2
IC1
A
B
D
Food F
•Suppose points A & B are on the same
indifference curve & therefore have the
same utility level.
•Let’s break up the move from A to B into 2
parts.
•AD: TU = C (MUC)
•DB: TU = F (MUF)
•AB:
0 = TU = C (MUC) + F (MUF)
•  C (MUC) = – F (MUF)
•  C/F = – MUF / MUC
So along an indifference curve, the slope or MRS is the
negative of the ratio of the marginal utilities (with the MU of
the good on the horizontal axis in the numerator).
MRS = – MUX / MUY
For example,
Clothing C
IC1 =90
IC2 = 96
A
6
B
5
D
7
9
Food F
•Suppose IC1 is the 90-util indifference
curve & IC2 is the 96-util indifference
curve.
•Point A is 7 units of food & 6 of clothing.
•B is 9 units of food & 5 of clothing.
•Since an additional unit of clothing
gives you 6 more utils of satisfaction,
the MU of clothing must be 6.
•Since an additional 2 units of food also
give you 6 more utils of satisfaction, the
MU of food must be 3.
•So, MRS = – MUF / MUC = -3/6 = -0.5 .
•You’d give up 2 units of food to get
1 units of clothing.
Budget Constraint or Budget Line
• This equation tells you what you can buy.
• For example, suppose you have $24, & there are two
goods.
• The price of the first good is $3 per unit & the price of
the second good is $4 per unit.
• So, if you buy X units of the first good for $3 each, you
spend 3X on that good.
• Similarly, if you buy Y units of the second good, you
spend 4Y on that good.
• Your total spending is 3X+4Y.
• If you spend all 24 dollars that you have, 3X+4Y=24.
• That equation is your budget constraint.
Example: Budget constraint for $24 of income,
and $3 & $4 for the prices of the two goods.
Y
(0,6)
0
If you spent all $24 on the 1st good,
you could buy 8 units.
If you spent all $24 on the 2nd good,
you could buy 6 units.
So we have the intercepts of the
budget constraint.
The slope of the line connecting these
two points is
Y/X = – 6/8 = – 3/4 = – 0.75 .
(8,0)
X
Let’s generalize. Keep in mind that income was $24
and the prices of the goods were $3 & $4. The equation of the
budget constraint in our example was 3X + 4Y = 24.
Y
(0,6)
0
So the budget constraint is p1X + p2Y = I
Solving for Y in terms of X, p2Y = I – p1X,
or Y = I /p2 – (p1/p2)X
So from our slope-intercept form, we see that
the intercept is I /p2, and the slope is –p1/p2 .
The intercept is income divided by the price of
the good on the vertical axis.
The slope is the negative of the ratio of
the prices, with the price of the good
on the horizontal axis in the
numerator.
(8,0)
X
We have the intercept is I /p2,
& the slope is –p1/p2 .
Y
(0,9)
(0,6)
0
•What if income increased?
•The slope would stay the same & the budget constraint
would shift out parallel to the original one.
•Suppose in our example with income of 24 & prices of
3 & 4, income increased to 36.
•
Our new y-intercept will be 36/4 =9
•
& the new X-intercept will be 36/3=12.
(8,0)
(12,0)
X
Suppose the price of the good on the X-axis increased.
Y
(0,6)
0
•If we bought only the good whose price
increased, we could afford less of it.
•If we bought only the other good, our purchases
would be unchanged.
•So the budget constraint would pivot inward
about the Y-intercept.
For example, if the price increased
from $3 to $4, our $24 would only
buy 6 units.
(6,0)
(8,0)
X
Similarly, if the price of the good on the Y-axis
increased, the budget constraint would pivot in
about the X-intercept.
Y
(0,6)
Suppose the price of the 2nd good
increased from $4 to $6. If you bought
only that good, with your $24, your $24
would only buy 4 units of it.
(0,4)
0
(8,0)
X
Let’s combine our indifference curves & budget
constraint to determine our utility maximizing
point.
IC3
Y
IC1
IC2
•Point A doesn’t maximize
our utility & it doesn’t
spend all our income. (It’s
below the budget
constraint.)
A
0
X
IC3
Y
IC1
IC2
•Points B & C spend all our income
but they don’t maximize our utility.
We can reach a higher indifference
curve.
B
C
0
X
•Point D is unattainable. We
can’t reach it with our budget.
IC3
Y
IC1
IC2
D
0
X
•Point E is our utility-maximizing point.
•We can’t do any better than at E.
•Notice that our utility is maximized at the
point of tangency between the budget
constraint & the indifference curve.
IC3
Y
IC1
IC2
E
0
X
Recall from Principles of Microeconomics, to maximize your
utility, you should purchase goods so that the marginal utility
per dollar is the same for all goods.
• If there were just two goods, that means that MU1/P1 = MU2/P2
• Multiplying both sides by P1/MU2, we have MU1/MU2 = P1/P2 .
• The expression on the right is the negative of the slope of the budget
constraint.
• The expression on the left is the negative of the slope of the
indifference curve.
• So the slope of the indifference curve must be equal to the slope of
the budget constraint.
• If at a particular point, two functions have the same slope, they are
tangent to each other.
• That means your utility-maximizing consumption levels are where
your indifference curve is tangent to the budget constraint.
• This is the same conclusion we reached using our graph.
Example: If TU = 10X + 24Y – 0.5 X2 – 0.5 Y2, the prices of the
two goods are 2 and 6, and we have $44, how much should you
consume of each good?
• Taking the derivatives of TU we have
MU1 = 10 – X and MU2 = 24 – Y
• Since MU1/MU2 = P1/P2 , we have
•
•
•
•
•
(10 – X) / (24 – Y) = 2 / 6 ,
or 60 – 6 X = 48 – 2Y ,
or 6X – 2Y = 12 .
This an equation with two unknowns.
Our budget constraint provides us with a 2nd equation.
Combining the two equations, we can solve for X & Y.
The budget constraint is 2X + 6Y = 44 .
So our two equations are
6X – 2Y = 12 and 2X + 6Y = 44
• Multiplying the second equation by 3 yields
6X + 18Y = 132 .
• Now we have 6X + 18Y = 132
6X – 2Y = 12
-------------------• So,
20Y = 120
• and
Y= 6.
• Plugging 6 in for Y in the 1st equation yields 6X – 12 = 12,
• or
6X = 24.
• So,
X= 4.
Let’s see if all this works.
•We had $44, the prices were 2 & 6, and
MU1 = 10 – X & MU2 = 24 – Y. We bought 4 units of the
1st good & 6 of the 2nd good.
•First, did we spend exactly what we had?
•We spent (2)(4) + (6)(6) = 8 + 36 = 44 Good.
•Is the marginal utility per dollar the same for both
goods?
•For the 1st good: MU1/P1 = (10 – X)/2 = (10-4)/2 = 3
•For the 2nd good: MU2/P2 = (24 – Y)/6 = (24-6)/6 = 3
•So they’re equal and things look fine.
What happens to consumption when income
rises?
• For normal goods, consumption increases.
• For inferior goods, consumption decreases.
• What does this look like on our graph?
Two Normal Goods
Y
IC3
IC2
IC1
C
Y3
•As income increases, the
budget constraint shifts out &
we are able to reach higher &
higher IC’s.
•The points of tangency are at
higher & higher levels of
consumption of both goods.
B
Y2
Y1
A
X1 X2 X3
X
Income-Consumption Curve
Y
IC3
IC2
IC1
C
Y3
•The curve that traces out
these points is called the
income-consumption curve.
•For two normal goods, the
curve slopes upward.
•It may be convex (as drawn
here), concave, or linear.
B
Y2
Y1
A
X1 X2 X3
X
One Normal Good & One Inferior
Good
Y
IC1
IC2
IC3
Y1
Y2
Y3
A
•Suppose the good on the
horizontal axis is normal &
the one on the vertical axis is
inferior.
•Then X will rise & Y will fall
as income increases.
B
C
X1
X2
X3
X
Income-Consumption Curve
•The result is a downward sloping
income-consumption curve.
Y
IC1
IC2
IC3
Y1
Y2
Y3
A
B
C
X1
X2
X3
X
Engel Curve
Income
C
I3
B
I2
I1
A
X1 X2 X3
X
•The Engel Curve shows
the quantity of a good
purchased at each income
level.
•The graph has income on
the vertical axis and the
quantity of the good on
the horizontal.
•It slopes up for normal
goods & down for inferior
goods.
We can also look at consumption levels of two
goods when the price of one of them changes.
Y
•Suppose there is an increase in the price of
the 1st good (the good on the X-axis).
•The budget constraint pivots inward.
•
Here we see X drop & Y increase.
•
In this case, our 2 goods are
substitutes.
Y3
Y2
Y1
X3
X2
X1
X
If we connect the points, we have the
price consumption curve.
•It shows the utility-maximizing
points when the price of a good
changes.
Y
Y3
Y2
Y1
X3
X2
X1
X
If we look at the price of a good & the amount of it
consumed, we have the demand curve for our
particular individual.
P
•As the price decreases the
quantity demanded increases &
vice versa.
P1
P2
P3
X1 X2 X3
X
We can separate the effect of a change in the price of a good on
its consumption level into two parts:
the income effect & the substitution effect.
Y
YB
YA
B
A
XB
XA
•Suppose the price of the first
good increases.
•The budget constraint was
originally the blue line and we
were at A consuming
quantities XA & YA.
•After the price change, the
budget constraint is the red
line, and we’re at B
consuming XB & YB .
X
We first want to capture the effect of the price change without
the effect of the change in income.
Y
H
YH
YB
YA
B
A
XB XH
XA
We draw a line parallel to the new budget
constraint and tangent to the old indifference
curve.
This will reflect the new relative prices, but since
we are tangent to the old indifference curve we
are just as well off as initially.
Under those circumstances we would be at point
H (for hypothetical).
Since the 1st good is now relatively more
expensive compared to the 2nd, we will
substitute, increasing Y & decreasing X.
X
The movement from A to H is the
substitution effect.
As a result of the increase in the relative
price of the 1st good, we reduce our
consumption of it and consume more of
the other good.
Y
H
YH
YB
YA
B
A
XB XH
XA
X
Now we move from H to B
Y
H
YH
YB
YA
B
A
XB XH
XA
Our purchasing power has been reduced by
the price change. That results in the income
effect.
In our graph, we now hold the relative prices
constant at the new level, but income has
fallen. Our budget constraint has shifted
inward.
If both goods are normal, as a result of the
change in income, we reduce our consumption
of both goods, and X & Y fall.
This is the income effect of the price change.
X
Total Effect of Price Increase
The total effect is to move from A to B.
X has fallen.
Both the substitution & income effects led to a
drop in X.
Y has increased in this case.
The substitution effect increased consumption
of the 2nd good, but the income effect reduced
it by less than the substitution effect increased
it.
Y
H
YH
YB
YA
B
A
XB XH
XA
X
Let’s do a price decrease.
The budget constraint moves from the blue line
to the red line.
We draw a line parallel to the new budget
constraint and tangent to the old indifference
curve.
H is the tangency of the hypothetical budget
constraint with the old indifference curve.
The substitution effect is the movement from A
to H.
We substitute increasing X & decreasing Y.
Y
B
YB
YA
A
H
YH
XA XH XB
X
The movement from H to B is the income effect.
As a result of the higher income (greater
purchasing power), we consume more of both
goods, if they are normal goods.
Y
B
YB
YA
A
H
YH
XA XH XB
X
Total Effect
The total effect is to move from A to B.
X has increased.
Both the substitution & income effects led to an
increase in X.
Y has also increased in this case.
The substitution effect decreased consumption
of the 2nd good, but the income effect
increased it by more than the substitution
effect decreased it.
Y
B
YB
YA
A
H
YH
XA XH XB
X
Income and Substitution Effects, in words
The income effect is the result of the change in purchasing
power.
If the price of a normal good increases, you feel poorer,
and the income effect is to consume less.
If the price of a normal good decreases, you feel richer, and
the income effect is to consume more.
The substitution effect is the result of a change in relative
prices.
If the price of a good increases, the substitution effect is to
consume less of it & more of the other goods that are
now relatively cheaper.
If the price decreases, the substitution effect is to consume
more of it & less of the goods that are now relatively
more expensive.
What if the price changed of an inferior good?
The substitution effect would be the same
but the income effect would be the opposite.
Price increase for an inferior good
Income effect:
Your purchasing power has decreased. You feel poorer.
So you consume more of the inferior good.
Substitution effect:
The good is now relatively more expensive than other goods,
so you consume less of it and more of other goods.
Notice the IE & SE are in opposite directions in this case.
If the SE is larger than the IE, you will consume less of the
good.
If the IE is larger than the SE, you will consume more of the
good.
An inferior good for which the IE is larger than
the SE is called a Giffen good.
•It is a good for which consumption rises when
the price increases, and consumption falls when
the price decreases.
Price decrease for an inferior good
Income effect:
Your purchasing power has increased. You feel richer. So
you consume less of the inferior good.
Substitution effect:
The good is now relatively more cheaper than other goods,
so you consume more of it and less of other goods.
Again the IE & SE are in opposite directions in this case.
If the SE is larger than the IE, you will consume more of the
good.
If the IE is larger than the SE, you will consume less of the
good.
We previously looked at the demand curve for
individuals.
How do we get the market demand curve
from the demand curve for individuals?
We just horizontally sum up the individual
demand curves.
Market Demand Curve: 3-person example
At a price of $1, person A will buy 4 units of a good, B will buy 2
units, & C will buy 3 units. So at a price of $1, the quantity
demanded by the entire 3-person market is 9 units.
P
P
P
P
2
2
2
2
1
1
1
1
2
4
Person A
Q
1 2
Q
Person B
1
3
Person C
Q
4
Market
9
Q
Market Demand Curve: 3-person example
At a price of $2, person A will buy 2 units of a good, B will buy 1
units, & C will buy 1 units. So at a price of $2, the quantity
demanded by the entire 3-person market is 4 units.
P
P
P
P
2
2
2
2
1
1
1
1
2
4
Person A
Q
1 2
Q
Person B
1
3
Person C
Q
4
Market
9
Q
Market Demand Curve: 3-person example
Continuing the process, we get the market demand curve.
P
P
P
P
2
2
2
2
1
1
1
1
2
4
Person A
Q
1 2
Q
Person B
1
3
Person C
Q
4
Market
9
Q
The Demand for a product can be expressed as a
function of
1.
2.
its price
(changes in which lead to movements along the
demand curve), and
other determinants such as income, prices of related
goods, & expectations (changes in which lead to shifts
of the demand curve).
So we have QDX = g(PX, Psubst, Pcomp, Inc., Expect.)
A particular demand curve QDX = g(PX) shows the
relation between the quantity demanded of a product
and its price when we hold all the factors constant.
This is also sometimes written as P= f(Q).
Total Revenue
• TR = PQ
Average Revenue
• total revenue per unit of output
• AR = TR / Q
•
= (PQ) / Q
• =P
• AR & P are the same function of Q.
Marginal Revenue
• The additional revenue associated with an
additional unit of output
• MR = dTR / dQ
Example: Horizontal Demand Curve
(price is a constant function)
P
P = f(Q) = 10
AR = P = 10
10
D = AR =MR
P
Q
TR
slope = MR = 10
Q
TR = PQ = 10 Q
MR = dTR / dQ = 10
So D, AR, & MR are the same
horizontal function.
TR is an upward sloping line with
a constant slope.
Implications for revenue:
Every time you sell another unit
of output, revenue increases by
the price, which is constant.
Example:
linear, downward-sloping Demand curve
P
8
D = AR
MR
Q
P
TR
Q
P = f(Q) = 8 – 3Q
AR = P = 8 – 3Q
TR = PQ = (8 – 3Q) Q
= 8Q – 3Q2
MR = dTR / dQ = 8 – 6Q
D & MR have the same vertical
intercept. MR is twice as steep
as D. (The slope of MR is -6;
the slope of D is -3)
Implications for revenue:
Revenue increases more &
more slowly & then decreases
more & more quickly.
Example:
quadratic, downward-sloping Demand curve
P
20
D = AR
MR
Q
P
TR
Q
P = f(Q) = 20 – Q2
AR = P = 20 – Q2
TR = PQ = (20 – Q2) Q
= 20Q – Q3
MR = dTR / dQ = 20 – 3Q2
MR is a quadratic; TR is a cubic.
Elasticity
•Responsiveness or sensitivity of one variable to
a change in another variable
•
(% change in X)
ε = -----------------------(% change in Y)
Price Elasticity of Demand
•
(% change in quantity demanded)
ε = -----------------------------------------------(% change in price)
Two methods of calculating elasticity
• Arc elasticity: measures responsiveness
between 2 points
• Point elasticity: measures responsiveness at a
single point for an infinitesimally small change
Arc Elasticity
• ΔQ/(avg Q)
[Q2 – Q1] / [(Q1+Q2)/2]
• --------------- = -------------------------------• ΔP/(avg P)
[P2 – P1] / [(P1+P2)/2]
Arc Elasticity Example
• Calculate the price elasticity of demand if in response to a increase in
price from 9 to 11 dollars, the quantity demanded decreases from
60 to 40 units.
• P: 9→11 Q: 60→40
•
•
ΔQ/(avg Q)
[Q2 – Q1] / [(Q1+Q2)/2]
ε = --------------- = -------------------------------ΔP/(avg P)
[P2 – P1] / [(P1+P2)/2]
[40 – 60] / [(60+40)/2]
-20 / 50
-0.4
= ------------------------------ = ------------ = ------- = -2.0
[11 – 9] / [(9+11)/2]
2 / 10
0.2
• The negative sign indicates that price & quantity move in opposite
directions.
• The negative sign is sometimes dropped with the understanding that price
& quantity are still moving in opposite directions.
Point Elasticity
•
dQ / Q
dQ P
ε = ---------- = ----- ---dP / P
dP Q
Point Elasticity Example
• If the demand function is Q = 245 – 3.5 P, find the price
elasticity of demand when the price is 10.
• When P = 10, Q = 245 – 3.5(10) = 210
• The derivative dQ / dP = = – 3.5
•
dQ / Q
dQ P
10
ε = ---------- = ----- --- = - 3.5 ------ = - 0.167
dP / P
dP Q
210
Categories of
Price Elasticity of Demand
• Demand is elastic if
• Demand is inelastic if
• Demand is unit elastic if
|ε| > 1
|ε| < 1
|ε| = 1
What is the relationship between elasticity &
the slope of the demand curve?
•
•
dQ / Q
dQ P
1
P
ε = ---------- = ----- ---- = -------- ---dP / P
dP Q
dP/dQ Q
= (1/slope) (P/Q)
• So, if 2 demand curves pass through the same point
(& therefore have the same values of P & Q at that
point), the flatter curve (curve with the smaller slope)
has the greater elasticity at that point.
Example
P
E
D1
D2
Q
At point E, D1 has greater elasticity than D2 .
Elasticity on a Linear Demand
Curve
P
Above the midpoint, |ε| > 1 (elastic)
At the midpoint, |ε| = 1 (unit elasticity)
Below the midpoint, |ε| < 1 (inelastic)
Q
Let’s show in an example that |ε| = 1
at the midpoint of a linear demand curve.
demand curve: P = 24 – 4Q
When Q = 0, P = 24. So that’s our vertical intercept.
When P = 0, Q = 6. That’s our horizontal intercept.
The midpoint then is (3,12).
The slope is dP/dQ = – 4 .
We found earlier that ε = (1/slope) (P/Q)
P
24
12
|ε| = 1
MR
D
3
6 Q
So ε = (1/-4) (12/3)
= (1/-4) (4)
= -1
Relationship between Elasticity
& Total Revenue
•
•
•
•
Price increase:
|ε| > 1: P   Q   TR
|ε| < 1: P   Q   TR 
|ε| = 1: P   Q   TR unchanged
Relationship between Elasticity
& Total Revenue
•
•
•
•
Price decrease:
|ε| > 1: P   Q   TR 
|ε| < 1: P   Q   TR 
|ε| = 1: P   Q   TR unchanged
The most profitable place to be is in the elastic
portion of the demand curve.
• In the inelastic portion of the demand curve,
marginal revenue is negative (additional units
of output lower total revenue).
• While this is true in general, we can
demonstrate it in the linear demand case.
Recall that for a linear demand curve,
marginal revenue is twice as steep as the demand curve.
P
So MR reaches the horizontal axis
when the demand curve is only halfway
there.
|ε| > 1
|ε| = 1
MR
|ε| < 1
So when MR = 0 at the midpoint, |ε| = 1.
D
Q
P
From the graph, we can see that above
the midpoint where |ε| > 1, MR > 0
& TR is increasing.
Below the midpoint, where |ε| < 1 ,
MR < 0 & TR is decreasing.
TR
Q
Special Elasticity Cases
|ε| = 
Infinite Elasticity
Perfectly Elastic
P
D
Q
|ε| = 0
Zero Elasticity
Perfectly Inelastic
P
D
Q
|ε| = 1
Unit Elasticity
P
D
P=k/Q
where k is a constant
Q
Facts about Price Elasticity of
Demand
1. The more substitutes there are for a product, the
more elastic the demand.
2. An individual firm’s product has a more elastic
demand than the entire industry’s product.
3. The longer the time period, the greater the
elasticity of demand, because the greater the
adjustments that are possible.
4. Products (like salt) that are a small part of the
budget have low elasticities of demand.
Some examples of
estimated price elasticities of demand
Commodity
Price Elasticity of Demand
wheat
0.08
cotton
0.12
potatoes
0.31
beef
0.92
haddock
2.20
movies
3.70
Notice that the demands for wheat & cotton are not very responsive to price
changes, whereas the demands for haddock and movies are very responsive.
So far the only elasticity that we
have discussed is price elasticity of
demand.
• There are other types of elasticities.
• Each type can be computed as
arc elasticity or point elasticity.
Income Elasticity of Demand
•
(% change in quantity demanded)
εI = -----------------------------------------------(% change in income)
Categories of Income Elasticity of Demand
•Normal Goods:
•εI > 1
income elastic
•εI = 1
unit income elastic
•0 < εI < 1 income inelastic
•Luxury items have high income elasticities of demand,
while necessities have low income elasticities of demand.
•Inferior Goods:
•εI < 0
Some examples of
estimated income elasticities of demand
Commodity
Income Elasticity of Demand
flour
-0.36
margarine
-0.20
milk
0.07
butter
0.42
books
1.44
restaurant
consumption
1.48
Note that flour & margarine are inferior goods, milk is not very responsive
to income changes, & books & restaurant consumption are income elastic.
Cross Elasticity of Demand
•
εYX
(% change in quantity demanded of Y)
= --------------------------------------------------(% change in price of X)
Categories of Cross Elasticity of Demand
•Substitutes:
εYX > 0
•The price of X & the quantity demanded of Y
move in the same direction. When the price of X
increases, you consume less of X and more of the
goods that you can use in place of X.
•Complements: εYX < 0
•The price of X & the quantity demanded of Y
move in opposite directions. When the price of X
increases, you consume less of X and less of the
goods that you use with X.
Some examples of
estimated cross elasticities of demand
Commodity
Cross Elasticity with
respect to price of
Cross elasticity
Pork
Beef
+0.14
Beef
Pork
+0.28
Butter
Margarine
+0.67
Margarine
Butter
+0.81
Notice that the cross elasticity of demand for Y with respect to
the price of X is not necessarily equal to the cross elasticity of
demand for X with respect to the price of Y.
Price Elasticity of Supply
•
(% change in quantity supplied)
εS = -----------------------------------------------(% change in price)
Categories of
Price Elasticity of Supply
• Supply is elastic if
• Supply is inelastic if
• Supply is unit elastic if
εS > 1
εS < 1
εS = 1