Chapter Fifteen

1
Chapter Fifteen

• Market Demand

2
From Individual to Market Demand Functions

• Think of an economy containing 2 consumers
  denoted A and B.
• Consumer is ordinary demand function for
  commodity 1 is

3
From Individual to Market Demand Functions

• The market demand function for commodity 1 is the
  sum of the demands of all individual consumers
• If both consumers are identical (same preferences
  and income) where total income M 2m.

4
From Individual to Market Demand Functions

• The market demand curve is the horizontal sum
  of the individual consumers demand curves.
• This is because, for every price level, we add
  the quantity demanded for each consumer to get
  the market demand.
• For example, consider the following two demand
  curves for consumers A and B.

5
From Individual to Market Demand Functions
p1
p1
p1
p1
p1
p1
20
15
As demand
Bs demand
Derive the Market Demand Curve.
6
From Individual to Market Demand Functions
p1
p1
p1
p1
p1
p1
20
15
p1
If p1gtp1 market demand is zero. Neither A nor B
buy the good.
p1
7
From Individual to Market Demand Functions
p1
p1
p1
p1
p1
p1
20
15
p1
If p1 lt p1lt p1 only B consumes the good and
market demand is the same as Bs demand.
p1
p1
8
From Individual to Market Demand Functions
p1
p1
p1
p1
p1
p1
20
15
p1
p1lt p1 both A and B consume the good and market
demand is the sum of the two consumers demands.
p1
p1
35
9
From Individual to Market Demand Functions
p1
p1
p1
p1
p1
p1
20
15
p1
Market Demand is the horizontal sumof the
demand curvesof individuals A and B.
p1
p1
35
10
Example 15.1

• Suppose there are only two consumers of cokes.
• Bettys demand for cokes is given by D(p)1-p.
• Ethels demand for cokes is given by D(p)2-p.
• Draw the individual demands and derive and draw
  the market demand.

11
Elasticities

• Elasticity measures the sensitivity of one
  variable with respect to another.
• The elasticity of variable X with respect to
  variable Y is

12
Arc and Point Elasticities

• An average own-price elasticity of demand for
  commodity x over an interval of values for px is
  an arc-elasticity, usually computed by a
  mid-point formula.
• In this class, we will compute elasticity for a
  single value of px or a point elasticity.

13
Economic Applications of Elasticity

• Economists use elasticities to measure the
  sensitivity of quantity demanded to changes in
  various variables.
• own-price elasticity of demand is the percentage
  change in quantity demanded of commodity x with
  respect to a percentage change in the price of
  commodity x

14
Economic Applications of Elasticity

• cross-price elasticity of demand The percentage
  change in the demand for commodity x with respect
  to a percentage change in the price of commodity
  y.

15
Economic Applications of Elasticity

• income elasticity of demand the percentage change
  in the demand for x with respect to a percentage
  change in income

16
Example 15.2

• We can calculate this elasticity from the demand
  function.
• Suppose the demand for good 1 is given by the
  following x1(p1,p2,m)p2 - 2p1m / 100
• Calculate the own price elasticity, cross price
  elasticity, and income elasticity.
• Is x1 ordinary or giffen?
• Are x1 and x2 gross substitutes or complements?
• Is x1 normal or inferior?

17
Elastic, Inelastic, Unit Elastic, Perfectly
Elastic, Perfectly Inelastic

• Elastic E gt1
• Inelastic E lt1
• Unit elastic E 1
• Perfectly elastic E infinity
• Perfectly inelastic E 0

18
Elastic, Inelastic, Unit Elastic, Perfectly
Elastic, Perfectly Inelastic

• More inelastic demands/supplies will be steeper.
• Perfectly inelastic demands/supplies will be
  vertical lines.
• Perfectly elastic demands/supplies are horizontal
  lines

19
Properties of Linear Demand

• Suppose the demand for good 1 is given by the
  following
• x a bPx
• Inverse demand is Px a/b - x/b
• The price elasticity of x is then

Tip Always plug in demand function for x here.
20
Properties of Linear Demand
px a/b x/b
px
a/b
Xi
a
Elasticity changes along the demand curve.
21
Properties of Linear Demand
Px a/b x/b
pi
a/b
Xi
a
22
Properties of Linear Demand
pi
a/b
Xi
a
23
Properties of Linear Demand
pi
a/b
a/2b
Xi
a
a/2
Price elasticity is exactly -1 at midpoint of
the linear demand curve.
24
Properties of Linear Demand
Elasticity changes along the linear demand curve.
pi
a/b
a/2b
Xi
a/2
a
25
Properties of Linear Demand
Elasticity changes along the linear demand curve.
pi
a/b
price elastic
a/2b
price inelastic
Xi
a/2
a
26
Constant Own Price Elasticity Demand

• In general, elasticity changes along the demand
  curve.
• However there is one type of demand function
  whose elasticity does not change along the curve.

27
Constant Own Price Elasticity Demand
There is one type of demand function where
elasticity never changes. The general form for
the demand is
Then
28
Constant Own Price Elasticity Demand

• For example, suppose that the demand for good x
  is given by
• x 4px-2
• Then the price elasticity is given by
• E x,px dx/dpx px/x
• 4(-2)px-3 px/ 4px-2
• -2

29
Constant Own Price Elasticity Demand
pi
everywhere alongthe demand curve.
Xi
30
Revenue and Own-Price Elasticity of Demand

• Own price Elasticty of demand can also inform us
  as to how revenue may change when prices change.
• Revenue PQuantity Sold
• If P increases, Quantity demanded decreases.
• The change in revenues will depend on which
  effect dominates.

31
Revenue and Own-Price Elasticity of Demand

• Suppose P increases,
• If ?P gt ? quantity demanded revenues
  increase. Rearranging this implies that
• ? quantity demanded / ?P lt 1.
• In other words then if demand is inelastic, then
  a price increase will cause sellers revenue to
  increase.
• If ?P lt ? quantity demanded revenues
  decrease. Rearranging this implies that
• ? quantity demanded / ?P gt 1.
• In other words then if demand is elastic, then a
  price increase will cause sellers revenue to
  decrease.
• Revenue is maximized when demand is unit elastic.

32
Revenue and Own-Price Elasticity of Demand
Another way to get this result is to
use calculus Let q(p) denote the quantity
demanded as a function of price (p) Sellers
revenue as a function of price
33
Revenue and Own-Price Elasticity of Demand
Sellers revenue is
So using the multiplication rule, we can
calculate
34
Revenue and Own-Price Elasticity of Demand
Sellers revenue is
So
35
Revenue and Own-Price Elasticity of Demand
Sellers revenue is
So
36
Revenue and Own-Price Elasticity of Demand
Since E is always negative (for
ordinary goods-which are assumed), we see that
revenue will increase with a price increase when
demand is inelastic. ie.
37
Revenue and Own-Price Elasticity of Demand
then
So if
If demand is unit elastic, an increase in
price does not alter sellers revenue.
38
Revenue and Own-Price Elasticity of Demand
then
And if
If demand is elastic, a price increase reduces
sellers revenue.
39
Revenue and Own-Price Elasticity of Demand
In summary
Own-price inelastic demandprice rise causes
rise in sellers revenue.
Own-price unit elastic demandprice rise causes
no change in sellersrevenue.
Own-price elastic demandprice rise causes fall
in sellers revenue.
40
Revenue and Own-Price Elasticity of Demand

• Revenue is maximized when the price elasticity of
  demand -1.
• For a linear demand curve, this occurs at the
  midpoint of the demand.

41
Marginal Revenue and Own-Price Elasticity of
Demand

• A sellers marginal revenue is the rate at which
  revenue changes with the number of units sold by
  the seller.
• If we have revenue as a function of quantity sold
  (q), then we can derive MR (q)

42
Marginal Revenue and Own-Price Elasticity of
Demand
p(q) denotes the sellers inverse demand
function i.e. the price at which the seller can
sell q units. Then
so
43
Marginal Revenue and Own-Price Elasticity of
Demand
An example with linear inverse demand.
Then
and
44
Marginal Revenue and Own-Price Elasticity of
Demand
p/MR
a/b
a
q
a/2
For a linear demand, MR has the same vertical
Intercept and is twice as steep.
45
Marginal Revenue and Own-Price Elasticity of
Demand

• What is the revenue maximizing quantity and
  price?
• Revenue is maximized when MR(q)0

46
Marginal Revenue and Own-Price Elasticity of
Demand
p
a/b

a
q
a/2
R(q)
q
a
a/2
47
Marginal Revenue and Own-Price Elasticity of
Demand

• What is the revenue maximizing quantity and
  price?
• Revenue is maximized when MR(q)0

Solving for q
48
Marginal Revenue and Own-Price Elasticity of
Demand

• What is the revenue maximizing quantity and
  price?
• Revenue is maximized when MR(q)0

Solving for q
Solving for p using the inverse demand
49
Marginal Revenue and Own-Price Elasticity of
Demand
p
a/b
E-1
a/2b

a
q
a/2
R(q)
q
a
a/2
50
Example 15.3

• Suppose demand is given by D(p)(p3)-2
• Calculate the price elasticity of demand.
• At what price and quantity does this elasticity
  -1?
• Derive revenue as a function of price.
• At what price and quantity is revenue maximized?

51
Example 15.4

• Suppose the demand function is given by D200-10P
• Derive the inverse demand function.
• Derive revenue as a function of quantity.
• Calculate marginal revenue.
• At what price and quantity is revenue maximized.