Pricing
Strategies
Some pricing strategies that we will explore:
1.
2.
3.
4.
5.
6.
7.
8.
Price discrimination – 1st, 2nd, & 3rd degree
“Two-part Tariff” pricing
Bundling
Advertising
Cost-plus markup pricing
Product Lines
Peak-Load pricing
Transfer Pricing
Price Discrimination
• Charging different prices to different
consumers for the same product.
• Enables firms to charge some consumers
higher prices, and to capture consumer
surplus.
Under what conditions is
price discrimination possible?
a. Firm must have some control over price.
b. The firm can identify different submarkets.
c. The submarkets have different price
elasticities of demand.
d. The firm can prevent arbitrage
(the purchase of an item for immediate resale
in order to profit from the price discrepancy).
3 Types of Price Discrimination
First Degree
Each customer is charged the maximum price that
they are willing to pay.
Second Degree – involves self-selection
One type of 2nd degree price discrimination is block
pricing or quantity discounts in which firms charge
different prices depending on volume of usage.
Third Degree or multi-market (most common)
Markets distinguished by other factors.
Since under first degree price discrimination,
each customer is charged the maximum
price that they are willing to pay, consumer
surplus is zero.
First Degree Price Discrimination
Note: Each time the firm
sells another unit, it
increases its revenues by
the price for which it sells
that unit.
Unlike the usual situation, it
doesn’t need to lower to
price to all the other
customers in order to sell to
the additional one.
So P=MR & the Demand &
MR curves are the same,
with 1st degree price
discrimination.
$
D = MR
Q
First degree price discrimination
Example:
Suppose the demand curve for a monopolist’s product is:
P = 9 – 0.005 Q.
The average total cost curve is a horizontal line:
ATC = MC = 1.5
(1) Determine the price, quantity, consumer surplus,
producer surplus (profit), & the sum of consumer &
producer surplus, if the firm does NOT price
discriminate.
(2) Determine the quantity, consumer surplus, producer
surplus (profit), & the sum of consumer & producer
surplus, if the firm does price discriminate.
First, if the firm does NOT price discriminate:
We have the demand curve for a monopolist’s product is:
P = 9 – 0.005 Q.
& the average total cost curve is:
ATC = MC = 1.5
TR = PQ =(9 – 0.005 Q) Q = 9 Q – 0.005 Q2 .
Then MR = dTR/dQ = 9 – 0.01 Q .
Setting MR = MC, we have 9 – 0.01Q = 1.5
or 7.5 = 0.01 Q .
So Q = 750,
& P = 9 – 0.005 Q = 9 – 0.005 (750) = 5.25.
Profit or producer surplus = TR – TC = PQ – (ATC)Q
= (5.25)(750) – (1.5)(750) = 2812.5
Without Price Discrimination
Consumer Surplus
= (1/2)(750)(3.75)
= 1406.25
$
9
P*= 5.25
profit
ATC =MC =1.5
MR
D
Q
Q*= 750
Combined consumer & producer surplus is
CS + PS = 1406.25 + 2812.5 = 4218.75
Now if the firm does
1st degree price discrimination:
We had the demand curve: P = 9 – 0.005 Q,
& the average total cost curve: ATC = MC = 1.5
MR is now the same as the demand function,
so MR = 9 – 0.005 Q.
Setting MR = MC,
we have 9 – 0.005Q = 1.5
or 7.5 = 0.005 Q .
So Q = 1500.
With 1st Degree Price Discrimination
$
9
Profit
= (1/2) (1500) (7.5)
= 5625
ATC =MC =1.5
D=MR
Q*= 1500
Q
Combined consumer & producer surplus is
CS + PS = 0 + 5625 = 5625
2nd Degree Price Discrimination: Block Pricing
Price is based on volume of usage of the good.
Those who consume large quantities are charged
a lower price.
Those consuming small quantities are charged a
higher price.
Second Degree Price Discrimination Example
Suppose there are 100 high-volume consumers who value the 1st
unit of a good at $15 & a 2nd unit at $10.
There are also 100 low-volume consumers who value the 1st unit
at $12.
The total cost of production is TC = 6 Q.
(So ATC = TC/Q = 6Q/Q = 6.)
Determine the total revenue, total cost, producer surplus (profit),
consumer surplus, & sum of the producer & consumer surplus
for the following four options:
1. No price discrimination – one unit sells for $15.
2. No price discrimination – one unit sells for $12.
3. No price discrimination – one unit sells for $10.
4. Offer two sizes of packages, 1 unit for $12 & 2 units for $20.
High-volume consumers
value the 1st unit of a good at
$15 & the 2nd unit at $10.
$
Low-volume consumers
value the 1st unit at $12.
$
15
12
10
6
6
ATC
1
2
Q
ATC
1
Q
Suppose firm sells all units individually for $15.
The 100 low-volume
The 100 high-volume
consumers will buy 0 units.
consumers will buy 1unit.
$
$
15
12
10
profit
6
6
ATC
1
2
Q
ATC
1
TR = PQ = 15(100) = 1500,
TC = 6 Q = 6 (100) = 600, &
Producer Surplus or  = TR – TC = 1500 – 600 = 900.
Consumer surplus = 0(100) = 0.
PS + CS = 900 + 0 = 900
Q
Suppose firm sells all units individually for $12.
The 100 low-volume
The 100 high-volume
consumers will buy 1 unit.
consumers will buy 1unit.
$
$
15
12
10
CS
12
profit
profit
6
6
ATC
1
2
Q
ATC
1
TR = 12(200) = 2400,
TC = 6 Q = 6(200) = 1200, &
Producer Surplus or  = TR – TC = 2400 – 1200 = 1200.
Consumer Surplus = 3(100) + 0(100) = 300
PS + CS = 1200 + 300 = 1500
Q
Suppose firm sells all units individually for $10.
The 100 low-volume
The 100 high-volume
consumers will buy 1 units.
consumers will buy 2 units.
$
$
15
12
10
CS
10
profit
CS
profit
profit
6
6
ATC
ATC
1
2
Q
1
TR = PQ = 10(300) = 3000,
TC = 6 Q = 6 (300) = 1800, &
Producer Surplus or  = TR – TC = 3000 – 1800 = 1200.
Consumer surplus = 5(100) + 2(100) = 700.
PS + CS = 1200 + 700 = 1900
Q
Suppose firm sells 1-unit packs for $12 & 2-unit packs for $20.
The low-volume consumers
The high-volume consumers
will buy a 1-unit pack.
will buy a 2-unit pack.
$
$
15
12
CS
10
profit
profit
ATC
6
1
2
Q
6
ATC
1
TR = PQ = 12(100) + 20(100)= 3200,
TC = 6 Q = 6 (300) = 1800, &
Producer Surplus or  = TR – TC = 3200 – 1800 = 1400.
Consumer surplus = 5(100) + 0(100) = 500.
PS + CS = 1400 + 500 = 1900
Q
In our 2nd degree price discrimination case, the firm offered two
sizes of packages, 1 unit for $12 & 2 units for $20.
The 100 high-volume consumers value the 1st unit of a good at
$15 & the 2nd unit at $10.
However, notice that if the firm tried to charge $25 for the 2-pack,
the high-volume consumers would only buy a 1-pack. This is
because they would be better off with consumer surplus of
$15 – $12 = 3 with a 1-pack than consumer surplus of
$25 – $25 = 0 with a 2-pack.
The profit with 2nd order price discrimination is more than the
profit for the one-price options. PS+CS is the same as for the
unit price of $10, but the producer has captured the $200 lowvolume CS as PS or profit.
Third Degree Price Discrimination
Charging different prices to different groups.
Example: Charging lower movie admissions to
students & senior citizens than to other moviegoers.
Third Degree Price Discrimination
For each group, the firm produces such that
MR = MC .
The group with the lowest elasticity pays the
highest price.
Example: Students & senior citizens may
have more limited incomes, and therefore be
more responsive to changes in movie prices.
Other movie-goers may be less responsive to
changes in movie prices.
Suppose the demand functions for two groups of consumers are
D1: P = 101 – 13Q and D2: P = 53 – 7 Q.
Notice that D1 is steeper and so less elastic than D2 .
(So group 1 will pay a higher price than group 2.)
The total cost function is TC = 90 + 128Q – 22Q2 + Q3 .
If the firm is able to price discriminate between the two groups,
determine the prices that should be charged, the quantities that
will be purchased, total revenue, total cost, and profit.
We need to equate the two MR functions to the MC function.
MC = dTC/dQ = 128 – 44Q + 3Q2.
Group 1: TR1 = PQ = (101 – 13Q)Q = 101Q – 13Q2 , and
MR1 = dTR1/dQ = 101 – 26 Q
Group 2: TR2 = PQ = (53 – 7Q)Q = 53Q – 7Q2 , and
MR2 = dTR2/dQ = 53 – 14 Q
Our Group 1 Demand function was P = 101 – 13 Q, and
the MR function was MR1 = 101 – 26 Q.
The MC function was MC = 128 – 44Q + 3Q2.
Set MR1 = MC: 101 – 26 Q = 128 – 44Q + 3Q2
0 = 3Q2 – 18Q + 27
Dividing by 3 to simplify: 0 = Q2 – 6Q + 9
0 = (Q – 3) (Q – 3)
Q–3=0
So for Group 1, Q = 3
From Group 1’s demand function, P1 = 101 – 13 (3) = 62.
The revenue from Group 1 will be PQ = (62)(3) = 186.
Our Group 2 Demand function was P = 53 – 7 Q, and
the MR function was MR2 = 53 – 14 Q.
The MC function was MC = 128 – 44Q + 3Q2.
Set MR2 = MC: 53 – 14 Q = 128 – 44Q + 3Q2
0 = 3Q2 – 30Q + 75
Dividing by 3 to simplify: 0 = Q2 – 10Q + 25
0 = (Q – 5) (Q – 5)
Q–5=0
So for Group 2, Q = 5
From Group 2’s demand function, P2 = 53 – 7 (5) = 18.
The revenue from Group 2 will be PQ = (18)(5) = 90
Adding the revenues from the two groups together, we
get TR = 186 + 90 = 276.
Since we produced 3 units for Group 1 and 5 for
Group 2, our production level is 8.
Plugging 8 into our total cost function,
TC = 90 + 128Q – 22Q2 + Q3 = 218.
So our profit is  = TR – TC = 276 – 218 = 58.
The “Two-Part Tariff”
There are two components to the price: a unit price
(P) for each unit consumed, & a “tariff” (T) for
entry into the market.
Examples include BJ’s, telephone service, health
clubs, etc.
The tariff enables the firm to capture some
consumer surplus.
Suppose that a firm has constant average and marginal
costs as shown.
Also, each customer has the indicated
demand curve.
P
Suppose that the firm charges price P* per
unit.
Based on the per unit charge, the firm earns
revenues equal to the area of the blue box.
P*
ATC=MC
D
Q
Q*
The firm can also pick up the consumer surplus,
P
if it charges a membership fee
equal to the area of the green
triangle.
P*
ATC=MC
D
Q
Q*
The firm’s total revenue (from each customer) is the
combined areas of the blue box and the green triangle.
Recall that ATC = TC/Q.
P
So, TC = ATC•Q.
So, the total cost (from each
customer) is the purple box.
P*
ATC=MC
D
Q
Q*
The firm’s profit
(per customer) is
TR - TC which is
the orange figure.
The firm’s profit from this two-part tariff strategy
will be greatest if it produces where the Demand
and ATC = MC curves intersect, or P = ATC = MC.
P
P*
ATC=MC
D
Q
Q*
Two-Part Tariff Example
Suppose a firm’s TC function is TC = 5Q.
Suppose also that each of the firm’s customers
has this demand curve: P = 35 – Q .
Determine the appropriate unit price and
membership fee for a two-part tariff pricing
strategy.
Also determine the quantity purchased, total
revenue, total cost, and profit per customer.
Demand function (for each person): P = 35 – Q
Total cost function: TC = 5Q
As we indicated previously, the firm’s profit will be
greatest if it produces where P = ATC = MC.
ATC = MC = 5
So, P = 35 – Q = 5 ,
So, 30 = Q.
So revenue per person from per unit sales is
PQ = (5)(30) = 150 .
Next we need to determine the appropriate
membership fee.
The membership fee is the consumer surplus.
P
35
That is the area of the orange triangle,
which is (1/2)(30)(30) = 450.
So the membership fee should be $450.
5
ATC=MC
D
30
Q
Combining the membership fee of $450 with the
per unit sales revenues of $150 that we found
earlier, we have total revenues per customer of
$450 + $150 = $600.
From the total cost function, the total production
cost for the 30 units per customer is
TC = 5Q = 5(30) = 150.
So our profit per customer is
 = TR – TC = 600 – 150 = $450.
Bundling
Bundling is packaging two or more products to
gain a pricing advantage.
Conditions necessary for bundling to be the
appropriate pricing alternative:
Customers are heterogeneous.
Price discrimination is not possible.
Demands for the two products are negatively
correlated.
Consider the following reservations prices,
for two buyers: Alan and Beth
Stereo
TV
Sum of
reservation
prices
Alan
$225
$375
$600
Beth
$325
$275
$600
Maximum price for
both to buy the good
$225
$275
To get both people to buy both goods without bundling, you
can only charge $225 + $275 = $500, & each person would
have consumer surplus of $600 – $500 = $100.
If you bundle, you can charge $600 & consumer surplus = 0.
The effectiveness of bundling as a
pricing strategy depends upon the
degree of negative correlation between
the demands for the two goods.
Advertising:
How does a firm determine
the profit-maximizing advertising level?
The ratio of the firm’s advertising to its sales
revenue should equal the negative of the ratio
of the advertising & price elasticities of
demand.
That is, A/(P*Q) = - A / D
So you should advertise a lot if the elasticity of
demand
(1) with respect to advertising is high, &
(2) with respect to price is low.
Example: Suppose that elasticity of demand with
respect to advertising is 0.10, and elasticity of
demand with respect to price is -0.50. What
percent of sales revenues should the advertising
budget should be?
A/(P*Q) = - A / D = -0.10 / -0.50 = 0.20 or 20%
Cost-Plus Pricing
The price charged by the firm is the average total
cost of production plus a percentage of that cost.
Example: If the average total cost of production
is $50, and the firm uses a 10% markup, the firm
will sell the product for $55.
Pricing using Product Lines
A firm may have several lines of a product, such as
(1) a regular line,
(2) an economy product (for people who want to save money), &
(3) a top-of-the-line product (for people who want “the best”).
To maximize profit, the firm sets MR = MC for each product line.
Product-Line Pricing Example: A company has 3 product lines.
deluxe: TC = 70 + 40Q + Q2 & demand function is P = 90 – 4Q
regular: TC = 65 + 30Q + Q2 & demand function is P = 84 – 2Q
economy: TC = 50 + 20Q + Q2 & demand function is P = 60 – Q
Determine the profit-maximizing price for each line.
For each product line, we want MR = MC. So for each line, we need
to calculate MC = dTC/dQ, TR = PQ, & MR = dTR/dQ.
deluxe:
MC = 40 + 2 Q
TR = (90 – 4Q)Q = 90 Q – 4Q2
MR = 90 – 8Q
regular:
MC = 30 + 2 Q
TR = (84 – 2Q)Q = 84 Q – 2Q2
MR = 84 – 4Q
economy: MC = 20 + 2 Q
TR = (60 – Q)Q = 60 Q – Q2
MR = 60 – 2Q
For our product-line pricing example, we have so far:
deluxe: P = 90 – 4Q, MR = 90 – 8 Q MC = 40 + 2 Q,
regular: P = 84 – 2Q, MR = 84 – 4Q MC = 30 + 2 Q,
economy: P = 60 – Q, MR = 60 – 2Q MC = 20 + 2 Q,
For each line we set MR = MC. So,
For the deluxe line,
90 – 8 Q = 40 + 2 Q
50 = 10 Q & Q = 5.
From the demand function, the deluxe price is
P = 90 – 4 Q = 90 – 4(5) = 90 – 20 = 70.
For the regular line,
84 – 4 Q = 30 + 2 Q
54 = 6Q & Q = 9.
The regular price is P = 84 – 2(9) = 84 – 18 = 66.
For the economy line,
60 – 2 Q = 20 + 2 Q
40 = 4 Q & Q = 10.
The economy price is P = 60 – 10 = 60 – 10 = 50.
Peak-Load Pricing
When demand is not evenly distributed, a firm needs to
have facilities to accommodate periods of high
demand.
Even with large facilities, the firm may experience times
when the demand is greater than can be handled.
Then the firm may experience costly computer system
crashes.
During off-peak times (periods of lower demand), there is
excess capacity.
The firm charges less at off-peak times.
Example: More phone calls are made during business
hours than in the evenings and on weekends. So the
phone companies charge more during business hours.
Peak-Load Pricing Example:
Suppose the demand function for a firm’s service is
Peak times (days):
P = 74 – 5 Q
Off-peak times (nights): P = 26 – 5 Q
The marginal cost of providing the service is MC = 2 + 2Q .
Determine the day & night profit-maximizing prices.
We need to find when MR = MC for days & for nights.
For days,
TR = PQ = (74 – 5 Q) Q = 74 Q – 5 Q2
So MR = dTR/dQ = 74 – 10 Q .
MR = MC implies 74 – 10 Q = 2 + 2 Q ,
or 72 = 12 Q.
So Q = 6
& peak price is P = 74 – 5 Q = 74 – 5(6) = $44 per unit.
Next we need to do the same thing for nights to find the
off-peak price.
We had these demand functions:
Peak times (days):
P = 74 – 5 Q
Off-peak times (nights): P = 26 – 5 Q
and the marginal cost function was MC = 2 + 2Q .
For nights,
TR = PQ = (26 – 5 Q) Q = 26 Q – 5 Q2
So MR = dTR/dQ = 26 – 10 Q .
MR = MC implies 26 – 10 Q = 2 + 2 Q ,
or 24 = 12 Q.
So Q = 2
& off-peak price is P = 26 – 5 Q = 26 – 5(2) = $16 per unit
(instead of $44 per unit as it was for peak times).
Transfer Pricing
Sometimes firms are organized into separate
divisions.
One division may produce an intermediate
product and supply it to another division to
produce the final product.
How does the firm determine the efficient price
at which the intermediate product should be
sold. That is, what is the transfer price?
The Simplest Case
The firm has 2 divisions: E and A
Division E produces the intermediate product (engine)
for Division A which produces the final product
(automobile).
Division E does not sell engines to anyone but division
A, and division A does not buy engines from anyone
but division E.
Each unit of output (automobile) requires one unit of
the input (engine).
The goal is to maximize the firm’s profit.
How do we determine the optimal quantity & price
for the final product (the auto)?
First, find the company’s (total) marginal cost MCT,
which is the marginal cost of division E’s producing an
engine (MCE) plus the marginal cost of division A’s
producing an auto (MCA).
That is, MCT =
MCA + MCE .
Then, produce the amount of output (autos) so that the
marginal revenue from selling an auto (MR) is equal to
the marginal cost of production (MCT).
The appropriate price of the auto for that quantity of
output is determined from the demand curve for the
firm’s autos.
So what is the transfer price at which division E sells
the intermediate product to division A?
If the company determines the price of the engine,
then division E is a price taker. So, PE and MRE will be
equal.
The firm should set the price of the intermediate
product (the engine) so that PE = MRE = MCE at the
profit-maximizing output level previously determined.
Transfer Pricing Example
A company has 2 divisions: production & marketing.
The production division’s total cost function is
TCp = 70,000 + 15Q + 0.005 Q2.
The marketing division’s total cost function is
TCm = 30,000 + 10 Q .
The demand function for the final marketed product is
Pf = 100 – 0.001 Q .
What should be the price that transfers the product from
production to marketing?
Also determine the price of the final product and the firm’s profit.
The marginal cost functions for the 2 divisions are
Production: MCp = dTCp/dQ = 15 + 0.01 Q
Marketing: MCm = dTCm/dQ = 10
So the combined MC = MCp + MCm = 25 + 0.01 Q
TR = PfQ = (100 – 0.001 Q) Q = 100 Q – 0.001 Q2
So MR = dTR/dQ = 100 – 0.002 Q
Continuing, we have
demand for the final product: Pf = 100 – 0.001 Q .
TCp = 70,000 + 15Q + 0.005 Q2 ; TCm = 30,000 + 10 Q .
MCp = dTCp/dQ = 15 + 0.01 Q;
MCm = dTCm/dQ = 10
MC = MCp + MCm = 25 + 0.01 Q ; MR = 100 – 0.002 Q
Equating MR & MC, we have
100 – 0.002 Q = 25 + 0.01 Q .
So 75 = 0.012 Q
& Q = 75/0.012 = 6,250 .
So the price of the intermediate product is
Pi = MCp = 15 + 0.01 (6,250) = $77.50 .
The price of the final product is
Pf = 100 – 0.001 Q = 100 – 0.001 (6,250) = $93.75 .
Plugging the quantity 6,250 into the two total cost functions &
adding, we find TC = TCp + TCm = $451,562.50 .
The total revenue is TR = Pf Q = (93.75) (6250) = $585,937.50 .
So the firm’s profit is TR – TC = $134,375 .