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					 Pricing
Strategies
Some pricing strategies that we will explore:

1.   Price discrimination – 1st, 2nd, & 3rd degree
2.   “Two-part Tariff” pricing
3.   Bundling
4.   Advertising
5.   Cost-plus markup pricing
6.   Product Lines
7.   Peak-Load pricing
8.   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         D = MR
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,                       Q
with 1st degree price
discrimination.
        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)
         9            = 1406.25


P*= 5.25
             profit
                                               ATC =MC =1.5
                                           D
                            MR
                                               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
 $
               Profit
     9         = (1/2) (1500) (7.5)
               = 5625


                             ATC =MC =1.5
                            D=MR

                             Q
                 Q*= 1500

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                  Low-volume consumers
    value the 1st unit of a good at        value the 1st unit at $12.
    $15 & the 2nd unit at $10.
$                                     $


15
                                      12
10
    6                                 6
                              ATC                              ATC


            1       2         Q                1                  Q
        Suppose firm sells all units individually for $15.
 The 100 high-volume               The 100 low-volume
 consumers will buy 1unit.         consumers will buy 0 units.
$                                  $

15
                                   12
      profit
10

 6                                 6
                             ATC                           ATC

               1   2        Q               1                Q
     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
        Suppose firm sells all units individually for $12.
 The 100 high-volume               The 100 low-volume
 consumers will buy 1unit.         consumers will buy 1 unit.
$                                  $

15
       CS                          12
12
10    profit                            profit

 6                                 6
                             ATC                             ATC

               1   2        Q                    1             Q
     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
        Suppose firm sells all units individually for $10.
 The 100 high-volume               The 100 low-volume
 consumers will buy 2 units.       consumers will buy 1 units.
$                                      $

15
       CS                             12
                                            CS
10                                    10
      profit       profit                  profit

 6                                     6
                                ATC                          ATC


               1            2   Q                   1          Q
     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
Suppose firm sells 1-unit packs for $12 & 2-unit packs for $20.
 The high-volume consumers          The low-volume consumers
 will buy a 2-unit pack.            will buy a 1-unit pack.
 $                                 $

15
       CS                          12
10                                      profit
            profit
 6                           ATC   6                         ATC


             1       2      Q                    1             Q
     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
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 low-
  volume 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 movie-
goers.
   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 D 2 .
          (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 = Q 2 – 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 = Q 2 – 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.
                 Suppose that the firm charges price P* per
       P         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,

                if it charges a membership fee
        P       equal to the area of the green
                triangle.


      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

For each person, TR = PQ = (35 – Q)Q = 35 Q – Q2.
So, MR = dTR/dQ = 35 – 2 Q
MC = dTC/dQ = 5
Setting MR = MC: 35 – 2 Q = 5
                    30 = 2 Q
                    15 = Q
Solving for P from the demand function,
          P = 35 – 15 = 20.
So revenue per person from per unit sales is
  PQ = (20)(15) = 300 .
Next we need to determine the appropriate
  membership fee.
     The membership fee is the consumer surplus.

P
40              That is the area of the green triangle,
                which is (1/2)(15)(15) = 112.50.
35              So the membership fee should be
                $112.50.
20



                       D
                                 Q
           15
Combining the membership fee of $112.50 with the
per unit sales revenues of $300 that we found
earlier, we have total revenues per customer of
$112.50 + $300 = $412.50

From the total cost function, the total production
cost for the 15 units per customer is
TC = 5Q = 5(15) = 75.
So our profit per customer is
 = TR – TC = 412.50 – 75 = $337.50.
                  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
                                               Sum of
                       Stereo       TV       reservation
                                               prices
        Alan            $225       $375        $600

       Beth             $325       $275        $600
 Maximum price for
                        $225       $275
both to buy the good
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.
           Cost-Plus Pricing
Cost-plus pricing can result in the profit-maximizing
price, if the firm uses marginal cost instead of
average total cost and the markup is
                       1
                          .
                      1 
 In other words,
                            1    
              P  MC 1         
                       1       
For example, suppose that MC = $10 and the
elasticity of demand is ε = -3. Then,
                               1 
               P  MC 1            
                             1   
                                     
                            1 
                 10 1         
                       1  ( 3)  
                                    
                        1 
                  10 1     
                        2  
                 10(1  0.5)  15
So the MC is $10 and the price is the MC plus an
additional 50% markup or $15.
       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 + Q 2 & demand function is P = 84 – 2Q
economy: TC = 50 + 20Q + Q 2 & 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 MC T,
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 Q 2 ;      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 .

				
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