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					              Chapter 4


Product and Pricing Strategies
           for the
  Multiproduct Monopolist

       Industrial Organization: Chapter 4   1
                              Introduction

• A monopolist can offer goods of different varieties
   – multiproduct firms
• The “big” issues:
   – pricing
   – product variety: how many?
   – product bundling:
       • how to bundle
       • how to price
       • whether to tie the sales of one product to sales of another
• Price discrimination



                          Industrial Organization: Chapter 4           2
                                                                       4-1
                      Price Discrimination
• This is a natural phenomenon with multiproduct firms
   – restaurant meals: table d’hôte or à la carte
   – different varieties of the same car
   – airline travel
       • “goods” of different quality are offered at very different prices
• Note the constraints
   – arbitrage
       • ensuring that consumers buy the “appropriate” good
   – identification
• How to price goods of different quality?



                         Industrial Organization: Chapter 4                  3
             Price discrimination and quality
• Extract all consumer surplus from the low quality good
• Use screening devices
   – Set the prices of higher quality goods
       • to meet incentive compatibility constraint
       • to meet the constraint that higher price is justified by higher quality
• One interesting type of screening: crimping the product
   – offer a product of reasonably high quality
   – produce lower quality by damaging the higher quality good
       • student version of Mathematica
       • different versions of Matlab
       • the “slow” 486SX produced by damaging the higher speed 486DX
   – why?
       • for cost reasons

                         Industrial Organization: Chapter 4                        4
         A Spatial Approach to Product Variety
• Approach to product quality in Chapter 3 is an example of
  vertical product differentiation
   – products differ in quality
   – consumers have similar attitudes to quality: value high quality
• An alternative approach
   – consumers differ in their tastes
   – firm has to decide how best to serve different types of consumer
   – offer products with different characteristics but similar qualities
• This is horizontal product differentiation




                         Industrial Organization: Chapter 4                5
    A Spatial Approach to Product Variety (cont.)
• The spatial model (Hotelling) is useful to consider
   – pricing
   – design
   – variety
• Has a much richer application as a model of product
  differentiation
   – “location” can be thought of in
       • space (geography)
       • time (departure times of planes, buses, trains)
       • product characteristics (design and variety)




                         Industrial Organization: Chapter 4   6
    A Spatial Approach to Product Variety (cont.)
• Assume N consumers living equally spaced along Main
  Street – 1 mile long.
• Monopolist must decide how best to supply these
  consumers
• Consumers buy exactly one unit provided that price plus
  transport costs is less than V.
• Consumers incur there-and-back transport costs of t per
  unit
• The monopolist operates one shop
   – reasonable to expect that this is located at the center of Main Street



                        Industrial Organization: Chapter 4                7
                          The spatial model
     Price                                Suppose that the monopolist Price
               p1 + t.x                                      p
                                             sets a price of p1 + t.x
                                                               1


     V                                                                              V

All consumers within                  t               t
distance x1 to the left                        p1
                                                                     What determines
and right of the shop
                                                                           x1?
 will by the product



      x=0                  x1               1/2                 x1            x=1
                                          Shop 1
                          p1 + t.x1 = V, so x1 = (V – p1)/t

                           Industrial Organization: Chapter 4                           8
                            The spatial model
        Price                                                                    Price
                 p1 + t.x                                             p1 + t.x
                                                        Suppose the firm
        V                                               reduces the price              V
                                                             to p2?

Then all consumers                              p1
 within distance x2
                                                   p2
of the shop will buy
    from the firm


         x=0           x2   x1               1/2                 x1       x2     x=1
                                          Shop 1



                            Industrial Organization: Chapter 4                             9
                        The spatial model
• Suppose that all consumers are to be served at price p.
   – The highest price is that charged to the consumers at the ends of
     the market
   – Their transport costs are t/2 : since they travel ½ mile to the shop
   – So they pay p + t/2 which must be no greater than V.
   – So p = V – t/2.
• Suppose that marginal costs are c per unit.
• Suppose also that a shop has set-up costs of F.
• Then profit is p(N, 1) = N(V – t/2 – c) – F.




                        Industrial Organization: Chapter 4                  10
        Monopoly Pricing in the Spatial Model
• What if there are two shops?
• The monopolist will coordinate prices at the two shops
• With identical costs and symmetric locations, these prices
  will be equal: p1 = p2 = p
   – Where should they be located?
   – What is the optimal price p*?




                      Industrial Organization: Chapter 4       11
                        Location with Two Shops
                                                                        Delivered price to
             Suppose that the entire market is                       to be served the
                                                                         consumers at
                                                                       market center equals
                           Price                                      their reservation price Price
If there are two shops
  they will be located
                            V                                                                       V
    symmetrically a
  distance d from the
                                                                                                    p(d)
The maximum price p(d)
    end-points of the
 the firm can charge
         market                                       What determines
is determined by the
        Now raise the price                                p(d)?
   consumers at the
            at each shop
      Start with market
center of the a low price
         at each shop                    d                     1/2                 1-d
                           x=0                                                               x=1
                                       Shop 1                                     Shop 2
         Suppose that                             The shops should be
           d < 1/4                                  moved inwards
                                Industrial Organization: Chapter 4                             12
                                   Delivered Shops
                        Location with Twoprice to
                                                 consumers at the
The maximum price                               end-points equals
                                              their reservation price
the firm can charge
                             Price                                                 Price
 is now determined
  by the consumers
   at the end-points         V                                                           V
     of the market
                        p(d)                                                             p(d)

                                                                  Now what
       Now raise the price                                      determines p(d)?
           at each shop
 Start with a low price
      at each shop                                  d           1/2      1-d
                             x=0                                                   x=1
                                                  Shop 1                Shop 2
      Now suppose that                             The shops should be
          d > 1/4                                   moved outwards
                                 Industrial Organization: Chapter 4                 13
It follows that         Location with Two Shops
shop 1 should                                           Price at each
 be located at                                          shop is then
                            Price                                                Price
1/4 and shop 2                                          p* = V - t/4
     at 3/4
                          V                                                            V

                      V - t/4                                                       V - t/4
Profit at each shop
 is given by the
   shaded area             c                                                           c



                           x=0                1/4              1/2       3/4     x=1
                                             Shop 1                     Shop 2

               Profit is now p(N, 2) = N(V - t/4 - c) – 2F
                                Industrial Organization: Chapter 4                14
                                                               By the same argument
                              Three Shops                      they should be located
   What if there                                                 at 1/6, 1/2 and 5/6
  are three shops?
                      Price                                                   Price


                    V                                                               V
Price at each   V - t/6                                                             V - t/6
shop is now
   V - t/6



                     x=0         1/6                    1/2             5/6   x=1
                                Shop 1                 Shop 2          Shop 3

            Profit is now p(N, 3) = N(V - t/6 - c) – 3F
                          Industrial Organization: Chapter 4                   15
               Optimal Number of Shops
• A consistent pattern is emerging.
Assume that there are n shops.
They will be symmetrically located distance 1/n apart.
We have already considered n = 2 and n = 3. How many
When n = 2 we have p(N, 2) = V - t/4         shops should
                                               there be?
When n = 3 we have p(N, 3) = V - t/6
It follows that p(N, n) = V - t/2n
 Aggregate profit is then p(N, n) = N(V - t/2n - c) – n.F



                     Industrial Organization: Chapter 4     16
             Optimal number of shops (cont.)
Profit from n shops is p(N, n) = (V - t/2n - c)N - n.F

and the profit from having n + 1 shops is:
               p*(N, n+1) = (V - t/2(n + 1)-c)N - (n + 1)F

Adding the (n +1)th shop is profitable if p(N,n+1) - p(N,n) > 0

This requires tN/2n - tN/2(n + 1) > F
which requires that n(n + 1) < tN/2F.




                      Industrial Organization: Chapter 4      17
                          An example
Suppose that F = $50,000 , N = 5 million and t = $1
          Then t.N/2F = 50
So we need n(n + 1) < 50. This gives n = 6
There should be no more than seven shops in this case: if
n = 6 then adding one more shop is profitable.

But if n = 7 then adding another shop is unprofitable.




                     Industrial Organization: Chapter 4     18
                      Some Intuition
• What does the condition on n tell us?
• Simply, we should expect to find greater product variety
  when:
• there are many consumers.
• set-up costs of increasing product variety are low.
• consumers have strong preferences over product
  characteristics and differ in these.




                    Industrial Organization: Chapter 4       19
            How Much of the Market to Supply
• Should the whole market be served?
   – Suppose not. Then each shop has a local monopoly
   – Each shop sells to consumers within distance r
   – How is r determined?
       •   it must be that p + tr = V so r = (V – p)/t
       •   so total demand is 2N(V – p)/t
       •   profit to each shop is then p = 2N(p – c)(V – p)/t – F
       •   differentiate with respect to p and set to zero:
       •   dp/dp = 2N(V – 2p + c)/t = 0
   – So the optimal price at each shop is p* = (V + c)/2
   – If all consumers are to be served then price is p(N,n) = V – t/2n
• Only part of the market should be served if p(N,n) > p*
• This implies that V > c + t/n.
                           Industrial Organization: Chapter 4            20
                    Partial Market Supply
• If c + t/n > V supply only part of the market and set price
  p* = (V + c)/2
• If c + t/n < V supply the whole market and set price
  p(N,n) = V – t/2n
• Supply only part of the market:
   – if the consumer reservation price is low relative to marginal
     production costs and transport costs
   – if there are very few outlets




                        Industrial Organization: Chapter 4           21
                                           Are there too
                      Social Optimum
                                          many shops or
What number of shops maximizes total surplus? too few?

Total surplus is consumer surplus plus profit
Consumer surplus is total willingness to pay minus total revenue
Profit is total revenue minus total cost
Total surplus is then total willingness to pay minus total costs
Total willingness to pay by consumers is N.V

    Total surplus is therefore N.V - Total Cost

   So what is Total Cost?

                      Industrial Organization: Chapter 4       22
                    Social optimum (cont.)
      Assume that
         there
      are n shops   Price                                                         Price


                    V                                                 Transport cost for
                                                                                       V
Consider shop                                                       each shop is the area
      i                                                             of these two triangles
                                                                         multiplied by
                                                                      consumer density
  Total cost is                  t/2n                        t/2n
 total transport
cost plus set-up    x=0                   1/2n       1/2n                         x=1
      costs                                   Shop i
                                                               This area is t/4n2


                        Industrial Organization: Chapter 4                         23
                    Social optimum (cont.)
Total cost with n shops is, therefore: C(N,n) = n(t/4n2)N + n.F

                                            = tN/4n + n.F
                                        If t = $1, F = $50,000,
                                       There5should be (n+1).F
Total cost with n + 1 shops is: C(N,n+1) ==tN/4(n+1)+ fivethis
                                        N       million then shops:
                                              n = 4 adding another
                                        withC(N,n + tells us
Adding another shop is socially efficient ifcondition 1) < C(N,n)
                                              shop is efficient
                                            that n(n+1) < 25
This requires that tN/4n - tN/4(n+1) > F
which implies that n(n + 1) < tN/4F

    The monopolist operates too many shops and, more
       generally, provides too much product variety

                        Industrial Organization: Chapter 4        24
Monopoly, Product Variety and Price Discrimination
• Suppose that the monopolist delivers the product.
   – then it is possible to price discriminate
• What pricing policy to adopt?
   –   charge every consumer his reservation price V
   –   the firm pays the transport costs
   –   this is uniform delivered pricing
   –   it is discriminatory because price does not reflect costs
• Should every consumer be supplied?
   –   suppose that there are n shops evenly spaced on Main Street
   –   cost to the most distant consumer is c + t/2n
   –   supply this consumer so long as V (revenue) > c + t/2n
   –   This is a weaker condition than without price discrimination.
   –   Price discrimination allows more consumers to be served.

                         Industrial Organization: Chapter 4            25
      Price Discrimination and Product Variety
• How many shops should the monopolist operate now?
Suppose that the monopolist has n shops and is supplying
the entire market.
Total revenue minus production costs is N.V – N.c
Total transport costs plus set-up costs is C(N, n)=tN/4n + n.F
So profit is p(N,n) = N.V – N.c – C(N,n)
But then maximizing profit means minimizing C(N, n)
The discriminating monopolist operates the socially
optimal number of shops.



                    Industrial Organization: Chapter 4       26
                            Bundling
• Firms sell goods as bundles
   – selling two or more goods in a single package
   – complete stereo systems
   – fixed-price meals in restaurants
• Firms also use tie-in sales: less restrictive than bundling
   – tie the sale of one good to the purchase of another
   – computer printers and printer cartridges
   – constraining the use of spare parts
• Why?
• Because it is profitable to do so!



                       Industrial Organization: Chapter 4       27
                 Bundling: an example
• Two television stations offered two old Hollywoodcan
                                         How much films
                                  How much can
   – Casablanca and Son of Godzilla       be charged for
                                  be
              If the films are sold charged for
• Arbitrage is possible between the stationsGodzilla?
                 separately total Casablanca?
• Willingness revenue is $19,000
               to pay is:
                                                       $7,000
              Willingness to Willingness to
                 pay for        pay for                  $2,500
               Casablanca      Godzilla

 Station A        $8,000                     $2,500

 Station B        $7,000                     $3,000

                    Industrial Organization: Chapter 4       28
                                How much can
                Bundling: an example
                                  Bundling for
                                be charged is profitable
        Now suppose                  package?
                                 thebecause it exploits
    that the two films are    aggregate willingness
              If the sold
      bundled and films are sold
             Willingness to Willingness to
                                        pay       Total
                  a package total pay for
               as pay for
        as a package                           Willingness
              revenue is $20,000Godzilla
               Casablanca                        to pay

Station A      $8,000                     $2,500       $10,500

Station B      $7,000                     $3,000       $10,000


                   $10,000

                  Industrial Organization: Chapter 4             29
                       Bundling (cont.)
• Extend this example to allow for
   – costs
   – mixed bundling: offering products in a bundle and separately




                       Industrial Organization: Chapter 4           30
                                               has
                                 Consumer y Each consumer
           Bundling: another example py1
                               reservation price the firm
                      are
  Suppose that thereAll consumers in           that
                                     Supposebuys exactly one
                                        sets All py2 1 for
                                for good 1 and consumers in
                                             price p
    two goods and that region B buy             unit of a good
R2                                    good 1 2
                                    for good and price buy
                                               region A p2
                        only good 2
    consumers differ in Consumer x has good 2 that price
                                            provided
                                           for both goods
 their reservation pricesreservation price px1is less than her
           B                 A
      for these goods      for good 1 and px2
                                              reservation price
py2
                               for good
                     Allyconsumers in 2         All consumers in
 p2                    region C buy               region D buy
               x                                  Consumers
px2                    neither good                only good 1
                                                                split into
           C                             D                     four groups


               px1       p1 py1                           R1



                     Industrial Organization: Chapter 4                  31
     Bundling: the example (cont.)
                            Now consider pure
R2                                       at some
                             bundling consumers in
                                     All
                                   price pB E buy
                  Consumers in these two regions
                                       region
pB                 can buy each good even though
                                           the bundle
                  their reservation price for one of
                     Ethe goods is less than its
                                                 Consumers
                         All marginal cost
                             consumers in
                            region F do not           now split into
      F                     buy the bundle             two groups
c2


       c1                 pB                     R1



            Industrial Organization: Chapter 4                    32
                    Mixed Bundling
                  In this region         Now consider mixed
                 consumers buy
            Consumers in Good 1 is sold bundling
                             this
   R2           either the bundle
             region buy only price p
                               at         1
                   or product 2
    pB                                      Good Consumers in this
                  good 2 Consumers in this 2 is sold
                                                  region
                                  region also at price p are willing to
                                                  This 2leaves
                               buy the bundle buy both goods. They
    p2                                            two regions
                                                     buy the bundle
                                                        Consumers split
                                                          In this region
                             Consumers in this           consumers buy
                                                        into four groups:
                                                       either the bundle
                            region buy nothing in thisbuy the bundle
                                      Consumers
pB - p1                              The bundle is sold
                                        region < p only or product 1
                                    at price pBbuy 1 + pbuy only good 1
                                                         2
                                            good 1
                                                              buy only good 2
                                                                buy nothing
          pB - p2      p1             pB                 R1



                    Industrial Organization: Chapter 4                   33
              Mixed Bundling (cont.)
                     Similarly, all
                     consumers in
   R2               this region buy
    pB              only product 2                          The consumer
                                                            x will buy only
                                                               product 1
                                              Consider consumer x with
    p2                                                     All consumers
                                               reservation prices p for in
                                       Which is this surplus from 1x
                                        Consumer             surplus from
                                                 Consumerthis region buy
                                                 product 1 and p2x for
                                        measure Her aggregate willingness
                                                   product only
                                                            1 bundle is
                                          buyingbuying theis 2product 1
                                                        product
pB - p1                                             to pay for
                                                 p1x -pp1 + p thep bundle is
                                                                 - B
                                                        1x    2x
   p2x                                      x               p1x + p2x


          pB - p2      p1             pB p1x             R1
                                                  p1x+p2x


                    Industrial Organization: Chapter 4                  34
                  Mixed Bundling (cont.)
• What should a firm actually do?
• There is no simple answer
   – mixed bundling is generally better than pure bundling
   – but bundling is not always the best strategy
• Each case needs to be worked out on its merits




                       Industrial Organization: Chapter 4    35
                      An Example
Four consumers; two products; MC1 = $100, MC2 = $150

               Reservation            Reservation        Sum of
 Consumer       Price for              Price for       Reservation
                 Good 1                 Good 2           Prices

    A             $50                     $450            $500

    B            $250                     $275            $525

    C            $300                     $220            $520

    D            $450                       $50           $500

                  Industrial Organization: Chapter 4                 36
           The example (cont.)
        Good 1: Marginal Cost $100
Price    Quantity              Consider
                          Total revenue simple
                                           Profit
$450         1                monopoly pricing
                              $450         $350
$300         2                 $600        $400
                Good 1 should be sold
$250         3                $750         $450
                at $250 and good 2 at
$50          4                $200         -$200
                  $450. Total profit
                    is $450 $150
        Good 2: Marginal Cost + $300
Price    Quantity          $750
                         = Total revenue          Profit
$450         1                          $450      $300
$275         2                          $550      $200
$220         3                          $660      $210
$50          4                          $200      -$400

             Industrial Organization: Chapter 4            37
              The example (cont.)
                                          Now consider pure
                                              bundling

             Reservation     Reservation               Sum of
Consumer                      Price bundle
              Price forThe highest for               Reservation
                               Good 2
               Good 1 price that can be                Prices
                      considered is $500
                               will
           All four consumers$450buy
   A             $50                                    $500
             the bundle and profit is
   B        4x$500 - 4x($150 + $100)
                $250          $275                      $525
                     = $1,000
   C            $300          $220                      $520

   D           $450                       $50           $500

                Industrial Organization: Chapter 4                 38
                      The example (cont.)
                                          Now consider mixed
Take the monopoly prices p1 = $250; p2 = $450 and a bundle price pB = $500
                                                bundling
              All four consumers buy
              something and profit is
                    Reservation        Reservation        Sum of
   Consumer       $250x2 + $150x2
                      Price for
          Can the seller improve Price for             Reservation
                       Good 1
                         = $800          Good 2            Prices
                  on this?
        A               $50                     $450         $500

        B               $250                    $275         $500
                                                             $525

        C               $250
                        $300                    $220         $520

        D               $450
                        $250                      $50        $500

                        Industrial Organization: Chapter 4           39
                      The example (cont.)
Try instead the prices p1 = $450; p2 = $450 and a bundle price pB = $520
                          This is actually
                         the       that the
  All four consumers buy best Reservation
                  Reservation                                  Sum of
  Consumer
    and profit is $300 + firm can do for
                   Price for        Price                    Reservation
                       Good 1                  Good 2          Prices
        $270x2 + $350
           = $1,190
       A            $50                         $450
                                                $450            $500

       B               $250                     $275            $520
                                                                $525

       C               $300                     $220            $520

       D                $450
                       $450                       $50           $500

                        Industrial Organization: Chapter 4                 40
                        Bundling (cont.)
• Bundling does not always work
• Requires that there are reasonably large differences in
  consumer valuations of the goods
• What about tie-in sales?
   – “like” bundling but proportions vary
   – allows the monopolist to make supernormal profits on the tied
     good
   – different users charged different effective prices depending upon
     usage
   – facilitates price discrimination by making buyers reveal their
     demands



                       Industrial Organization: Chapter 4                41
                            Tie-in Sales
• Suppose that a firm offers a specialized product – a
  camera? – that uses highly specialized film cartridges
• Then it has effectively tied the sales of film cartridges to
  the purchase of the camera
   – this is actually what has happened with computer printers and ink
     cartridges
• How should it price the camera and film?
   – suppose that marginal costs of the film and of making the camera
     are zero (to keep things simple)
   – suppose also that there are two types of consumer: high-demand
     and low-demand



                       Industrial Organization: Chapter 4                42
                   Suppose that Example
                Tie-In Sales: anthe Profit is $72 from each
                                         type of consumer
                       firm leases the Low-Demand
       High-Demand
        Consumersproduct for $72 perSo this gives profit of
            Is this the best             Consumers
                           period      $144 per pair of high-
                   that
      Demand: P = 16 - Q               Demand:low-demand
                                         and P = 12 - Q
           the firm can do?
$                                 $         consumers
$16
                                         $12                      Low-demand
                                                                  consumers are
                                 High-demand                     willing to buy 12
                               consumers buy 16                        units
         $128                        units
                                                  $72


                         16                                      12
            Quantity                                        Quantity
                       Industrial Organization: Chapter 4                      43
                  Tie-In Sales: an Example
                   Suppose that          Profit is $70 from each
       High-Demandfirm sets a
                  the                    low-demand consumer:
                                            Low-Demand
                 price of firm can set a Consumers
                    So
        Consumers the$2 per                     $50 + $20
                     lease charge of $50and $78 from each
                        unit
      Demand: P = 16 - Q each type of Demand: P = 12 - Q
                        to                   Consumer consumer:
                                        high-demandsurplus
$                                                low-demand
                    consumer: it cannot for$50 + $28
                                    $
$16             Consumer surplus             consumers is $50
                         discriminate giving $148 per pair of
                 for high-demand
                                   $12    high-demand and low-
                 consumers is $98
                             High-demand             Low-demand
                           consumers buy 14       demand buy 10
                                                   consumers
                                        units                             units
        $98                                          $50

$2                                           $2

                         14 16                                   10 12
              Quantity                                         Quantity
                          Industrial Organization: Chapter 4                      44
                   Tie-In Sales: an Example
                    Suppose that
                   the firm can         Profit is $72 from each
       High-Demand                         Low-Demand
                                        low-demand consumer
                 bundle the two
        Consumers                            Consumers
                                          and $80 from each
                  goods instead
                  a produce
        ProduceSobundled a second
      Demand: P = 16 - Q them
                    of tieof camera plus  Demand: P = consumer
                                        high-demand 12 - Q
         product of camera
                bundle
$
                                        giving $150 per pair of
       plus 12-shot cartridge $
                    16-shot cartridgehigh-demand and low-
$16                 High-demand
                       High-demand
                 consumers get $48               demand
                                    pay
                     consumers will$12
                   consumer surplus                                  Low-demand
                     $80 for this bundled
                    from buying it                                 consumers can be
                     camera ($128 - $48)
         $48                                                       sold this bundled
                                                                    product for $72
       $72                                               $72
                      $8
                     12     16                                      12
               Quantity                                        Quantity
                          Industrial Organization: Chapter 4                      45
                  Complementary Goods
• Complementary goods are goods that are consumed
  together
   – nuts and bolts
   – PC monitors and computer processors
• How should these goods be produced?
• How should they be priced?
• Take the example of nuts and bolts
   – these are perfect complements: need one of each!
• Assume that demand for nut/bolt pairs is:
           Q = A - (PB + PN)


                      Industrial Organization: Chapter 4   46
              Complementary goods (cont.)
This demand curve can be written individually for nuts and bolts
For bolts: QB = A - (PB + PN)
For nuts: QN = A - (PB + PN)
These give the inverse demands: PB = (A - PN) - QB
                                PN = (A - PB) - QN
These allow us to calculate profit maximizing prices
Assume that nuts and bolts are produced by independent firms
Each sets MR = MC to maximize profits
MRB = (A - PN) - 2QB
                                    Assume MCB = MCN = 0
MRN = (A - PB) - 2QN
                     Industrial Organization: Chapter 4     47
              Complementary goods (cont.)
Therefore QB = (A - PN)/2
and PB = (A - PN) - QB = (A - PN)/2
by a symmetric argument PN = (A - PB)/2

The price set by each firm is affected by
     the price set by the other firm

 In equilibrium the price set by the two
        firms must be consistent




                     Industrial Organization: Chapter 4   48
      Complementary goods (cont.)
                                                PB = (A - PN)/2
                                                PN = (A - PB)/2
 PB    Pricing rule for
                                               PN = A/2 - (A - PN)/4
           the Nut
 A        Equilibrium is                           = A/4 + PN/4
          Producer: rule for
              Pricing
                                               3PN/4 = A/4
         = (A - Pthe two
      PN where theseBolt
                  B)/2
           pricing rules                       PN = A/3
                 Producer:
             intersect                        PB = A/3
A/2          PB = (A - PN)/2
                                              PB + PN = 2A/3
A/3                                           Q = A - (PB+PN) = A/3
                                              Profit of the Bolt Producer
                                                       = PBQB = A2/9
      A/3 A/2                 A      PN
                                              Profit of the Nut Producer
                                                       = PNQN = A2/9

             Industrial Organization: Chapter 4                         49
               Complementary goods (cont.)
What happens if the two goods are produced by the same firm?
                     Merger for a two firms
The firm will set a price PNB of thenut/bolt pair.
                        A - PNB consumers
Demand is now QNB =results inso that PNB = A - QNB
                         Why? Because the
                          being charged
                                  $
 MRNB = A - 2QNB      merged firm is the firm
                    lower prices and able to
  MR = MC = 0         coordinate the profits
                      making greaterprices of
                               A
 Q = A /2                 the two goods
     NB

 PNB = A /2
                                     A/2
Profit of the nut/bolt producer
is PNBQNB = A2/4                                              Demand
                                                 MR

                                                        A/2     A      Quantity
                        Industrial Organization: Chapter 4                50
Industrial Organization: Chapter 4   51
                   Product variety (cont.)

  d < 1/4

 We know that p(d) satisfies the following constraint:
 p(d) + t(1/2 - d) = V
 This gives: p(d) = V - t/2 + t.d
           p(d) = V - t/2 + t.d
 Aggregate profit is then: p(d) = (p(d) - c)N
                                      = (V - t/2 + t.d - c)N

This is increasing in d so if d < 1/4 then d should be increased.


                      Industrial Organization: Chapter 4       52
                   Product variety (cont.)

  d > 1/4

 We now know that p(d) satisfies the following constraint:
 p(d) + t.d = V
 This gives: p(d) = V - t.d

 Aggregate profit is then: p(d) = (p(d) - c)N
                                = (V - t.d - c)N


This is decreasing in d so if d > 1/4 then d should be decreased.


                      Industrial Organization: Chapter 4     53

				
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