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Course outline I Monopoly _quantity setting_ Decision problem

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Course outline I Monopoly _quantity setting_ Decision problem Powered By Docstoc
					Stratégies des entreprises




               Course outline I
                 Introduction
                 Game theory
                 Price setting
                  –   monopoly
                  –   oligopoly
                                             Homogeneous
                 Quantity setting
                                             goods
                  –   monopoly
                  –   oligopoly
                 Process innovation
                                                           1




               Monopoly (quantity setting)
                 Inverse demand function
                 Revenues, costs, profits
                 Profit maximizing quantity
                  –   Basic model
                  –   Price discrimination
                  –   Several factories
                 Double marginalization
                 Welfare analysis
                 Executive summary
                                                           2




               Decision problem




                          X                        Π
                                                           3




Harald Wiese                                                   1
Stratégies des entreprises




               Inverse demand function
                       p

                                demand
                   p           function



                                               inverse demand
               p( X )                                 function

                                                                X
                           X ( p)               X
                                                                                     4




               Revenue, costs and profit
                 Revenue:           R( X ) = p ( X ) ⋅ X
                 Costs:             C(X )
                 Profit:            Π ( X ) = R( X ) − C ( X )




                                                                                     5




               Marginal revenue with respect to
               quantity
                dR                   dp( X )
                   = MR X = p( X ) +         X
                dX                    dX
               When a firm increases the quantity by one unit, revenue
                 goes up by p (the price of the last unit),
                 but goes down by dp/dX X (the quantity increase diminishes
                 the price and this price decrease is applied to all units)
                                                     ⎛         X    dp ( X ) ⎞
               Amoroso-Robinson MR X = p ( X ) ⎜ 1 +                         ⎟
               relation:                       ⎝     p( X )           dX ⎠
                                                  ⎛    1      ⎞      ⎛    1      ⎞
                                             = p ⋅⎜ 1+        ⎟ = p ⋅⎜ 1−        ⎟
                                                  ⎜ ε         ⎟      ⎜ ε         ⎟
                                                  ⎝    X ,p   ⎠      ⎝    X ,p   ⎠
                                                                                     6




Harald Wiese                                                                             2
Stratégies des entreprises




               Marginal revenue = price?
               dR            dp ( X )
                  = p( X ) +          X
               dX             dX


                        dp ( X )
               1.                =0 ⇒      i.e. horizontal demand curve, perfect
                         dX                competition

               2. X=0               ⇒      sale of first unit
                                    ⇒      first degree price discrimination



                                                                                   7




               Linear demand curve in a
               monopoly
                 Demand:                            p( X ) = a − bX

                 Revenue:                           R ( X ) = aX − bX 2

                 Marginal revenue:                  MR = a − 2bX




                                                                                   8




               Exercise (Depicting the linear
               demand curve p( X ) = a − bX )
                Slope of demand curve: ....
                Slope of marginal revenue curve: ....
                The ............................             has the same
                vertical intercept, ..., as the demand curve.
                Economically,
                    –   the vertical intercept is .................,
                    –   the horizontal intercept is ................. .
                                                                                   9




Harald Wiese                                                                           3
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               Depicting demand and marginal
               revenue
                 p
                 a

                                           1
                                                b
                              2b


                                       1                   p( X )
                                               MR
                                                                           X
                                           a/(2b)                   a/b
                                                                               10




               First order condition
                dΠ ( X )              dp ( X ) dC ( X ) !
                         = p( X ) + X         −        =0
                 dX                    dX       dX

                                               MR X                 MC X
                 Notation:                     MR := MRX
                                               MC := MC X


                                                                               11




               First order condition, alternative
               formulations
                   !         ⎛    1                 ⎞
                MC = MR = p ⋅⎜ 1−                   ⎟
                             ⎜ ε                    ⎟
                             ⎝    X ,p              ⎠
                     !
                              1                  ε X ,p
                p=                     MC =               MC
                         1−
                                   1           ε X ,p − 1
                              ε X ,p

                p − MC ! 1
                       =                       (price-cost margin)
                   p     ε X ,p
                                                                               12




Harald Wiese                                                                        4
Stratégies des entreprises




               Depicting the Cournot monopoly
                        p



                                                             MC
                                       Cournot
                    M
                    p                    point



                                                                 p( X )

                                             MR
                                                                                X
                                       XM
                                                                                     13




               Exercise (Quantity)
               Consider a monopolist facing the inverse demand
               function p(X)=24-X. Assume that the average and
               marginal costs are given by AC=2.
               Find the profit-maximizing quantity!
                S. : X M = 11




                                                                                     14




               Profit in a monopoly
                Marginal point of view:                  Average point of view:
                p                                            p



                                                                    p(X)
                                                         M
               pM                                    p
                                                                                    MC
                                        MC

                                      p(X)
                                                                                    AC
                                 MR
                                                 X                                   X
                            XM                                        XM   MR
                                                                                     15




Harald Wiese                                                                              5
Stratégies des entreprises




                Exercise (monopoly)
                Consider a monopoly facing the inverse demand
                function p(X)=40-X2. Assume that the cost function
                is given by C(X)=13X+19.
                Find the profit-maximizing price and calculate the
                profit.
                S. : Π M = 35




                                                                                    16




                Price discrimination
                    First degree price discrimination:
                     Every consumer pays a different price which is equal to his
                     or her willingness to pay.

                    Second degree price discrimination:
                     Prices differ according to the quantity demanded and sold
                     (quantity rebate).

                    Third degree price discrimination:
                     Consumer groups (students, children, ...) are treated
                     differently.

                                                                                    17




                Monopolistic price
                discrimination (two markets)
                                                    p



                                               p1
                                                    p2
                    x1 (p1 )                                           x 2 (p 2 )
                                       MR 1                MR 2
                                                                                MC

                                   *
               x1                 x1        total output          x*
                                                                   2                x2
                                                                                    18




Harald Wiese                                                                             6
Stratégies des entreprises




                Inverse elasticities rule for third
                degree price discrimination
               Supplying a good X to two markets results in the inverse demand
               functions p1(x1) and p2(x2).
               Profit function:                    Π ( x1 ,x2 )= p1 ( x1 )⋅x1 + p2 ( x2 )⋅x2 −C ( x1 + x2 )
                                                                  4 4 4 3
                                                                1 2 3 1 24
                                                                   R1 (x1 )      R2 (x2 )
                                                   ∂ Π ( x1 ,x2 )                                  !
               First order conditions:                            =MR1 ( x1 )− MC ( x1 + x2 )=0
                                                       ∂ x1


                                                                                            =
                                                   ∂ Π (x1 ,x2 )                                   !
                                                                  = MR2 ( x2 )− MC ( x1 + x2 )=0
                                                       ∂x 2
               Equating the marginal revenues (using the Amoroso-Robinson relation) leads to:
                                                         ⎛        1 ⎞!              ⎛        1 ⎞
                                                         ⎜ ε ( x ) ⎟= p2 ( x 2 )⋅⎜1− ε ( x ) ⎟
                                               p1 ( x1 )⋅⎜1−            ⎟           ⎜             ⎟
                                                         ⎝      1 1 ⎠               ⎝      2    2 ⎠
                                                ε 1 ( x1 ) < ε 2 ( x2 ) ⇒ p1 ( x1 ) > p2 ( x2 )
                                                                                                              19




               Exercise (two markets or one)
               A monopoly sells in two markets:
               p1(x1)=100-x1 and p2(x2)=80-x2.
               a) Calculate the profit-maximizing quantities and the profit
               at these quantities, if the cost function is given by C(X)=X2.
               b) Calculate the profit-maximizing quantities and the profit
               at these quantities, if the cost function is given by
               C(X)=10X.
               c) What happens if price discrimination between the two
               markets is not possible anymore? Consider C(X)=10X.
               Hint: Differentiate between quantities below and above 20.
                S. : a) Π M = 1400, b) Π M = 3250, c) Π M = 3200                                              20




                Solution III (one market)

                                               p

                                         100

                                          90
                                          80




                                          50

                                                             MR                             p( X )


                                          10                                                MC
                                                     20          50          80       100
                                                                                               X


                                                                                                              21




Harald Wiese                                                                                                       7
Stratégies des entreprises




               One market, two factories
                 Profit function:
                Π ( x1 , x2 ) = p ( x1 , x2 )( x1 + x2 ) − C1 ( x1 ) − C2 ( x2 )
                 First order conditions:
                ∂Π ( x1 , x2 )                               !
                               = MR ( x1 , x2 ) − MC1 ( x1 ) = 0
                   ∂x1
                                        =




                ∂Π ( x1 , x2 )                                !
                               = MR ( x1 , x2 ) − MC 2 ( x2 ) = 0
                   ∂x2
                                           !
                ⇒              MC1 ( x1 ) = MC 2 ( x2 )
                                                                                        22




               One market, two factories II
                          factory 2                           factory 1
                                                      p
                           MC 2
                                                                          MC1




                x2                     *                       *                   x1
                                      x2                      x1

                                               total output
                                                                                        23




               Double marginalization - idea
                 Retailer, not producer sells to consumers.
                 Assumptions:
                     –   Zero costs for retailing.
                     –   Producer decides on a quantity and charges a price
                         ppro to the retailer.
                     –   ppro is the retailer‘s marginal cost.
                     –   The retailer‘s MR=MC condition defines the
                         producer‘s demand function.


                                                                                        24




Harald Wiese                                                                                 8
Stratégies des entreprises




                Double marginalization - linear case
                    p(x)=a-bX
                    MCpro=ACpro=c
                    Second stage: First order condition for the retailer:
                                                   !
                                     MC = p pro = a − 2bX = MR
                    First stage: First order condition for the producer:
                                                       !
                                         MC pro = c = a − 4bX = MR pro
                          a−c                         a+c
                  X =                    ⇒ p pro* =
                           4b                           2
                                                   3a + c
                                              p* =
                                                      4
                                                                                       25




                Double marginalization - depicting
                the solution
                         p                                            Producer decides
                         a                                            on quantity and
                                                           retailer   announces ppro*
                                                                      to retailer.
                p * = 3a4+ c                               producer
                                                                      Retailer decides
                                                                      on quantity (fore-
               p pro* = a 2 c
                          +                                           seen by producer).



                                MR pro            MR = p pro ( X )
                           c

                                          a −c              a                      a
                                                                                       X
                                           4b               2b                     b
                                                                                       26




                Double marginalization - exercise
                p(X)=110-X
                c=10

                a) Calculate the price the consumers have to pay!
                b) What price would they pay if the producer sold
                directly to the consumers?

                S. : a) p M = 85
                     b) p M = 60

                                                                                       27




Harald Wiese                                                                                9
Stratégies des entreprises




               Welfare Analysis
                    Evaluation of economic policy measures
                    Welfare = consumer surplus (CS)
                            + producer surplus (PS)
                            + taxes - subsidies
                    CS = willingness to pay - price
                    PS = revenue - variable costs
                       = profit + fixed costs

                                                                                         28




               CS, PS - graphically
               p



                                                supply (=MC)
                      CS

                                                           ï       Welfare is
                      PS
                                                demand             maximized at
                                                                   the equilibrium.

                0
                                                           X

                                                                                         29




               The deadweight loss of a monopoly
                           p    Without price discrimination a monopoly
                                realizes a deadweight loss.

                                                           MC


                      pM
                                                                           Perfect
                      p∗                                       p = MC      competition
                               MR = MC
                                                                  p( X )
                                                MR
                                           XM        X∗
                                                                                X
                                                                                         30




Harald Wiese                                                                                  10
Stratégies des entreprises




               Exercise (deadweight loss)
                  Consider a monopoly where the demand is
                  given by p(X)=-2X+12. Suppose that the
                  marginal costs are given by MC=2X.
                  Calculate the deadweight loss
                       –       without price discrimination,
                       –       with perfect price discrimination.
                   S. :          DLwopd = 2
                                 DLwpd = 0
                                                                                          31




               Exercise (price cap in a monopoly)
                       p
                                                How does a price cap influence the
                                                demand and the marginal revenue curves?


                                                                   MC
               pM
               p cap

                                                                p( X )

                                                  MR
                                        X   M                                   X
                                                                                          32




               Right or wrong? Why?
                           p




                                                                     MC
                   cap
                 p
                 pM
                                                                p( X )

                                                            MR
                                                   XM                            X

                                                                                          33




Harald Wiese                                                                                   11
Stratégies des entreprises




                      Price cap and welfare
                               p

                                             additional welfare

                                                                     MC
                          pM
           p   cap
                     =p   M ,cap



                                                                    p( X )


                                                                                   X
                                              X M X M ,cap    MR
                                                                                                34




                      Taxes on profits
                                   p
                                                                                       C( X )
                                                                     R( X )


                                                             Π(X)

                                        MR
                                   pM
                                                                                         MC
                      Π(X)(1−t)
                                                                              p( X )

                                                        XM
                                                                                           X
                                                                                                35




                      Taxes on profits and welfare
                             Quantity / price unchanged
                             CS is constant
                             PS decrease; in the same extent net public
                             revenues increase
                             deadweight loss is zero




                                                                                                36




Harald Wiese                                                                                         12
Stratégies des entreprises




                Additional deadweight loss due
                to quantity tax
                     p
                                                           consumer’s surplus: A+B+C→A
                                                           producer’s surplus: T+E+F →E+B
                                          additional       net public revenue: 0 →T
                                          deadweight loss
                                     A
                pT                        C
                p
                             B                                                        MC + t
                                                                                      MC
                                     E    F

                                 T                                           p( X )
                                                 MR
                                                                                           X
                                         XT X
                                                                                                                            37




                Exercise (quantity taxes)
               A monopolist is facing a demand curve given by p(X)=a-X.
               The monopoly’s unit production cost is given by c>0.
               Now, suppose that the government imposes a specific tax
               of t dollars per unit sold.
               a) Show that this tax would raise the price paid by
               consumers by less than t.
               b) Would your answer change if the market inverse demand
               curve is given by p(X)=-ln(X)+5.
               c) If the demand curve is given by p(X)=X-1/2, what is the
               influence on price?
                      1                                                                        Industrial Organization ; Oz Shy
               S. : a) , b) 1, c) 2
                      2                                                                                                     38




                Illustrating the solutions
                         p
               a)                                                   b) p



               pM(c+t)
                                                                   pM(c+t)
                 pM(c)                                   c+t                                                c+t

                                                p( X )               p (c)
                                                                       M                              p( X )
                                                                                        MR                    c
                                                               c

                                                                   X
                                              MR
                                                                                                                            X
                                                                                                                            39




Harald Wiese                                                                                                                      13
Stratégies des entreprises




               Lerner index of monopoly power
                First order condition:
                           !
                                                   dp    ⎛      1 ⎞
                MC ( X ) = MR ( X ) = p( X ) + X      = p⎜1 −        ⎟
                                                   dX    ⎜    ε X ,p ⎟
                                                         ⎝           ⎠
                Lerner index:
                                ⎛    1        ⎞
                          p − p ⎜1 −          ⎟
                 p − MC         ⎜ ε           ⎟
                        !
                        =       ⎝    X,p      ⎠= 1
                    p             p             εX ,p

                                                                      40




               Monopoly profits
               and monopoly power
                       p
                               AC             pM = AC ⇒ ΠM = 0


                                    Cournot
                  pM                  point


                                                             MC

                                                    p( X )
                                         MR
                                                              X
                                    XM
                                                                      41




               Executive summary
                A profit-maximizing monopolist always sets the
                quantity in the elastic region of the demand curve.
                Monopolistic power: price will be set above the
                marginal cost by a profit maximizer.
                If the demand curve is tangent to the average cost
                curve, the profit-maximizing price is set above
                marginal cost and equal to average cost.
                ⇒ monopolistic power and zero profits
                Monopolistic quantities without price
                discrimination (!) lead to a welfare loss.
                A quantity tax leads to a welfare loss, a tax on
                profits does not.                                     42




Harald Wiese                                                               14

				
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