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Pricing Of Multiple Products

  Pricing of Multiple Products

Illustration 89

Presume an industry in cattle raising and manufacturing joint products Ham and hides. The demand functions of two commodities are provided as under:

Ham: Rx = 120 – 0.5Vx and Hides: Ry = 140 – 4Vy where Rx is price of ham and Ry is rate of hides and Vx and Vy are volume of the two commodities respectively.

Moreover, presume that marginal cost of manufacturing of joint commodities of ham and hides is given by the following:

                        MC      =          80 + V

How many units of commodities be manufactured and what prices must be fixed for ham and hides?

Solution

Let X stand for Ham and Y be Hides. In order to ascertain the profit optimising level of productivity of the package of joint commodities we have to procure marginal revenues of the two commodities.

Therefore,

                        TRx     =          RxVx  =          (120 – 0.5Vx) Vx

                                                           =          120Vx – 0.5Vx^2

                        MRx    =          dTRx   =          120 – Vx                                 ….. (1)
                                                 dVx

                        TRy     =          RyVy   =          (140 – 4Vy) Vy

                                                           =          140Vy – 4Vy^2

                        MRy    =          dTRy   =          140 – 8Vy                               ….. (2)
                                                 dVy

Summing up vertically the two marginal revenues, we have

                        MRt     =          MRx + MRy

                                    =          120 – Vx + 140 – 8Vy

                                    =          260 – Vx – 8Vy

As the two are joint commodities we can write the package of Vx and Vy as V, hence,

                                    =          260 – 9V

To procure profit optimising productivity of the package of joint commodity we fix,

                                                MRt     =          MC

                                                MC      =          80 + V

Therefore,                                MRt     =          MC

                                    260 – 9V         =          80 + V

                                                180      =          10V

                                                V*       =          180 / 10           =          18

Or, 18 units of the commodity package.

Substituting the value of symmetry volume V* in the demand functions of the two commodities, we have

                                                Rx       =          120 – 0.5(18)

                                                            =          120 – 9

                                                            =          111

                                                Ry       =          140 – 4(18)

                                                            =          140 – 72

                                                            =          68

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