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Stackelberg's Model

   Stackelberg's Model

Illustration 83

Provided the demand function, R = 30 – V and the two industries A and B in a firm manufacturing a commodity. Marginal cost of manufacturing of every industry is zero. Presume industry A performs as a Stackelberg’s leader and industry B as its follower.

What productivity will be manufactured by them and what price will be fixed?

Solution

Follower industry B makes its decision after leader industry ‘A’ presuming leader industry’s productivity as provided as invariable. Here the invariable term of the demand function x = 30 and incline of the demand function y = 1.

Reaction function of the follower industry is:

                        V2       =          x – yV1
                                                    2y

Or                    V2       =          30 – V1
                                                     2

                                    =          15 – ½ V1                               ….. (1)

Aggregate Revenue or Total Revenue TR of the leader industry A is

TRa     =          R.V1   =          30V1 – V1. (V1+V2)

                                    =          30V1 – V1^2 – V1V2                        ….. (2)

Substituting V2 = 15 – ½ V1 in total revenue function equation (2) of the leader industry we have

                        TRa     =          30 V1 – V1^2 – V1 (15 – ½ V1)

                                    =          30V1 – V1^2 – 15V1 + ½ V1^2

                                    =          15V1 – ½ V1^2                      ….. (3)

Marginal Revenue MRa          d (TRa) of the leader industry A.
                                                d V1

                                    =          15 – V1                                   ….. (4)

Predetermining MR1 equal to marginal cost which is null, we have,

                        15 – V1           =          0

                                    V1*     =          15

Now, substituting the value of leader productivity V1 in the reaction function equation (1) of the follower industry (2), we have

                                    V2*     =          15 – ½ V1

                                                =          15 – ½ (15)

                                                =          15 – 7.5

                                    V2*     =          7.5

Therefore, the leader industry’s manufacture is 15 and adherent industry’s productivity is 7.5. Therefore, total productivity V* is 15 + 7.5 = 22.5.

To procure price we substitute the total productivity of 22.5 in the provided demand function. Therefore,

                        R         =          15 – V

                                    =          30 – 22.5         =          7.5

Illustration 84

The market demand curve for a Stackelberg’s leader and adherent is provided by R = 20 – V. If each one has a marginal cost of 4, what will be the symmetry volume and rate for each commodity?

Solution

In this exercise we are provided a positive marginal cost instead of zero marginal cost.

Follower’s Reaction Function: Let us first derive follower’s reaction function. The manufacturer B is the adherent.

                                    R         =          20 – V

Or,                               R         =          20 – (V1 + V2)                       ….. (1)

Incrementing the demand function equation (1) by the productivity V2 of manufacturer B to procure aggregate revenue function, we have,

                                    TRb     =          RV2    =          20V2 – V1V2 – V2^2

                                    MRb    =          20 – V1 – 2V2
Fixing MRb equal to marginal cost we get the reaction function for the adherent. Therefore,

                                    20 – V1 – 2V2                        =          4

                                                            2V2     =          20 – 4 – V1

                                                            V2       =          16 – V1
                                                                                         2

                                                            V2       =          8 – ½ V1         ….. (2)

Equation (2) explains the reaction function equation for the adherent.

Leader’s symmetry: Demand function for the leader is

            R         =          20 – (V1 + V2)

Incrementing both sides by productivity V1 of leader we have

            TRa     =          RV1    =          20V1 – V1^2 – V1V2                        ….. (3)

Substituting the value of V2   =          8 – ½ V1 in TRa equation (3), we get,

            TRa     =          20V1 – V1^2 – V1 (8 – ½ V1)

                        =          20V1 – V1^2 – 8V1 + ½ V1^2

                        =          12V1 – ½ V1^2

            MRa    =          d TRa  =          12 – V1                                   ….. (4)
                                      dV

Fixing MRa equal to marginal cost MC, We get,

                        12 – V1           =          4

                                    V1       =          12 – 4  =          8

Therefore, the leader manufactures 8 units of the commodity.

Substituting V1 = 8 in the adherent’s function equation (2), we get,

                        V2       =          8 – ½ (8)          =          4

Therefore, the adherent will manufacture 4 units of the commodity. Total Productivity 8 + 4 = 12. Substituting the total productivity manufactured in the provided demand function we get price at which the commodity will be sold.

Therefore,      R         =          20 – V = 20 – 12         =          8.

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