# 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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