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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