Question

Suppose that the duopolists Max and Sam face an inverse demand function for apples of P = 100 - 30, where Q = QM + Qs is the total number of apples that reach the market and P is the price of apples. Suppose further that Max's cost function is CM(QM) = 4QM and Sam's cost function is Cs(Qs) = 16Qs. In the Cournot-Nash equilibrium, Sam's production is ____.

Answer 1

The answer is 6

Step-by-step explanation:

From the question given we find the solution to Sam's production

Recall that,

P = 100-3Qm - 3Qs

The max

TRm= PQm = 100 Qm -3Qm²- 3QmQs

MRm = 100 - 6Qm - 3Qs

Cm (Qm) = 4Qm

MCm = 4

Then,

We put MRm =MCm

100 - 6Qm - 3Qs = 4

96- 3Qs = 6Qm

16-0.5Qs = Qm This is taken as the equation (1)

Thus,

We solve for SAM

Sam

TRs = PQs - 3QmQs -3Qs²

MRs = 100 -3Qm -6Qs

Cs (Qs) = 16Qs

Now,

MCs =16

We put MRs =MCs

100 -3Qm -6Qs =16

84 - 3Qm = 6Qs

14 -0.5Qm = 6Qs This is the equation (2)

Now,

From the both equations (1) and (2) respectively,

Qs = 14 -0.5Qm = 14 -0.5 (16 -0.5 Qs)

Qs = 14- 8 + 0.25Qs

Thus,

0.75Qs = 6

Therefore Qs = 6

Sam's production is 6