Lopez Industries has developed a new product called Gizmo and now must decide wh

Question

Lopez Industries has developed a new product called Gizmo and now must decide whether to build a factory with the capacity to produce one or two units of the Gizmo. Its total cost function is CA=8qA + q2A. Acme estimates market demand will be Q=10-0.5P. Lopez expects that its rival, Cogswell Cogs, will very soon develop a copy-cat version of Gizmo and will very likely build a factory to produce either one or two units of their version as well. Total demand Q = qA + qC. Acme knows that Cogswell will have the same costs as it does and will play as a Cournot duopolist. What capacity will Lopez build? What capacity will Cogswell build?

Explanation / Answer

Q = qA+qC here as bost cost and price are same so qA=qC=q

TC = 8q+q^2

MC = dTC/dq = 8+2q

P = 20-2(qA+qC) (Inverse market demand)

Revenue for firm A

R = P*qA=20*qA-2qA^2-2qAqC

MR=dR/dqA = 20-4qA-2qC

Similarly marginal revenue for firm C

MR = 20 - 4qC-2qA

For firm A

MC=MR

8+2qA=20-4qA-2qC

qA = 2-qC/3

similarly from firm C

we get

qC=2-qA/3

So qA=qC=1.5

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