Question

Suppose that the inverse demand for a downstream firm is P = -82 − 2Q. Its upstream division produces a critical input with costs of CU(Qd) = 3(Qd)2. The downstream firm's cost is Cd(Q) = 2Q. When there is no external market for the downstream firm's critical input, the downstream firm should produce:

Answer 1

Q=8 Units

Explanation:

FIrst from the Critical cost if we take first derivative we get marginal upcost i-e

`CU(Qd) = 3(Qd)^2`

taking first derivative we get

`MC_U = 3 \cdot2 Q`

`MC_U = 6`

similarly taking derivative of downstream we get

`MC_d = 2Q`

`MC_d = 2`

`MR= 2+6Q`

`P = 82-2Q`

`Revenue = P*Q = 82Q-2Q^2`

So
`Revenue = P*Q = 82Q-2Q^2`

`82-4Q = 2+6Q`

`10Q=82-2`

`Q=(80)/(10)`

`Q=8~Units\\`