Question

Suppose that the inverse demand for a downstream firm is P = 150 − Q. Its upstream division produces a critical input with costs of CU(Qd) = 5(Qd)2. The downstream firm's cost is Cd(Q) = 10Q. When there is no external market for the downstream firm's critical input, the marginal revenue for the downstream firm is:

Answer 1

The marginal revenue is
`MT_d = 116.66`

Explanation:

From the question we are told that

The inverse demand for a downstream firm is
`P = 150 -Q`

The cost of the critical input produced by upstream division is
`CU(Q_d) = 5(Q_d)^2`

The cost of the critical input produced by downstream firm is Cd(Q) = 10Q

Generally

The marginal revenue of the downstream firm - The marginal cost of the downstream firm = Net marginal revenue of downstream

i.e

`MR_d - MC_d = MT_d`

Also

The marginal revenue of the downstream firm - The marginal cost of the downstream firm = Marginal upstream cost

i.e

`MR_d - MC_d = MC_u`

So

`MR_d - MC_d = MC_u`

Generally the total revenue of downstream firm is mathematically represented as

`TR = P * Q`

Here Q stands for quantity produced by the downstream firm and TR is the total revenue

`TR = [150 - Q] * Q`

=>
`TR = 150Q - Q^2`

Generally the marginal revenue of the downstream firm is mathematically evaluated as

`MR_d =(d (TR))/(d Q) = 150 - 2Q`

Generally marginal downstream first cost is mathematically evaluated as

`MC_d = (d(Cd(Q)) )/(dQ) = 10`

Generally the net marginal revenue of the downstream firm is mathematically represented as

`150 - 2Q -10 = MT_d`

=>
`MT_d = 140 - 2Q`

Generally the marginal upstream cost is mathematically represented a

`MC _u =(d [CU(Q_d)])/(dQ_d) = 10(Q_d)`

Generally
`Q_d = Q` this because
`Q_d` represents the quantity produced by the downstream firm and also Q is associated with the cost of the downstream quantity

So

`MC _u =(d [CU(Q)])/(dQ) = 10Q`

=>
`10(Q) = 140 -2Q`

=>
`Q = 11.67`

So the net marginal revenue of the downstream firm is mathematically represented as

=>
`MT_d = 140 - 2(11.67)`

=>
`MT_d = 116.66`