Answer:
The marginal revenue is [tex]MT_d = 116.66[/tex]
Step-by-step explanation:
From the question we are told that
The inverse demand for a downstream firm is [tex]P = 150 -Q[/tex]
The cost of the critical input produced by upstream division is [tex]CU(Q_d) = 5(Q_d)^2[/tex]
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
[tex]MR_d - MC_d = MT_d[/tex]
Also
The marginal revenue of the downstream firm - The marginal cost of the downstream firm = Marginal upstream cost
i.e
[tex]MR_d - MC_d = MC_u[/tex]
So
[tex]MR_d - MC_d = MC_u[/tex]
Generally the total revenue of downstream firm is mathematically represented as
[tex]TR = P * Q[/tex]
Here Q stands for quantity produced by the downstream firm and TR is the total revenue
[tex]TR = [150 - Q] * Q[/tex]
=> [tex]TR = 150Q - Q^2[/tex]
Generally the marginal revenue of the downstream firm is mathematically evaluated as
[tex]MR_d =\frac{d (TR)}{d Q} = 150 - 2Q[/tex]
Generally marginal downstream first cost is mathematically evaluated as
[tex]MC_d = \frac{d(Cd(Q)) }{dQ} = 10[/tex]
Generally the net marginal revenue of the downstream firm is mathematically represented as
[tex]150 - 2Q -10 = MT_d[/tex]
=> [tex]MT_d = 140 - 2Q[/tex]
Generally the marginal upstream cost is mathematically represented a
[tex]MC _u =\frac{d [CU(Q_d)]}{dQ_d} = 10(Q_d)[/tex]
Generally [tex]Q_d = Q[/tex] this because [tex]Q_d[/tex] represents the quantity produced by the downstream firm and also Q is associated with the cost of the downstream quantity
So
[tex]MC _u =\frac{d [CU(Q)]}{dQ} = 10Q[/tex]
=> [tex]10(Q) = 140 -2Q[/tex]
=> [tex]Q = 11.67[/tex]
So the net marginal revenue of the downstream firm is mathematically represented as
=> [tex]MT_d = 140 - 2(11.67)[/tex]
=> [tex]MT_d = 116.66[/tex]
