10.
a. To solve this problem, we find the MR and MC functions, set them equal to each other,
and solve for the optimal Q. Using this Q, we then find the optimal P.
Given functions as follows
TC = 500,000 + 0.85Q + 0.015Q2
Q = 14,166 – 16.6P
So, P = 853.37 - 0.06Q
TR = PQ = Q(853.37 – 0.06Q) = 853.37Q - 0.06Q2
MR = δTR/δQ = δ(853.37Q - 0.06Q2)/δQ = 853.37 - 0.12Q
MC = δTC/δQ = δ(500,000 + 0.85Q + 0.015Q2)/δQ = 0.85 + 0.03Q
Short run profit maximizing rule is MR = MC, So we can write
853.37 - 0.12Q = .85 + 0.03Q
Q* = 5683.466 (Profit maximizing output level)
P = 853.37 - 0.06(5683.466)
P* = $512.36 (profit maximizing price)
b.
$512.36 = P*
D
MR
AC
$
AVC
MC
3000
0
4000
Q* =
5683.47
5000
Q
6000
7000