Find the producer's surplus at Q = 9 for the following supply functions:
(a) P = 12 + 2Q
(b) P = 20√Q + 15
a) =P=12+2QP=12+2QP=12+2Q
At Q=9,P=12+18=30
Therefore producer's surplus =P.Q−∫PdQP. Q-\int PdQP.Q−∫PdQ
=(30)(9)−∫(12+2Q)dQ=(30)(9)-\int(12+2Q)dQ=(30)(9)−∫(12+2Q)dQ
=270−[12Q+Q2]09=270-[12Q+Q^2] ^9_0=270−[12Q+Q2]09
=270−[108+81]=270-[108+81]=270−[108+81]
=81=81=81
b) P=20Q+15P=20\sqrt{Q}+15P=20Q+15
At Q=9,P=209+15=75P=20\sqrt{9}+15 =75P=209+15=75
Producer's Surplus =PQ−∫PdQPQ-\int PdQPQ−∫PdQ
=(75)(9)−∫20Q+15dQ=(75)(9)-\int 20\sqrt{Q}+15 dQ=(75)(9)−∫20Q+15dQ
=675−[20Q3232+15Q]09=675-[20\frac {Q^{\frac {3} {2} }}{\frac {3} {2}} +15Q]^9_0=675−[2023Q23+15Q]09
=675−495=675-495=675−495
=180=180=180
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