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This 0 page Class Notes was uploaded by Jamir Zieme Sr. on Monday November 2, 2015. The Class Notes belongs to ECON4306 at California State University - East Bay taught by AnthonyLima in Fall. Since its upload, it has received 15 views. For similar materials see /class/234381/econ4306-california-state-university-east-bay in Economcs at California State University - East Bay.
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Date Created: 11/02/15
Econormcs 4306 Envlmnmmtal Economics Ton Lima Apnl 28 2009 Mathematical Details of the TwoiPeriud Dynamic 0 39 ization Process Used in the Appendices to Chapters 2 and 7 EVERYTHING IN THIS DOCUMENT IS OPTIONAL I have put it on the course web site at the request of several en 1 The simple twperind mndel R 0 b4 MC c Economics 4306 Twoperiod dynamic optimization model So total surplus is the area to the left of q above MC and below the WTP line The text calculates this as total spending area aquO minus total cost cqt We know the price at qo will be a 7 bqo from the inverse demand curve Therefore the area of the rectangle is a 7 bqo X qo 7 0 aqo 7 bqoz The area ofthe triangle is a 7 a 7 bq0 X qo 7 02 bqozZ We just need to add these two up The following equations include the calculations of both areas bqoxqo 0 2 ago 47902 faabqo MoO z 2 2 Total WTP 2 ago 7112 bqoz ago Lq02 2 2 2 I otdeIPzaqo b gf Total cost cqO Net bene t 2 ago quoz cq0 Let s first look at the twoperiod case We know that the total consumed must equal q1 qz Therefore Q qz Qq1 Economics 4306 Twoperiod dynamic optimization model We ll use the last equation to help us maximize the function With only two periods we can reduce the model to a single decision variable q1 b 2 Penod 1 net bene ts aq1 1 cq1 b 2 aqz 62 0 12 Period 2 net benefits T 1quot Substituting q2 Q q1 2 I Q q aQ CQql Period 2 net benefits 1 r 2 2 aQWh WCQC 1r ZaQ Zaq1 sz 2qu1 bql2 ZCQ 2ch 2 1 r 2quot sz ZCQ 2 2qu1 bq12 2ch 2 1 r The present value of net benefits is 2aQbQ2 ZCQ2aq1 2qu1 bq12 20 39 21r Taking the derivative with respect to q1 gives dNPV abql C ZbQ Za qu1 20 dql 21r 7Q a bq1 c 1r 2 NPVaq1 b cq1 a bql c Economics 4306 Twoperiod dynamic optimization model Now we just set that derivative equal to zero and solve for q1 bQ a bq1 6 0 1r a c1rbQ ac 96111rbq1 1r 1r a c1rbQ acbql1rbq1 a car crbQ acqubqlrbq1 a crbQq1b2r a bql c a crbQ a c rb 1 sz b2rQ Qb2r a cr bQ 12 b2r 2bQer a cr bQ 12 b2r q bQer a cr b2r Economics 4306 Twoperiod dynamic optimization model Let s check to make sure we re consuming all the Q a crbQ I bQer a cr 611 612 b2r b2r 2bQer 12 b2r Qb2r q1q2 b2r Q
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