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NRE02 03 A non renewable resource multi period model - YouTube
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we now switch to continuous time
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framework in order to consider multiple
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periods and we we define we P as the net
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price directly to keep a bit to the
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mathematic samples the other things we
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do now is that we consider a generic fan
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of functions demand function we are
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abandoning the assumption of linearity
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we made in the two periods model which
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also in the previous slides and given
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that we can state that the utility as a
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functions of the expression of of the
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resource is just the integral of from
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zero to the level of extraction of the
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impacts demand functions in terms of the
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net price if we go back a bit back a bit
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is basically when we consider the net
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price all this price is go down all the
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price of the extraction so it become
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apparent up to here this one become the
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area but we are considering from here
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till till here in times of net price and
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so if we differentiate both sides with
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respect to the extraction of the natural
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resource we end up as a conditions for
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the maximizations
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that the marginal social utility of the
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resource must be equal to the net price
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and if you see this one in this flow
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shot we see pretty clearly that the
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trade-off again is to choose how well to
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stop
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to extract the resource and we stop at
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each moment in time when the benefits
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were we obtained from the marginal
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benefit that we obtained from extracting
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the resource that is the marginal
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utility in this very simple flow chart
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it's equal to the value of the resource
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in situ with so with the net value net
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of extraction cost when we apply to this
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problem intertemporal a social welfare
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functions of Italian forms we define
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welfare the integral from zero to a
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given time T of the utility at each
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moment in time so here actually we have
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a double integral we have one integral
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over time and here we said that the
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utility is an integral over over the
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resource extractions but we don't need
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to consider this one we just considered
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a but here is whatever functions of the
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level of extraction are in any given
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point in x discounted by the social
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discount rate and our objective is to
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find the level of extraction of the
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resources at any given point in time as
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to maximize this valpha and as before we
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have the constraints that the wall
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initial stock of resources must be
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extracted in this given framework in
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this given time period and so the
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devaluation of the stock must be equal
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to the negative of the extractions and
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it is important to notice that
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differently from our previous lessons
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the in the problem of what we are
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examine a ting here the time at which it
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is optimal to stop extracting the
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resource it is endogenous so we have
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also to determine so before in the
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problem on the on the previous lesson we
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had to choose the level of extractions
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in x and because we had specific
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explicit production function we had also
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to choose the level of our consumption
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in x but we had time fixes as infinite
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yeah we don't have this one as a control
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variable we have left only we with the
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resource but we have conversely to
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determine which is exactly this amount
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of time that maximizes the wildfire
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