Ordering the Extraction of Polluting Nonrenewable Resources

Transcription

1 Web Appendi for Ordering the Etraction of Polluting Nonrenewable Resources Appendi A: Maimizing umulative Etraction of oal along a otelling Path implies that the Stock of Pollution is Always at the eiling For,, we have a common scarcity rent λ ( p, p.onsider ( ( ρt the otelling path d( λ ( e, written in reduced form as d ( t, over the time interval, Δ during which p( t ρt λ ( e increases from ( λ to p. Then ln p lnλ ( Δ. Let the etraction sequence be given by{, } such ρ that + d( t. We show that among paths starting from Z Z and satisfying the constraint Z( t Z, Z( Δ Z, maimizing the etraction of coal given a stock of gas implies that the ceiling constraint is always binding. That is, we must have Z ( t Z over the entire interval, Δ ]. Define ˆ ( t and ˆ ( t to be the etraction rates of gas and coal when the pollution stock is at the ceiling as defined in equation (3. The maimization problem can be written as: Maimize Δ ( t dt subject to ( t θ ( d( t ( t + θ ( t Z( t, Z α Z ZZ, Zt (, t, ], Z( Z Δ Δ, d ( t ( t, ( t.

2 The Lagrangian can be written as L π + γ. + θ( d + θ αz] + ν Z Z] + γ d ] The first order conditions are L + (A π θ θ] γ γ, (A L π π απ + ν, Z (A3 ν, ν Z Z ], (A4 γ, γ ( d, and (A5 γ, γ. The shadow price of pollutionπ must be non-positive and continuous. First we show that ˆ ( t and ( t,t (, ˆ Δ satisfy the above first order conditions (A-A5. Since ˆ ( t (,d( t, both γ ( t, γ ( t equal zero. Then (A implies π <. Substituting in (A and using the fact thatπ is a constant yields α ν >,t (, Δ. We show that if Z < Z over any interval ( t, t, Δ] such that along its boundary, ( t Z Z( t, it leads to a contradiction. From (A and (A3, we have Z α ( t t (A6 π t π( t e, t ( t,t ( There are five possible cases:

3 (i. t and t 3 : t < t and t < t 3 Δ such that Z( t Z for t,t ] t,t ]. t 3 Then by definition, ( t ˆ ( t henceπ( t for t t,t ] t,t3 ]. By α( t t (A6, we have lim π( t e t t < θ θ, which implies a discontinuity in the path of π at t t < Δ, a contradiction since it must be continuous. (ii. t and t3 : t < t3 Δ with Z( t Z,t t,t3 ]. We then have π ( t e θ θ αt π where ( π is written asπ. Thus for (,t, α ( t t α ( t t t + π( t θ θ ] + π( t e ( θ θ e >. From (A, (A4 and (A5, γ ( t > hence ( t d( t > since Z Z. This implies Z( t > Z,t (,t a contradiction. (iii. t : t < t with Z( t Z,t t,t ], t Δ. By (A6, π( t e α ( t t so + π ( t θ θ ] < hence γ ( t > so that by (A5, ( t >. Since Z( t Z and ( t d( t <, t ( t, t, we have lim Z( t Z < t Δ which violates the terminal condition Z( Δ Z. αt (iv. t,t Δ. This yields π( t π e,t (, Δ. Suppose π >. Then π θ θ] + >. Since π is continuous (or by (A6 there eistsε > such that + π ( ε θ θ ] >. By (A, γ ( t and we get the same contradiction as in part (ii. > Now considerπ. Then + π ( t θ θ] < so that γ ( t > and ( t,t (,. But ( t can be positive over some subinterval of, Δ and Δ ( since we maimize its integral over, Δ ], it must be. ence a contradiction. 3

4 (v. Suppose Z ( t < Z ecept at a finite number of discrete points t t.. tn < Δ such that Z( t i Z,i,,.., n. At time t n the problem becomes the same initial problem but restricted to the interval t, ] The argument from case (iv applies. Then proceed by backward induction. n Δ Appendi B: haracterization of Optimal otelling Paths Starting from the eiling in Zone II onsider point in Fig. B. It is chosen so that the stock of coal at is higher than at A. Let A A be the translation of the AA curve through. One possible path from is to etract both resources while keeping the stock of pollution at the ceiling, Z ( t Z. This is a otelling path in which the common scarcity rent λ corresponds to the initial aggregate stock at. This rent must equal the one starting from point D on the AA curve, since the global aggregate stock is equal for both and resources are perfect substitutes and the paths are otelling. From, etraction proceeds along the A curve. At any point on this curve, etraction rates are eactly equal to the corresponding points on the AA curve obtained by drawing a 45 line as shown for. This program ends at at A which has the same aggregate stock A is p, although the residual coal stock is lower than from as in point A. The price of energy. In general, any path A to the origin may now be followed provided the ceiling constraint is not violated ρt and + d( λ e. Fig. B here] Since the vector of endowments at is under the AA curve, there eist alternative etraction sequences that will not violate the ceiling. For eample, we may use only natural gas at first. Since the aggregate endowment at is strictly lower than ( lies left of the 45 line through B, scarcity rent will be higher at and the 4

5 etraction rate of natural gas ( t d( ( lower than. This path may cross the λ AA curve and reach a point such as E. As the price of the resource increases, etraction of gas decreases, and the stock of pollution also decreases. At E the stock of pollution is lower than the ceiling and thus smaller than on the path A. Since the aggregate resource stock is higher at E relative to A, the common shadow price is lower. oal can be used at rates higher than beginning from E to go back to the AA curve at point F. The stock of pollution will rise from E towards point F. The lengths of these two periods can be so chosen that the pollution stock at F is eactly Z. The remaining path may follow the curve AA until A followed by coal use until ehaustion. Yet another alternative may be to use gas until point G on the AA curve where Z ( t < Z, then use coal until some location where Z ( t Z. From there etraction can follow the translation of the AA curve through. Alternative sequences are possible including single or joint use of the two resources such that Z is not eceeded. Once the vector of stocks achieves the boundary AB of zone I, the proportion of each resource that can be used in response to the common scarcity rent is no longer restrained. For instance, from location A, only coal can be used until ehaustion, and the ceiling will not be violated. This is not possible for initial coal stocks larger than such as from point. An important feature of etraction from any location in zone II is that the residual vector of stocks must stay either on or above the A curve for some initial period. Paths such as JK are not allowed since they imply etraction of the polluting resource at rates higher than and violation of the ceiling constraint. Appendi : Determining Optimal Paths for Initial Endowments in Zones III, IV and V. Initial Endowments in Zone III 5

6 On the line A B (see Fig. aggregate stocks of the two resources must sum to. The common value of the initial scarcity rent λ equals p. onsider point D in zone III, with stocks D ( D,D and point D ( D,D on line A B with D > D. Then starting from D gas is consumed first at the maimal rate over a time D D interval Δ until D is reached. The price of energy at D is p. The initial value ρδ of the common scarcity rent is λ p e. In this first period, λ ( t p( t p θ, we have λ( t μ( t p e Δ t. Since ρ ( Δ t p e ], t, Δ μ( t θ. t Δ, + All points on any line parallel to A B must have the same scarcity rent as well as the same length of the first period when only gas is etracted. The further right the location of this line, the lower the scarcity rent, the longer is the period of gas etraction and higher in absolute terms the starting value of μ ( t. Fig. here]. Initial Endowments in Zone V onsider an initial endowment F (Fig. detailed in Fig., with endowments (,F such that F >. oal is used at the maimum rate until the stock decreases to. The energy price is constant at p. The length of this phase is given F ρδ by Δ. The initial scarcity rent of coal isλ p e. The larger the value of F, the longer is the duration of this phase and smaller the scarcity rent of coal. The lncr ln p net phase is pure otelling of duration Δ. For a phase with joint use to ρ occur at the beginning, initial resource endowments must be higher than at F. 6

7 Fig. here] Suppose Δ is the duration of this first phase starting from G. Then the additional stocks required for the segment G to F are given by i ( t dt,i, where i ( t are Δ given by (3. The maimum length of this phase is ln p ln ρ p because the initial price of energy is p and the final price p. onsider point with a higher stock of coal. Then starting from, the duration of the intermediate phase Δ will be longer. Moreover, consider point K with the same stock of gas as in G. The duration of the phase from K to is eactly the same as from G to F. The consumption of the two resources is eactly equal and the stock of pollution is at the ceiling. owever, the resource prices and ln p ln p scarcity rents are not equal. The maimum length of this phase is which ρ corresponds to starting stocks at F and G. Thus the curve is a vertical translation of F F and of A which of course has no intermediate phase Δ. During the period when resources are jointly etracted, their marginal cost must be equal, i.e., λ ( t μ( t θ λ( t μ( t θ. Define the terminal time for this phase as Δ. Then p( t p e Δ t. As in (3, we can write θ d( p e θ Δ t θ ] and θ d( p e Δ t ] d. Thus <,lim ( t and t Δ dt d dt,lim t ( t > Δ. Note that etraction depends upon the time variable Δ t and not on calendar time. Equating the marginal costs of gas and coal at time Δ givesλ ( Δ μ( Δ θ λ( Δ μ( Δ θ. Since the shadow prices and μ all grow at the rate of discount, we have λ λ μ θ. Substituting the initial value of ( θ Δ + Δ the scarcity rent of coal given by λ p e, we get 7

8 ρδ ρδ ρδ pe ( e μ( Δ e θ μ. For points located on the A curve where Δ, we have μ so that λ regulation is non-binding. λ. Both resources are perfect substitutes and Finally we show that coal is used eclusively beyond Δ, i.e., the marginal cost of gas is higher than p in the interval ( Δ, Δ + Δ. In this period, we have λ( t μ( t θ p λ( Δ e μ( t θ p λ( Δ μ( Δ ( θ θ ]e μ( Δ e θ p λ ( Δ μ( Δ θ ]e p e p p ( e >. In the final p interval ( Δ + Δ, Δ + Δ + regulation is no longer active hence μ ( t. The Δ marginal cost of the resource is its scarcity rent. Since λ t > ( t, coal is cheaper than natural gas. ( λ onsider the vertical line through 3. Initial Endowments in Zone IV (points such as A,F, in Fig. where the resource price is p and the phase of joint use at the ceiling is of maimum duration and points with a higher stock of gas, such as ( E,F. The path must be dynamically E consistent and we already know the etraction sequence from location F. We show that only gas is consumed at its maimum rate from E to F. The duration of this phase is E given by Δ. We only need to show that the marginal cost of coal is higher than that of gas which equals p in this period. The proof mimics the one above but on the interval, Δ. We have λ ( t μ( t θ p λ ( Δ e μ( t θ λ ( Δ + μ( Δ ( ]e p μ( Δ e θ p λ ( Δ μ( Δ θ ]e p e p p ( e >. p 8

9 Figures for the Web Appendi A E F A G D K J 45 B B Fig.B. otelling paths starting from Zone II D D D A D D B D B D Δ Fig.. Iso-scarcity rent loci in Zone III D 9

10 I Zone V Zone IV I K F E K F E G F F G A G K E I Fig.. Endowments in zones IV and V