Appendix. M. Germain, A. Magnus, and V. van Steenberghe. December 2004.

Transcription

1 Appendix to Should developing countries participate in the Clean Development Mechanism under the Kyoto Protocol? The low-hanging fruits and baseline issues by M. Germain, A. Magnus, and V. van Steenberghe. December We give here the mathematical derivation of some propositions and statements of the paper. Contents. Preliminaries: the second period imization problem The absolute baseline case a. Proof of proposition 2.a (existence, unicity and characterization of λ a ) a.. The equation for λ a : the imum condition a.2. Existence of λ a : the first order condition a.3. Unicity of the solution of the first order condition b. Proof of Proposition 2.b (behaviour of e,, and y with respect to τ ) The relative baseline case a. Proof of proposition 3.a (existence, unicity and characterization of λ r ) b. Discussion of Result 3.b (behaviour of e,, and y with respect to τ ) c. Proof of Proposition 3.c (comparison of y and under both baselines) d. Proof of Proposition 3.d (comparison of Π under both baselines)... 8

2 Appendix to low hanging fruits etc. 2. Preliminaries: the second period imization problem. We see that the full problem (7) contains several times the same subproblem: we first have to imize with respect to a variable e 0 an expression of the form Ae γ Be+C, with A,B > 0, and γ <. The derivative γae γ B has exactly one positive zero: [ γa /[ γ e =, (A.) B which leads to Be = γae γ, so that the imum is [ γae γ + C = γ A /[ γ B γ/[ γ + C, where γ = [ γγ γ/[ γ. (A.2) We apply now this to Π 2. fig A. In Π 2 (λ,τ 2 ), for a given τ 2, A and B depend on λ 2 : A = λ β 2, B = p e+τ 2 +p λ 2, and so β[p e+τ 2 fig. A.2 β[p e+τ 2 F λ 2 Π 2 λ Π 2 (λ,τ 2 ) = λ 2 λ F(λ 2 ;τ 2 ) + C, where F(λ 2 ;τ 2 ) := e2>0 eγ 2 λ 2 [p e + τ 2 + p λ 2 e 2 = γ λ β/[ γ 2 [p e + τ 2 + p λ 2 γ/[ γ, (A.3) F(λ 2 ;τ 2 ) increases between λ 2 = 0 and λ 2 = β[p e + τ 2 /[, and decreases afterwards (see fig. A.). Therefore, when λ β[p e + τ 2 /[, the imum over λ 2 λ of F(λ 2 ;τ 2 ) is the imal value of F, i.e., F ( ) β[pe + τ 2 ;τ 2 = γ [p e + τ 2 α/[ γ p β/[ γ αp, (A.4) where γ = γ β β/[ γ α α/[ γ /γ γ/[ γ = [ γβ β/[ γ α α/[ γ ; whereas, when λ β[p e + τ 2 /[, the imum over λ 2 λ of F(λ 2 ;τ 2 ) is merely F(λ ;τ 2 ). Thus: Π 2 (λ,τ 2 ) = γ λ β/[ γ [p e + τ 2 + p λ γ/[ γ + C, if λ β[p e + τ 2 /[, (A.5) = γ [p e + τ 2 α/[ γ p β/[ γ + C if λ β[p e + τ 2 /[, Remar that Π 2 (λ,τ 2 ) is a non increasing function of λ (see figure A.2). This also holds for the integral term of (7), as this term is a positive linear combination of functions Π 2 (λ,τ 2 ) for an interval of values of τ 2.

3 Appendix to low hanging fruits etc The absolute baseline case. 2.a. Proof of proposition 2.a (existence, unicity and characterization of λ a ). 2.a.. The equation for λ a : the imum condition. We now come to the full problem (4), with an absolute baseline: e 0,λ 0 eγ λβ [p e + p λ e + τ [e e τ Emphasizing the dependence in e, using (2): λ 0 e 0 f(τ 2 ) e γ 2 e 2 0 λβ 2 [p e + p λ 2 e 2 + τ 2 [ẽ 2 δ[e e e 2 λ 2 λ e γ λβ [p e + p λ + τ ρδ τ 2 e τ f(τ 2 ) λ 2 λ e 2 0 which is first solved with respect to e : Π a (λ,τ ) := F(λ ;τ ρδ τ 2 ) + ρ λ 0 dτ 2. e γ 2 λβ 2 [p e + p λ 2 + τ 2 e 2 dτ 2 + τ e + τ 2 ρ[ẽ 2 δe τ f(τ 2 )[ F(λ 2 ;τ 2 )dτ 2 + τ e + τ 2 ρ[ẽ 2 δe, λ 2 λ using the definition of F in (A.3). Recall that the first term is an increasing function of λ as long as λ λ ref, where (A.6) λ ref := β[p e + τ ρδ τ 2, (A.7) and that the second term contains only non-increasing functions of λ. Moreover, if λ is small enough, so that the condition seen above λ β[p e + τ 2 /[ holds for all the τ 2 s in the integral, i.e., if λ β[p e + /[, all the Π 2 s are constant functions of λ, and the net result is a function Π a behaving lie the first term F plus a constant. case (2) fig. A.3 case () F+const. β[p e+ Π a λ λ ref λ a λ ref β[p e+ fig. A.4 F+const. Π a λ When λ > β[p e + /[, the sum is F plus an actually decreasing function of λ, and is smaller than the continuation of the rule F plus a constant (shown by a dashed line in the figures A.3 and A.4 nearby, whereas the actual Π a [solid line is even smaller). Therefore, () If τ τ 2min + ρδ τ 2, Π a reaches its imum value at λ = λ ref = β[p e + τ ρδ τ 2 /[, as this latter value is smaller than β[p e + /[, which is the place where a more complicated behaviour occurs, but we don t have to care, as the imum has already been encountered. (2) If τ > + ρδ τ 2, the actual imum of Π a occurs between β[p e + /[ and λ ref = β[p e +τ ρδ τ 2 /[, as the λ derivative is still positive at the first bound (F is still increasing, and the integral term still contains constant values with respect to λ ), and is negative at the second bound (the integral now contains actually decreasing functions of λ ).

4 Appendix to low hanging fruits etc. 4 We have therefore established parts (i) and (iii) of Proposition 2.a, i.e., that the imum is reached at λ a = λ ref in case (); that β[p e + τ 2min < λ a < λ ref in case (2). 2.a.2. Existence of λ a : the first order condition. In order to establish more properties of λ a when τ > τ 2min + ρδ τ 2 (case (2) above) we get a closer loo at the equation for λ a from the λ derivative Π a (λ,τ ) = τ2 F(λ ;τ ρδ τ 2 ) + ρ f(τ 2 )[ F(λ 2 ;τ 2 )dτ 2 = 0, λ λ λ 2 λ or F(λ ;τ ρδ τ 2 ) αp λ /β p e + ρ f(τ 2 ) F(λ ;τ 2 ) dτ 2 = 0, λ τ 2min λ as we saw in (A.5) that Π 2 (λ,τ 2 ) is a constant function with respect to λ while λ β[p e + τ 2 /[, and leaves therefore no trace in the integral of the λ derivative, so that we only have to consider the derivative when λ > β[p e + τ 2 /[, or τ 2 < λ /β p e in the integral. Remar that the upper bound in the integral is actually larger than the lower bound, i.e., λ /β p e > τ 2min, as λ > β[p e + /[ (case (2) above). We now perform the λ derivative of F(λ;τ) = const. λ β/[ γ 2 [p e + τ + p λ 2 γ/[ γ : βλ β/[ γ αp λ /β p e [p e + τ + p λ ρδ τ 2 γ/[ γ γp λ β/[ γ [p e + τ + p λ ρδ τ 2 γ/[ γ f(τ 2 )βλ β/[ γ [p e +τ 2 +p λ γ/[ γ γp λ β/[ γ [p e +τ 2 +p λ γ/[ γ dτ 2 = 0, The condition on λ amounts to [β[p e + τ ρδ τ 2 λ [p e + τ + p λ ρδ τ 2 /[ γ fig. A.5 G β[p e+ λ + ρ αp λ /β p e f(τ 2 )[β[p e + τ 2 λ [p e + τ 2 + p λ /[ γ dτ 2 = 0. Remar that the first term is positive, while the second term is negative when β[p e + τ 2min /[ < λ < λ ref = β[p e + τ ρδ τ 2 /[. Managing to eep only positive terms: G(λ,τ ) =, where λ ref αp λ /β p e λ β[p e + τ 2 [p e + τ + p λ ρδ τ 2 /[ γ G(λ,τ ) = ρ f(τ 2 ) β[p e + τ ρδ τ 2 λ [p e + τ 2 + p λ /[ γ dτ 2. (A.8) G(λ,τ ) = 0 when λ = β[p e + /[, and G(λ,τ ) + when λ β[p e + τ ρδ τ 2 /[, so there must be at least one intermediate value of λ where G = (see fig. A.5). This is another way to ensure parts (i) and (iii) of Proposition 2.a, i.e., the existence of a solution β[p e + /[ < λ < λ ref = β[p e + τ ρδ τ 2 /[.

5 Appendix to low hanging fruits etc. 5 2.a.3. Unicity of the solution of the first order condition. Π a? fig. A.6 λ When λ imizes (7), the equation G = is certainly solved. The converse is not clear, as solutions of G =, i.e., of Π a / λ = 0, may be any ind of stationary point for (7) (see fig. A.6). The situation is non ambiguous when G is a monotonous function of λ, as G = can then have at most one root. We will loo for a condition ensuring that G(λ,τ ) is an increasing function of λ. Thus, we have to consider [ G(λ,τ ) λ αp λ /β p e = ρ f(τ 2 )[τ ρδ τ 2 τ 2 p γ αβ[ γ [ λ β[p e + τ 2 [β[p e + τ ρδ τ 2 λ [p e + τ + p λ ρδ τ 2 [p e + τ 2 + p λ λ β[p e + τ 2 [p e + τ + p λ ρδ τ 2 /[ γ β[p e + τ ρδ τ 2 λ [p e + τ 2 + p λ /[ γ dτ 2 τ ρδ τ 2 τ 2 is positive in the whole integration interval, as τ 2 < λ /β p e, so λ > β[p e + τ 2 /[, and we saw that λ < β[p e + τ ρδ τ 2 /[. αp We also see that the ratio λ β[p e + τ 2 is positive for τ β[p e + τ ρδ τ 2 λ 2 in the (open) integration interval. To be sure that G is increasing, it remains to show that the big intermediate factor is positive too, or that αβ[ γ > λ β[p e + τ 2 β[p e + τ ρδ τ 2 λ p e + τ 2 + p λ p e + τ + p λ ρδ τ 2 on the whole τ 2 interval, i.e., at τ 2 = τ 2min, as the right-hand side is a decreasing function of τ 2. And as we do not now more about λ that β[p e + τ 2min /[ < λ < β[p e + τ ρδ τ 2 /[, we tae the first fraction of the right hand side at its largest possible value, which is reached at λ = β[p e +τ ρδ τ 2 /[, and so for the second fraction, but this one at λ = β[p e + /[. The sufficient condition of unicity is then αβ[ γ > α 2 β 2 [τ ρδ τ 2 2 [α[p e + τ 2min + β[p e + τ ρδ τ 2 [β[p e + τ 2min + α[p e + τ ρδ τ 2 which depends only on nown values. The condition can be rewored as where x = τ ρδ τ 2 p e +, or γ < stronger sufficient condition γ > αβx 2 [γ + βx[γ + αx, + αβ x 2, or, as αβ is always smaller than γ 2 /4, a γ 2 + x γ < + x + x + x 2 /4. (A.9) For a given x (or, equivalently, τ ), returns to scale must be sufficiently decreasing (γ sufficiently low), and conversely. So that part (ii) of Proposition 2.a is established.

6 Appendix to low hanging fruits etc. 6 2.b. Proof of Proposition 2.b (behaviour of e,, and y with respect to τ ). We now discuss e,, and y, which are positive powers of λ divided by powers of p λ + p e + τ ρδ τ 2 : constant λ a /[p λ + p e + τ ρδ τ 2 b, where the coefficients a and b are a b e γ β γ α y γ β γ Remar that a < b. Things are simple when τ + ρδ τ 2, as exact formulas are available: λ = β[p e + τ ρδ τ 2 /[, so that λ a [p λ + p e + τ ρδ τ 2 b = constant [p e + τ ρδ τ 2 a b is obviously a decreasing function of τ. When τ > τ 2min + ρδ τ 2, things are less explicit. We discuss first variations of λ with respect to τ. As G must eep a fixed value G =, dg = [ G/ λ dλ + [ G/ τ dτ = 0, so, dλ /dτ = [ G/ τ /[ G/ λ. As seen above, G being an increasing function of λ, G/ λ > 0 (see fig. A.5). For a fixed λ, G(λ,τ ) will increase or decrease according to the sign of G/ τ which, from (A.8), is the sign of [p e + τ + p λ ρδ τ 2 /[ γ = γ[β[p e +τ ρδ τ 2 [ βp λ [p e + τ + p λ ρδ τ 2 γ/[ γ τ β[p e + τ ρδ τ 2 λ [β[p e + τ ρδ τ 2 λ 2. So, we see that G decreases when τ increases for a fixed λ, as long as λ β[p e +τ ρδ τ 2 /[p [ β. Therefore, λ will increase with τ as long as λ β[p e + τ ρδ τ 2 /[p [ β, and decreases if λ is lower than this bound. Let us derivate λ a /[p λ + p e + τ ρδ τ 2 b with respect to τ. The sign of the derivative is the sign of [[a bp λ + a[p e + τ ρδ τ 2 dλ dτ bλ. For e, a = β and b =, the sign of dλ /dτ is exactly the sign of [ βp λ β[p e +τ ρδ τ 2, as seen above. So, e is a decreasing function of τ, and this settles the first part of Proposition 2.b. For and y, it will be shown here that these functions of τ are smaller than their values at λ ref = β[p e + τ ρδ τ 2 /[, i.e., that [λ a a [p λ a + p e + τ ρδ τ 2 b < λ a ref [p λ ref + p e + τ ρδ τ 2 b, nowing that λ a < λ ref. Indeed, λ a /[p λ + p e + τ ρδ τ 2 b is an increasing function of λ up to λ = β[p e + τ ρδ τ 2 /[ if a[p λ + p e + τ ρδ τ 2 bp λ = [[a bβ/α + a[p e + τ ρδ τ 2 0, or aγ bβ, which holds for and y. Now, the upper bound λ a ref /[p λ ref +p e +τ ρδ τ 2 b = const. [p e + τ ρδ τ 2 a b is a decreasing function of τ, so that the remaining part of Proposition 2.b is established.

7 [ + τ e y /[ γ Appendix to low hanging fruits etc. 7 We rewrite (4) with a relative baseline: e 0,λ 0 eγ λβ [p e + p λ e + τ τ 3. The relative baseline case. [ e e y y Emphasizing the dependence in e, using () and (2): [ e + τ λ 0 e 0 y τ f(τ 2 ) e γ 2 e 2 0 λβ 2 [p e + p λ 2 e 2 + τ 2 [ẽ 2 δ[e e e 2 λ 2 λ e γ λβ [p e + p λ + τ ρδ τ 2 e f(τ 2 ) λ 2 λ which is first solved with respect to e : λ 0 Π r (λ,τ ) := e 2 0 dτ 2. e γ 2 λβ 2 [p e + p λ 2 + τ 2 e 2 dτ 2 + τ 2 ρ[ẽ 2 δe [ e /[ γ + τ F(λ,τ ρδ τ 2 ) y τ f(τ 2 ) F(λ 2,τ 2 ) dτ 2 + τ 2 [ẽ 2 δe (A.0) λ 2 λ where, as before, F is given by (A.3). Optimal emissions of period with respect to λ and τ are given by [ e r = λ β [ + τ e /y γ /[ γ p e + τ + p λ ρδ τ 2 Note that e and y are the solutions of the elementary problem when τ = 0: λ = β[p e ρδ τ 2 /[, e γ = γλ β /[p e + p λ ρδ τ 2 = βλ β /p, or e/y = e γ λ β = β/[p λ, or e = α [ β/[ γ β β/[ γ [p e ρδ τ 2 [β /[ γ p β/[ γ, (A.) e y = α p e ρδ τ 2. 3.a. Proof of proposition 3.a (existence, unicity and characterization of λ r ). (A.2) As seen before, the integral does not depend on λ if τ τ 2min + ρδ τ 2, and the imum is reached at λ = λ ref = β[p e + τ ρδ τ 2 /[, as in the absolute case. Fig. A.7 G λ r λ a λ ref When τ > + ρδ τ 2, we consider the λ derivative which is [ e /[ γ + τ [β[p e + τ ρδ τ 2 λ [p e + τ + p λ ρδ τ 2 /[ γ λ y αp λ /β p e + ρ f(τ 2 )[β[p e + τ 2 λ [p e + τ 2 + p λ /[ γ dτ 2 = 0,

8 Appendix to low hanging fruits etc. 8 or G(λ,τ ) = [ e /[ γ + τ, with the same G as above, in (A.8). As before, G(λ,τ ) = 0 when λ = β[p e + /[, and G(λ,τ ) + when λ β[p e + τ ρδ τ 2 /[, so there must be at least one intermediate value of λ where the equation for λ is satisfied: β[p e + /[ < λ < β[p e +τ ρδ τ 2 /[ (see fig. A.7). As the right-hand side is larger than, and if G is an increasing function of λ (remember the sufficient condition (A.9) γ [ + x/[/x + x 2 /4, which is still valid here), we have β[p e + τ 2min /[ < λ a < λ r < β[p e + τ ρδ τ 2 /[, and the whole Proposition 3.a is established. 3.b. Discussion of Result 3.b (behaviour of e,, and y with respect to τ ). e, = λ e, and y = e γ λβ now contain a power of + τ e y too, so e,, and y write lie constant λ a [p λ + p e + τ ρδ τ 2 [ b + τ e c, y where the coeficients a, b, c are a b c e γ β γ α y γ β γ γ We loo at the variation with respect to τ assuming λ = β[p e + τ ρδ τ 2 /[, which is exact when τ τ 2min + ρδ τ 2, and found numerically to be very close for larger τ s. Then, the above expresion for e,, y becomes: constant [p e + τ ρδ τ 2 a b [ + τ e /y c, = constant [p e + τ ρδ τ 2 a b [p e + ατ ρδ τ 2 c. The τ derivative at τ = 0 has the sign of a b+αc = γ,0, and α[γ for e,, and y. Whereas for large τ, the behaviour is τ a b+c always increasing in the long run (β, α, and β), which are indications of a U shape behaviour of e,, and y with respect to τ. This U shape behaviour is indeed confirmed numerically. Finally, e and y, computed assuming λ r = λ ref, are minimal when [a b + αc[p e ρδ τ 2 + α[a b + cτ = 0, so argmin y = α argmin e. This closes the discussion of result 3.b. 3.c. Proof of Proposition 3.c (comparison of y and under both baselines). We already saw that and y are increasing functions of λ, so r a and y r y a hold, as λ a λ r λ ref = β[p e + τ ρδ τ 2 /[, and as there is a further factor + τ e /y > in the relative case. Note that for e, the expected result is not established, but the factor + τ e /y present in the formula for e r maes the inequality e r e a valid in most cases. Indeed, the inequality is verified numerically. 3.d. Proof of Proposition 3.d (comparison of Π under both baselines). Π r versus Π a : We show that Π r Π a while τ is smaller than the smallest positive root of p e ρδ τ 2 X γ [ [ + αx /[ γ [ + X α/[ γ = 0. (A.3) α From (A.6), (A.0), and (A.3), we may write for a given τ : Π a (λ ) = H a (λ ) + K(λ ), where H a (λ ) = F(λ ;τ ρδ τ 2 ) + τ e ; y

9 Appendix to low hanging fruits etc. 9 [ and Π r (λ ) = H r (λ ) + K(λ ), where H r (λ ) = + τ e /[ γ y F(λ ;τ ρδ τ 2 ) ; and where K(λ ) is the integral term K(λ ) = ρ τ f(τ 2 ) λ2 λ F(λ 2,τ 2 )dτ 2 + τ 2 ρ[ẽ 2 δe. Then, Π a (λ a ) Π r (λ r ) [Π a (λ ref ) Π r (λ ref ) = = 0, λ a Π a(λ )dλ + λr λ a Π a(λ )dλ λ r Π r(λ )dλ λ r [H a(λ ) + K (λ )dλ + λ r Π a(λ )dλ + λ r Π r(λ )dλ λ r [H r(λ ) + K (λ )dλ = λ r [H r(λ ) H a(λ )dλ as Π a is negative between λ a and λ r, Π a = H a + K being a decreasing function of λ when λr λ > λ a (see fig. A.4), so that Π a(λ )dλ 0. λ a Finally, H r and H a are positive as H r and H a, both of the form const. F+ const. (see section ), reach their imum at λ = λ ref, and H r = [ + τ e /y /[ γ H a > H a. So, Π a (λ a ) Π r (λ r ) Π a (λ ref ) Π r (λ ref ). We even have an equality while τ τ 2min + ρδ τ 2, as λ a = λ r = λ ref in this case. We compute now the right-hand side in order to estimate the τ interval where this lower bound is positive. It is [ [ e /[ γ Π a (λ ref ) Π r (λ ref ) = τ e + τ γ λ β/[ γ [p e + τ + p λ ρδ τ 2 γ/[ γ y = τ α [ β/[ γ β β/[ γ [p e ρδ τ 2 [β /[ γ p β/[ γ [ [ ατ /[ γ + γ λ β/[ γ p e ρδ τ ref [p e + τ + p λ ref ρδ τ 2 γ/[ γ 2 [ [ τ = const. γ ατ /[ γ [ τ α/[ γ + + p e ρδ τ 2 α p e ρδ τ 2 p e ρδ τ 2 = const. X γ [ [ + αx /[ γ [ + X α/[ γ, (A.4) α from (A.), (A.2), and λ = β[p e ρδ τ 2 τ, and where X =. p e ρδ τ 2 The lower bound Π a (λ ref ) Π r (λ ref ) is positive for small positive τ, as the Taylor expansion of α[2 γ (A.4) with respect to X about X = 0 starts with const. 2[ γ X2 > 0, but the bound behaves lie X [ α/[ γ < 0 for large X, so there is a finite lowest positive root, however comfortably large if γ is not too close to. An empirical formula for a valid interval is 0 < X < 6 γ.