CDM domino Текст научной статьи по специальности «Математика»

CDM Domino5

2 Kyoto Institute of Economic Research, Kyoto University Yoshida, Sakyo-Ku, Kyoto, Japan e-mail address: imai@kier.kyoto-u.ac.jp 3 Graduate School of Economics and Management, Tohoku University Kawauchi, Aoba-Ku, Sendai, Japan e-mail address: akita@econ.tohoky.ac.jp 4 School of Economics, University of Hyogo,

Gakuen-Nishi-Machi, Nishi-Ku, Kobe, Japan e-mail address: niizawa@econ.u-hyogo.ac.jp

Abstract. Clean Development Mechanism (CDM) is a newly adopted scheme to give incentives to emission reduction projects in developing countries under Kyoto Protocol. We consider its implication under the demads for the products produced by firms engaging in CDM project are interrelated. In particular, we try to give examples where an adoption of a CDM project by one firm enhances the incentive of other firms to follow. What we found is that the condition for this to take place is rather stringent, indicating that the external push may be desirable

for one to promote CDM activities in these situations.

Keywords: Kyoto Protocol, Clean Development Mechanism, related goods

oligopoly, complements.

Introduction

CDM is a scheme introduced in Kyoto Protocol (1997) as the first attempt to convert emission reduction in developing country (and, hence, without an assigned limit) into the amount of emission toward fulfillment of assignment on the part of developed countries (more exactly, signatory of Kyoto Protocol). The scheme is more complicated than a mere subsidy scheme for emission reduction, and contains immense conceptual difficulty, which made some people dubious of the functioning of this mechanism (see [Bohm and Carlen, 2009] for example). Actually, after years

5 This work was supported by the French Foundation for Fundamental Researches under grants

No.99-01-00146 and 96-15-96245.

of trial and errors, number of registered projects surpassed 400 (as of 2006) and now, inclusive of proposed projects (called ones in the pipeline), expected credits may reach 2 billion tones, (as of 2007) which may be already sufficient to fill the gap between demand and supply in the upcoming emission trade scheme under Kyoto Protocol (2008-2012) according to some speculation. (For a general overview of the current state of CDM, see [Capoor and Ambrosi, 2007] for instance.) Many developing countries were rather skeptical of the mechanism when it was proposed, but now they seem to find more interests in this mechanism and the category of eligible “projects” as CDM projects tend to be expanded (to include “program” CDM and further “policy” CDM or “sector” CDM is proposed). By contrast, some parties start criticizing complicated procedure of CDM as a burden, and propose to replace the mechanism by simpler schemes. In short, even though CDM seems to have launched successfully, but there remains a room for farther controversy.

One issue we have raised concerning CDM is the baseline methodologies [Imai and Akita, 2001]. Then we analyze the same issue for a private firm operating in an imperfectly competitive industry [Imai, Akita, and Niizawa, 2008]. In this paper, we again consider a private firm in an imperfectly competitive industry, but now our focus is on the incentive to undertake CDM projects for firms whose decisions are related through market demands. In particular, our interests are on if an early adoption of a CDM project by some firm affects the incentives of other firms to do the same, and if so, positively or negatively. The answer turns out to be very simple that under the most modes of oligopolistic competition the effect is negative. This itself has a significant meaning in terms of policy implication and we shall investigate this issue in depth in another opportunity.

Here, we shall pursue the possibility for this effect to be positive. Not surprisingly, this is the case if firms’ demands are positively related, i.e. goods are complements. In this case, there is a positive externality across firms, and there one firm’s enhancement of own demand or production level tends to raise demand for goods produced by other firms and so there would be an incentive to follow the suit.

Below, by means of an example, we show a case where adoption of CDM projects gives momentum for others to do so, which we call “CDM domino”, although actual domino (CDM is adopted by firm, one after another) can occur under very stringent assumptions. We chose this way because such an example would exemplify the effect beautifully, and we can discuss welfare implication as well as some issues concerning mechanism design options in CDM.

1. Related Goods Oligopoly

We shall consider a group of firms 1 through n where relations among demands qi(i = 1,n) are characterized by the complement relations. Specifically, given prices Pi(i = 1,n), the demand for good i is given by

qi = 1 - Pi + b ^ pj (= 1 - (1 - b) pi + bP, where P = ^ pj) with |b| < 1/n.

j=i j

Each firm has the unit costs ci(< 1) with 0 < c\ < c2 < ... < cn. For the sake of simplicity, we assume that there is no fixed costs (of production). (In the concluding remark we discuss the case where fixed cost may have some relevance.)

Firms decide their prices given the demand schedule which is determined by other firms’ prices as well as its own, so as to maximize its profits: pi (1 — pi + b ^ pj) which

j=i

yields the best response function:

_ (l+frSpj+a) _ 1+a+bP-i (-w]lere p_. _ J^Pj) which in turn yields P =

j=i

f + f + Miiilp (where C = Ecj), or

P=—r---------------r[« + C] n + C

Thus,

ь(п-і) 1 2 — 6(n — 1)

r> r> n + C

P-*=P-ft=2-6(n-l)-^

1 + + b2-b(n-l) b

Рг = ----2---------2ft'

And, hence, the Nash equilibrium is

Pi= „А . гл

b(n+C)

2 —(n —1)Ь -1- “Г 4, ~r 2 —b(n —1)

2(1 + |) 2 + 6 With equilibrium output level

1 _i_ r’- _i_ Kw+C) „

= i-d-&) + Vn+c

and using notation

2 + b 2 — b (n — 1)

1 _ i-b (•n + C) _ (1-6)

2 + 6 2- (n- 1)6 2 + 6 *’

в = ъ' l + c

2 — (n — 1) b the equilibrium profits are

1 + + b (2-(n-l)b)

2 — (и — 1, ,,

-------— X

2 + 6

x (1 + b—L+^_ _ (1 + 6)1 + + 6

2 — (n — 1) b v ' 2 + b

180 Haruo Imai, Jiro Akita, Hidenori Niizawa

+ Л+в_(1+ь)1 + « + в

2+b 7V 2+b

1 + ci + B B \ ( 1 + b . ( 1 + b

-^ + 7Г—Л[^-7Г—Л1 + с1)+В 1-

2 + b 2 + b^ V 2 + o ' V 2 + b

1 1 + b B 4 2

--- — ----------Cj ---------

2 + b 2 + b 2 + b,

1 Л „ b b (2 + C_i) 4 2

1 — (1 + b) Cj + -----------p-----ЇТТС* ^

(2 + b)2 V 2 — (n — 1)b ‘ ' (2 — (n — 1) b)

1 Л-Ь+ь- ‘^TU+ b(1+c-)

(2 + b)2 V I 2 — (n — 1)bj ’ 2 — (n — 1)b)'

where C-i = E Cj.

j=i

Next, consider a CDM project requiring investment costs for firm i, I > 0 which results in a reduction of unit costs to ci (< ci). A typical project which would produce this type of cost change would be energy switch projects. With an investment in equipments, energy source is switched to the one with less emission (with more or less costly fuel price). If it is less costly, then it must be the case that the increase in equilibrium profits due to the introduction of the CDM projects should be less than I, in order to meet the additionality requirement.

From above, we obtain that the change in profits of firm i when its unit cost shifts from ci to ci’ given the firms’ costs c-i = (ci, c2,..., ci-1, ci+i,..., cn) is given by

(ni (ci, C-i) — ni (ci, C-i)) (2 + b)2 =

2 — (n — 2) b — (n — 1) b2

= 2 (ci — c^ j 1 + = (ci — ci) \ —2 —

2 — (n — 1) b

b (1 + C-i)

(ci — ci + ci

2 — (n — 1) b b + bCi — (2 — (n — 2) b — (n — 1) b2) (ci + ci)

2 — (n — 1) b y

and its sign can be confirmed by

—4 + 2(n — 1) b — b — bCi + (2 — (n — 2) b — (n — 1) b2) (ci + ci) < 0. Now, suppose that conditions

A'Ki (ci, £, C-i) > 1 > Ani+i (ci+l, £, C-i+i)

are met where

£ = ci — ci; Ani (c^ £, C-1) = (ni (ci, C-i) — ni (ci — £, C-i)) !

—£2b

rv — ---------^----------------! Cj-Li — Cj ~\~ K.

(2 + b) (2 — (n — 1) b) 1+1 1

Further, assume that firms make decisions sequentially in the order of their suffixes. Then we obtain the sequential adoption of CDM projects by n firms. Obviously, this is a sort of a forced realization of CDM domino. Our question is if this could take place even without the condition on ordering.

One easy answer is that firms are myopic. That is, when making decision on adoption of a CDM project, firms take other firms’ decisions as given. However, appealing to irrationality on the side of firms may not be a very attractive story. In fact, if firms are given a chance to make this decision simultaneously, rational firms would decide to adopt the project as it is one of the Nash equilibria. So, we can rephrase our question in the following form: if this could occur with rational firms even if firms have a chance to make decisions simultaneously (and sequentially). To answer this question affirmatively, we shall appeal to the possibility of incomplete information on the side of firms.

2. Incomplete Information

In order to produce a more plausible example of the CDM domino, we introduce a noise to the investment costs of each firm. This could be justified by the presence of the so, called capacity building costs, i.e. firms may lack employees equipped with sufficient knowledge, experiences, and capability to adapt to new machines or new methods of production, and this cost may not be observable to outsiders.

Let us write: ni (ci, c-i) = IIi (ci, C-i) and q — ci = £.

We assume that each firm’s investment cost could be high or low, and accordingly we call them type H and type L for each firm. In particular we assume that for each firm i type L:

ni4 (n — 1) ci — (i — 1) — ni (ci, (n — 1) ci — (i — 1) £) > Ii >

> ni (ci, (n — 1) ci — (i — 2) £) — ni (ci, (n — 1) ci — (i — 2) £);

and for type H:

ni (n — 1) c'j — -fj-i (ci, (n — 1) c'j) < I'j'j Each firm i suspects firm j (= i) s type to be L with a probability nj, so that

nj | ni (ci, (n — 1) ci — (i — 1) £) — ni (ci, (n — 1) ci — (i — 1) £)| + (1 — Vj )

l^fli (ci, (n — 1) ci — (i — 2) £) — IIi (ci, (n — 1) ci — (i — 2) £^ < Ij.

Under these suppositions, we consider a game played in n discrete periods. Each firm can make a (irreversible) decision to adopt a CDM project, at each period. As direct consequences of the above assumptions, we obtain the following properties:

1. Every firm of type L adopted the project simultaneously is not a Bayesian equilibrium.

2. For firm 1 of type L, adoption at the inception in the sequential decision problem does not result in a loss while for firm j > 1, adoption of the project yields expected loss.

3. Given firms 1 through i — 1 adopt the project, type L of firm I does not incur a loss by the adoption of the project, in the next period, while other firm j > 1, it suffers from an expected loss.

4. Given n periods, firm I of type L should adopt the project as soon as it finds the project profitable.

Theorem 1. The following strategy profile with a belief system forms a perfect Bayesian equilibrium. If all the types are L, then CDM domino takes place. (i) Each type H never adopts the project. (ii) Type L of firm i adopts the project if and only if already i — 1 or more firms have adopted the project. (iii) Belief on firm j is given by H~{j} until period j, and after period j, the belief on firm j to be of type L is set at 0, while for all firms who have adopted the project, belief becomes 1.

Proof.

The proof is almost straightforward due to the assumptions made. (i) is optimal by the assumption we made. (ii) is optimal given belief, i.e. given m firms adopting the project at period i with m < i — 1 and firm i having not doing so yet, then given the belief in (iii), for firm i (and firm j with j > i) no adoption is optimal.(Note that type L of firm j < m who has not adopted yet would do so (and is optimal to do so), but other firms believe that this firm is of type H. In some literature, the property of belief given in (iii) that the possibility of being type L vanished at one time ressurects by adoption is untenable. Here since adopted firm never acts again, so power of this criticism may be weak if valid.)

3. A simple dynamic model

As another alternative, we may provide a simple dynamic model incorporating learning and spillover effect with respect to capacity building. Now, suppose firm iis costs of investment Ii decreases both with time and the number of firms already adopted the projects. Time is continuous and the rate of decline in investment costs given number of firms adopted the project is ^ >. Firm i's investment cost at t when m firms have adopted the project 5me-rt (Ii0 + Iie-^mt) where Ii0 > 0, 0 < 5 < 1 and m is the number of firms adopted the project and 0 < < ^1 < • • • < , we

assum that 5mIj0 < f^° nm (cl, • • • , dm-1, c'm, cm+1, • • • , cn) e-rtdt < 5m-1Ii0 with i = m to make sure that the firms except for I may not have an incentive to adopt.

Consider the last firm’s problem, provided that the last firm is firm n, and firm 1 • • • n — 1 adopted the project in that order (at ti for firm i).

tn pOO

(c1, • • • , 4-1, cn) dt + / e-rtnn (c1, • • • ,c'n) dt

’ tn — 1 J tn

— e-rtn ^5n-1In0 + e-^dti-ti—l)Ir)j (With to = 0). The first order condition yields

-rtn (Hn (c1, • • • ,c'n) — nn (c1, • • • , c'n-1, cn)) =

/ n \ n / \

= re-rtn( 5n-1In0 + e-^l{u-u—l) In) + Vne-rtn x e-i=i ti-t In.

Rearranging, we have

(nn (c'1? • • • , cJn) — nn (c'1? • • • , <_l7 cn)) — 1/no _ e~1) (r H- f^n)

or

n- 1

№i

t*n — tn-1 — --- (ti ~ U-l) —

Vn

— log

nn (c1, • • • ,c'n) — nn ( (ci, • • • , Cn-1, cn)) — 5n 1In0

^ ( r + Mn) ^n

n- 1

n

n1

| nw (Cj; • • • ;<4) Hw ((c[, ■ ■ ■ , c'n_1, Cw)) (5” 1In0 y (r + Hn) In

From this, we obtain that

dtn _ Vi+1 — v

dti V'

> 0 (but < 1), for i ^ n — 1.

n

Next, we consider firm i's problem with i < n.

Firm i maximizes

r ti f tj+i

/ e-rtn (c1, • • • , c'n-1, cn) dt + ^ / e-rtn (c[, • • • , cj, cj+1, • • • ,cn) dt—

j=1J tj

— Tt- I si-1 T - -ti—l)T

e rti 5 1 Iio + e j=1 Ii

Firm n chooses tn to maximize

(with tj = t* for i < j S n and tn+1 = to).

Usually this condition is not so tractable. When n = 2, the payoff for firm1 becomes:

ft- j' t2 (ti) i'^

/ e-rtn1 (c, c) dt + / e-rtn1 (c', c) dt + / e-rtn1 (c', c1) dt—

•JO J11 J 12 (ti )

—e-rtl (I10 + e-^111) =0, and the first order condition is

e-rtl (Ili (c', c) - n, (c, c)) + (n, (c', c') - n, (c', c)) =

dt1

= e-rt1 (rIw + (r + V1) e-^1tI1) .

Writing t2 (ti) = j t\ — A and dt^-1 = 1 — we have

dt2 (tl) c-rt2(tl) e-r{(l-%)ti-A}

dt1 V M2 / ’

and so

[II, (c', c)] - r/10 - (r + /xi) /ie_Mltl H---(IIi (c'; c') - IIi (c'; c)) 1 = 0

V2

determines t* (and, hence, t*2 = t2 = t2 (t*), the detail of which depends upon

V4, V2 and r.) To obtain a tractable solution, we may adapt the assumption that the

complementation is unilateral. I.e. adoption of the firm i affects the cost of firm j > i, but not vice versa. (In terms of demand functions, p1 affects demand of firm j but pj does not affect firm i’s demand for i < j.)

Under this assumption, for i > j, tj does not enter into the expression for profits of firm i. Thus, the first order condition becomes like that one for the n-th firm. I.e. writing

c1 — {ni (c1, • • • ,ci, • • • , cn) ni {c1, • • • , ci—1, ci, • • • , cn) } :

4 = r5i-1 Iio, the first order condition for firm i becomes

i-1

Viti = — log (c\ — c\) + log (r + Vi) Ii + ^ (Vj+1 — Vj) tj

j=1

In particular, > 0.

4. Discussion

4.1. Baseline Methodology

Earlier we compared several baseline setting methods in their effects on incentives and overall performances. The CDM credits in this paper are computed through the method which we called the “ex post” baseline. That is, the credits are computed as the gap between the emission level with old and new technology of production provided that the output level is given by the actually observed output level.

One alternative method is to define credits as the gap between the actual emission level and the level obtained when the old technology is used with the output level forecast ex ante under that situation. In particular, under the stationary environment, forecast output level could be given by the actual output level observed before the project and so before the new technology is introduced. These two methods could give rise to quite distinctive incentives to the firm when output level is controlled by the form itself (see [Imai and Akita, 2003] and [Laurikka, 2002]). Also see [Fischer, 2005]. We adopted the ex post method in the main part because it is the chiefly utilized method in reality probablly both for its practicality and intuitive appealingness, in such CDM projects that emission level can be broken down into the output level and emission coefficients. However, there is a history that initially COP and the EB considered the ex ante method as one alternative, and possibly more legitimate method [cf. Marrakech Accord, 2001].

In the static version of the above model, we assumed that the “effective” marginal costs decrease as a result of an adoption of a CDM project. This is based upon several presumptions, but at least the ex post method contributes to the reduction of the effective marginal costs because for each unit produced, revenue from the sales of an additional reduction (compared to the old technology) brings a reduction of marginal costs (inclusive of the proceeds of credits). By contrast, under the ex ante baseline, credits decreases with the output (after CDM). Thus, an adoption of a CDM project works to increases the effective marginal costs.

Since a CDM project is adopted only if the firm expects a positive return from the engagement, it may be natural to think that the firm’s effective marginal costs are still lower than the pre-CDM level even under the ex ante baseline. In that case, our argument above continues to hold although the range for which the assumption is valid may be narrower. However, in the extreme, one may imagine a case where this effective marginal costs go up as a whole as a consequence of the adoption of a CDM project (while the profitability of the project is assured by a decline in fixed costs, whose presence we had not assumed in our earlier model). In this case, the story is completely reversed. Under complementarity in demands CDM by one firm would induce a contraction of its output (due to ex ante baseline) which reduces the incentive for other firms to follow the suit (due to complementarity in demand). Rather the industry where firms’ demands are characterized by the substitution relation becomes the suitable case for CDM domino.

4.2. Additionality

Another issue frequently raised related to CDM is the issue of additionality. Again consider the static model in the main text. The additionality constraint would be naturally given by that the profit difference without CDM credits is not sufficient to induce the firm to adopt the project. Letting p be the (expected) price of credits, this constraint can be expressed as ni(c'1,..., ci + p, ci+1,..., cn) > Ii where the opportunity costs of investment are supposedly included in Ii. We have implicitly assumed that this constraint was met. As a matter of fact, this constraint only affects the decision over if this project is accepted or not, and does not affect the level of profits. In this sense, this constraint does not directly affect arguments for the CDM domino. However, once one takes into consideration the possibillity of the timing when the calculation of profits is made on which additionaly test is conducted, we get some insights on the importance of timing in CDM. For instance, suppose that the firm may adopt the project simultaneously or independently. If the additionality test compares the profits before the adoption and the profits the firm would earn when the firm adopts the project by itself, then there could be the case where the adoption by one firm alone is not profitable without credits, but if many firms adopt together, the project is profitable even without the credits. This provides an incentive for firms with sufficient foresight to adopt earlier expecting that other firms to adopt later, and to this effect all the firms may adopt at an earlier stage.

4.3. Double Counting

A related issue is that of double counting. Admittedly, externality among firm’s demands does not raise a serious issue concerning double counting because this is not related to emission accounting or baseline itself. However, theoretically one could argue that the ensuing adoption of a CDM project upon an adoption by one firm may indicate that the firm adopting earlier may claim a part of subsequent emission reduction for itself as the fact of the adoption is attributable to the decision made by the first firm and, hence, may enlarge its project boundary. In some marginal situation, such enlargement would be crucial to induce the firm to employ the CDM project (provided that such enlargement does not affect the adoption decision of the firms adopting later). These issues can be oversome by a suitably packaged program CDM.

4.4. Contactual Form

We assumed that firms maximize their profits (inclusive of credits). The exact incentives of firms vary depending upon the details of the contract which the firm has with the contractual partners if any.

In the case of unilateral CDM, our profit maximization assumption holds without question. With a contractwith a buyer of the credits at the spot price with a fixed quantity without a penalty on non-delivery, or fixed price with a buyer’s guarantee to purchase whole quantity delivered, our assumption is still valid. Otherwise, an obvious modification is necessary.

4.5. Welfare & Environment

It is well recognized that the Kyoto mechanism workes given the quota levied on each or any unit in terms of GHG emission.

And this implies that any reduction brought about by the CDM project would be used up by those units which emission level is bound by the quota. (There is some exceptional case where emission credits are bought by some entity which intentionally let the credit retire, or some entity banks (i.e. somes) for the usage for an indefinite future although this possibility is explicitly restricted by Marrakech Accord) to a certain extent.

Thus emission reduction under CDM project is merely a replacement of emission reduction somewherelse, where reduction costs are supposedly much higher. In this sense, the sheer effect of CDM would be the costs saved relative to other opportunities which would be given by the gap between the value of credits (evaluated at the emission price given by the emission trade market of Kyoto mechanism or EU-ETS, minus the actual costs, which in turn is the gross profits of CDM project.

Another effect of CDM often raised is the promotion of technology transfer. Generally speaking, a free transfer of profit creating technology is unlikely to take place. However, through several channels, like learning by watching or communications through employed workers, this may be realized.

The example developed in this paper is apparently extreme and making gevite convinient assumption, and, thus, not suited for empirical purpose, live as indicating a way to incorporate those factors like technology transfer and capacity building through CDM, we claim this is a really pioneering attempt.

Acknowlegments

The authors are grateful for the financial aid from the MEXT Grants-in-Aidfor Scientific Research 18310031.

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