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Thin versus Thick CO2 Market
Journal of Environmental Economics and Management 41, 295᎐311 Ž2001. doi:10.1006rjeem.2000.1144, available online at http:rrwww.idealibrary.com on
Thin versus Thick CO 2 Market 1 Matti Liski Helsinki School of Economics, P.O. Box 1210, 00101 Helsinki, Finland E-mail: [email protected] Received January 28, 1999; revised June 24, 1999; published online October 24, 2000 The idea that trading is more costly the thinner the market is is common in most studies of market exchange with frictions. Surprisingly, this element is lacking from previous attempts to allow for frictions in pollution permit markets. This paper considers a CO 2 cap-and-trade model where trading costs develop endogenously as a function of the market size. The pre-trade allocation of permits determines whether the market size can be strongly influenced by expectations that have a role because of adjustment costs. The pre-trade allocation also sets preconditions for endogenously vanishing trading costs and thus has nonstandard effects on long-run trading levels and market allocations. 䊚 2001 Academic Press
1. INTRODUCTION The market mechanisms built into the Kyoto Protocol have the potential of significantly reducing the costs of limiting greenhouse gases Že.g., w1, 3, 24x., but the extent to which these gains will be achieved depends on the degree of friction in the CO 2 market.2 Given the economic stakes involved in this market, it is important that trading institutions are designed to support the development of frictionless CO 2 trading.3 However, there has been no formal attempt to identify the determinants of such a development. While it is straightforward to consider the effect of frictions on pollution permit markets by introducing trading costs that are formally equivalent to a tax or transportation cost w27x, this approach has no hope of explaining the level of these costs: they are not endogenous in equilibrium. In particular, the common idea that trading is more costly the thinner the market is is lacking Žsee, e.g., w5, 11, 12x.. I consider a dynamic model of CO 2 trading where trading costs are not a priori absent Žas in w21x. nor exogenously given Žas in w27x. but rather develop endogenously over time as a function of the market size. Endogenous trading costs can be used to analyze the determinants of a frictionless CO 2 market. A critical determinant is the pre-trade allocation of permits, the 1
This paper is based on ideas developed in Chapter 2 of my thesis which benefited from comments by Pertti Haaparanta, Michael Hoel, Larry Karp, Matti Pohjola, Seppo Salo, and Olli Tahvonen. I also thank Cees Withagen, anonymous referees, an associate editor, and the participants at the workshops ‘‘Economic Issues Related to Climate Change’’ in Oslo 1998 and ‘‘Design of Energy Markets and Environment’’ in Copenhagen 1999 and at the 1999 EAERErEEA annual conferences. 2 The Kyoto Protocol covers six types of greenhouse gases, but because each type of greenhouse gas can be converted into CO 2 -equivalent units, I interpret the emission limits as CO 2 limits. 3 References w27x and w8x discuss the empirical evidence of trading costs in earlier pollution trading experiments. 295 0095-0696r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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key policy variable, which alters the basic economic properties of tradable pollution permits in two important ways. First, the pre-trade allocation determines whether self-fulfilling prophecies can strongly influence the market size: polluters’ expectation of low trading volumes can lead to expectation of a thin CO 2 market, which discourages polluters from undertaking emissions reductions that would leave permits for sale, thereby fulfilling the original expectation. Second, the pre-trade allocation sets preconditions for endogenously vanishing trading costs and thus has nonstandard effects on long-run trading levels and market allocations of emissions. Trading costs in permit markets can be justified in a number of ways. Stavins w27x identifies two potential sources of trading costs incurred by firms: Ž1. search and information and Ž2. bargaining and decision. Type-1 costs are best understood if trades are conducted by middlemen, whose brokerage fees constitute a direct financial cost of trading. Type-2 costs involve, for example, the cost of entering into negotiations and fees for legal and insurance services. However, both types of costs can be interpreted in terms of costs due to a lack of information that is underprovided by thin markets. Thick markets with numerous transactions are likely to generate positive informational externalities that absorb part of the trading costs Žsee also w27, p. 144x.. For example, the number of middlemen can rise as the trading activity increases; a larger pool of brokers means a lower cost of finding a trading partner, thereby reducing the amount of real resources taken up by trading. Alternatively, if polluters conduct trades themselves the transaction cost may reflect the cost of gathering information on the price level. Such costs are likely to be lower the thicker the market is Žsee w23, p. 121x..4 To my knowledge, the implications of trade externalities of this type have not been addressed in connection with the global CO 2 or pollution trading in general.5 The implications of trade externalities are cast in a North᎐South CO 2 trading model, where the North and the South are asymmetric in two respects.6 First, there is the usual asymmetry that the North is more productive in using a unit of CO 2 . In this sense, the costs of reducing emissions are higher in the North. Second, resources committed to activities that change the emission level move more sluggishly in the South because of higher adjustment costs.7 Costs of shifting resources arise, for example, from temporary drops in output caused by the phase-out of old machinery and the installation of new machinery. Phase-out periods may be longer and contracting costs higher because of a shortage of skilled labor, corruption, political instability, and insufficient infrastructure in general. 4 The evidence from trading under the Clean Air Act Amendments of 1990 supports these observations: middlemen’s commissions have been continually falling with trading activity, as has the uncertainty regarding the allowance price w13, pp. 683᎐684x.. 5 Such trade externalities are common in search theoretic models Že.g., w5, 22x.. My approach is not search theoretic but is closest to w12x, which assumes that trading costs depend on the market size. 6 Given the current state of the climate treaty, this setting is still quite hypothetical. First, the studies addressing the current trading potential often view the CO 2 market as an East᎐West market Že.g., w3, 24x.. But if the long-run treaty covers the major economies that are currently not participating, this market can reasonably be viewed as a North᎐South market. Second, the problems of monitoring and enforcement are likely to pose serious problems to the development of a global cap-and-trade system w26x. 7 Reference w7x makes a similar type of assumption in a North᎐South trade model, where the distinguishing feature of the South is its lower speed of factor mobility Žp. 178.. The inertia in the South may be due to the duality of its economy, which can be explained by differences in the nature of rationality w16x or in technology and information w2x between the Southern sectors.
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Also, a given reduction in emissions may require a greater shift in labor between sectors in the Southern economy. Adjustment costs reflect costs due to congestion, a period of unemployment, or retraining of labor. As in the investment theory, it is difficult to distinguish polluters’ internal and external adjustment costs, but both types of costs can lead to the same formal model Žsee w9, 19x.. Adjustment costs create a role for a forward-looking element that is a key to the analysis of whether polluters end up in a thin or thick market equilibrium. The expected returns on emissions reductions can be seen to depend on the thinness of the future CO 2 market, which in turn depends on the pre-trade allocation of permits. I find that while a relatively large pre-trade allocation in the low-productivity South is needed to rationalize expectations about a costless and thick CO 2 market, such an allocation can also allow a role for expectations that can strongly influence the market size. The strong dependence on expectations is likely, for example, when adjustment costs are low, implying a fast reallocation of emissions and thus a fast fulfillment of the original expectation. That the trading level can depend on expectations indicates a potential expectational role for public forecasts in tradable permit experiments Žas in aggregate demand management, see w6x.. For the above reasons, the choice of one allocation over another should be defended not only from an equity point of view but also from an economic point of view. This is in sharp contrast with studies that use the CO 2 market as a frictionless transfer mechanism to evaluate the distributional effects of different allocation rules of the pre-trade CO 2 cap Žsee, e.g., w4, 15x.. By some estimates these transfers could exceed the current level of development aid Žsee w15, p. 34x.. I find that when trade frictions are endogenous, the CO 2 market can be a frictionless transfer mechanism only for pre-trade allocations that are significantly different from the long-run market allocation that would be achieved if trading were costless by assumption. The next section sets up the model and derives a dynamic system whose solution can be used to characterize the equilibrium. Section 3 identifies the set of long-run equilibria for various pre-trade allocations, and Section 4 considers equilibrium paths. The latter section explains the role of expectations and shows which equilibrium can actually be chosen. Section 5 isolates nonstandard long-run effects of the pre-trade allocations by focusing on those stationary equilibria to which a dynamic equilibrium potentially converges.
2. THE MODEL Permits. I postulate a two-region model ŽNorth᎐South. with large and equal populations of polluters within both regions. All Northern and Southern polluters are respectively identical. The global cap on CO 2 emissions remains constant over time and is implemented by issuing in each period a given fixed number of CO 2 rights Žpermits. that are usable only in the period of issuance Žthe banking or borrowing of permits is not allowed.. The regional shares of these permits remain constant over time and are distributed free of charge. Thus, in each period t the equation e s en q e s
Ž 1.
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holds, where e n and e s are the pre-trade allocations of permits per polluter in the North and the South, respectively, and e is the global per polluter cap on emissions. Technology. The revenue function for a polluter of type i Žs n, s . is R i Ž x i . s a i q bi x i y Ž c ir2. x i2 , where x i is the emission level and a i , bi , c i are positive parameters. This function can be thought of as arising from a relationship between the polluter’s output and a vector of energy-intensive inputs Žsee, e.g., w15, p. 22x.. The linear-quadratic form is not necessary, but it will help to keep the model parsimonious. It proves useful to write the productivity gap as depending upon x s , D Ž x s . s RXn Ž e y x s . y RXs Ž x s . s a q bx s ,
Ž 2.
where x n s e y x s , a ' bn y bs y c n e and b ' c n q c s . I shall maintain the assumptions Ži. that the pre-trade valuation of additional units of CO 2 is higher for the representative Northern polluter, i.e., DŽ e s . ) 0, and Žii. that there is an allocation Ž x n , x s . s Ž e y x s , x s . of e that eliminates the productivity gap, i.e., DŽ x s . s 0 for x s s x 1s ' yarb ) 0. The first assumption ensures that it is the Northern trader who takes the role of a long-run buyer, and the second rules out the unrealistic equilibrium where all Southern permits are reallocated to the North. Trading cost. Buyers of CO 2 permits face a transaction cost ␦ Ž z i . hŽ z . where z i Ž) 0. is the number of permits purchased by a source of type i Žs n, s . and z is the global average trading volume. Because the number of traders is fixed, I shall use the words average and total trading volume interchangeably. For ease of presentation, I assume that the trading cost is characterized by
␦ Ž zi . s
½
z i2 for z i G 0 0 otherwise 1 2
and
hŽ z . s
½
zU y z for z F zU 0 otherwise
Ž 3.
where the parameter zU is positive. Note first that sellers pay no trading cost, i.e., ␦ Ž z i . s 0 for z i - 0.8 Note next that, by the large number of traders, hŽ z . is an exogenous parameter from an individual trader’s point of view, implying that the trading cost is increasing in the size of private purchase, ␦ X Ž z i . ) 0. The assumption that the trading cost is decreasing in the total trading volume, hX Ž z . F 0, formalizes the idea that it is more costly to operate in a thin than in a thick market. Because the trading volume z is endogenous in equilibrium, the level of the trading cost will also be endogenous in equilibrium. This aspect of transaction costs in connection with pollution permits has been suggested, but not formalized, by w27, p. 142x: ‘‘ . . . at the market level, decreasing marginal transaction costs could also be due to positive information externalities Žassociated with larger trading volumes. that systematically lower transaction costs’’ Žsee also w27, p. 144x.. Note that private scale economies in trading ␦ Y Ž z i . - 0, would not be unreasonable w27x. For 8
All the equilibria described in this paper would remain the same even if we assumed that the seller pays the transaction cost Žas in w27, p. 140x..
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tractability, I exclude this case but include market-level scale economies by hX Ž z . F 0.9 The trading externality eliminates the trading cost entirely whenever the total trading volume exceeds the level zU Žsee Ž3... To keep the role of this externality interesting, the following restrictions are imposed: e y x 1s ) zU ) b.
Ž 4.
The quantity e y x 1s should be seen as the largest conceivable trading volume that can be achieved. It is the quantity sold by a representative Southern polluter when its pre-trade allocation is extremely large, e s s e, and when trading eliminates the productivity gap, DŽ x 1s . s 0. The first inequality thus allows a potential role for frictionless trading, because hŽ z . s 0 for z s e y x 1s . The second inequality in Ž4. will rule out an inevitable development of frictionless CO 2 markets by setting a lower bound for the trading level that is needed for costless trading. To make sure that there exists a positive equilibrium price for CO 2 for any given allocation of emissions, the following assumption will be maintained: RXn Ž e y x s . y ␦ X Ž e s y x s . h Ž < e s y x s < . ) 0.
Ž 5.
By Ž5., the Northern buyer’s valuation of additional CO 2 units net of trading costs remains positive for all conceivable allocations Ž x n , x s . s Ž e y x s , x s . of e. Because all permits are positively valued, no permits will be left unused in equilibrium. Northern profits. By the large population of polluters, the representative Northern firm takes the price pŽ t . of a CO 2 unit and the total trading volume z Ž t . at each time t as given and maximizes the discounted value of its profits by choosing its emission and trading levels Žsuppressing the dependence on time.: max
⬁
H x ,z 0 n
R n Ž x n . y pz n y ␦ Ž z n . h Ž z . 4 ey t dt
Ž 6.
n
s.t. x n F en q z n ,
Ž 7.
where is the constant discount rate. If the polluter is currently a buyer, its profit flow is the revenue accrued by using polluting inputs in production minus the cost of purchasing z n Ž) 0. permits Žprice plus the transaction cost.. If the polluter is a seller, it receives revenues from selling z n Ž- 0. permits but, by assumption, pays no trading cost. Since there is no state equation, the problem Ž6. ᎐ Ž7. is a static one. By Ž5., the equilibrium will be characterized by interior necessary conditions that can be combined to yield the demand for permits p s RXn Ž x n . y ␦ X Ž x n y e n . h Ž z . .
Ž 8.
Y 9 One may argue that ␦ Ž z i . ) 0 can be avoided simply by splitting the quantity z i into smaller trades w27x. But this assumes that contacting new trading partners is costless, which is not reasonable in a less than perfect market. Indeed, in most studies of market exchange with frictions, potential traders meet sequentially in a time-taking process w5, 6, 11, 22x.. This alternative approach models the source of friction explicitly, but it introduces another problem: matching models have difficulties in considering trade with divisible objects such as the permit endowment w28x. My approach has no problems with divisibility, but is less explicit about individual buyer᎐seller relationships Žas is w12x..
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In view of Ž8., the effect of the trading cost can be seen as a drop in the productivity of the Northern firm: it will purchase additional units of emission rights up to the point where their marginal contribution to its revenue net of transaction costs equals the price, while in the absence of trading costs the price would indicate the true marginal valuation of CO 2 permits in production. Southern profits. The distinguishing feature of the South, besides the productivity gap DŽ e s . ) 0, is its lower speed of factor mobility. Factors move only slowly because it is costly for them to move. This ‘‘moving cost’’ is assumed to be quadratic, cŽ r . s 2 r 2 , where r is the rate of change in the Southern emissions x s and is a parameter.10 Unless otherwise stated, I assume that s 1. The cost cŽ r . occurs in the form of real resources used up in the process of increasing or decreasing emissionsᎏas explained in the Introduction there are a number of reasons why these costs are likely to be high in the South. As soon as the moving cost is introduced, the decision to change the emission level becomes an investment decision, the profitability of which depends not only on the current market price pŽ t . and trading volume z Ž t . but also on the anticipated future values of these variables. Thus, given the rationally expected pŽ t . and z Ž t ., the representative Southern polluter chooses a capacity adjustment plan r Ž t . and a permit trading plan z s Ž t . to maximize the present value of its profit flow, max r, zs
⬁
H0
R s Ž x s . y pz s y ␦ Ž z s . h Ž z . y c Ž r . 4 ey t dt
Ž 9.
s.t.
˙x s s r ,
Ž 10 .
x s F es q zs ,
Ž 11 .
e G x s Ž 0.
Ž 12 .
Ž x s Ž0. given by history.. As in the North, the polluter’s profit flow gross of adjustment costs accrue from output sales and from CO 2 trading, which has a positive or negative impact depending on whether the firm is a selling or a buying source. The initial emission level x s Ž0., which is assumed to be less than the global per polluter gap e, is the Southern emission level at the outset of CO 2 trading. It is the emission capacity determined by past decisions to increase emissions and, by the presence of adjustment costs, a given fixed variable at t s 0. Appendix 1 shows that in equilibrium the interior necessary conditions for Ž9. ᎐ Ž12. are satisfied. They can be used to yield
m s m ˙ q RXs Ž x s . y p y ␦ X Ž z s . h Ž z . 4 , 10
Ž 13 .
This is not a cost of developing or changing the abatement technology but merely an adjustment cost. While cŽ r . vanishes in a steady state, it can alter the choice between steady states ŽSection 4 below.. There are also other ways to introduce adjustment costs. First, it would be realistic to assume positive adjustment costs for both regions. However, I do not believe that this is central to the argument; the assumption of one-sided adjustment costs keeps issues comprehensive and produces the realistic equilibrium outcome that both emission levels x n and x s are adjusted gradually. Second, the cost could be made asymmetric w.r.t. increases and decreases in emissions by assuming that cŽ r . ' Žr2. r 2 q ␥ r, where ␥ is a positive or negative parameter depending on whether increasing or decreasing emissions is respectively more difficult. Third, the identities of the North and South could be changed such that cŽ r . is incurred only when x n is adjusted, i.e., adjustment costs would be higher in the North. The last two of these changes would cause nontrivial changes in polluters’ payoffs, but rather obvious qualifications to the results below.
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where m should be seen as the shadow price of having a marginal increase in the Southern emission capacity. Because of the moving cost cŽ r ., increasing x s uses up real resources that could be used elsewhere. In particular, if equals the world interest rate, the required rate of return on the investment of size m is, of course, m. If the amount m is invested to increase the Souther emission capacity by one marginal unit, the required rate of return equals the rate of capital gain m ˙ plus the current net earnings for a unit of emissions RXs Ž x s . y p y ␦ X Ž z s . hŽ z .. The term Ž p q ␦ X Ž z s . hŽ z .. is the current cost of purchasing an emission right if the firm is a net buyer and the opportunity cost of using a permit in the South if the firm is a net seller. In equilibrium, a marginal increase in x s indeed requires an investment of size m because emissions are changed at a rate given by the equality of the marginal moving cost cX Ž r . and the shadow price m Žsee Appendix 1.. By this latter equality, we have
˙x s Ž t . s m Ž t . s
⬁
Ht
RXs Ž x s . y p y ␦ X Ž z s . h Ž z . 4 ey Žyt . d ,
Ž 14 . Ž 15 .
where Ž15. reexpresses m as the discounted value of net earnings RXs Ž x s . y p y ␦ X Ž z s . hŽ z . along the equilibrium path Žfrom the integration of Ž13... If m ) 0, emissions in the South are increased; if m - 0, they are decreased. Equilibrium. Clearly, polluter’s compliance strategies will depend on their expectations about the CO 2 price pŽ t . and the thinness of the market z Ž t ., both altering the profitability of using emission rights in the South. To pin down equilibrium expectations, I derive next a dynamic system that fully characterizes all conceivable equilibria. To this end, note that in equilibrium the private marginal trading cost is given by ␦ X Ž z n . hŽ z . s Ž e s y x s . hŽ e s y x s . if the Northern firm is the buyer Ž x s F e s ., and by ␦ X Ž z s . hŽ z . s Ž x s y e s . hŽ x s y e s . if the Southern firm is the buyer Ž x s ) e s .. For a given initial allocation e s , define T Ž x s . ' Ž es y x s . h Ž < es y x s <. ,
Ž 16 .
which gives the marginal Northern purchasing cost when x s F e s . The corresponding Southern cost is just yT Ž x s . when x s ) e s . Using the fact that the equilibrium price is given by Ž8. and the definitions Ž2. and Ž16., the equation Ž13. can be rewritten as m ˙ s m q DŽ xs . y T Ž xs . .
Ž 17 .
The equations Ž14. and Ž17. jointly define a planar dynamical system Ž ˙ xs, m ˙ . on w0, e x = wy⬁, ⬁x that can be used to characterize the equilibrium for a given pre-trade allocation of emissions. Note that the system treats the level of the private marginal trading cost as an endogenous variable T Ž x s ., although hŽ z . is an exogenous parameter from the private point of view.11 As is demonstrated below, the properties of the system depend on the Southern emission share e s . 11
w25x.
The method of deriving the equilibrium dynamics in the presence of externalities is the same as in
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FIG. 1. Thin and thick market steady states.
3. THIN VERSUS THICK MARKET STEADY STATES A stationary allocation of emissions eliminates all incentives to reallocate emissions, which can hold only if m s 0 Žsee Ž14... Using this in Ž17. implies that the steady state allocation must also satisfy DŽ x s . s T Ž x s ., i.e., the productivity gap must be equalized with the private marginal trading cost. Allocations satisfying the latter condition are best characterized by using Fig. 1, which shows on its horizontal axis the Southern emission level x s and on its vertical axis the values of DŽ x s . and T Ž x s .. For any given x s , the Northern emission level is given by the residual e y x s . To make the shape of the graph for T Ž x s . comprehensive, suppose first that the Southern emissions are just equal to the pre-trade allocation, x s s e s . Because no permits are left for sale or purchased, the marginal trading cost is zero, i.e., T Ž e s . s 0, as depicted. As the trading volume increases Žwe move horizontally to the left from x s s e s . the cost T Ž x s . first increases and then decreases.12 The increasing part is explained by the fact that the trading cost is increasing in the size of private trades, and the decreasing part by the trading externality that systematically lowers the private trading cost as the total trading volume increases. In fact, for x s F e s y zU trading becomes costless because hŽ< e s y x s <. s 0; the graph of T Ž x s . is given by the horizontal axis in this range. There are at most three steady states Žsee Fig. 1..13 First there is the ‘‘thick market’’ steady state x 1s , where the relatively high trading volume e s y x 1s eliminates both the productivity gap and the trading cost. Second, there are the ‘‘thin market’’ steady states x s2 and x s3, where trading volumes are relatively low and where neither the productivity gap nor the trading cost is eliminated, i.e., DŽ x s3 . s T Ž x s3 . ) DŽ x s2 . s T Ž x s2 . ) 0. The existences of thin and thick market steady states depend on the allocation e s . This dependence is shown in Fig. 2, where e1 ' x 1s q zU and e 2 ' x 1s q Ž b q zU . 2r4b. Consider first the thick market steady state x 1s , which must exist whenever 12
We can confine our attention to emission levels x s F e s because the South can never be the long-run buyer of CO 2 . Formally, T Ž x s . is nonpositive for all x s ) e s , which rules out steady states in this region. 13 I shall ignore the knife-edge cases where the graphs of DŽ x s . and T Ž x s . are tangent to each other.
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FIG. 2. Existence of steady states.
e s G e1. Economically, such an allocation allows for trading volumes that are large enough to eliminate the trading cost through the trading externality. Geometrically, the allocation e s G e1 stretches the horizontal part of the graph T Ž x s . from the origin to a point that is not less than x 1s . By Ž4., the interval w e1 , e x is not empty. Consider then the thin market steady states x s2 and x s3. These equilibria exist for any given e s g Ž e1, e 2 ., and they are given by the roots of Ž a q bx s . y Ž e s y x s .Ž zU y Ž e s y x s .. s 0 Žby DŽ x s . y T Ž x s . s 0.,14 x s2, 3 s e s y
1
bqz 2½
U
" Ž b q zU . y 4 Ž a q be s . 2
1r2
5.
Ž 18 .
Geometrically, e s g Ž e1 , e 2 . is the case depicted in Fig. 1. Economically, such an e s dictates the existence of emission levels x s for which the productivity gap falls short of the marginal trading cost Žthe interval Ž x s2 , x s3 . in Fig. 1.. This makes it possible to eliminate incentives for further trading not only at x 1s but also at x s2 and x s3. Increasing e s above e 2 rules out the occurrence of the latter two equilibria Žsee Fig. 2.: the total number of permits purchased for any given x s becomes larger, which reduces the trading cost through the training externality and thereby keeps this cost below the productivity gap. Geometrically, the inverted U-shaped part of the graph for T Ž x s . dips below the graph for DŽ x s .. Formally, for e s larger than e 2 , the roots given by Ž18. are not real, leading to the uniqueness of x 1s . This policy of eliminating the existence of thin market equilibria through a large allocation e s can be pursued, however, only if the total emission cap is relatively large, i.e., if e ) e 2 . If e - e 2 the interval Ž e 2 , e x in Fig. 2 does not exist, meaning that the possibility of a thin market steady state prevails for all e s ) x 1s .15 Finally, for small allocations e s g Ž x 1s , e1 ., the largest conceivable trading volume e s y x 1s is less than the volume needed for costless trading, meaning that the possibility of the thick market equilibrium x 1s is eliminated; the only allocation satisfying DŽ x s . s T Ž x s . is x s3 , a thin market equilibrium. Formally, x s3 is given by Ž18., whereas the smaller root x s2 is less than x 1s and therefore no longer characterizes a steady state.
For e s s e1 , these roots satisfy x s3 s x 1s q zU y b ) x s2 s x 1s , and for e s s e 2 , x s2 s x s3 s x 1s q Ž zU q b .Ž zU y b .r4 b ) x 1s , where the inequalities follow from zU ) b in Ž4.. Using these and the fact that the term under the square root in Ž18. is decreasing in e s show that the roots are real, positive, and greater than x 1s for e s g Ž e1 , e 2 .. 15 Using the definitions of e 2 and x 1s , e ) e 2 is seen to hold iff e ) Ž bs y bn .Ž1rc s . q Ž c s q c n q U 2 z . Ž1r4c s .. While e is a given fixed constant throughout this paper, it is of interest to note that reducing e may cause economic costs not only by increasing abatement costs but also by increasing trading costs if the possibility of thick market equilibrium is eliminated. 14
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4. THE ROLE OF EXPECTATIONS Having identified how the set of long-run market allocations depends on the pre-trade allocation e s , we must next determine which one of these equilibria is actually chosen and how. To this end, I consider equilibrium paths leading to a steady state and first confine attention to the case e s g Ž e 1, e 2 .. Equilibrium paths are best characterized by using Fig. 3 Žtop, middle and bottom. which are phase planes for the planar system Ž ˙ xs, m ˙ .. The figures show on their horizontal axes the emission level x s and on their vertical axes the shadow price m which, by Ž14., determines the direction of change in x s . Also, a change in the Southern emissions is matched by a change in the Northern emissions, ˙ x s s yx˙n . Note next that the m ˙ s 0 locus is given by the equation m s T Ž x s . y DŽ x s .4r which is just the difference between the graphs in Fig. 1 divided by the discount rate. The shape of the m ˙ s 0 locus is thus driven by the factors discussed in the previous section.16 The intersections of this locus with the horizontal axis, the ˙ x s s 0 locus, characterize the steady states. For a given fixed e s , the steady states in Fig. 3 coincide with those in Fig. 1. Appendix 2 shows that for any given x s an equilibrium path always exists, and that generic equilibrium paths can be globally characterized as follows. First, the equilibrium of type I leads to the thick market equilibrium x 1s , and the path originates from the vertical axis ŽFig. 3.. Second, the equilibrium of type II also leads to the thick market equilibrium x 1s , but this path emanates either from the thin market equilibrium x s2 ŽFig. 3 Žtop and bottom.. or from the vertical line e ŽFig. 3 Žmiddle... Third, the equilibrium of type III heads for the thin market equilibrium x s3, and it emanates either from x s2 ŽFig. 3 Žtop and middle.. or from the vertical axis ŽFig. 3 Žbottom... Finally, the equilibrium of type IV also leads to the thin market equilibrium x s3 , and it originates from the vertical line e ŽFig. 3.. What is the economic meaning of the above equilibrium paths? Recall that the private cost of trading is determined by two factors pulling in opposite directions: the size of private trades and the thinness of the CO 2 market depending on how much CO 2 other traders are providing for sale. To see how these factors interact in different equilibria, suppose that the initial Southern emission level is from the interval ⍀ that denotes a common domain of any two equilibrium paths in Fig. 3. Given that a polluter believes that the future CO 2 market will be thin, meaning costly future trading opportunities, it makes sense to increase emissions in the South and reduce the number of permits left for sale Žpath III in Fig. 3 Žtop... If a polluter believes that the future CO 2 market is thick, meaning frictionless future trading, it is rational to commit resources to emissions reduction in the South in order to provide more permits for sale as long as the productivity gap is positive Žpath II in Fig. 3 Žtop..; Figure 3 Žmiddle, bottom. shows that self-fulfilling expectations of this type can choose not only among paths II and III but also among I and III ŽFig. 3 Žbottom.. and among II and IV ŽFig. 3 Žmiddle... What are the factors that determine whether the above type of leeway regarding the future trading activity exists? For a given x s Ž0., this amounts to finding the The kink is caused by the kink in T Ž x s . at x s s e s y zU Žsee Fig. 1.. Despite the kink, the usual existence᎐uniqueness results can be applied to the system Ž ˙ xs, m ˙ . because T Ž x s . is locally Lipschitz: K < x y y < G < T Ž x . y T Ž y .< holds for all Ž x, y . in a neighborhood of x s s e s y zU , where K is the right-hand derivative of T Ž x s . at x s s e s y zU . By the differentiability, the r.h.s. of Ž17. is trivially Lipschitz outside the kink. 16
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FIG. 3. Equilibrium paths, e1 - e s - e 2 .
parameters that determine the size of the interval ⍀. If ⍀ is empty the equilibrium depends on history; for each initial level of Southern emissions there is a unique equilibrium path that determines the future trading level. In such cases the future CO 2 market becomes thick provided that the initial trading level is greater than e s y x s2 or thin provided that the initial trading level is less than e s y x s2 . On the other hand, when ⍀ is not empty and when x s Ž0. g ⍀ the equilibrium is given by
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expectations that may encourage polluters to reach trading levels that are significantly different from the current levels.17 Given that the system Ž ˙ xs, m ˙ . is nonlinear, the size of ⍀ is difficult to address Žsee w20, p. 647x for reasons.. However, the following argument regarding the adjustment cost cŽ r . s Žr2. r 2 can be made. Consider Fig. 3 Žtop. and suppose that the interval ⍀ is empty, i.e., assume that the paths of type II and III do not overlap for any x s . Then it is not difficult to show that the adjustment cost parameter must be higher than a given cutoff value U ; if - U the paths II and III must overlap. Similarly, given the adjustment parameter , it is possible to define a cutoff value for the interest rate such that the interval ⍀ can be empty only for ) U ; if - U the overlap must exist Žsee Appendix 3 for a proof.. As in w14x, the presence of external economies creates a strong interdependence among private decisions: the future returns on emissions reductions will depend on how much other polluters’ leave permits for sale. This explains the decisive role of discounting for the existence of the overlap ⍀; the less the future is discounted, the more polluters care about the thinness of the future CO 2 market, i.e., about the actions of other polluters. The logic for the adjustment cost is similar: the lower is the faster the reallocation of emissions and thus the fulfillment of the original expectation. In pollution trading the role of expectations depends not only on the above parameters but also on the pre-trade allocation e s that is a policy instrument. To see this, suppose now that e ) e 2 and that e s g Ž e 2 , e x. In view of Fig. 2, the thick market equilibrium x 1s becomes unique, which eliminates the role of expectations: for any given initial emission level, there is a unique equilibrium path in Fig. 3 leading to the thick market equilibrium Žsee Appendix 2.. Geometrically, for e s g Ž e 2 , e x, the m ˙ s 0 locus intersects the horizontal axis only at x 1s and is thus positive for all x s - x 1s and negative for all x s ) x 1s . Another type of pre-trade allocation that rules out the role of expectations is e s g Ž x 1s , e1 ., which ensures the existence of a unique equilibrium path to the thin market equilibrium x s3 Žsee Appendix 2.. In this case, the m ˙ s 0 locus intersects the horizontal axis only at x s3 and is thus positive for all x s - x s3 and negative for all x s ) x s3. What we can now see is that there is no decisive role for expectations in equilibrium selection if e s is sufficiently large Ž e s ) e 2 . or small Ž e s - e1 .. If e s is large external economies will be large enough to rationalize straightforward expectations about costless trading. If e s is small such expectations can be immediately ruled out because there is not much scope for external economies. It is only the medium allocation e s g Ž e1 , e 2 . which can allow for a sufficiently strong interdependence among private decisions that makes expectations about thin and thick CO 2 markets equally rational.
17
Analogously, in the process of industrialization the economy may not be able to escape from a low level of development if the equilibrium depends on history, whereas the takeoff is possible if the equilibrium depends on expectations w20x. Interestingly, if low x s is viewed as being an indicator of a large development gap between the North and South, thick CO 2 market is associated with a large development gap. But, of course, development is an intricate process involving important aspects that are not included here Že.g., the adoption of technology..
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5. LONG-RUN EFFECTS OF PRE-TRADE ALLOCATIONS Our attention can now be confined to those long-run equilibria to which a dynamic equilibrium potentially converges, i.e., to the steady states x 1s and x s3. I shall next isolate three nonstandard steady state effects of the pre-trade allocation on Ž1. the level of the trading cost; Ž2. trading levels, and Ž3. market allocations of emissions reductions. Consider the effect on the steady state trading cost T Ž x s . and note first that the thin market trading cost satisfies T Ž x s3 . s DŽ x s3 . s a q bx s3 for e s g Ž x 1s , e 2 ., where x s3 is the larger root in Ž18.. In Fig. 4, a q bx s3 is graphed as a function of e s Žif e 2 ) e, the domain of the graph extends to e .. Note next that the thick market trading cost T Ž x s . is given by the horizontal axis for e s g w e1, e x. For ease of discussion, recall that x 1s is the after-trade allocation that would be achieved if trading were costless by assumption Ž x 1s eliminates the productivity gap.. Now, in view of Fig. 4, it can be seen that no pre-trade allocation that is close to the market allocation x 1s can support the emergence of a frictionless long-run market: for e s g Ž x 1s , e 1 . the trading cost is necessarily positive. Also, at least initially, the larger the departure of e s from x 1s the greater the actual thin market trading cost. On the other hand, pre-trade allocations that significantly depart from the market allocation x 1s have chances of supporting costless long-run trading. This can succeed in two ways. First, if the departure can be made sufficiently large Ži.e., e s ) e 2 for e ) e 2 . the long-run trading cost will vanish for sure. Second, for medium departures from the allocation x 1s Ži.e., for e s g Ž e 1, e 2 .. the policy succeeds if polluters coordinate their expectations in favor of thick markets or if history dictates the emergence of thick markets Žas explained in Section 4.. Consider next the thin and thick market trading levels that are depicted in Fig. 5 for all feasible allocations e s . Note first that in a thick market steady state the trading level is given by the 45⬚ line: because trading is costless, the entire quantity e s y x 1s will flow to the high-productivity North. In the thin market steady state x s3 the trading level is given by e s y x s3, which falls short of the 45⬚ line Žsee the graph defined for e s g Ž x 1s , e 2 . in Fig. 5.. Given the trading cost schedules depicted in Fig. 4, this pattern of steady state trading levels is not surprising. Consider finally the effect on the market allocations of emissions. The above pattern of trading levels indicates that no pre-trade allocation that is close to the
FIG. 4. Long-run marginal trading cost.
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FIG. 5. Long-run trading level.
after-trade allocation x 1s can actually support the achievement of x 1s as a market allocation: for e s g Ž x 1s , e 1 . the productivity gap is not eliminated because the thin market trading level falls short of the 45⬚ line in Fig. 5. The pattern of trading levels also indicates that allocations e s which significantly depart from x 1s can actually achieve x 1s as a market allocation: In view of Fig. 5, the thick market trading level that supports the elimination of the productivity gap can be achieved if e s g w e1 , e x. These results potentially have important implications for studies that see the CO 2 market as a frictionless transfer mechanism Že.g., w4, 15x.. In these papers the pre-trade allocation of emissions has no effect on the market allocation of emissions, which is why the pre-trade allocation directly determines trade flows that redistribute the global income. In my model, pre-trade allocations can have this property only if they are significantly different from the long-run market allocation that could be achieved if trading were costless. 6. CONCLUSIONS This paper provides the first step toward analyzing explicitly the gradual development of frictions in a tradable pollution permit market. It incorporated adjustment costs and the common idea that the ease with which trades can be conducted depends on the market size. The principal result is that the key policy variable, the pre-trade allocation of permits, is a critical determinant of the frictionless long-run permit market. More specifically, pre-trade allocations that are close to those after-trade allocations that could be achieved if trading were frictionless have poor chances of supporting the actual development of frictionless trading. On the other hand, pre-trade allocations that are significantly different from the market allocations achieved in a perfect market provide the most favorable basis for the actual emergence of a perfect market. But the policy of using pre-trade allocations for the latter purpose may involve a risk: the pre-trade allocation that supports the development of a thick market can also be the one that has the potential of supporting the emergence of a costly thin market as a self-fulfilling prophecy. On a theoretical level, there are also other ways of analyzing the development of friction in tradable permit schemes than the current one. First, my objective has
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merely been to demonstrate the working of a market-based policy instrument that is incomplete. To obtain further refinements of this instrument, one should consider either the Pareto ranking of the various outcomes provided here or the full social optimum. In the latter case it might be optimal to subsidize the trading activity because of the positive externality from the market size. Second, while there are trading costs that are likely to vanish as soon as the market for permits becomes thick enough, some sources of trading costs may have more to do with the history of trades than with the current state of the market. For example, a trader may better see its options in trading, or a brokerage firm may find it easier to take the role of a market maker the longer the historical data about the relevant variables Že.g., past prices. are. This approach is complementary rather than competitive to the one in this paper in that it is another way of introducing market-level scale economies in trading. However, it could provide very different insights, e.g., because trading might not be costly in a market that is currently thin, provided it has been thick in the past. Third, although the approach taken here is able to endogenize the level of the trading cost, it does not explain how the burden of this cost is shared among buyers and sellers. To this end, one should explicitly consider the bargaining over the division of the value of individual exchange opportunities. As in search theory Že.g., w22x., the division should depend on polluters’ outside options, e.g., on the difficulties in finding other trading partners, that are determined endogenously in equilibrium. This allocative effect could be of key importance when considering the permit market as a transfer mechanism. The analysis of this effect calls for a search theoretic approach Žsee w18x.. APPENDIX 1: SOLUTION OF PROBLEM (9) ᎐ (12) Let LŽ x s , z s , r . s H Ž x s , z s , r . q Ž e s q z s y x s . be the Lagrangian, where H Ž x s , z s , r . s R s Ž x s . y pz s y ␦ Ž z s . hŽ z . y cŽ r . q mr is the Hamiltonian; the multipliers m and are in the current value form. Interior necessary conditions include ŽTheorems 6.5.1 and 9.3.1 w17x. L z s s yp y ␦ X Ž z s . h Ž z . q s 0,
Ž 19 .
L r s ycX Ž r . q m s 0,
Ž 20 .
L s e s q z s y x s s 0,
Ž 21 .
m ˙ s m y RXs Ž x s . q .
Ž 22 .
Note that for all Ž x x , z s . satisfying Ž21., the price p is positive by Ž5., implying that ) 0. By the strict convexity of cŽ r ., Ž20. must hold, which by Ž10. gives the equation ˙ x s s m. Using Ž19. in Ž22. yields Ž13. which leads to the equation m ˙ s m q DŽ x s . y T Ž x s ., as explained in the text. A solution of Ž ˙x s , m ˙ . defines a mapping t ¬ Ž p, z .. Denote this mapping by ⌫ i Ž⭈; x s Ž0., mŽ0.., where i s I, II, III, and IV correspond to solution paths identified in Section 4. For each ⌫ i, take the pair Ž p, z . : t g w0, ⬁.4 and insert it into the problem Ž9. ᎐ Ž12.. Then, by the construction of Ž ˙ xs, m ˙ ., the polluter chooses functions r Ž t . and z s Ž t . that are the same that define ⌫ i for each i, and that are optimal up to the necessary conditions. The concavity of L in Ž x s , z s , r . jointly and the fact that the suggested equilibrium paths converge imply that the sufficient conditions for Ž9. ᎐ Ž12. hold for each ⌫ i ŽTheorem 9.3.1 of w17x..
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APPENDIX 2: THE EXISTENCE OF EQUILIBRIUM PATHS Local existence. The linearization of Ž ˙ xs, m ˙ . in a steady state yields < J < s Ž dmrdx s . , where J is the Jacobian of the linearized system and dmrdx s is the slope of the m ˙ s 0 locus. The trade of J is trŽ J . s ) 0. The signs of < J < and trŽ J . evaluated in a steady state determine the local stability properties of the system. In the case e s g Ž e1 , e 2 ., the slope of the m ˙ s 0 locus in Fig. 3 indicates that the steady states Ž x 1s , 0. and Ž x s3 , 0. are locally saddlepoints Ž< J < - 0. and that the steady state Ž x s2 , 0. is locally a source Ž< J < ) 0, trŽ J . s ) 0.. In the cases e s g Ž x 1s , e 1 . and e s g Ž e 2 , e x, the steady states are the saddlepoints Ž x s3,0. and Ž x 1s , 0., respectively. Thus, by the local saddlepoint theorem there exists a local equilibrium path to a steady state for each allocation e s . Global existence. The following property of Ž ˙ xs, m ˙ . is global: There is no closed orbit on the x s y m plane. This follows from Bendixson’s criterion, which holds because ⭸ ˙ x sr⭸ x s q ⭸ mr ˙ ⭸ m s is a constant ŽTheorem 1.8.2 of w10x.. This property, together with the Poincare᎐Bendixson Theorem Ž1.8.1 of w10x., implies ´ the global existence of equilibrium paths to the steady states Ž x s3, 0. and Ž x 1s , 0. in cases e s g Ž x 1s , e1 . and e2 g Ž e 2 , e x, respectively. Then consider the case e s g Ž e1, e 2 .. The nonexistence of a closed orbit implies that either a path of type II or III is connected to the steady state Ž x s2 , 0., or that they are both connected to Ž x s2 , 0.. Suppose the contrary, that neither of these equilibrium types originates from Ž x s2 , 0.. Because the unstable saddlepoints emanating from Ž x 1s , 0. and Ž x s3 , 0. cannot cross the paths II and III nor lead to Ž x s2 , 0., they must define a closed curve around Ž x s2 , 0., a contradiction. These characteristics, together with the Poincare᎐Bendixson Theorem, imply the classification presented by Fig. 3. The ´ nongeneric case of a heteroclinic orbit Ža path connecting steady states, see w10, p. 45x. should be seen as a subcase of Fig. 3 Žmiddle, bottom.. Thus, for any given x s , there exists an equilibrium path to a steady state. APPENDIX 3: THE EXISTENCE OF THE OVERLAP The interval ⍀ can be empty only if the roots for the system obtained by linearizing Ž ˙ xs, m ˙ . at Ž x s2 , 0. are real. The roots are real Žimaginary, resp.. iff 2 trŽ J . ) 4 < J < Ž- 4 < J <, resp... This inequality can be rewritten as 2 ) 4 < J < s Ž4 r .Ž dmrdx s ., where dmrdx s is the slope of the m ˙ s 0 locus at Ž x s2 , 0.. Thus, U the roots are real iff ) ' Ž4r .Ž dmrdx s . and imaginary iff - U . Similarly, the cut-off value for the interest rate is U ' Ž4r .Ž dmrdx s .. REFERENCES 1. S. Brown, D. Kennedy, C. Polidiano, K. Woffenden, G. Jakeman, B. Graham, F. Jotzo, and B. S. Fisher, Economic impacts of the Kyoto Protocol, Res. Rep. 99.6, Abare Canberra Ž1999.. 2. A. Banerjee and A. Newman, Information, the dual economy, and development, Re¨ . Econom. Studies 65, 631᎐653 Ž1998.. 3. S. Barrett, Political economy of the Kyoto Protocol, Oxford Re¨ . Econom. Policy 14, 20᎐39 Ž1998.. 4. P. Bohm and B. Larsen, Fairness in a tradable-permit treaty for carbon emissions reductions in Europe and the former Soviet Union, En¨ iron. Res. Econom. 4, 219᎐239 Ž1994.. 5. P. Diamond, Aggregate demand management in search equilibrium, J. Polit. Economy 90, 881᎐894 Ž1982..
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6. P. Diamond and D. Fudenberg, Rational expectations business cycles in search equilibrium, J. Polit. Economy 97, 606᎐620 Ž1989.. 7. D. Dollar, Technological innovation, capital mobility, and the product cycle in the North᎐South trade, Amer. Econom. Re¨ . 76, 177᎐190 Ž1986.. 8. V. Foster and R. Hahn, Designing more efficient markets: Lessons from Los Angeles smog control, J. Law Econom. 38, 19᎐48 Ž1995.. 9. J. Gould, Adjustment costs in the theory of investment of the firm, Re¨ . Econom. Studies 35, 47᎐56 Ž1968.. 10. J. Guckenheimer and P. Holmes, ‘‘Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,’’ Springer-Verlag, New York Ž1993.. 11. P. Howitt and P. McAfee, Costly search and recruiting, Internat. Econom. Re¨ . 28, 89᎐107 Ž1987.. 12. P. Howitt and P. McAfee, Stability of equilibria with externalities, Quart. J. Econ. 103, 261᎐277 Ž1988.. 13. P. Joskow, R. Schmalensee, and E. Bailey, The market for sulfur dioxide emissions, Amer. Econom. Re¨ . 88, 669᎐685 Ž1998.. 14. P. Krugman, History versus expectations, Quart. J. Econom. 106, 651᎐667 Ž1991.. 15. S. Kverndokk, ‘‘Essays on Cost-Effectiveness, Efficiency and Justice in International CO 2 Agreements,’’ dissertation, University of Oslo Ž1994.. 16. A. Lewis, Economic development with unlimited supplies of labor, Manchester School 28, 139᎐191 Ž1954.. 17. D. Leonard and N. Van Long, ‘‘Optimal Control and Static Optimization in Economics,’’ Cambridge Univ. Press, Cambridge, UK Ž1992.. 18. M. Liski and J. Virrankoski, Bilateral CO 2 trading, mimeo, Helsinki School of Economics Ž1999.. 19. R. Lucas, Adjustment costs and the theory of supply, J. Polit. Economy 75, 321᎐334 Ž1967.. 20. K. Matsuyama, Increasing returns, industrialization, and indeterminacy of equilibrium, Quart. J. Econom. 106, 617᎐650 Ž1991.. 21. W. Montgomery, Markets in licenses and efficient pollution control programs, J. Econom. Theory 5, 395᎐418 Ž1972.. 22. D. Mortensen, Property rights and efficiency in mating, racing and related games, Amer. Econom. Re¨ . 72, 968᎐980 Ž1982.. 23. R. Noll, Implementing marketable emission permits, Amer. Econom. Re¨ . 72, 120᎐124 Ž1982.. 24. W. Nordhaus and J. Boyer, Requim for Kyoto: An economic analysis of the Kyoto Protocol, Energy J. ŽKyoto Special Issue., 74᎐130 Ž1999.. 25. P. Romer, Increasing returns and long-run growth, J. Polit. Economy 94, 1002᎐1037 Ž1986.. 26. R. Schmalensee, Greenhouse policy architectures and institutions, in ‘‘Economic and Policy Issues in Climate Change: Evaluations of the Intergovermental Panel Report’’ ŽW. Nordhaus, Ed.., Resources for the Future, Washington, D.C. Ž1998.. 27. R. Stavins, Transaction costs and tradeable permits, J. En¨ iron. Econom. Management 29, 133᎐148 Ž1995.. 28. A. Trejos and R. Wright, Search, bargaining, money, and prices, J. Polit. Economy 103, 118᎐141 Ž1995..
Thin versus Thick CO 2 Market 1 Matti Liski Helsinki School of Economics, P.O. Box 1210, 00101 Helsinki, Finland E-mail: [email protected] Received January 28, 1999; revised June 24, 1999; published online October 24, 2000 The idea that trading is more costly the thinner the market is is common in most studies of market exchange with frictions. Surprisingly, this element is lacking from previous attempts to allow for frictions in pollution permit markets. This paper considers a CO 2 cap-and-trade model where trading costs develop endogenously as a function of the market size. The pre-trade allocation of permits determines whether the market size can be strongly influenced by expectations that have a role because of adjustment costs. The pre-trade allocation also sets preconditions for endogenously vanishing trading costs and thus has nonstandard effects on long-run trading levels and market allocations. 䊚 2001 Academic Press
1. INTRODUCTION The market mechanisms built into the Kyoto Protocol have the potential of significantly reducing the costs of limiting greenhouse gases Že.g., w1, 3, 24x., but the extent to which these gains will be achieved depends on the degree of friction in the CO 2 market.2 Given the economic stakes involved in this market, it is important that trading institutions are designed to support the development of frictionless CO 2 trading.3 However, there has been no formal attempt to identify the determinants of such a development. While it is straightforward to consider the effect of frictions on pollution permit markets by introducing trading costs that are formally equivalent to a tax or transportation cost w27x, this approach has no hope of explaining the level of these costs: they are not endogenous in equilibrium. In particular, the common idea that trading is more costly the thinner the market is is lacking Žsee, e.g., w5, 11, 12x.. I consider a dynamic model of CO 2 trading where trading costs are not a priori absent Žas in w21x. nor exogenously given Žas in w27x. but rather develop endogenously over time as a function of the market size. Endogenous trading costs can be used to analyze the determinants of a frictionless CO 2 market. A critical determinant is the pre-trade allocation of permits, the 1
This paper is based on ideas developed in Chapter 2 of my thesis which benefited from comments by Pertti Haaparanta, Michael Hoel, Larry Karp, Matti Pohjola, Seppo Salo, and Olli Tahvonen. I also thank Cees Withagen, anonymous referees, an associate editor, and the participants at the workshops ‘‘Economic Issues Related to Climate Change’’ in Oslo 1998 and ‘‘Design of Energy Markets and Environment’’ in Copenhagen 1999 and at the 1999 EAERErEEA annual conferences. 2 The Kyoto Protocol covers six types of greenhouse gases, but because each type of greenhouse gas can be converted into CO 2 -equivalent units, I interpret the emission limits as CO 2 limits. 3 References w27x and w8x discuss the empirical evidence of trading costs in earlier pollution trading experiments. 295 0095-0696r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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key policy variable, which alters the basic economic properties of tradable pollution permits in two important ways. First, the pre-trade allocation determines whether self-fulfilling prophecies can strongly influence the market size: polluters’ expectation of low trading volumes can lead to expectation of a thin CO 2 market, which discourages polluters from undertaking emissions reductions that would leave permits for sale, thereby fulfilling the original expectation. Second, the pre-trade allocation sets preconditions for endogenously vanishing trading costs and thus has nonstandard effects on long-run trading levels and market allocations of emissions. Trading costs in permit markets can be justified in a number of ways. Stavins w27x identifies two potential sources of trading costs incurred by firms: Ž1. search and information and Ž2. bargaining and decision. Type-1 costs are best understood if trades are conducted by middlemen, whose brokerage fees constitute a direct financial cost of trading. Type-2 costs involve, for example, the cost of entering into negotiations and fees for legal and insurance services. However, both types of costs can be interpreted in terms of costs due to a lack of information that is underprovided by thin markets. Thick markets with numerous transactions are likely to generate positive informational externalities that absorb part of the trading costs Žsee also w27, p. 144x.. For example, the number of middlemen can rise as the trading activity increases; a larger pool of brokers means a lower cost of finding a trading partner, thereby reducing the amount of real resources taken up by trading. Alternatively, if polluters conduct trades themselves the transaction cost may reflect the cost of gathering information on the price level. Such costs are likely to be lower the thicker the market is Žsee w23, p. 121x..4 To my knowledge, the implications of trade externalities of this type have not been addressed in connection with the global CO 2 or pollution trading in general.5 The implications of trade externalities are cast in a North᎐South CO 2 trading model, where the North and the South are asymmetric in two respects.6 First, there is the usual asymmetry that the North is more productive in using a unit of CO 2 . In this sense, the costs of reducing emissions are higher in the North. Second, resources committed to activities that change the emission level move more sluggishly in the South because of higher adjustment costs.7 Costs of shifting resources arise, for example, from temporary drops in output caused by the phase-out of old machinery and the installation of new machinery. Phase-out periods may be longer and contracting costs higher because of a shortage of skilled labor, corruption, political instability, and insufficient infrastructure in general. 4 The evidence from trading under the Clean Air Act Amendments of 1990 supports these observations: middlemen’s commissions have been continually falling with trading activity, as has the uncertainty regarding the allowance price w13, pp. 683᎐684x.. 5 Such trade externalities are common in search theoretic models Že.g., w5, 22x.. My approach is not search theoretic but is closest to w12x, which assumes that trading costs depend on the market size. 6 Given the current state of the climate treaty, this setting is still quite hypothetical. First, the studies addressing the current trading potential often view the CO 2 market as an East᎐West market Že.g., w3, 24x.. But if the long-run treaty covers the major economies that are currently not participating, this market can reasonably be viewed as a North᎐South market. Second, the problems of monitoring and enforcement are likely to pose serious problems to the development of a global cap-and-trade system w26x. 7 Reference w7x makes a similar type of assumption in a North᎐South trade model, where the distinguishing feature of the South is its lower speed of factor mobility Žp. 178.. The inertia in the South may be due to the duality of its economy, which can be explained by differences in the nature of rationality w16x or in technology and information w2x between the Southern sectors.
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Also, a given reduction in emissions may require a greater shift in labor between sectors in the Southern economy. Adjustment costs reflect costs due to congestion, a period of unemployment, or retraining of labor. As in the investment theory, it is difficult to distinguish polluters’ internal and external adjustment costs, but both types of costs can lead to the same formal model Žsee w9, 19x.. Adjustment costs create a role for a forward-looking element that is a key to the analysis of whether polluters end up in a thin or thick market equilibrium. The expected returns on emissions reductions can be seen to depend on the thinness of the future CO 2 market, which in turn depends on the pre-trade allocation of permits. I find that while a relatively large pre-trade allocation in the low-productivity South is needed to rationalize expectations about a costless and thick CO 2 market, such an allocation can also allow a role for expectations that can strongly influence the market size. The strong dependence on expectations is likely, for example, when adjustment costs are low, implying a fast reallocation of emissions and thus a fast fulfillment of the original expectation. That the trading level can depend on expectations indicates a potential expectational role for public forecasts in tradable permit experiments Žas in aggregate demand management, see w6x.. For the above reasons, the choice of one allocation over another should be defended not only from an equity point of view but also from an economic point of view. This is in sharp contrast with studies that use the CO 2 market as a frictionless transfer mechanism to evaluate the distributional effects of different allocation rules of the pre-trade CO 2 cap Žsee, e.g., w4, 15x.. By some estimates these transfers could exceed the current level of development aid Žsee w15, p. 34x.. I find that when trade frictions are endogenous, the CO 2 market can be a frictionless transfer mechanism only for pre-trade allocations that are significantly different from the long-run market allocation that would be achieved if trading were costless by assumption. The next section sets up the model and derives a dynamic system whose solution can be used to characterize the equilibrium. Section 3 identifies the set of long-run equilibria for various pre-trade allocations, and Section 4 considers equilibrium paths. The latter section explains the role of expectations and shows which equilibrium can actually be chosen. Section 5 isolates nonstandard long-run effects of the pre-trade allocations by focusing on those stationary equilibria to which a dynamic equilibrium potentially converges.
2. THE MODEL Permits. I postulate a two-region model ŽNorth᎐South. with large and equal populations of polluters within both regions. All Northern and Southern polluters are respectively identical. The global cap on CO 2 emissions remains constant over time and is implemented by issuing in each period a given fixed number of CO 2 rights Žpermits. that are usable only in the period of issuance Žthe banking or borrowing of permits is not allowed.. The regional shares of these permits remain constant over time and are distributed free of charge. Thus, in each period t the equation e s en q e s
Ž 1.
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holds, where e n and e s are the pre-trade allocations of permits per polluter in the North and the South, respectively, and e is the global per polluter cap on emissions. Technology. The revenue function for a polluter of type i Žs n, s . is R i Ž x i . s a i q bi x i y Ž c ir2. x i2 , where x i is the emission level and a i , bi , c i are positive parameters. This function can be thought of as arising from a relationship between the polluter’s output and a vector of energy-intensive inputs Žsee, e.g., w15, p. 22x.. The linear-quadratic form is not necessary, but it will help to keep the model parsimonious. It proves useful to write the productivity gap as depending upon x s , D Ž x s . s RXn Ž e y x s . y RXs Ž x s . s a q bx s ,
Ž 2.
where x n s e y x s , a ' bn y bs y c n e and b ' c n q c s . I shall maintain the assumptions Ži. that the pre-trade valuation of additional units of CO 2 is higher for the representative Northern polluter, i.e., DŽ e s . ) 0, and Žii. that there is an allocation Ž x n , x s . s Ž e y x s , x s . of e that eliminates the productivity gap, i.e., DŽ x s . s 0 for x s s x 1s ' yarb ) 0. The first assumption ensures that it is the Northern trader who takes the role of a long-run buyer, and the second rules out the unrealistic equilibrium where all Southern permits are reallocated to the North. Trading cost. Buyers of CO 2 permits face a transaction cost ␦ Ž z i . hŽ z . where z i Ž) 0. is the number of permits purchased by a source of type i Žs n, s . and z is the global average trading volume. Because the number of traders is fixed, I shall use the words average and total trading volume interchangeably. For ease of presentation, I assume that the trading cost is characterized by
␦ Ž zi . s
½
z i2 for z i G 0 0 otherwise 1 2
and
hŽ z . s
½
zU y z for z F zU 0 otherwise
Ž 3.
where the parameter zU is positive. Note first that sellers pay no trading cost, i.e., ␦ Ž z i . s 0 for z i - 0.8 Note next that, by the large number of traders, hŽ z . is an exogenous parameter from an individual trader’s point of view, implying that the trading cost is increasing in the size of private purchase, ␦ X Ž z i . ) 0. The assumption that the trading cost is decreasing in the total trading volume, hX Ž z . F 0, formalizes the idea that it is more costly to operate in a thin than in a thick market. Because the trading volume z is endogenous in equilibrium, the level of the trading cost will also be endogenous in equilibrium. This aspect of transaction costs in connection with pollution permits has been suggested, but not formalized, by w27, p. 142x: ‘‘ . . . at the market level, decreasing marginal transaction costs could also be due to positive information externalities Žassociated with larger trading volumes. that systematically lower transaction costs’’ Žsee also w27, p. 144x.. Note that private scale economies in trading ␦ Y Ž z i . - 0, would not be unreasonable w27x. For 8
All the equilibria described in this paper would remain the same even if we assumed that the seller pays the transaction cost Žas in w27, p. 140x..
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tractability, I exclude this case but include market-level scale economies by hX Ž z . F 0.9 The trading externality eliminates the trading cost entirely whenever the total trading volume exceeds the level zU Žsee Ž3... To keep the role of this externality interesting, the following restrictions are imposed: e y x 1s ) zU ) b.
Ž 4.
The quantity e y x 1s should be seen as the largest conceivable trading volume that can be achieved. It is the quantity sold by a representative Southern polluter when its pre-trade allocation is extremely large, e s s e, and when trading eliminates the productivity gap, DŽ x 1s . s 0. The first inequality thus allows a potential role for frictionless trading, because hŽ z . s 0 for z s e y x 1s . The second inequality in Ž4. will rule out an inevitable development of frictionless CO 2 markets by setting a lower bound for the trading level that is needed for costless trading. To make sure that there exists a positive equilibrium price for CO 2 for any given allocation of emissions, the following assumption will be maintained: RXn Ž e y x s . y ␦ X Ž e s y x s . h Ž < e s y x s < . ) 0.
Ž 5.
By Ž5., the Northern buyer’s valuation of additional CO 2 units net of trading costs remains positive for all conceivable allocations Ž x n , x s . s Ž e y x s , x s . of e. Because all permits are positively valued, no permits will be left unused in equilibrium. Northern profits. By the large population of polluters, the representative Northern firm takes the price pŽ t . of a CO 2 unit and the total trading volume z Ž t . at each time t as given and maximizes the discounted value of its profits by choosing its emission and trading levels Žsuppressing the dependence on time.: max
⬁
H x ,z 0 n
R n Ž x n . y pz n y ␦ Ž z n . h Ž z . 4 ey t dt
Ž 6.
n
s.t. x n F en q z n ,
Ž 7.
where is the constant discount rate. If the polluter is currently a buyer, its profit flow is the revenue accrued by using polluting inputs in production minus the cost of purchasing z n Ž) 0. permits Žprice plus the transaction cost.. If the polluter is a seller, it receives revenues from selling z n Ž- 0. permits but, by assumption, pays no trading cost. Since there is no state equation, the problem Ž6. ᎐ Ž7. is a static one. By Ž5., the equilibrium will be characterized by interior necessary conditions that can be combined to yield the demand for permits p s RXn Ž x n . y ␦ X Ž x n y e n . h Ž z . .
Ž 8.
Y 9 One may argue that ␦ Ž z i . ) 0 can be avoided simply by splitting the quantity z i into smaller trades w27x. But this assumes that contacting new trading partners is costless, which is not reasonable in a less than perfect market. Indeed, in most studies of market exchange with frictions, potential traders meet sequentially in a time-taking process w5, 6, 11, 22x.. This alternative approach models the source of friction explicitly, but it introduces another problem: matching models have difficulties in considering trade with divisible objects such as the permit endowment w28x. My approach has no problems with divisibility, but is less explicit about individual buyer᎐seller relationships Žas is w12x..
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In view of Ž8., the effect of the trading cost can be seen as a drop in the productivity of the Northern firm: it will purchase additional units of emission rights up to the point where their marginal contribution to its revenue net of transaction costs equals the price, while in the absence of trading costs the price would indicate the true marginal valuation of CO 2 permits in production. Southern profits. The distinguishing feature of the South, besides the productivity gap DŽ e s . ) 0, is its lower speed of factor mobility. Factors move only slowly because it is costly for them to move. This ‘‘moving cost’’ is assumed to be quadratic, cŽ r . s 2 r 2 , where r is the rate of change in the Southern emissions x s and is a parameter.10 Unless otherwise stated, I assume that s 1. The cost cŽ r . occurs in the form of real resources used up in the process of increasing or decreasing emissionsᎏas explained in the Introduction there are a number of reasons why these costs are likely to be high in the South. As soon as the moving cost is introduced, the decision to change the emission level becomes an investment decision, the profitability of which depends not only on the current market price pŽ t . and trading volume z Ž t . but also on the anticipated future values of these variables. Thus, given the rationally expected pŽ t . and z Ž t ., the representative Southern polluter chooses a capacity adjustment plan r Ž t . and a permit trading plan z s Ž t . to maximize the present value of its profit flow, max r, zs
⬁
H0
R s Ž x s . y pz s y ␦ Ž z s . h Ž z . y c Ž r . 4 ey t dt
Ž 9.
s.t.
˙x s s r ,
Ž 10 .
x s F es q zs ,
Ž 11 .
e G x s Ž 0.
Ž 12 .
Ž x s Ž0. given by history.. As in the North, the polluter’s profit flow gross of adjustment costs accrue from output sales and from CO 2 trading, which has a positive or negative impact depending on whether the firm is a selling or a buying source. The initial emission level x s Ž0., which is assumed to be less than the global per polluter gap e, is the Southern emission level at the outset of CO 2 trading. It is the emission capacity determined by past decisions to increase emissions and, by the presence of adjustment costs, a given fixed variable at t s 0. Appendix 1 shows that in equilibrium the interior necessary conditions for Ž9. ᎐ Ž12. are satisfied. They can be used to yield
m s m ˙ q RXs Ž x s . y p y ␦ X Ž z s . h Ž z . 4 , 10
Ž 13 .
This is not a cost of developing or changing the abatement technology but merely an adjustment cost. While cŽ r . vanishes in a steady state, it can alter the choice between steady states ŽSection 4 below.. There are also other ways to introduce adjustment costs. First, it would be realistic to assume positive adjustment costs for both regions. However, I do not believe that this is central to the argument; the assumption of one-sided adjustment costs keeps issues comprehensive and produces the realistic equilibrium outcome that both emission levels x n and x s are adjusted gradually. Second, the cost could be made asymmetric w.r.t. increases and decreases in emissions by assuming that cŽ r . ' Žr2. r 2 q ␥ r, where ␥ is a positive or negative parameter depending on whether increasing or decreasing emissions is respectively more difficult. Third, the identities of the North and South could be changed such that cŽ r . is incurred only when x n is adjusted, i.e., adjustment costs would be higher in the North. The last two of these changes would cause nontrivial changes in polluters’ payoffs, but rather obvious qualifications to the results below.
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where m should be seen as the shadow price of having a marginal increase in the Southern emission capacity. Because of the moving cost cŽ r ., increasing x s uses up real resources that could be used elsewhere. In particular, if equals the world interest rate, the required rate of return on the investment of size m is, of course, m. If the amount m is invested to increase the Souther emission capacity by one marginal unit, the required rate of return equals the rate of capital gain m ˙ plus the current net earnings for a unit of emissions RXs Ž x s . y p y ␦ X Ž z s . hŽ z .. The term Ž p q ␦ X Ž z s . hŽ z .. is the current cost of purchasing an emission right if the firm is a net buyer and the opportunity cost of using a permit in the South if the firm is a net seller. In equilibrium, a marginal increase in x s indeed requires an investment of size m because emissions are changed at a rate given by the equality of the marginal moving cost cX Ž r . and the shadow price m Žsee Appendix 1.. By this latter equality, we have
˙x s Ž t . s m Ž t . s
⬁
Ht
RXs Ž x s . y p y ␦ X Ž z s . h Ž z . 4 ey Žyt . d ,
Ž 14 . Ž 15 .
where Ž15. reexpresses m as the discounted value of net earnings RXs Ž x s . y p y ␦ X Ž z s . hŽ z . along the equilibrium path Žfrom the integration of Ž13... If m ) 0, emissions in the South are increased; if m - 0, they are decreased. Equilibrium. Clearly, polluter’s compliance strategies will depend on their expectations about the CO 2 price pŽ t . and the thinness of the market z Ž t ., both altering the profitability of using emission rights in the South. To pin down equilibrium expectations, I derive next a dynamic system that fully characterizes all conceivable equilibria. To this end, note that in equilibrium the private marginal trading cost is given by ␦ X Ž z n . hŽ z . s Ž e s y x s . hŽ e s y x s . if the Northern firm is the buyer Ž x s F e s ., and by ␦ X Ž z s . hŽ z . s Ž x s y e s . hŽ x s y e s . if the Southern firm is the buyer Ž x s ) e s .. For a given initial allocation e s , define T Ž x s . ' Ž es y x s . h Ž < es y x s <. ,
Ž 16 .
which gives the marginal Northern purchasing cost when x s F e s . The corresponding Southern cost is just yT Ž x s . when x s ) e s . Using the fact that the equilibrium price is given by Ž8. and the definitions Ž2. and Ž16., the equation Ž13. can be rewritten as m ˙ s m q DŽ xs . y T Ž xs . .
Ž 17 .
The equations Ž14. and Ž17. jointly define a planar dynamical system Ž ˙ xs, m ˙ . on w0, e x = wy⬁, ⬁x that can be used to characterize the equilibrium for a given pre-trade allocation of emissions. Note that the system treats the level of the private marginal trading cost as an endogenous variable T Ž x s ., although hŽ z . is an exogenous parameter from the private point of view.11 As is demonstrated below, the properties of the system depend on the Southern emission share e s . 11
w25x.
The method of deriving the equilibrium dynamics in the presence of externalities is the same as in
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FIG. 1. Thin and thick market steady states.
3. THIN VERSUS THICK MARKET STEADY STATES A stationary allocation of emissions eliminates all incentives to reallocate emissions, which can hold only if m s 0 Žsee Ž14... Using this in Ž17. implies that the steady state allocation must also satisfy DŽ x s . s T Ž x s ., i.e., the productivity gap must be equalized with the private marginal trading cost. Allocations satisfying the latter condition are best characterized by using Fig. 1, which shows on its horizontal axis the Southern emission level x s and on its vertical axis the values of DŽ x s . and T Ž x s .. For any given x s , the Northern emission level is given by the residual e y x s . To make the shape of the graph for T Ž x s . comprehensive, suppose first that the Southern emissions are just equal to the pre-trade allocation, x s s e s . Because no permits are left for sale or purchased, the marginal trading cost is zero, i.e., T Ž e s . s 0, as depicted. As the trading volume increases Žwe move horizontally to the left from x s s e s . the cost T Ž x s . first increases and then decreases.12 The increasing part is explained by the fact that the trading cost is increasing in the size of private trades, and the decreasing part by the trading externality that systematically lowers the private trading cost as the total trading volume increases. In fact, for x s F e s y zU trading becomes costless because hŽ< e s y x s <. s 0; the graph of T Ž x s . is given by the horizontal axis in this range. There are at most three steady states Žsee Fig. 1..13 First there is the ‘‘thick market’’ steady state x 1s , where the relatively high trading volume e s y x 1s eliminates both the productivity gap and the trading cost. Second, there are the ‘‘thin market’’ steady states x s2 and x s3, where trading volumes are relatively low and where neither the productivity gap nor the trading cost is eliminated, i.e., DŽ x s3 . s T Ž x s3 . ) DŽ x s2 . s T Ž x s2 . ) 0. The existences of thin and thick market steady states depend on the allocation e s . This dependence is shown in Fig. 2, where e1 ' x 1s q zU and e 2 ' x 1s q Ž b q zU . 2r4b. Consider first the thick market steady state x 1s , which must exist whenever 12
We can confine our attention to emission levels x s F e s because the South can never be the long-run buyer of CO 2 . Formally, T Ž x s . is nonpositive for all x s ) e s , which rules out steady states in this region. 13 I shall ignore the knife-edge cases where the graphs of DŽ x s . and T Ž x s . are tangent to each other.
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FIG. 2. Existence of steady states.
e s G e1. Economically, such an allocation allows for trading volumes that are large enough to eliminate the trading cost through the trading externality. Geometrically, the allocation e s G e1 stretches the horizontal part of the graph T Ž x s . from the origin to a point that is not less than x 1s . By Ž4., the interval w e1 , e x is not empty. Consider then the thin market steady states x s2 and x s3. These equilibria exist for any given e s g Ž e1, e 2 ., and they are given by the roots of Ž a q bx s . y Ž e s y x s .Ž zU y Ž e s y x s .. s 0 Žby DŽ x s . y T Ž x s . s 0.,14 x s2, 3 s e s y
1
bqz 2½
U
" Ž b q zU . y 4 Ž a q be s . 2
1r2
5.
Ž 18 .
Geometrically, e s g Ž e1 , e 2 . is the case depicted in Fig. 1. Economically, such an e s dictates the existence of emission levels x s for which the productivity gap falls short of the marginal trading cost Žthe interval Ž x s2 , x s3 . in Fig. 1.. This makes it possible to eliminate incentives for further trading not only at x 1s but also at x s2 and x s3. Increasing e s above e 2 rules out the occurrence of the latter two equilibria Žsee Fig. 2.: the total number of permits purchased for any given x s becomes larger, which reduces the trading cost through the training externality and thereby keeps this cost below the productivity gap. Geometrically, the inverted U-shaped part of the graph for T Ž x s . dips below the graph for DŽ x s .. Formally, for e s larger than e 2 , the roots given by Ž18. are not real, leading to the uniqueness of x 1s . This policy of eliminating the existence of thin market equilibria through a large allocation e s can be pursued, however, only if the total emission cap is relatively large, i.e., if e ) e 2 . If e - e 2 the interval Ž e 2 , e x in Fig. 2 does not exist, meaning that the possibility of a thin market steady state prevails for all e s ) x 1s .15 Finally, for small allocations e s g Ž x 1s , e1 ., the largest conceivable trading volume e s y x 1s is less than the volume needed for costless trading, meaning that the possibility of the thick market equilibrium x 1s is eliminated; the only allocation satisfying DŽ x s . s T Ž x s . is x s3 , a thin market equilibrium. Formally, x s3 is given by Ž18., whereas the smaller root x s2 is less than x 1s and therefore no longer characterizes a steady state.
For e s s e1 , these roots satisfy x s3 s x 1s q zU y b ) x s2 s x 1s , and for e s s e 2 , x s2 s x s3 s x 1s q Ž zU q b .Ž zU y b .r4 b ) x 1s , where the inequalities follow from zU ) b in Ž4.. Using these and the fact that the term under the square root in Ž18. is decreasing in e s show that the roots are real, positive, and greater than x 1s for e s g Ž e1 , e 2 .. 15 Using the definitions of e 2 and x 1s , e ) e 2 is seen to hold iff e ) Ž bs y bn .Ž1rc s . q Ž c s q c n q U 2 z . Ž1r4c s .. While e is a given fixed constant throughout this paper, it is of interest to note that reducing e may cause economic costs not only by increasing abatement costs but also by increasing trading costs if the possibility of thick market equilibrium is eliminated. 14
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MATTI LISKI
4. THE ROLE OF EXPECTATIONS Having identified how the set of long-run market allocations depends on the pre-trade allocation e s , we must next determine which one of these equilibria is actually chosen and how. To this end, I consider equilibrium paths leading to a steady state and first confine attention to the case e s g Ž e 1, e 2 .. Equilibrium paths are best characterized by using Fig. 3 Žtop, middle and bottom. which are phase planes for the planar system Ž ˙ xs, m ˙ .. The figures show on their horizontal axes the emission level x s and on their vertical axes the shadow price m which, by Ž14., determines the direction of change in x s . Also, a change in the Southern emissions is matched by a change in the Northern emissions, ˙ x s s yx˙n . Note next that the m ˙ s 0 locus is given by the equation m s T Ž x s . y DŽ x s .4r which is just the difference between the graphs in Fig. 1 divided by the discount rate. The shape of the m ˙ s 0 locus is thus driven by the factors discussed in the previous section.16 The intersections of this locus with the horizontal axis, the ˙ x s s 0 locus, characterize the steady states. For a given fixed e s , the steady states in Fig. 3 coincide with those in Fig. 1. Appendix 2 shows that for any given x s an equilibrium path always exists, and that generic equilibrium paths can be globally characterized as follows. First, the equilibrium of type I leads to the thick market equilibrium x 1s , and the path originates from the vertical axis ŽFig. 3.. Second, the equilibrium of type II also leads to the thick market equilibrium x 1s , but this path emanates either from the thin market equilibrium x s2 ŽFig. 3 Žtop and bottom.. or from the vertical line e ŽFig. 3 Žmiddle... Third, the equilibrium of type III heads for the thin market equilibrium x s3, and it emanates either from x s2 ŽFig. 3 Žtop and middle.. or from the vertical axis ŽFig. 3 Žbottom... Finally, the equilibrium of type IV also leads to the thin market equilibrium x s3 , and it originates from the vertical line e ŽFig. 3.. What is the economic meaning of the above equilibrium paths? Recall that the private cost of trading is determined by two factors pulling in opposite directions: the size of private trades and the thinness of the CO 2 market depending on how much CO 2 other traders are providing for sale. To see how these factors interact in different equilibria, suppose that the initial Southern emission level is from the interval ⍀ that denotes a common domain of any two equilibrium paths in Fig. 3. Given that a polluter believes that the future CO 2 market will be thin, meaning costly future trading opportunities, it makes sense to increase emissions in the South and reduce the number of permits left for sale Žpath III in Fig. 3 Žtop... If a polluter believes that the future CO 2 market is thick, meaning frictionless future trading, it is rational to commit resources to emissions reduction in the South in order to provide more permits for sale as long as the productivity gap is positive Žpath II in Fig. 3 Žtop..; Figure 3 Žmiddle, bottom. shows that self-fulfilling expectations of this type can choose not only among paths II and III but also among I and III ŽFig. 3 Žbottom.. and among II and IV ŽFig. 3 Žmiddle... What are the factors that determine whether the above type of leeway regarding the future trading activity exists? For a given x s Ž0., this amounts to finding the The kink is caused by the kink in T Ž x s . at x s s e s y zU Žsee Fig. 1.. Despite the kink, the usual existence᎐uniqueness results can be applied to the system Ž ˙ xs, m ˙ . because T Ž x s . is locally Lipschitz: K < x y y < G < T Ž x . y T Ž y .< holds for all Ž x, y . in a neighborhood of x s s e s y zU , where K is the right-hand derivative of T Ž x s . at x s s e s y zU . By the differentiability, the r.h.s. of Ž17. is trivially Lipschitz outside the kink. 16
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FIG. 3. Equilibrium paths, e1 - e s - e 2 .
parameters that determine the size of the interval ⍀. If ⍀ is empty the equilibrium depends on history; for each initial level of Southern emissions there is a unique equilibrium path that determines the future trading level. In such cases the future CO 2 market becomes thick provided that the initial trading level is greater than e s y x s2 or thin provided that the initial trading level is less than e s y x s2 . On the other hand, when ⍀ is not empty and when x s Ž0. g ⍀ the equilibrium is given by
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expectations that may encourage polluters to reach trading levels that are significantly different from the current levels.17 Given that the system Ž ˙ xs, m ˙ . is nonlinear, the size of ⍀ is difficult to address Žsee w20, p. 647x for reasons.. However, the following argument regarding the adjustment cost cŽ r . s Žr2. r 2 can be made. Consider Fig. 3 Žtop. and suppose that the interval ⍀ is empty, i.e., assume that the paths of type II and III do not overlap for any x s . Then it is not difficult to show that the adjustment cost parameter must be higher than a given cutoff value U ; if - U the paths II and III must overlap. Similarly, given the adjustment parameter , it is possible to define a cutoff value for the interest rate such that the interval ⍀ can be empty only for ) U ; if - U the overlap must exist Žsee Appendix 3 for a proof.. As in w14x, the presence of external economies creates a strong interdependence among private decisions: the future returns on emissions reductions will depend on how much other polluters’ leave permits for sale. This explains the decisive role of discounting for the existence of the overlap ⍀; the less the future is discounted, the more polluters care about the thinness of the future CO 2 market, i.e., about the actions of other polluters. The logic for the adjustment cost is similar: the lower is the faster the reallocation of emissions and thus the fulfillment of the original expectation. In pollution trading the role of expectations depends not only on the above parameters but also on the pre-trade allocation e s that is a policy instrument. To see this, suppose now that e ) e 2 and that e s g Ž e 2 , e x. In view of Fig. 2, the thick market equilibrium x 1s becomes unique, which eliminates the role of expectations: for any given initial emission level, there is a unique equilibrium path in Fig. 3 leading to the thick market equilibrium Žsee Appendix 2.. Geometrically, for e s g Ž e 2 , e x, the m ˙ s 0 locus intersects the horizontal axis only at x 1s and is thus positive for all x s - x 1s and negative for all x s ) x 1s . Another type of pre-trade allocation that rules out the role of expectations is e s g Ž x 1s , e1 ., which ensures the existence of a unique equilibrium path to the thin market equilibrium x s3 Žsee Appendix 2.. In this case, the m ˙ s 0 locus intersects the horizontal axis only at x s3 and is thus positive for all x s - x s3 and negative for all x s ) x s3. What we can now see is that there is no decisive role for expectations in equilibrium selection if e s is sufficiently large Ž e s ) e 2 . or small Ž e s - e1 .. If e s is large external economies will be large enough to rationalize straightforward expectations about costless trading. If e s is small such expectations can be immediately ruled out because there is not much scope for external economies. It is only the medium allocation e s g Ž e1 , e 2 . which can allow for a sufficiently strong interdependence among private decisions that makes expectations about thin and thick CO 2 markets equally rational.
17
Analogously, in the process of industrialization the economy may not be able to escape from a low level of development if the equilibrium depends on history, whereas the takeoff is possible if the equilibrium depends on expectations w20x. Interestingly, if low x s is viewed as being an indicator of a large development gap between the North and South, thick CO 2 market is associated with a large development gap. But, of course, development is an intricate process involving important aspects that are not included here Že.g., the adoption of technology..
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5. LONG-RUN EFFECTS OF PRE-TRADE ALLOCATIONS Our attention can now be confined to those long-run equilibria to which a dynamic equilibrium potentially converges, i.e., to the steady states x 1s and x s3. I shall next isolate three nonstandard steady state effects of the pre-trade allocation on Ž1. the level of the trading cost; Ž2. trading levels, and Ž3. market allocations of emissions reductions. Consider the effect on the steady state trading cost T Ž x s . and note first that the thin market trading cost satisfies T Ž x s3 . s DŽ x s3 . s a q bx s3 for e s g Ž x 1s , e 2 ., where x s3 is the larger root in Ž18.. In Fig. 4, a q bx s3 is graphed as a function of e s Žif e 2 ) e, the domain of the graph extends to e .. Note next that the thick market trading cost T Ž x s . is given by the horizontal axis for e s g w e1, e x. For ease of discussion, recall that x 1s is the after-trade allocation that would be achieved if trading were costless by assumption Ž x 1s eliminates the productivity gap.. Now, in view of Fig. 4, it can be seen that no pre-trade allocation that is close to the market allocation x 1s can support the emergence of a frictionless long-run market: for e s g Ž x 1s , e 1 . the trading cost is necessarily positive. Also, at least initially, the larger the departure of e s from x 1s the greater the actual thin market trading cost. On the other hand, pre-trade allocations that significantly depart from the market allocation x 1s have chances of supporting costless long-run trading. This can succeed in two ways. First, if the departure can be made sufficiently large Ži.e., e s ) e 2 for e ) e 2 . the long-run trading cost will vanish for sure. Second, for medium departures from the allocation x 1s Ži.e., for e s g Ž e 1, e 2 .. the policy succeeds if polluters coordinate their expectations in favor of thick markets or if history dictates the emergence of thick markets Žas explained in Section 4.. Consider next the thin and thick market trading levels that are depicted in Fig. 5 for all feasible allocations e s . Note first that in a thick market steady state the trading level is given by the 45⬚ line: because trading is costless, the entire quantity e s y x 1s will flow to the high-productivity North. In the thin market steady state x s3 the trading level is given by e s y x s3, which falls short of the 45⬚ line Žsee the graph defined for e s g Ž x 1s , e 2 . in Fig. 5.. Given the trading cost schedules depicted in Fig. 4, this pattern of steady state trading levels is not surprising. Consider finally the effect on the market allocations of emissions. The above pattern of trading levels indicates that no pre-trade allocation that is close to the
FIG. 4. Long-run marginal trading cost.
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FIG. 5. Long-run trading level.
after-trade allocation x 1s can actually support the achievement of x 1s as a market allocation: for e s g Ž x 1s , e 1 . the productivity gap is not eliminated because the thin market trading level falls short of the 45⬚ line in Fig. 5. The pattern of trading levels also indicates that allocations e s which significantly depart from x 1s can actually achieve x 1s as a market allocation: In view of Fig. 5, the thick market trading level that supports the elimination of the productivity gap can be achieved if e s g w e1 , e x. These results potentially have important implications for studies that see the CO 2 market as a frictionless transfer mechanism Že.g., w4, 15x.. In these papers the pre-trade allocation of emissions has no effect on the market allocation of emissions, which is why the pre-trade allocation directly determines trade flows that redistribute the global income. In my model, pre-trade allocations can have this property only if they are significantly different from the long-run market allocation that could be achieved if trading were costless. 6. CONCLUSIONS This paper provides the first step toward analyzing explicitly the gradual development of frictions in a tradable pollution permit market. It incorporated adjustment costs and the common idea that the ease with which trades can be conducted depends on the market size. The principal result is that the key policy variable, the pre-trade allocation of permits, is a critical determinant of the frictionless long-run permit market. More specifically, pre-trade allocations that are close to those after-trade allocations that could be achieved if trading were frictionless have poor chances of supporting the actual development of frictionless trading. On the other hand, pre-trade allocations that are significantly different from the market allocations achieved in a perfect market provide the most favorable basis for the actual emergence of a perfect market. But the policy of using pre-trade allocations for the latter purpose may involve a risk: the pre-trade allocation that supports the development of a thick market can also be the one that has the potential of supporting the emergence of a costly thin market as a self-fulfilling prophecy. On a theoretical level, there are also other ways of analyzing the development of friction in tradable permit schemes than the current one. First, my objective has
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merely been to demonstrate the working of a market-based policy instrument that is incomplete. To obtain further refinements of this instrument, one should consider either the Pareto ranking of the various outcomes provided here or the full social optimum. In the latter case it might be optimal to subsidize the trading activity because of the positive externality from the market size. Second, while there are trading costs that are likely to vanish as soon as the market for permits becomes thick enough, some sources of trading costs may have more to do with the history of trades than with the current state of the market. For example, a trader may better see its options in trading, or a brokerage firm may find it easier to take the role of a market maker the longer the historical data about the relevant variables Že.g., past prices. are. This approach is complementary rather than competitive to the one in this paper in that it is another way of introducing market-level scale economies in trading. However, it could provide very different insights, e.g., because trading might not be costly in a market that is currently thin, provided it has been thick in the past. Third, although the approach taken here is able to endogenize the level of the trading cost, it does not explain how the burden of this cost is shared among buyers and sellers. To this end, one should explicitly consider the bargaining over the division of the value of individual exchange opportunities. As in search theory Že.g., w22x., the division should depend on polluters’ outside options, e.g., on the difficulties in finding other trading partners, that are determined endogenously in equilibrium. This allocative effect could be of key importance when considering the permit market as a transfer mechanism. The analysis of this effect calls for a search theoretic approach Žsee w18x.. APPENDIX 1: SOLUTION OF PROBLEM (9) ᎐ (12) Let LŽ x s , z s , r . s H Ž x s , z s , r . q Ž e s q z s y x s . be the Lagrangian, where H Ž x s , z s , r . s R s Ž x s . y pz s y ␦ Ž z s . hŽ z . y cŽ r . q mr is the Hamiltonian; the multipliers m and are in the current value form. Interior necessary conditions include ŽTheorems 6.5.1 and 9.3.1 w17x. L z s s yp y ␦ X Ž z s . h Ž z . q s 0,
Ž 19 .
L r s ycX Ž r . q m s 0,
Ž 20 .
L s e s q z s y x s s 0,
Ž 21 .
m ˙ s m y RXs Ž x s . q .
Ž 22 .
Note that for all Ž x x , z s . satisfying Ž21., the price p is positive by Ž5., implying that ) 0. By the strict convexity of cŽ r ., Ž20. must hold, which by Ž10. gives the equation ˙ x s s m. Using Ž19. in Ž22. yields Ž13. which leads to the equation m ˙ s m q DŽ x s . y T Ž x s ., as explained in the text. A solution of Ž ˙x s , m ˙ . defines a mapping t ¬ Ž p, z .. Denote this mapping by ⌫ i Ž⭈; x s Ž0., mŽ0.., where i s I, II, III, and IV correspond to solution paths identified in Section 4. For each ⌫ i, take the pair Ž p, z . : t g w0, ⬁.4 and insert it into the problem Ž9. ᎐ Ž12.. Then, by the construction of Ž ˙ xs, m ˙ ., the polluter chooses functions r Ž t . and z s Ž t . that are the same that define ⌫ i for each i, and that are optimal up to the necessary conditions. The concavity of L in Ž x s , z s , r . jointly and the fact that the suggested equilibrium paths converge imply that the sufficient conditions for Ž9. ᎐ Ž12. hold for each ⌫ i ŽTheorem 9.3.1 of w17x..
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APPENDIX 2: THE EXISTENCE OF EQUILIBRIUM PATHS Local existence. The linearization of Ž ˙ xs, m ˙ . in a steady state yields < J < s Ž dmrdx s . , where J is the Jacobian of the linearized system and dmrdx s is the slope of the m ˙ s 0 locus. The trade of J is trŽ J . s ) 0. The signs of < J < and trŽ J . evaluated in a steady state determine the local stability properties of the system. In the case e s g Ž e1 , e 2 ., the slope of the m ˙ s 0 locus in Fig. 3 indicates that the steady states Ž x 1s , 0. and Ž x s3 , 0. are locally saddlepoints Ž< J < - 0. and that the steady state Ž x s2 , 0. is locally a source Ž< J < ) 0, trŽ J . s ) 0.. In the cases e s g Ž x 1s , e 1 . and e s g Ž e 2 , e x, the steady states are the saddlepoints Ž x s3,0. and Ž x 1s , 0., respectively. Thus, by the local saddlepoint theorem there exists a local equilibrium path to a steady state for each allocation e s . Global existence. The following property of Ž ˙ xs, m ˙ . is global: There is no closed orbit on the x s y m plane. This follows from Bendixson’s criterion, which holds because ⭸ ˙ x sr⭸ x s q ⭸ mr ˙ ⭸ m s is a constant ŽTheorem 1.8.2 of w10x.. This property, together with the Poincare᎐Bendixson Theorem Ž1.8.1 of w10x., implies ´ the global existence of equilibrium paths to the steady states Ž x s3, 0. and Ž x 1s , 0. in cases e s g Ž x 1s , e1 . and e2 g Ž e 2 , e x, respectively. Then consider the case e s g Ž e1, e 2 .. The nonexistence of a closed orbit implies that either a path of type II or III is connected to the steady state Ž x s2 , 0., or that they are both connected to Ž x s2 , 0.. Suppose the contrary, that neither of these equilibrium types originates from Ž x s2 , 0.. Because the unstable saddlepoints emanating from Ž x 1s , 0. and Ž x s3 , 0. cannot cross the paths II and III nor lead to Ž x s2 , 0., they must define a closed curve around Ž x s2 , 0., a contradiction. These characteristics, together with the Poincare᎐Bendixson Theorem, imply the classification presented by Fig. 3. The ´ nongeneric case of a heteroclinic orbit Ža path connecting steady states, see w10, p. 45x. should be seen as a subcase of Fig. 3 Žmiddle, bottom.. Thus, for any given x s , there exists an equilibrium path to a steady state. APPENDIX 3: THE EXISTENCE OF THE OVERLAP The interval ⍀ can be empty only if the roots for the system obtained by linearizing Ž ˙ xs, m ˙ . at Ž x s2 , 0. are real. The roots are real Žimaginary, resp.. iff 2 trŽ J . ) 4 < J < Ž- 4 < J <, resp... This inequality can be rewritten as 2 ) 4 < J < s Ž4 r .Ž dmrdx s ., where dmrdx s is the slope of the m ˙ s 0 locus at Ž x s2 , 0.. Thus, U the roots are real iff ) ' Ž4r .Ž dmrdx s . and imaginary iff - U . Similarly, the cut-off value for the interest rate is U ' Ž4r .Ž dmrdx s .. REFERENCES 1. S. Brown, D. Kennedy, C. Polidiano, K. Woffenden, G. Jakeman, B. Graham, F. Jotzo, and B. S. Fisher, Economic impacts of the Kyoto Protocol, Res. Rep. 99.6, Abare Canberra Ž1999.. 2. A. Banerjee and A. Newman, Information, the dual economy, and development, Re¨ . Econom. Studies 65, 631᎐653 Ž1998.. 3. S. Barrett, Political economy of the Kyoto Protocol, Oxford Re¨ . Econom. Policy 14, 20᎐39 Ž1998.. 4. P. Bohm and B. Larsen, Fairness in a tradable-permit treaty for carbon emissions reductions in Europe and the former Soviet Union, En¨ iron. Res. Econom. 4, 219᎐239 Ž1994.. 5. P. Diamond, Aggregate demand management in search equilibrium, J. Polit. Economy 90, 881᎐894 Ž1982..
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