Technology transfers for climate change by May Elsayyad and Florian Morath Max Planck Institute for Tax Law and Public Finance and Munich Graduate School of Economics, Germany; Max Planck Institute for Tax Law and Public Finance, Germany
February 27, 2015 forthcoming in: International Economic Review
Abstract: This paper considers the transfer of cost-reducing technology in the context of contributions to climate protection. We analyze a two-period public goods model where later contributions can be based on better information, but delaying the mitigation e¤ort is costly because of irreversible damages. Investments in technology a¤ect the countries’timing of contributing. We show that countries have an incentive to provide cost-reducing technology as this can lead to an earlier contribution of other countries and can therefore reduce a country’s burden of contributing to the public good. Our results provide a rationale for the support of technology sharing initiatives. Keywords: Environmental public goods; Private provision of public goods; Technology sharing; Uncertainty; Irreversibility JEL Codes: H41; Q52; D62; D83; F53 We thank Reyer Gerlagh, Bård Harstad, Kai Konrad, Johannes Münster, Karen Pittel, Salmai Qari, Monika Schnitzer, Jan-Peter Siedlarek, Marcel Thum, participants of the ESI Workshop in Munich, the SFB conference in Caputh 2012, the APET Conference in Lisbon 2013, the IIPF Congress 2013, the CESifo Area Conference on Energy and Climate Economics 2013, the Annual Congress of the Verein für Socialpolitik 2014, seminar participants at LMU Munich, University of Augsburg and TU Dortmund, the three referees and the co-editor for helpful comments and suggestions. Please address correspondence to: Florian Morath, Department of Public Economics, Max Planck Institute for Tax Law and Public Finance, Marstallplatz 1, 80539 Munich, Germany. E-mail: ‡
[email protected].
1
1
Introduction
Getting countries to commit to new post-Kyoto binding CO2 emission reduction targets has hitherto remained an elusive goal. A continued success on an international scale, however, has been the support of renewable technology initiatives. For example, the Cancún Summit in 2011 declared the start of a $1 billion new initiative and fund for the exchange of climate change technology. Technology transfer mechanisms have always been a dimension of climate change agreements. Article 4.5 of the United Nations Framework Convention on Climate Change states that countries “shall take all practicable steps to promote, facilitate and …nance, as appropriate, the transfer of, or access to, environmentally sound technologies and know-how to other Parties.”1 In fact, recent studies tracking the development of clean technologies show their steady and persistent rise (see, for instance, UNEP 2011). This development is not surprising, given the strong national policies in support of renewable technologies which are being implemented, most notably, by the US and the EU.2 However, this support is often controversially debated. Investments in technology can be pro…table if they are perceived as investments in new markets. But in the public good framework of environmental protection a particularly persistent argument has been that unilateral investments in technology hurt the investing country, as other countries can reduce their e¤ort on climate protection in return.3 Given the strong international support for technology sharing initiatives, this paper provides an argument in favor of sharing cost-reducing technologies. A country may provide a new technology because it can induce other countries not to delay their e¤orts but instead contribute to climate protection today. To develop this rationale, three distinctive features, which in‡uence the decision to contribute to climate protection, are taken into consideration. First, e¤orts to mitigate global warming are, to a large 1
Chapter 16 of the Stern Review (Stern 2007) identi…ed technology-based schemes as an indispensable strategy to tackle
climate change. 2
See Moselle et al. (2010) for an overview.
3
For the e¤ects of unilateral actions in a public goods framework see Hoel (1991), Buchholz and Konrad (1994, 1995),
Buchholz et al. (2005), and Beccherle and Tirole (2011).
2
extent, private contributions to a global public good. As such, the strategic interaction between countries causes strong incentives to delay one’s own contribution since, in reaction to the high e¤ort of one country, other countries can reduce their e¤ort on climate protection. Second, international coordination is hampered by the fact that there is uncertainty with regard to the (country-speci…c) need for climate protection. The uncertainty connected with climate protection stems from the fact that the costs and bene…ts of environmental damage and its reduction remain largely uncertain. The assessment of the impact of climate change is not only highly reliant on projections of the impact of CO2 concentrations on temperatures, but even for a given rise in global temperature there are substantial uncertainties about the economic and social consequences for a country.4 Consequently, such strong uncertainties should push policy-makers toward a later contribution to climate protection, that is, after the uncertainty has been resolved. Third, greenhouse gas emissions have irreversible consequences and cause damages that may possibly be mitigated only at a very high cost. Therefore, delaying the …ght against global warming may prove to be expensive. For example, the accumulation of CO2 emissions in the atmosphere is di¢ cult to reduce, and the damage to the ecosystems from an increase in global temperatures, from acidi…ed lakes and streams or from the clear-cutting of forests, can be permanent.5 Our contribution is twofold. First, we extend a standard model of private provision of a public good to a framework that incorporates the important trade-o¤ that countries face when deciding on climate policies: uncertainty versus irreversibility of damages. Our model builds on the classic concept of irreversible investments and the option value of information; however, we consider investments that exhibit a positive externality and therefore a¤ect other players’bene…t from investing.6 We derive the equilibrium 4
See Allen et al. (2009) for a summary of CO2 impact projections and their variability.
5
The 2007 IPCC report on climate change clearly outlines the long-term cost of a ‘business-as-usual’CO2 emissions path
(see Chapter 3 of the IPCC Synthesis Report). For an overview of di¤erent aspects of climate protection policies see, for instance, Aldy et al. (2001). 6
The results of a standard one-shot public goods game and a model of irreversible investments are obtained as special
cases of our model.
3
contributions to climate protection and identify the main mechanisms driving the timing of the countries’ contribution decisions. In a two-country model, we show that for low degrees of irreversibility both countries would like to wait until the uncertainty has been resolved, while for high degrees of irreversibility countries prefer a full early provision. For intermediate ranges of irreversibility, an alternating equilibrium emerges where one country chooses a ‘partial’early contribution and the other country might contribute in a later period of the game; a result strongly in line with empirical observations. Second, building on these results, we analyze how an investment in cost-reducing technology by one country alters the timing of both countries’ decisions to contribute to climate protection. We consider an investment in technology in the context of technology sharing where both countries have access to the cost-reducing technology. Here, we identify two scenarios where, by a targeted use of cost-reducing technology, one country can induce the other country to increase its current contribution and in this way reduce the own burden of contributing. This free-riding incentive for investments in technology is in sharp contrast to the usual argument that unilateral investments only increase the own burden of contributing. Our model is related to the literature on the timing of environmental policy adoption. Mainly developed by Arrow and Fisher (1974) and Henry (1974) for the case of irreversible investments, this literature analyzes the trade-o¤ between uncertainty and irreversibility in a one-player setting and shows that there is an option value to waiting until the uncertainty has been resolved.7 Our paper takes up the timing issue of policy adoption and introduces the notions of irreversibility and uncertainty in a standard twoplayer model where investments are contributions to a public good. This allows us to isolate the e¤ects of uncertainty and irreversibility in the strategic context of contribution considerations.8 Methodologically, our study is related to the literature on the private provision of a public good 7
See also Conrad (1980) , Epstein (1980), McDonald and Siegel (1986), Pindyck (1991), Kolstad (1996), Ulph and Ulph
(1997), Fisher (2000), Gollier et al. (2000), and Pindyck (2002). 8
Issues of timing have continued to play a role in the environmental literature with the recent struggles of international
coordination in the post-Kyoto era. See Schmidt and Strausz (2011) and Beccherle and Tirole (2011) who analyze the impacts of delayed negotiations.
4
in a static framework; seminal papers are Bergstrom et al. (1986), Cornes and Sandler (1985), and Varian (1994). Our dynamic two-period model reinforces the free-riding incentives as countries can also free-ride on the other players’future contributions, similar to the results of Fershtman and Nitzan (1991) and Admati and Perry (1991) in the context of dynamic contributions. Lockwood and Thomas (2002) use the notion of irreversibility in the context of contributions to a public good where, in their model, irreversibility refers to the fact that investments in previous periods cannot be taken back, a feature which is also present in our model. Gradstein (1992) introduces incomplete information into a dynamic two-period model of contributions to a public good and shows that there is an ine¢ cient delay of individual contributions. Bramoullé and Treich (2009) examine a framework with risk-averse countries where the e¤ect of pollution emissions is uncertain. In their model, uncertainty leads to higher climate protection e¤orts, while in our case there is an informational advantage to delaying contributions, which causes current contributions to be lower. In a related study, Boucher and Bramoullé (2010) analyze international cooperation when climate protection bene…ts are uncertain.9 To our knowledge, our study next to Morath (2010) is the …rst to simultaneously analyze the e¤ects of uncertainty and irreversibility in a context of private contributions to a public good.10 Focusing on the interaction between technology and contributions to climate protection, Buchholz and Konrad (1994, 1995) and Buchholz et al. (2005) show that the public good nature of environmental protection might induce countries to be ‘less green’in order to strengthen their bargaining position in the environmental policy coordination game; see also the results of Shah (2010) in the context of negotiations of emission caps. This argument has been further generalized by Beccherle and Tirole (2011) and still holds true when introducing uncertainty or dynamics (Harstad 2012, forthcoming; Konrad and Thum 2014). This result, however, stands in strong contrast to the steady rise of investments in renewable 9
For aspects of the formation of international environmental treaties under uncertainty see also Na and Shin (1998) and,
more recently, Kolstad and Ulph (2008) and Glazer and Proost (2012); see also the literature review in Barrett (2003, 2007). 10
Morath (2010) analyzes countries’ incentives to acquire information about the cost of climate change and shows that
there can be a strategic advantage to remaining uninformed.
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energy. Our model considers technology investments in the context of technology transfer mechanisms; we identify scenarios where investments in green technology can actually reduce a country’s burden of contributing to the public good.11 Our model abstracts from bargaining over a cooperative outcome and highlights the public goods nature of mitigation policies. By a¤ecting the time pattern of contributions, technology sharing can, in a non-cooperative approach, lead to a rise in current contributions to climate protection.
2
Model framework
Basic setup: We consider a framework with two countries A and B and two periods t and t + 1. In each period, countries simultaneously choose a contribution to a public good where xi; 2 R+ denotes country i’s contribution in period , i 2 fA; Bg and
2 ft; t + 1g. Moreover, we denote by x = (xA; ; xB; ) the
vector of contributions in period . The marginal contribution costs in the two periods are assumed to be constant and identical for both countries and are denoted by ct > 0 and ct+1 > 0. The contribution costs depend on the technology available to the countries, as explained below. Individual contributions in the two periods sum up to the total amount contributed to the public good. Country i’s payo¤ is equal to
(1)
i (xt ; xt+1 )
=
if
P
P
xk;
k=A;B =t;t+1
!
ct xi;t
ct+1 xi;t+1 ; i 2 fA; Bg :
Here, function f translates climate protection e¤ort into a mitigation outcome. As usual, f is assumed to be strictly increasing and concave, f 0 > 0, f 00 < 0.12 Due to the concavity of the production function, 11
See Golombek and Hoel (2005, 2011) for international agreements and cooperation on investments in technology when
technology investments have spillover e¤ects. Note also that there is a literature in industrial organization which considers the timing of technology adoption and …rst-mover advantages (seminal contributions include Reinganum 1981 and Fudenberg and Tirole 1985; for a survey see Hoppe 2002). 12
To simplify the exposition, we will assume that f 0 (0) is su¢ ciently large for all
> 0 to ensure that all types
>0
will prefer a strictly positive total amount of the public good. Note that we abstract from discounting. One can argue
6
mitigation e¤ort exhibits decreasing returns to scale, that is, the marginal cost of achieving an additional unit of mitigation outcome is increasing. Strategic interdependencies emerge through the production function f . The assumption of quasi-linear payo¤ functions is mainly made for tractability and will be discussed in Section 4.4.2.
Uncertainty: Countries di¤er in their valuations of the public good, denoted by
A
and
B.
The
heterogeneity in this valuation captures all country di¤erences in the cost-bene…t ratio of climate protection e¤orts (hence including di¤erences in the cost of e¤ort). The country-speci…c valuations of the public good are independent draws from two commonly known continuous distribution functions B
with support [0; ]. The functions
A
and
B
A
and
are assumed to be continuously di¤erentiable on (0; ).
We will restrict the analysis to probability distributions with the following reverse hazard rate:
Assumption 1
0( i i(
) )
1
for all
2 (0; ), i = A; B:
This assumption ensures that the countries’maximization problems in period t are well-behaved and that the objective function is concave.13 In period t, there is uncertainty about the valuations ( in period t+1; both countries’valuations
A
and
B
A; B )
of the public good, which will be resolved
become commonly known only between periods t and
t + 1. Overall, no country has private information on its bene…t from climate policy: Country-speci…c di¤erences with respect to the costs and bene…ts of climate protection are typically observable since research such as the studies by the IPCC and the UNFCCC and estimates of regional impacts are usually publicly accessible. The uncertainty in the model thus re‡ects the di¢ culty of assessing the cost-bene…t ratio and, hence, the valuation of climate protection.14 that discounting is already captured by the di¤erence in marginal contribution costs ct and ct+1 necessary to produce one contribution unit. 13
Assumption 1 is su¢ cient but not necessary for obtaining our results and it simpli…es the equilibrium analysis consid-
erably. It holds, for instance, for uniform or exponential probability distributions. 14
For example, in a review of impact estimates of climate change, Jamet and Corfee-Morlot (2009) identify …ve sources of
7
Contribution cost and irreversibility: The aspect of the irreversibility of foregone mitigation e¤orts is re‡ected in the contribution costs. We assume that the contribution cost in t + 1 per unit of e¤ ective mitigation outcome is strictly larger than contribution cost in t per unit of mitigation outcome:
Assumption 2 (‘Irreversibility condition’)
ct+1 > ct :
This assumption is built on the fact that CO2 is a stock pollutant and hence current emissions cause long-term costs. A general increase in the average world temperature cannot be easily reduced, regardless of how advanced the abatement technology is.15 CO2 stocks in the atmosphere dissipate very slowly and their impact can have considerable e¤ects on the ecosystem. Therefore, due to the irreversibility of emissions, delaying mitigation e¤orts makes reaching a given climate target f (X) more expensive. Similar arguments apply to other environmental damages like deforestation, acidi…ed rain and lakes, and the melting of polar ice caps and glaciers; compare also the discussions of scientists and environmentalists about the ‘point of no return’in climate change. The cost of delay will be particularly high if it turns out that climate change imposes great risks to economic development as well as to human health and security.16 Assumption 2 is a shortcut for the problem of irreversibility of delayed action against climate change. Since xi;t and xi;t+1 have the same marginal impact on public good provision, di¤erences in costs ct and uncertainty: greenhouse gas emission projections, the accumulation of emissions in the atmosphere and how these emissions a¤ect global temperatures, the physical impacts of a given increase in temperature, the valuation of physical impacts in terms of GDP and the risk of abrupt climate change. While there is also substantial uncertainty about variables which a¤ect all countries in a similar way, such as the relation between greenhouse gas concentration and the rise in global temperature, a given rise in temperature does not impose the same cost on all countries. Our model focuses on uncertainty about country-speci…c factors rather than common factors, that is, on the di¤erential impact of climate change on countries. 15
For an economic analysis of the cost of stabilization of CO2 concentration see Chapters 9 to 11 of the Stern Review
(2007) and the discussions in Mendelsohn (2008) and Dietz and Stern (2008). 16
For predictions of the e¤ects of delaying abatement e¤orts on cumulative emissions see, e.g., McKinsey (2009). In
addition to the direct e¤ect of delay on cumulative emissions, there can be a ‘lock-in e¤ect.’
8
ct+1 measure di¤erences in the e¤ectiveness of early changes in the emission path compared to delayed changes: Earlier e¤ort (e.g., improving energy e¢ ciency, clean power generation, reforestation) is relatively more e¤ective than delayed e¤ort in achieving a given reduction of greenhouse gas concentration compared to ‘business as usual.’ Intuitively, Assumption 2 on the relative contribution cost implies the following: If climate change turns out not to have severe consequences for a country (
i
is low) then this country’s
total contribution cost will not be much di¤erent today or tomorrow. But for a country that …nds out it is strongly a¤ected by climate change a delay, together with the irreversibility of emissions, will result in much higher mitigation cost when greenhouse gas concentration increases in the meantime (even taking technological progress into account). Section 3.1 shows how the equilibrium prediction changes if the (expected) future contribution cost per unit of mitigation outcome is lower than today’s contribution cost.
Investments in technology: Our main focus is on the implications of investments in cost-reducing technology for the equilibrium climate protection outcome. We concentrate on the notion that the developed abatement technology is shared between countries. Generally, successful investments in R&D have strong spillover e¤ects, for example, through trade magazines and reverse engineering by competitors; in addition, patent protections for new inventions and innovations only have a limited time frame. In the case of abatement technologies, such spillovers are further encouraged through large technology transfer initiatives. Thus, we consider investments in green technologies that a¤ect both countries’marginal costs in the same way. We assume contribution costs ct and ct+1 to be functions of the ‘technology level,’ which we denote by , and we analyze whether an ex ante improvement of the available technology (from a status quo level
0
to a level ~ ) will change the structure of the equilibrium contributions in periods t
and t + 1. Note that the transfer of technologies which reduce today’s contribution cost typically builds on progress achieved during the last decades, such as in the case of solar energy technology. Put di¤erently, technologies that reduce ct are su¢ ciently developed to be transferred immediately. Renewable energies 9
are, in fact, a good example of this type of technology. Making solar cell technology with high energy conversion e¢ ciency available (or subsidizing its transfer) reduces other countries’cost of increasing the share of renewable energies, both today and possibly in the future, even though the impact on future cost is less certain due to the expected innovation in the energy sector.
The game: Our analysis solves for the subgame-perfect Nash equilibrium of the following game. In stage 0, investments in technology take place and a¤ect the contribution costs of both countries in both periods, due to technology transfer. Our main analysis does not explicitly model the countries’investment choices but considers how changes in technology a¤ect the equilibrium contribution pattern. An example will illustrate the consequences of the results obtained for a strategic game of technology investment. In stage 1 of the game (period t), countries A and B simultaneously choose their contributions xA;t and xB;t
0. Then, both contributions are observed. At the beginning of stage 2 (period t + 1), the
country-speci…c valuations
A
and
B
are drawn from the distribution functions
A
and
B,
respectively,
and become publicly observable. Then, countries simultaneously choose their contributions xA;t+1 and xB;t+1
0. The choice xi;t+1 is a function of the observed valuations (
A; B )
t contributions (xA;t ; xB;t ); hence, the stage 2 strategy of country i is a mapping the set
0
of pairs of valuations
=(
A; B )
0
and the observed period i
:
R2+ ! R+ from
and the set R2+ of period t contributions xt = (xA;t ; xB;t ).
Finally, payo¤s are realized.
3
Contributions to climate protection
This section characterizes the equilibrium public good contributions in the two periods. The countries’ contributions depend not only on incentives to free-ride on the other country’s (current and future) contributions to climate protection but also on the trade-o¤ between uncertainty and irreversibility of damages. Before turning to the equilibrium analysis, two benchmark cases demonstrate the trade-o¤ between uncertainty and irreversibility.
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3.1
Benchmark cases: Uncertainty versus irreversibility
Suppose …rst that there is uncertainty about the valuations for climate protection but no irreversibility of foregone e¤orts to climate protection. When ct+1 approaches ct (or even becomes lower than ct ), a contribution in period t is strictly dominated. Both countries prefer to wait until the resolution of the uncertainty and a standard game of private provision of a public good will ensue in period t + 1, based on the realized valuations
A
and
B.
Now suppose instead that there is irreversibility but no uncertainty; that is, the variance of
A
and
goes to zero but the structure of the model remains unchanged. In the limit where the valuations are
B
already known in period t, the countries strictly prefer to contribute early, as contributions in t + 1 cause a higher marginal cost. Accordingly, countries only consider contributing in period t, as in a standard one-shot private provision game (based on the already known valuations). Thus, while uncertainty pushes the timing of the contribution to climate protection toward a later date, irreversibility pushes the timing toward an earlier date. After solving for the optimal quantity which balances the e¤ects of uncertainty and irreversibility we demonstrate how this trade-o¤ is a¤ected by the third important characteristic of the model: the public goods problem.
3.2
Equilibrium contributions in period t + 1
Solving the game through backward induction, consider …rst period t + 1. Here, the countries’valuations A
and
B
are common knowledge and the game is strategically equivalent to a standard private provision
game with a given contribution Xt = xA;t + xB;t . We de…ne country i’s preferred provision level of the public good in t + 1 as the quantity Qt+1 ( i ) that solves i’s …rst-order condition obtained from (1), if
0
(Qt+1 )
which yields
(2)
Qt+1 ( i ) :=
ct+1 = 0; i 2 fA; Bg ;
8 > > < (f 0 ) > > :
1
(ct+1 = i ) if 0
11
i
> 0;
otherwise.
Qt+1 ( i ) denotes the quantity up to which i would like to increase total contributions and is useful to characterize i’s equilibrium contribution: First, a country considers contributing in period t + 1 only if its preferred quantity Qt+1 ( i ) is higher than total early contributions Xt . Second, due to the quasi-linear payo¤ functions, only the country i with the higher preferred provision level Qt+1 ( i ) (or, equivalently, with the higher valuation
i)
may contribute in the equilibrium of the period t + 1 subgame.17 This
country i raises the contribution level up to its desired quantity Qt+1 ( i ), and country j 6= i free-rides and contributes zero. Thus, the equilibrium contributions in the period t + 1 subgame are as follows:18 8 > > < (max fQt+1 ( A ) Xt ; 0g ; 0) if A > B ; (3) xA;t+1 ; xB;t+1 = > > : (0 ; max fQt+1 ( B ) Xt ; 0g) if A < B :
Preferred provision levels in period t
3.3
3.3.1
Country i’s preferred early public good provision: The trade-o¤ between uncertainty and irreversibility
Inserting the equilibrium contributions in period t + 1 into country i’s decision problem in period t, country i chooses xi;t to maximize its expected payo¤ given xt+1 = xA;t+1 ; xB;t+1 and given xj;t . With choices in t + 1 denoted by
17
( ; xt ) = (
A(
; xt ) ;
i (xi;t ; xj;t )
=E [
B
( ; xt )), we de…ne
i (xi;t ; xj;t ;
( ; xt ))]
This is the standard textbook result for public goods models with quasi-linear utility functions; comparing the …rst-order
condition of both players shows that the equilibrium must be at a corner solution. 18
If
A
=
B
and Qt+1 (
A)
> Xt , there is a continuum of equilibria with xA;t+1 + xB;t+1 = Qt+1 (
to the assumption of continuous distribution functions
A
and
B,
however,
equilibrium selection in this case is inconsequential for the subsequent results.
12
A
=
B
A)
Xt . Due
occurs with probability zero; thus,
as country i’s expected payo¤ from the point of view of period t, given that the vector of strategies played in period t + 1 is
(4)
Z
0
Z
. Hence, with Xt = xi;t + xj;t ,
i (xi;t ; xj;t )
is equal to
i
( i f (maxfQt+1 ( i ) ; Xt g)
0
ct xi;t +
Z
0
Z
ct+1 max fQt+1 ( i )
i
Xt ; 0g) d
( i f (maxfQt+1 ( j ) ; Xt g)
j
( j) d
ct xi;t ) d
j
i ( i)
( j) d
i ( i) :
The expected payo¤ in (4) takes into account two possible cases: the case where it turns out that i has a higher valuation than j and may contribute in t + 1 (the …rst double integral in (4)); and the case where i has a lower valuation than j and can free-ride in t + 1 (the second double integral in (4)). In both cases, contributions in period t + 1 also depend on the early contribution Xt since i may only contribute in t + 1 if its preferred level Qt+1 ( i ) is larger than Xt . Using (2) and the fact that Qt+1 ( i ) is strictly increasing in
i
we can de…ne the inverse function of Qt+1 by ^ (Xt ) := ct+1 : f 0 (Xt )
(5)
^ denotes the critical valuation for which a country’s preferred provision level in t + 1 is exactly equal to Xt ; only countries with a realized valuation
> ^ may contribute in t + 1.
Now consider country i’s marginal expected payo¤ of an increase in xi;t . Suppose …rst that the given total contribution Xt is smaller than Qt+1 ( ) (where Qt+1 ( ) is the preferred provision level in t+1 for the highest possible valuation ). In this case, contributions in t + 1 occur with strictly positive probability. As we derive in Appendix A.1,
(6)
@ i @xi;t
= Xt
Z
0
^Z ^
+
if
0
(Xt )
ct d
j
( j) d
i ( i)
0
Z
^
Z
i
(ct+1
ct ) d
j
( j) d
0
i ( i)
+
Z
0
Z
maxf i ;^g
( ct ) d
j
( j) d
i ( i) :
First, if both countries’realized valuations are smaller than the critical valuation ^ (Xt ) then there is no contribution in t + 1. In this case, i’s marginal payo¤ of increasing the early contribution is the di¤erence between the marginal bene…t of public good consumption and the marginal contribution cost in t (the …rst term in (6)). Otherwise, if i’s realized valuation is greater than the critical valuation (i.e., 13
i
> ^),
then i would, in principle, be willing to make a contribution in t + 1, and its equilibrium contribution in t + 1 will depend on whether j has a lower or higher valuation for climate protection. The second term in (6) re‡ects the case where
i
>
j
and hence i’s equilibrium contribution in t + 1 is strictly positive.
Here, the marginal payo¤ of increasing xi;t is equal to the di¤erence in the contribution costs, ct+1
ct :
By increasing the early contribution, country i will save the higher contribution cost in t + 1. The third term in (6) represents i’s marginal payo¤ given that j has a higher valuation (and
j
> ^); in this case,
i’s marginal bene…t of increasing the early contribution is zero because this contribution would have been made by j in period t + 1 anyway, and a contribution only bears the marginal cost ct . Altogether, the three terms in (6) illustrate the trade-o¤ between uncertainty (unknown realization of the valuation) and irreversibility (higher contribution cost in t + 1) on the one hand and the incentives to free-ride on the other hand. While the e¤ect of irreversibility in the second term is always positive and the free-riding e¤ect in the third term is always negative, the sign of the …rst term depends on Xt . More precisely, the integrand in the …rst term in (6) is small and possibly negative for low realizations and increasing in
i.
If total early contributions Xt = xi;t + xj;t @ i @xi;t
(7)
i
= Xt Qt+1 ( )
Z
0
Qt+1 ( ) then
Z
if
0
(Xt )
ct d
j
( j) d
i ( i) :
0
Here, regardless of which valuation is revealed in t + 1, no additional contribution will take place. Therefore, incentives to increase xi;t are no longer driven by considerations of potential cost or saving of a contribution in t + 1. Instead, the expected marginal bene…t of increasing xi;t is simply equal to E ( i ) f 0 (Xt ) and the marginal cost is ct . Optimizing over xi;t yields country i’s preferred early provision level Qi;t , taking into account that equilibrium contributions in t + 1 are as in (3). Notice that Qi;t is not necessarily equal to i’s equilibrium contribution xi;t ; rather, Qi;t is the quantity up to which i would increase the public good contributions in t (for a given xj;t ). Compared to country i’s preferred provision level in t + 1, which directly depends on
i,
the preferred early provision Qi;t depends on i’s expectations of the realizations of 14
i
and
j
and the corresponding equilibrium contributions in the period t + 1 subgame. The following lemma characterizes the countries’preferred early provision levels. It is assumed, as for all following statements, that Assumption 1 holds.19
Lemma 1 The preferred early provision level Qi;t of country i 2 fA; Bg (i) is a ‘full’ provision equal to Qi;t = (f 0 )
1
(ct =E ( i ))
E ( i)
ct
(8)
Qt+1 ( ) if ;
ct+1
(ii) is a ‘partial’ provision and uniquely determined with 0 < Qi;t < (f 0 ) E ( i)
(9)
<
ct
j
1
(ct =E ( i )) if
( i )) ;
(iii) is equal to Qi;t = 0 if E i(
(10)
j
( i ))
ct ct+1
:
Lemma 1 shows that a country’s preferred early provision can be set in relation to properties of the distribution functions of
i
and
j
and the cost ratio ct =ct+1 . In part (i), country i prefers a ‘full’early
provision which is su¢ ciently high to crowd out all further contributions in t + 1; that is, Qi;t is higher than Qt+1 ( ), which is the preferred period t + 1 provision level for the highest possible valuation. In this case, Qi;t is determined irrespective of period t + 1 such that the marginal cost ct is equal to the expected marginal bene…t E ( i ) f 0 (Qi;t ). Condition (8) for a full early provision is equivalent to a positive marginal payo¤ of increasing Xt beyond the highest possible period t+1 provision Qt+1 ( ). Such a situation occurs if i’s expected valuation and/or the degree of irreversibility (the inverse cost ratio ct+1 =ct ) is high. For lower degrees of irreversibility (higher ratios ct =ct+1 ), country i would not make a full contribution in period t. Here, i balances today’s contribution cost against future expected contribution costs; the latter has to be incurred only if 19
j
<
i.
This causes the term E i (
j
( i )) to become important, which
The proof of Lemma 1 as well as the proofs of all subsequent results are relegated to the appendix.
15
is the expected probability that j has a lower valuation than i. If this probability is su¢ ciently high (as in (9)), then i prefers a ‘partial’ early provision, which is a quantity Qi;t < Qt+1 ( ) for which i might contribute again in period t + 1 (depending on the realized valuations). Otherwise, i does not want to make any early contribution, that is, Qi;t = 0. Condition (10) is equivalent to a negative marginal payo¤ of increasing Xt just above zero. Intuitively, the lower the degree of irreversibility (the relative cost advantage of contributing early), the lower a country’s preferred early provision is.
3.3.2
Which country prefers a higher early public good provision?
As Lemma 1(i) reveals, the ‘full’ provision level is an increasing function of E ( i ). Therefore, if both countries prefer a full early provision then Qi;t > Qj;t is equivalent to E ( i ) > E ( j ). Moreover, if i prefers a full early provision and j prefers a partial provision (de…ned in Lemma 1(ii)), this requires that E ( i) =
ct =ct+1 > E ( j ) = and hence E ( i ) > E ( j ), which implies that Qi;t = (f 0 )
strictly larger than (f 0 )
1
1
(ct =E ( i )) is
(ct =E ( j )) > Qj;t .
If both countries prefer a partial early provision, the comparison of the expected valuations E ( i ) and E ( j ) is no longer su¢ cient to determine which country would prefer a higher quantity to be provided early. For Xt < Qt+1 ( ), the di¤erence in the countries’incentive to contribute early can be expressed as (11)
@ i @xi;t
Xt
@ j @xj;t
=
^)
i(
h ^) E(
j(
Xt
i
j
i
^)
E(
+ ct+1
j
"Z
j
^
i ^) f 0 (Xt )
j
j
( i) d
i ( i)
Z
^
i ( j) d j
#
( j)
and is driven by two comparisons: …rst, by di¤erences in the expected bene…t from contributing conditional on there being no further contributions in t + 1 (the …rst term: conditional expected valuation multiplied by f 0 (Xt )); and second, by di¤erences in the expected contribution cost in period t + 1 (the second term: ct+1 multiplied by the probability that this cost has to be paid). Without making further assumptions on the distribution functions i
i
and
…rst-order stochastically dominates
j,
it is not straightforward when (11) is positive. If, for instance, j,
then (11) is positive for all Xt
0 and hence Qi;t > Qj;t . In
general, however, the …rst and the second term in (11) need not have the same sign. 16
3.3.3
The e¤ect of intertemporal free-riding on the preferred early provision
The comparison of the incentive to contribute early in (6) to the benchmark case of a single country isolates the free-riding e¤ect on the preferred early provision. Note …rst that in the case of a single country, the optimal period t + 1 contribution is simply xi;t+1 = max fQt+1 ( i )
xi;t ; 0g: The preferred quantity
Qt+1 ( i ) remains unchanged but xi;t+1 no longer depends on the other country’s realized valuation. Moreover, in period t, if xi;t < Qt+1 ( ), then i’s marginal payo¤ of increasing xi;t in the one-country benchmark is Z
(12)
^ if
0
(xi;t )
ct d
i ( i)
0
+
Z
^
(ct+1
ct ) d
i ( i) :
The …rst term in (12) describes the marginal payo¤ if there is no contribution in t+1 (because
i
^ (xi;t )
where ^ is as in (5)). A similar e¤ect emerges in the two-country case (the …rst term in (6)), but there, only with the probability that j also has a valuation below the critical valuation (
j
^). The second
term in (12) represents the savings in marginal contribution cost in case i’s valuation turns out to be high (
i
> ^). In (6), these savings are realized with lower probability due to the free-riding opportunities
whenever
j
>
i.
Finally, the two-country case identi…es an additional negative free-riding e¤ect (the
third term in (6)) which is not present in (12): In the two-country case, if it turns out that the other country has a high valuation, an increase in the early contribution would have caused an unnecessary cost. Corollary 1 Country i’s early provision in the one-country benchmark is strictly positive and (weakly) larger than i’s preferred early provision level Qi;t in the two-country case (strictly larger if and only if ct =ct+1 > E ( i ) = ). While the preferred early provision Qi;t in the two-country case can be zero (Lemma 1(iii)), it is always strictly positive in the one-country benchmark where incentives to free-ride on future contributions of the other country are absent. Moreover, whenever a country prefers a partial early provision, its preferred quantity is strictly higher in the benchmark of a single country. If, however, a country prefers a full early 17
provision, then intertemporal free-riding on future contributions becomes irrelevant since there will never be contributions in t + 1; thus, the preferred full provision in the one-country benchmark is as in Lemma 1(i). Note that the latter result holds for preferred provision levels but not necessarily for equilibrium contributions in period t, which we derive in Section 3.4 below. Overall, the strategic context of the public good problem and the possibilities to free-ride on other countries’ future contributions shift the timing of the contribution toward a later period.
3.3.4
A social planner’s choice
Equation (12) is also useful to determine the social optimum. In fact, if we set and replace
i
i
in (12) equal to
A+ B
by the probability distribution of the sum of the countries’valuations, this would yield the
marginal payo¤ of increasing the early contribution from a social planner’s point of view who maximizes the sum of expected payo¤s and can directly decide on the countries’ contributions. Compared to the benchmark of a single country above, the social planner takes the positive externality of the contributions to climate protection into account: A country’s emission reduction not only bene…ts the country itself but is also valuable for other countries. In t + 1, the social planner’s contribution choice is equal to max fQt+1 (
A
+
B)
Xt ; 0g and is,
hence, higher than what a single country would contribute. Despite this higher late contribution, the social planner would also choose a higher early provision.
Corollary 2 The early provision of a social planner is strictly higher than the early provision in the one-country benchmark and, hence, is also strictly higher than the preferred early provision level in the two-country case.
Taking into account optimal contributions in period t + 1 in the di¤erent cases, the preferred early provision is highest for a social planner who maximizes total expected payo¤s and accounts for the externality problem, lower in the benchmark of a single country in the absence of ‘intertemporal freeriding,’and lowest in the equilibrium of the two-country case where incentives to delay the contributions 18
are strongest.
3.4
Equilibrium contributions in period t
While Lemma 1 characterizes the countries’preferred early provision levels, the actual contributions in period t are determined in the strategic context of a private provision game. Due to the quasi-linear payo¤ functions, positive contributions by either country are perfect substitutes, similar to the subgame in period t + 1. As a consequence, only the country with the higher preferred early provision level will contribute in the equilibrium of period t. Proposition 1 For preferred early provision levels QA;t and QB;t as given in Lemma 1, the countries’ equilibrium contributions in period t are determined such that (i) if QA;t = QB;t = 0, then xA;t = xB;t = 0, 0, then xA;t = QA;t and xB;t = 0,
(ii) if QA;t > QB;t (iii) if QB;t > QA;t
0, then xA;t = 0 and xB;t = QB;t ,
(iv) if QA;t = QB;t > 0, then there is a continuum of equilibria with xA;t + xB;t = QA;t . In Proposition 1(i), both countries prefer not to contribute early but to instead wait until period t + 1. In this case, total expected equilibrium contributions to the public good are equal to E(Xt+1 ) = E [max fQt+1 (
A ) ; Qt+1 ( B )g]
since in t + 1 the country with the higher valuation will contribute. If at
least one country prefers a positive early provision (parts (ii) and (iii)), only the country with the higher preferred quantity contributes in t: Intuitively, if xi;t = Qi;t > Qj;t , then j’s best response is xj;t = 0, which makes xi;t = Qi;t optimal. Finally, if both countries prefer exactly the same positive early quantity (for instance, if
A
=
B ),
then there is a continuum of equilibria where the countries’contributions in
t sum up to this preferred quantity (part (iv)). The derived equilibrium contributions have several implications. First, it becomes clear that if there is a positive contribution to the public good in any period then it will be borne by only one country 19
(except for the special case in Proposition 1(iv)). Moreover, while the preferred early provision level is already a¤ected by the possibility of intertemporal free-riding, there is, in addition, free-riding within a period, which can cause countries not to contribute early even when the preferred early provision is positive.
4
Technology transfer and the timing of contributions
The focus on a situation in which all countries bene…t from the cost reductions caused by investments in green technology adds an interesting layer to the analysis of the equilibrium contribution pattern. Investments in cost-reducing technology can shift the major equilibrium burden of contributing from one country to the other; as a consequence, incentives to invest in technology are a¤ected by strategic considerations and free-riding incentives. We now analyze the e¤ect of an exogenous investment in cost-reducing technology on the equilibrium contribution pattern. We denote the level of technology by corresponding contribution costs ct = ct (
0)
and refer to
and ct+1 = ct+1 (
0 ))
0
as the status quo level (with
and to ~ as the level which results from
technology investments (with corresponding contribution costs c~t = ct (~ ) and c~t = ct (~ )). Accordingly, we denote by xi;t ( 0
0)
and xi;t (~ ) country i’s equilibrium contribution in period t under technology levels
and ~ , respectively.
4.1
Technology sharing to free-ride
This section identi…es two scenarios in which investments in technology are strategically advantageous by inducing ‘categorical changes’in the equilibrium contribution pattern. ‘Categorical changes’refer to the conditions of Lemma 1 and distinguish between whether in period t a country prefers no contribution, a ‘partial’ provision, or a ‘full’ provision. Since the conditions in Lemma 1 depend on the cost ratio, changes in the relative cost of contributing in the two periods have an impact on a country’s preferred early provision and hence on the equilibrium contribution pattern. The two following scenarios di¤er in
20
the equilibrium contributions in t at the status quo.
4.1.1
Scenario 1: No early contribution
Consider …rst a status quo scenario in which, without investments in technology, both countries prefer to delay their contribution to climate protection. By Lemma 1(iii), this occurs in equilibrium if
(13)
max E i (
j
( i )) ; E j (
ct
i ( j ))
ct+1
:
Proposition 2 Suppose that (13) holds such that, without investment in technology, equilibrium contributions in period t are zero. De…ne i 2 fA; Bg and j 6= i such that (14)
Then, there exists
E j(
i ( j ))
j
( i )) :
> 0 such that for all investments in cost-reducing technology ~ which lead to a cost
ratio
(15)
c~t 2 E j( c~t+1
i ( j ))
;E i (
j
( i )) ;
the resulting equilibrium contributions in period t satisfy xi;t (~ ) > 0 and xj;t (~ ) = 0. Proposition 2 shows that a targeted provision of cost-reducing technology by country j can raise i’s equilibrium contribution in period t from zero up to a strictly positive amount, while j free-rides. Even though both countries have access to the cost-reducing technology, there is a range of technology investments where the relative cost advantage of contributing early increases for both countries, but only one country prefers a positive early provision and hence contributes early. The strategic opportunity to bene…t from an early contribution of the other country exists for the country j that is less likely to bear the burden of contributing in case contributions are delayed (compare (14)). Note that it is unnecessary to distinguish between the type of positive contribution reached: j bene…ts if i prefers a partial or a full early provision of the public good. Condition (15) on the resulting cost ratio c~t =~ ct+1 is su¢ cient to
21
ensure that j’s preferred early provision is lower than i’s preferred quantity such that i contributes in the equilibrium with technology investments. In scenario 1, j not only bene…ts from reduced marginal contribution cost but also from i’s increased early contribution. But also i’s expected payo¤ increases with technology sharing: j’s early contribution is the same with and without technology investment (there is no crowding-out), and the contributions of the recipient country i are less costly based on the new technology. Thus, abstracting from the direct cost of providing the technology, both countries are strictly better o¤. In Section 4.3 we demonstrate the implications of Proposition 2 for a strategic game of technology investments in stage 0 of the model. To illustrate the result in Proposition 2, consider two key players to climate protection such as China (or India) and the United States; at the status quo both prefer to delay their (major) contributions. Given that China is more likely to have a higher valuation of climate protection, the United States have a strategic bene…t from developing green technology to be used domestically as well as transferred to China. Such considerations may be re‡ected in multilateral initiatives such as the Asia Paci…c Partnership on Clean Development and Climate (APP), with the explicit goal to facilitate the transfer of cost-e¤ective and cleaner technologies. While in retrospect the impact of the APP appears to be rather weak, it should be noted that a su¢ ciently high own (expected) valuation of climate protection is a prerequisite for a strong incentive to support the development and transfer of low-carbon technology. Put di¤erently, increasing (political) support for climate protection will also strengthen the incentive to initiate technology transfer mechanisms, as we will see next.
4.1.2
Scenario 2: Positive early equilibrium contribution
Consider now a status quo scenario where, without investment in technology, country j prefers a positive early provision while i prefers to delay the contribution. This implies that xi;t ( and holds if
(16)
E i(
j
( i ))
ct ct+1
i ( j )) ,
i 2 fA; Bg , j 6= i.
0)
= 0 and xj;t (
0)
>0
Here, at the status quo, the relative cost advantage of early contributions only makes j willing to contribute early. Investments in technology can, however, cause a ‘categorical change’ in the equilibrium contribution pattern and can lead to the opposite scenario in which the previously non-contributing country i now contributes a major share to climate protection.
Proposition 3 Suppose that (16) holds such that, without investment in technology, country j’s equilibrium contribution in period t is strictly positive. If E ( i ) > E ( j ) then there exists
> 0 such that for
all investments in cost-reducing technology ~ which lead to a cost ratio c~t E ( i) + ; < c~t+1
(17)
the resulting equilibrium contributions in period t satisfy xi;t (~ ) > 0 and xj;t (~ ) = 0. Proposition 3 captures a situation where the country that initially expects a higher potential saving from an early contribution is actually the country with the lower expected valuation of the public good. This country j with the lower expected valuation can still have a stronger incentive to contribute early: Preferred early provision levels trade o¤ the expected marginal bene…t from an early contribution and the expected marginal contribution cost in t + 1, which depends on the probability of having the higher valuation. Due to the heterogeneity in the probability distributions of the valuations for climate protection, a targeted reduction of the irreversibility ratio via cost-reducing technology can result in a shift of the burden of the early contribution to country i when the relative cost advantage of contributing early becomes su¢ ciently strong (condition (17)) such that the expected valuation becomes more important.20 The intuition behind Proposition 3 lies in the fact that the degree of uncertainty interacts with the optimal timing of the contributions. A key player like the European Union faces a rather low direct cost of climate change but also a low variance, while other countries have a higher expected valuation but also 20
Intuitively, this scenario can occur when the distribution function
i
with the higher expected value also exhibits
thicker tails. An example of distribution functions with E( i ) > E( j ) but E( and
j
Gamma(0:5; 8).
23
j ( i ))
< E(
i ( j ))
is
i
Gamma(1; 3)
face more uncertainties. For the latter types of countries, the incentive to delay the contribution can be stronger at the status quo since they still have the option to contribute in the future. But if the relative cost advantage of contributing early becomes su¢ ciently strong then their current contribution will be relatively high. Proposition 3 implies that the provision of green technology by, say, Europe generates a strategic advantage if it increases the early contribution of a country with a high expected valuation, say China, and in this way reduces Europe’s burden of contributing. Thus, Europe’s e¤orts to support the transfer of low-carbon technology such as, for instance, within the EU-China Partnership on Climate Change (or the EU-India action plan) are in line with the strategic relevance of Proposition 3: The bene…t of technology transfer is higher the higher the own willingness to contribute at the status quo. In scenario 2, technology transfer shifts the burden of an early contribution from j to i. While this clearly increases j’s expected payo¤, the crowding-out of j’s early contribution has a negative e¤ect on i’s expected payo¤, which is stronger the larger the amount j would have contributed without technology transfer. But technology investments also reduce i’s marginal contribution cost and lead to higher public good provision. Therefore, in scenario 2, country i would join the technology transfer mechanism (ignoring additional incentives) if and only if the positive e¤ects of lower marginal costs and increased climate protection outweigh the negative e¤ect of an increased burden of contributing; see also the example in Section 4.3.
4.2
Welfare implications
Investments in technology have direct and indirect e¤ects on ex ante expected welfare, de…ned as the sum of the countries’expected payo¤s. First, apart from the direct bene…t of reduced marginal contribution costs, there is a direct cost in terms of resources to be expended for developing the technology. In addition, however, there are indirect welfare e¤ects from investments in technology in situations where the countries’ timing of contributions is a¤ected (such as in Propositions 2 and 3). The following considerations focus on the indirect welfare e¤ects caused by changes in the equilibrium contribution pattern.
24
Corollary 3 (i) Total expected equilibrium contributions Xt + E
Xt+1 j Xt
are strictly increasing in
the early contribution Xt . (ii) Ex ante expected welfare is strictly increasing in the early contribution Xt . The result in Corollary 3(i) is straightforward: Higher early contributions strictly increase total equilibrium contributions to the public good since there is incomplete crowding-out of late contributions. Intuitively, while a marginal increase in Xt just results in an equivalent reduction of late contributions whenever at least one country’s valuation turns out to be high, it increases total contributions in case both countries’valuations turn out to be low (lower than the critical valuation ^ in (5)). Analytically, @ Xt + E @Xt
Xt+1 j Xt
=
^)
i(
^)
j(
0;
with strict inequality for all Xt > 0. Thus, even disregarding the direct e¤ect on the contribution costs, investments in technology cause total contributions to be higher when they a¤ect the timing of contributing as in Propositions 2 and 3. Corollary 3(ii) addresses the welfare e¤ects of such a change in the equilibrium contribution pattern. Even though higher early contributions increase the probability of an over-contribution from an individual country’s point of view (if its true valuation turns out to be low), a shift toward earlier contributions is welfare-improving for two reasons. First, due to the standard underprovision of the public good in the equilibrium of private provision, the resulting increase in total contributions Xt + E
Xt+1 j Xt
(as in
Corollary 3(i)) is welfare-improving. Second, higher early contributions mitigate the bias of the equilibrium contributions toward the late period caused by the uncertainty and the possibility of ‘intertemporal free-riding’(compare also Corollary 2 on the social planner’s choice). To summarize, investments in technology have direct costs and bene…ts: On the one hand, resources have to be expended to develop the technology; on the other hand, contribution costs are reduced. In addition, and more interestingly, there are indirect welfare e¤ects caused by the impact on the countries’ timing of equilibrium contributions. A shift towards early contributions increases welfare because it
25
mitigates the underprovision problem. Even when the investment cost exceeds the investing country j’s bene…t from providing the technology or if the recipient country i’s burden of contributing strongly increases (as in Proposition 3), welfare can still be higher if investments in technology are carried out and technology transfer mechanisms are implemented. In such situations very little support of technology sharing mechanisms at the supranational level may be needed to substantially improve the e¢ ciency of a non-cooperative game of contributions to climate protection.
4.3
An illustrative example
Propositions 2 and 3 show that investments in technology can be strategically advantageous whenever they increase the other country’s early contribution. The following example illustrates the results of Propositions 2 and 3 and their consequences for a technology investment game in stage 0 in which the countries make a simultaneous binary choice of whether or not to invest in technology ~ at a given cost. If at least one country invests, technology ~ is provided.21 Let f (X) = ln (X). Suppose that scale parameter 3 and
B
A
is a Gamma distribution on [0; 1) with shape parameter 1 and
is a Gamma distribution with shape parameter 0:5 and scale parameter 8. Due
to the support [0; 1), there will never be a ‘full’early provision (compare condition (8) in Lemma 1(i)). Moreover, E
A
(
B
(
A ))
0:522 and E
B
(
A ( B ))
0:478;
which intuitively means that the expected cost of delay is higher for A than for B even though E ( 1 3
B)
=
= 0:5 8.
Scenario 1: Suppose that ct ( 21
A)
0)
= 2 and ct+1 (
0)
= 3 at the status quo and that technology ~ would
This is, of course, a very simpli…ed version of a strategic game of technology investment, with the purpose of illustrating
the implications of Propositions 2 and 3. It abstracts from investment spillovers and joined e¤orts and considers only one possible type of technology. But the strategic bene…t of technology transfer will also appear for more general technology production functions involving, for instance, continuous and possibly complementary choices of the countries.
26
Figure 1: Example of equilibrium contributions and expected payo¤s depending on the cost ratio ct =ct+1 (for
A
= Gamma (1; 3),
B
= Gamma (0:5; 8), f (X) = ln (X)).
reduce both countries’contribution cost to c~t = 1 and c~t+1 = 2. Due to E
B
(
A ( B ))
<
c~t
A
(
B
(
A ))
<
ct ( ct+1 (
0) 0)
;
Lemma 1 implies that at the status quo there are no early contributions; at technology ~ , however, A prefers a partial early provision. Figure 1 summarizes the equilibrium outcome and shows that both countries’ expected payo¤s are higher under technology level ~ than under technology level
22 0.
In
analogy to Proposition 2, however, country j = B, which free-rides in period t under the improved technology, bene…ts more from technology ~ :
B
(~ )
B
(
0)
>
A (~ )
A ( 0) :
Hence, for a low cost of technology ~ the technology may be provided by either of the countries (the reduced-form game of technology provision has two pure strategy equilibria); for an intermediate cost of technology, however, there is a unique equilibrium in which country B (with the incentive to free-ride) 22
Preferred early provision levels are derived with (6) and ^ (Xt ) = ct+1 Xt for f (X) = ln (X). Expected payo¤s use the
resulting equilibrium contributions in periods t (from Proposition 1) and t + 1 (from (3)). Note that the negative expected payo¤ of A is due to ln (X) < 0 for X < 1.
27
provides the technology in stage 0. For a high cost of technology, no country invests in technology.23 Scenario 2: To illustrate that investments in technology can even shift the burden of contributing from an initially contributing country to the other country (and thus yield an even higher strategic bene…t), suppose instead that ct (
0)
= 1 and ct+1 (
0)
= 2 at the status quo. Consider the option to provide a
technology at level ~ which reduces both countries’contribution cost to c~t = 0:5 and c~t+1 = 1:25. Due to c~t
B
(
A ( B ))
<
ct ( ct+1 (
0) 0)
A
(
B
(
A )) ;
only country A prefers a positive early provision at the status quo, but both countries prefer a positive early provision at technology ~ . In the latter case, B’s preferred quantity is higher (see the equilibrium values in Figure 1). Hence, in analogy to Proposition 3, country j = A initially contributes early, but an investment in technology ~ changes the equilibrium contribution pattern and shifts the main burden of contributing to i = B. While both countries’expected payo¤s are higher under ~ than under
0,
country
A, which can free-ride, bene…ts much more from the cost-reducing technology. The consequences for the game of technology provision in stage 0 are similar to scenario 1. In particular, for an intermediate cost of technology ~ there is a unique equilibrium in which country A provides the technology. Since both countries are made better o¤ in terms of expected payo¤s, B will join the technology transfer mechanism even though doing so crowds out A’s early contribution. This holds in the example because the reduced contribution costs cause a su¢ ciently strong bene…t for B to compensate for the increased burden of public good provision. 23
For simplicity we assume that the cost of technology is the same for both countries, which is obviously a strong
assumption, not only because of di¤erences in R&D capacities but also because of country di¤erences in the cost of public funds.
28
4.4
Discussion
4.4.1
E¤ect of technology on the irreversibility ratio
Propositions 2 and 3 identify a strategic bene…t of technology sharing for technologies that increase the relative advantage of contributing early (reduce the cost ratio ct =ct+1 by reducing ct relatively more strongly). Intuitively, such technologies are relatively more e¤ective and useful in reducing current greenhouse gas emissions, given today’s information, but may be less e¤ective and useful with altered conditions or knowledge at a later date. One of the properties that a substantial share of low-carbon technologies has is that they are useful for reducing emissions today and in the near future; in light of potential technologies such as nuclear fusion and carbon capture and storage, the importance of many currently used abatement technologies for the distant future is rather uncertain.24 Implemented technology transfer initiatives often focus on transferring green technologies to be used immediately. But other types of technology investments (and multilateral initiatives such as the ITER fusion reactor) rather aim at long-term gains, providing a potential for future cost reductions. This section shows that technology transfer which reduces future contribution costs relatively more strongly is strategically disadvantageous and can even be ine¢ cient. The most interesting case to consider is a status quo scenario in which country i prefers a partial early provision (as in Lemma 1(ii)).
Corollary 4 (i) Suppose that E ( i ) =
ct =ct+1 < E i (
j
( i )). Then, investments in technology that
lead to a cost ratio c~t =~ ct+1 > ct =ct+1 strictly decrease country i’s preferred early provision level Qi;t and are thus strategically disadvantageous for country j. (ii) The negative welfare e¤ ect caused by lower early equilibrium contributions can outweigh the direct positive welfare e¤ ect caused by reduced contribution costs.
Technology investments with a strong impact on future contribution costs increase country i’s incentive 24
Examples include biofuels, hybrid electric vehicles, and coal-…red power generation technologies even though predictions
must be viewed with caution (compare the boom in natural gas due to hydraulic fracturing, for instance).
29
to delay the own contribution and hence reduce the quantity that i is willing to contribute early (Corollary 4(i)). Since this can result in a higher burden of contribution for country j, providing such types of technologies is strategically disadvantageous for j. An intuition for Corollary 4(i) can be seen in the conditions for preferred early quantities given in Lemma 1: If the inverse irreversibility ratio ct =ct+1 is increased (that is, ct+1 is reduced relatively more strongly) then a country’s preferred early provision level changes from a full provision to a partial provision or from a partial provision to no provision. Although countries also bene…t from lower marginal contribution costs, the negative strategic e¤ect can reduce the investing country j’s expected payo¤ (even disregarding the cost of technology). Moreover, while Corollary 3 addresses the indirect welfare e¤ects caused by delay and shows that lower early contributions Xt reduce ex ante expected welfare, there is also a direct positive welfare e¤ect when contribution costs are reduced. The total e¤ect can, however, be negative: Even disregarding the cost of technology, technology transfer which lowers future contribution costs relatively more strongly can have negative welfare consequences (Corollary 4(ii)). The proof of Corollary 4 contains an example in which technology investments which reduce future contribution costs relatively more strongly lead to lower total expected payo¤s, due to an increased incentive to delay the contributions. In general, the investment in cost-reducing technology has two e¤ects: a direct positive e¤ect of reducing the contribution cost and a strategic e¤ect on the equilibrium contribution pattern. Investments in technology which increase the other country’s early contribution are always strategically advantageous, while types of technologies which decrease the other country’s early contribution reduce the free-riding opportunities. Thus, a country’s incentive to support technology transfer is strongest for types of technologies which are of immediate use and increase the relative cost advantage of contributing early.
4.4.2
Contribution cost and equilibrium contribution pattern
The assumption of quasi-linear payo¤ functions keeps the analysis tractable by separating the two countries’optimization problems and reducing the problem to a comparison of the preferred quantities; this
30
has the advantage that the countries’ strategic considerations are most clearly highlighted. Besides, quasi-linear payo¤ functions are a good approximation in the range of investments in climate protection in which climate change policies are currently discussed and implemented.25 Due to the concavity of the production function f , the countries’objective functions exhibit decreasing returns in each dollar invested by the country itself as well as by the other country. While clearly being stylized, the one-sided contribution pattern which evolves in the two periods (compare (3) and Proposition 1) is qualitatively in line with climate protection e¤orts; the empirical pattern of implementing e¤ective climate change policies is, in reality, highly asymmetric and characterized by strong free-riding.26 Free-riding becomes weaker when assuming, for instance, convex contribution costs together with zero marginal contribution costs for the …rst unit contributed within a period, but the strategic considerations remain similar. Hence, assuming quasi-linearity is not crucial for the qualitative results obtained. We conclude this section by brie‡y discussing the case of convex contribution costs. If the …rst unit of climate protection e¤ort involves zero cost, then in equilibrium both countries contribute in both periods. 25
Empirical estimates of (country-speci…c) abatement cost curves are di¢ cult to obtain, due to behavioral reactions,
macroeconomic interactions, and many other factors. Moreover, marginal abatement cost curves have to incorporate the many di¤erent policy options available, including improving energy e¢ ciency, adopting more climate-friendly energies, lowering demand, and measures such as avoiding deforestation. Therefore, constant marginal cost is not unrealistic to assume unless abatement becomes very high. Also, instruments such as the Clean Development Mechanism (CDM) allow us to make use of regional di¤erences in abatement potential. For discussions and estimates of empirical abatement cost curves see, for instance, Wetzelaer et al. (2007), McKinsey (2009), and Kesicki and Ekins (2012). 26
In terms of timing, many European countries and (more recently) the United States, for instance, set goals to reduce
greenhouse gas emissions while other countries such as China or India do not currently choose comprehensive climate protection policies. But also among advanced economies there are clear asymmetries in the sense that some countries such as Germany and the UK, for instance, choose relatively high investments in renewable energies whereas other countries like Australia and Canada tend to delay abatement e¤orts. For recent comparisons of climate change policies across countries see, for instance, the Climate Change Performance Index at https://germanwatch.org/en/indices.
31
Non-linearity of the contribution costs generates additional interdependencies within and across periods; in particular, preferred provision levels are no longer independent of which country contributes how much. Moreover, an increase in the early contribution Xt (incompletely) crowds out future contributions, which reduces a country’s incentive to contribute early. Balancing the expected bene…t and cost of contributing early results in a similar trade-o¤ between uncertainty and irreversibility as with linear contribution costs. In particular, as with quasi-linear preferences, it is strategically most attractive to support the transfer of technologies which reduce today’s contribution costs relatively more strongly and, as a consequence, increase the other country’s early contribution. Again, the uncertainty regarding the bene…ts of climate protection captured by
A
and
B
is crucial for the incentive to delay the contribution and, hence, for
the countries’strategic bene…ts from technology transfer.
5
Conclusion
In this paper we have shown how the timing of the contribution to climate protection is a¤ected by uncertainty, irreversibility, and the possibility to free-ride. Uncertainty about the country-speci…c bene…t of climate protection creates an incentive to delay the contribution decision to a later date when the uncertainty has been resolved, while the irreversibility of damages makes an earlier contribution more desirable. Furthermore, the fact that mitigation e¤orts are contributions to a global public good and an anticipation of free-riding possibilities reduces the incentive to contribute early. In other words, the positive externalities caused by investments in climate protection increase a country’s option value of waiting. Investments in cost-reducing technology have an important impact on the trade-o¤ that countries face and, hence, on the timing of the contributions. In the game of private contributions to the public good with potentially asymmetric but known valuations for climate protection, the country with the highest valuation for climate protection will face the major burden of contributing. The fact that some countries are more likely than others to have the higher valuation makes them react di¤erently to changes in the degree of irreversibility caused by 32
investments in cost-reducing technology. The degree of irreversibility refers to the cost ratio of early and late mitigation e¤orts; the probability of having the higher valuation can be interpreted as the expected savings from an early contribution (since when having the higher valuation a country contributes in the late period at higher cost). Consequently, investments in cost-reducing technology can change the equilibrium contribution pattern. In particular, they can increase a country’s early equilibrium contribution and, thus, the other country’s free-riding opportunities. We identify this strategic bene…t of technology sharing in two main scenarios where investments in technology a¤ect the countries’timing of contributions. In the …rst scenario, at the status quo, the countries have a dominant strategy of not contributing until the uncertainty has been resolved; in the second scenario, one country j would contribute early even without technology transfer. In both cases, if the investment in technology changes the degree of irreversibility, one country will be more sensitive to this change and will prefer to contribute early. In turn, the other country can reduce its contribution. Our analysis puts the emphasis on types of technologies which are relatively more useful for current than for future abatement and, hence, lower the inverse irreversibility ratio ct =ct+1 . Being in line with implemented technology transfer initiatives, we show that the transfer of technologies which are relatively more e¤ective in reducing current contribution costs has a strategic bene…t. By strengthening the other countries’incentives to contribute early, providing such technologies may be bene…cial, due to the positive externality of others’ (early) contributions. On the other hand, sharing technologies with a stronger impact on future contribution cost is strategically disadvantageous and can lead to a higher own burden of contributing. The two-country model can be interpreted as the case of a strategic interaction between two key players (regions) that choose their contribution to climate protection and decide whether or not to implement technology sharing initiatives. This assumption, however, is not particularly restrictive. Assuming quasi-linear payo¤ functions, only the countries that potentially have the largest net bene…t will choose positive contributions in equilibrium. In a model with n > 2 countries, the equilibrium probability of
33
contributing will depend on all other countries’ (expected) valuations; but the main insights obtained from the two-country model and the resulting trade-o¤ between uncertainty and irreversibility carry over when considering more than two countries. In our model, investments in technology a¤ect the timing of contributions and can achieve a change in the equilibrium contribution pattern and, hence, in the investing country’s payo¤ (disregarding costs of technology investments). Moreover, in the two main scenarios considered, the cost-reducing technology strictly increases the quantity of the public good provided early and also the overall amount contributed to climate protection because early contributions only incompletely crowd out late contributions. Even if, ex post, a country has over-contributed from an individual perspective (because its early contribution was higher than what would have been optimal based on the true valuation), such over-contributions from an individual perspective are welfare-increasing due to the underprovision of the public good. In addition, early contributions are ine¢ ciently low due to ‘intertemporal free-riding.’ Thus, abstracting from the cost of providing cost-reducing technologies, the shift of the countries’equilibrium contributions toward early contributions has a positive e¤ect on welfare. Hindered by the large uncertainties and heterogeneity across countries, international agreements to increase climate protection e¤orts have been di¢ cult to implement and have remained rather ine¤ective. Our paper argues that technology sharing mechanisms can, in a non-cooperative setting, induce countries to increase their current contributions to climate protection and in this way make technology sharing bene…cial for the country that invests in green technology. Promoting technology sharing may thus be a promising approach in the …ght against climate change.
34
A A.1
Appendix Proof of Lemma 1
We …rst derive the marginal payo¤ of increasing xi;t . If 0 then the expected payo¤ Z
0
+ (18)
+
Therefore, if 0
i (xi;t ; xj;t )
^Z ^ 0
Z
^
if
(Xt ) d
j
^ < ),
in (4) is equivalent to
( j) d
i ( i)
+
Z
0
Z
Xt < Qt+1 ( ) (or, equivalently, 0
^Z
if
^
(Qt+1 ( j )) d
j
( j) d
i ( i)
i
[ i f (Qt+1 ( i ))
ct+1 (Qt+1 ( i )
Xt )] d
j
( j) d
i ( i)
0
Z
Z
^
if
(Qt+1 ( j )) d
j
( j) d
i ( i)
ct xi;t :
i
Xt < Qt+1 ( ), then
@ i = @xi;t
Z
0
^Z ^
0 i f (Xt ) d
j ( j) d
i ( i) +
0
Z
Z
^
i
ct+1 d
j
( j) d
i ( i)
ct +
0
@^ : @ ^ @xi;t
@
i
Since @
i
@^
Z
=
i f (Xt )
0
Z
0 ^ j ( )d i ( i )
0
Z
^ i f (Qt+1 ( ))
and Qt+1 (^) = Xt , we get @
^f (Qt+1 (^))
(19)
Moreover, in this case,
0
0
if
j
0
(Xt ) d
j
+
0 ^ i( )
( j)
^f (Qt+1 ( j )) d
^
i Xt ) d
j
( j)
j
Z
0
^Z ^
( j) d
if
00
0 ^ i( )
0 ^ i( )
i =@
^)(@ ^=@xi;t ) = 0 holds just as in an envelope
i ( i)
+
Z
^
Z
i
ct+1 d
j
( j) d
0
is concave in xi;t : Note that
@2 i = @x2i;t
( j)
Xt < Qt+1 ( ), then
0
i
Z
j
0 ^ i( )
( j)
^ = 0. (Intuitively, (@
theorem since ^ is set optimally.) Thus, if 0 ^Z ^
^f (Xt ) d
ct+1 (Qt+1 (^)
^f (Qt+1 ( j )) d i =@
^
0 ^ j ( )d i ( i )
^h
^
Z
+
^
0
Z
@ i = @xi;t
Z
^
(Xt ) d
j
( j) d
0
35
i ( i)
+
@^ @xi;t @ ^ @xi;t @2
i
i ( i)
ct :
and, using ^ = ct+1 =f 0 (Xt ), @2
i
@xi;t @ ^
Z
=
^ if
0
(Xt )
0
Z
=
if
0
(Xt )
(20) and hence @ 2
Z
0
=
Z
h
^Z ^
^f 0 (Xt ) d
^)
2 i =@xi;t
j
0 ^ i( )
( j)
^
ct+1 d
j
( j)
0
0 ^ i( )
0 ^ j ( )d i ( i ) :
00 i f (Xt ) d
j ( j) d
^f 00 (Xt ) =f 0 (Xt ). Altogether, i ( i) +
Z
^ 0 ^ j ( )d i ( i )
0 i f (Xt )
0
0
j(
Z
^
0
ct+1 f 00 (Xt ) = (f 0 (Xt ))2 =
Further, @ ^=@xi;t = =
+
^
0
@2 i @x2i;t
0 ^ j ( )d i ( i )
iZ ^ 0 (^) j
^ if
00
(Xt ) d
00 ^ f (Xt ) f 0 (Xt )
i ( i)
0
0 if Assumption 1 holds.
Now suppose instead that Xt
Qt+1 ( ). Then, using (4), @ i = E ( i ) f 0 (Xt ) @xi;t
(21)
Moreover, it follows immediately that @ 2
2 i =@xi;t
ct :
= E ( i ) f 00 (Xt ) < 0 in this case.
To show Lemma 1, note that using (21) (or also using (19)) we get @ i @xi;t
= E ( i ) f 0 Qt+1 ( )
ct = E ( i ) f 0
1
f0
ct+1 =
ct
0
Xt =Qt+1 ( )
if and only if E ( i ) =
ct =ct+1 . Therefore, due to the concavity of
i,
Qi;t
Qt+1 ( ) if and only if (8)
holds.27 In this case, Qi;t is uniquely determined by E ( i ) f 0 (Qi;t ) which yields Qi;t = (f 0 )
1
ct = 0;
(ct =E ( i )). Since (8) is equivalent to (f 0 )
1
(ct =E ( i ))
Qt+1 ( ), there is no
contribution in t + 1 (part (i)). If instead E ( i ) = < ct =ct+1 , then @
i =@xi;t jXt =Qt+1 ( )
< 0 and hence Qi;t < Qt+1 ( ). (Since Qt+1 ( )
is the highest type’s preferred contribution level in t+1, i might contribute again in t+1.) Moreover, Qi;t < 27
To be precise, if E ( i ) = = ct =ct+1 and @ 2
2 i =@xi;t X t
= 0 for Xt 2 Qt+1 ( )
; Qt+1 ( ) ,
> 0, then increasing
the early contribution within this interval does not change i’s expected payo¤. To simplify the exposition, we assume that Qi;t = Qt+1 ( ) in this special case.
36
(f 0 ) i
1
1
(ct =E ( i )): Note …rst that (f 0 )
> ^, @ Z
0
i =@xi;t
^Z ^
if
0
(ct =E ( i )) < Qt+1 ( ) and, using ct+1 = ^f 0 (Xt ) <
if
0 (X
t)
for
in (19) is strictly smaller than
(Xt ) d
j
( j) d
i ( i)
+
0
Z
Z
^
i
if
0
(Xt ) d
j
( j) d
ct < E ( i ) f 0 (Xt )
i ( i)
ct
0
for all Xt < Qt+1 ( ). Therefore, it holds in particular that @ i @xi;t
< E ( i) f 0 Xt =(f 0 )
1
f0
1
(ct =E ( i ))
ct = 0:
(ct =E( i ))
To determine whether or not Qi;t is positive, note that, with (19), it holds that Qi;t > 0 if @ i lim Xt !0 @xi;t or, equivalently, if E i (
j
=
Z
Z
0
Xt
i
ct+1 d
( j) d
j
i ( i)
ct > 0
0
( i )) > ct =ct+1 (part (ii)); in this case Qi;t is obtained by the …rst-order
condition based on (19). But if (10) holds, then Qi;t = 0.28 To complete the proof note that the range of parameters for which part (ii) may apply is non-empty: E(
j
( i )) > E ( i ) = . To see why, note that Assumption 1 is equivalent to @ @ i
(22)
Let z 2 (0; ]. Then, (22) implies that for all j
( i) i
j
0:
i
z,
i
(z) z
( i)
(z) z
j
(z) z
j i
j
i
=0
or, equivalently,
(23) 28
j
If ct =ct+1 = E i (
j
( i)
j
(z)
i
z
0:
( i )) and Assumption 1 holds with equality for
2 [0; 0 ],
0
> 0, then @
i =@xi;t
= 0 for all
Xt 2 [0; Qt+1 ( 0 )]. While in this case increasing Xt within this interval does not change i’s expected payo¤, i need not be indi¤erent between all contributions xi;t 2 [0; Qt+1 ( 0 )]; xi;t = 0 is at least weakly preferred to all contributions xi;t > 0. To simplify the exposition, we assume that in this case Qi;t = 0. If Assumption 1 holds with strict inequality, then Qi;t = 0 if and only if (10) is ful…lled.
37
Since (23) implies that Z
z j
( i)
j
i
(z)
0
setting z =
yields E (
j
( i ))
d
z
i ( i)
E ( i ) = , with strict inequality whenever Assumption 1 holds with
strict inequality on a non-empty interval within the support of
A.2
0;
j.
Proof of Corollary 1
De…ne by ^ i the expected payo¤ of i, given xj;t = xj;t+1 = 0 and xi;t+1 = max fQt+1 ( i ) xi;t < Qt+1 ( ), then @ ^ i =@xi;t is as in (12). If xi;t
Qt+1 ( ), then
@ ^i = E ( i ) f 0 (Xt ) @xi;t
(24)
xi;t ; 0g. If
ct ;
just as in the two-country case in (21), because in either case there are no contributions in t+1. Moreover, it is straightforward to verify that @ 2 ^ i =@x2i;t < 0 for all xi;t Suppose …rst that E ( i ) =
ct =ct+1 . Then, @ ^ i =@xi;t
0.
Xt =Qt+1 ( )
0 and the preferred early provi-
sion is determined using (24), which yields the same quantity as in Lemma 1(i). If instead E ( i ) = < ct =ct+1 , this implies that the preferred early provision is smaller than Qt+1 ( ), both in the one-country benchmark and in the two-country case. Here, @ ^ i =@xi;t in (12) can also be written as Z
0
^Z ^
if
0
(Xt ) d
j
( j) d
i ( i)
0
+
Z
0
^Z
^
if
0
(Xt ) d
j
( j) d
i ( i)
+
Z
^
ct+1 d
i ( i)
ct ;
which is strictly larger than Z
0
^Z ^ 0
if
0
(Xt ) d
j
( j) d
i ( i)
+
Z
^
Z
i
ct+1 d
j
( j) d
i ( i)
ct ;
0
that is, strictly larger than the marginal payo¤ in the two-country case (compare (19)). Finally, note that (12) approaches ct+1
ct > 0 if Xt ! 0. Thus, the early provision level in the benchmark of a
single country is always strictly positive; it is strictly larger than Qi;t as given in Lemma 1 whenever E ( i ) = < ct =ct+1 .
38
A.3
Proof of Corollary 2
De…ne by
SP
the sum of the countries’expected payo¤s from the point of view of period t, taking the
social planner’s optimal late contribution Xt+1 = max fQt+1 (
+
A
B)
Xt ; 0g as given. If Xt < Qt+1 ( )
or, equivalently, ^ < , then
(25)
@ SP = @Xt
Z
0
^Z ^
i
(
+
i
j) f
0
(Xt )
ct d
j
( j) d
i ( i)
0
Z
+
Z
as there is a contribution in t + 1 if and only if equivalent to ct+1 > Z
0
if
^Z
0 (X
t ),
+
j
ct d
j ( j) d
i ( i) +
i
using this relation, @
ct ) d
j
( j) d
i ( i) ;
i ;0g
> ^ (Xt ) where ^ is as in (5). Since
i
< ^ is
the second term in (25) is strictly larger than
0 i f (Xt )
^
i
(ct+1
maxf^
0
SP =@Xt
Z
^
Z
(ct+1
ct ) d
j
( j) d
i ( i) ;
0
in (25) is strictly larger than the marginal payo¤ in the one-country
benchmark in (12). If Qt+1 ( )
Xt < Qt+1 (2 ), there is a contribution in t + 1 with positive probability in the social
planner’s problem (but not in the one-country benchmark); this contribution occurs whenever
i+ j
>^
where ^ 2 [ ; 2 ). Thus, (26)
@ SP = @Xt
Z
0
Z
minf^
i;
g
(
i
+
j) f
0
(Xt )
ct d
j
( j) d
0
+
Z
Z
minf^
0
With ct+1 = ^f 0 (Xt )
f 0 (Xt ) Z
0
thus, @
SP =@Xt
if
Z
minf^
0 (X
t ),
(ct+1 i;
0
j
( j) d
i ( i) :
(Xt )
ct d
j
( j) d
i ( i) ;
g
in (26) is strictly larger than E ( i ) f 0 (Xt )
ct , which is the marginal payo¤ in the
one-country benchmark in (24). Finally, if Xt
ct ) d
g
the second term in (26) is strictly larger than if
i;
i ( i)
Qt+1 (2 ), there is no further contribution in t + 1 and @ SP = (E ( i ) + E ( j )) f 0 (Xt ) @Xt 39
ct > E ( i ) f 0 (Xt )
ct :
To sum up, due to the higher marginal payo¤ of increasing Xt (for any Xt ), the early provision in the social optimum is higher than in the one-country benchmark and, by Corollary 1, also higher than the preferred early provision in Lemma 1. Note that @ 2 early provision equal to (f 0 )
1
2 SP =@Xt
< 0 and that the social optimum is a ‘full’
(ct = (E ( i ) + E ( j ))) if and only if ct =ct+1
E ( i ) =(2 ). Otherwise, the
social optimum involves a positive but ‘partial’early provision, and further contributions in t + 1 with strictly positive probability.
A.4
Proof of Proposition 1
Note …rst that Qi;t as given in Lemma 1 is independent of xj;t . Due to the quasi-linear payo¤ functions, i’s preferred provision level does not depend on how this quantity is split into (xi;t ; xj;t ). Thus, country i’s best response to xj;t is xi;t = max fQi;t
xj;t ; 0g and in equilibrium only the country i with
the higher preferred early provision level will contribute in t. To see why, suppose that Qi;t > Qj;t and xj;t = xj;t > 0. Then, i’s best response is max fQi;t max fQj;t
max fQi;t
xj;t ; 0g. But then, j’s best response is
xj;t ; 0g ; 0g which is strictly smaller than xj;t (using Qi;t > Qj;t ), which yields a
contradiction. The equilibrium contributions are hence as follows. If Qi;t > Qj;t
0, then xi;t = Qi;t and xj;t = 0,
i; j 2 fA; Bg, j 6= i (parts (ii) and (iii)). If QA;t = QB;t = 0, then xA;t = xB;t = 0 (part (i)). If QA;t = QB;t > 0, then there is a continuum of equilibria where xA;t 2 [0; QA;t ] and xB;t = QA;t
xA;t
(part (iv)), all yielding the same total quantity Xt = QA;t = QB;t .
A.5
Proof of Proposition 2
First of all, c~t =~ ct+1 < E i (
j
( i )) implies that i prefers at least a partial early provision (i.e., Qi;t (~ ) > 0);
compare Lemma 1. Moreover, if (a) c~t =~ ct+1
E j(
i ( j )),
then Qj;t (~ ) = 0 < Qi;t (~ ), which yields
x ~i;t (~ ) > 0 = x ~j;t (~ ) at technology level ~ . If (b) c~t =~ ct+1 < E j (
i ( j )),
then j also prefers a positive
early provision. Since Qi;t and Qj;t are continuous in ct and ct+1 (compare (19)), there exists
40
> 0,
su¢ ciently small, such that 0 < Qj;t (~ ) < Qi;t (~ ) if c~t =~ ct+1 2 E j ( is smaller than but su¢ ciently close to E j (
i ( j )),
i ( j ))
;E j (
i ( j ))
: If c~t =~ ct+1
we get 0 < Qj;t (~ ) < Qi;t (~ ) and thus i contributes
in equilibrium. (For even lower cost ratios this can change, depending on the probability distributions and
j;
A.6
i
recall also the di¤erence in incentives to contribute early in (11).)
Proof of Proposition 3
Without investment in technology, j prefers a positive early provision while i prefers no early provision; hence xj;t ( c~t =~ ct+1 (f 0 )
1
0)
> 0 = xi;t (
0 ).
Consider …rst an investment in technology with corresponding cost ratio
E ( i ) = . In this case, i prefers a full early provision based on its expected valuation: Qi;t (~ ) =
(ct =E ( i )). Since (f 0 )
1
(ct =E ( i )) > (f 0 )
1
(ct =E ( j ))
Qj;t (~ ), we get xi;t (~ ) > 0 = xj;t (~ ).
By continuity of (19) and (21), this still holds if c~t =~ ct+1 2 E ( i ) = ; E ( i ) = +
,
> 0 but su¢ ciently
small. In other words, the investment in technology only needs to bring i’s preferred early quantity su¢ ciently close to a full early provision in order to guarantee that xi;t (~ ) > 0. Finally, note that E ( i ) > E ( j ) is su¢ cient but not necessary for obtaining a ‘categorical change’ in the equilibrium contribution pattern. If (16) holds and E ( i ) = < E ( j ) = < E i ( for cost ratios c~t =~ ct+1 2 E ( j ) = ; E i (
j
j
( i )) then
( i )) , both i and j prefer a partial early provision. Even if
E ( i ) < E ( j ), condition (11) can, depending on the shape of the distribution functions, be positive at Xt = Qj;t (~ ), in which case we would get Qi;t (~ ) > Qj;t (~ ) and xi;t (~ ) = Qi;t (~ ) > 0 = xj;t (~ ) at technology ~ .
A.7
Proof of Corollary 3
For part (i), note …rst that E Z
0
Z
maxf^; i g
(Qt+1 ( j )
Xt+1 j Xt is equal to Xt ) d
j ( j) d
i ( i) +
41
Z
^
Z
0
i
(Qt+1 ( i )
Xt ) d
j
( j) d
i ( i)
since j contributes in t + 1 if
> maxf^; i g and i contributes in t + 1 if
j
i
> ^ and
j
<
i.
Therefore,
@E Xt+1 j Xt =@Xt is equal to29 Z
0
Z
maxf^; i g
d
j
( j) d
Intuitively, when Xt is increased, E
Z
i ( i)
^
Z
i
d
j
( j) d
i ( i)
=
1
^)
^) :
i(
j(
0
Xt+1 j Xt decreases by the same amount, except if
i
and
j
are
both lower than the critical valuation ^ (in which case there is no contribution in period t + 1). Hence, we get @ Xt + E @Xt
Xt+1 j Xt
^)
=
^);
i(
j(
Xt =Xt
which is strictly positive for all Xt > 0 (i.e., ^ > 0). For part (ii), suppose …rst that Xt < Qt+1 ( ). With equilibrium contributions in t + 1 as in (3), the welfare e¤ect of an increase in Xt is @(
(27)
+ @Xt i
j)
=
Z
0
^Z ^
(
i
+
j) f
0
0
+
(Xt ) d Z
^
Z
j
( j) d
ct+1 d
i ( i)
j ( j) d
i ( i) +
0
Z
0
By part (i), public good provision increases if and only if
i
and
j
^Z
^
ct+1 d
j
( j) d
i ( i)
ct :
are both lower than ^ (Xt ); otherwise,
an increased early contribution results in cost savings. (Compared to a single country’s marginal payo¤ as in (6), the free-riding e¤ect disappears; compared to the social planner’s problem in (25), contributions in t + 1 are lower and occur with lower probability.) If Xt = 0, then @ (
i
+
j ) =@Xt
= ct+1
Xt > 0, suppose without loss of generality that Qi;t > Qj;t and hence Xt = xi;t . Since @ ( in (27) is strictly larger than @ @(
i
+
hence @ ( 29
j ) =@Xt jXt =X
t
i
+
i =@Xt
t
+
j ) =@Xt
in (19) and the latter is equal to zero at Xt = xi;t , we get
> 0. Finally, if Xt = xi;t
j ) =@Xt jXt =X
i
ct > 0. If
Qt+1 ( ), there are no contributions in t + 1 and
= 0 + E ( j ) f 0 (Xt ) > 0.
Note that ^ is a function of Xt but @E (Xt+1 j Xt ) =@ ^ = 0 using Qt+1 (^) = Xt , again as in an envelope theorem
(compare also the proof of Lemma 1).
42
A.8
Proof of Corollary 4
For part (i), note …rst that E ( i ) =
ct =ct+1 < E i (
j
( i )) implies that i prefers a partial early
provision Qi;t which is determined by setting the marginal payo¤ in (19) equal to zero or, equivalently, by ct+1
"Z
^ i
^)d
j(
^
0
i ( i)
+
Z
j
^
( i) d
i ( i)
ct ct+1
#
=0
where ^ = ^ (Qi;t ) = ct+1 =f 0 (Qi;t ). Since @ @^
Z
0
^ i
^
^)d
j(
i ( i)
+
Z
^
j
( i) d
!
i ( i)
^ =
^
^ j( )
0 ^ i( )
+
Z
0
0 (^)^ j
^ i
(^)2
^)
j(
d
i ( i)
^)
j(
0 ^ i( )
is negative if Assumption 1 holds, ^ (Qi;t ) must go down if ct =ct+1 is increased. Since ^ (Xt ) is increasing in Xt , i’s preferred early provision level at cost ratio c~t =~ ct+1 > ct =ct+1 must therefore be lower than i’s preferred early provision level at cost ratio ct =ct+1 . To show part (ii) on welfare (i.e., total expected payo¤s), consider the example of Section 4.3. Suppose that, at the status quo, ct = 0:5 and ct+1 = 1:25, with equilibrium contributions xA;t ( xB;t (
0)
4:567 and payo¤s
A ( 0)
5:113 and
B
(
0)
0)
= 0 and
4:881. Consider a technology investment by
j = A which reduces future contribution cost to c~t+1 = 1:1, leaving for simplicity c~t = ct = 0:5 unchanged (the results go through if c~t 2 (ct
"; ct ), " > 0 but su¢ ciently small). As a result of the improved future
climate protection opportunities, B’s early equilibrium contribution is reduced to xB;t (~ )
2:612. Since
xA;t = 0 remains unchanged, B must be better o¤ in the scenario with the lower future contribution costs: B’s expected equilibrium payo¤ increases to decreases to
A (~ )
B
(~ )
5:155. But A’s expected equilibrium payo¤
4:369, due to the reduced free-riding opportunities. Hence, it is strategically
disadvantageous for A to provide technology ~ . Moreover, the total welfare e¤ect of providing ~ is negative:
A (~ )
+
B
(~ ) <
A ( 0)
+
B
(
0 ).
43
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