Hejer Lasram
email: hejer.lasram@gmail.com

Didier Laussel
email: didier.laussel@outlook.fr

The determination of Public Tuition Fees in a Mixed Education System: A Majority Voting Model

We analyze the issue of majority voting equilibrium for the determination of public tuition fees. We consider a model with mixed education system where public and private universities coexist and compete in tuition fees. Since there is a private alternative, preferences for education are not single-peaked and no single-crossing condition holds. An equilibrium is shown to exist which is one of three types: high tuition fee (the "ends" are a majority), low tuition fee (the "middle" is a majority) or mixed (votes tie). the cost structure determines which equilibrium obtains. The equilibrium tuition is either greater (majority at the ends) or smaller (majority at the middle) than the optimal one.

Introduction

The determination of public universities' tuition fees has recently attracted a lot of attention following the important evolutions observed in some countries. [START_REF] Martin | Why Tuition Costs Are Rising So Quickly[END_REF], on the basis of 1995-2002 data, already noted an increase in tuition costs 1 much faster than inflation or family income. The same observation was more recently made on the period 2003-2013 by [START_REF] Desrochers | Trends in College Spending: 2003-2013[END_REF] for US colleges. The most striking recent case of a dramatic increase of the tuition fees is the United Kingdom where tuition fees were first introduced in September 1998 under the (New) Labour Government, with students being required to pay up to £1,000 a year for tuition. This amount was raised to a maximum of £3,000 in 2004. and to £9,000 a year from 2012, with 64 universities charging the maximum of £9,000 and the remaining 56 charging at 1 least £6,000 2 . Austria introduced in 2000 a 363 euros per semester tuition fee, following 30 years without tuition. Another example of a hike of tuition fees decided by the government is Quebec, where the 2013 increase though modest encountered a strong opposition. [START_REF] Martin | Why Tuition Costs Are Rising So Quickly[END_REF] attributed the increase in tuition fees to "the increasing time given to professors for scholarship and other activities", that is to "increases in noninstructional staff" 3 . This factor however cannot account for the wide cross-country differences in tuition fees which are currently recorded. [START_REF] Marcucci | Tuition Fee Policies in a Comparative Perspective: Theoretical and Political Rationales[END_REF] for instance distinguish between countries without public tuition fees (such as Brazil, Denmark, Germany, Norway, Sweden among others), countries with up-front tuition fees (including Belgium, Canada, Chile, Italy, Portugal, United States) and countries with deferred tuition fees (Australia, England, Wales..). According to [START_REF] Marcucci | Tuition Fee Policies in a Comparative Perspective: Theoretical and Political Rationales[END_REF], Table 1, page 29), medium public tuition fees for first degrees ranged from 0 (Hungary, 2001) to 9000 $ (USA, 2004/2005). Given the extent of these differences in tuition fees, which largely exceed possible differences in costs for similar countries, it is natural to look at political factors. More importantly, a pure cost approach cannot account for the dramatic increases in public tuition fees observed in some countries, as reported above, compared to their stability in many others. This is another reason for looking at political factors. More specifically, we will show here that, in some range of parameter values, a small variation of costs may lead, through a "tilting of majority" effect to an important increase in public tuition fees, when it increases the attractiveness of the private university in such a way as to lead to a majority reversal.

In order to understand the political factors contributing to the determination of tuition fees, we consider a model with a mixed education system where there is a private university which offers higher quality than its public competitor, incurring thus a higher unit cost per trained student. In this vertical differentiation framework, we assume that the agents are differentiated with respect to the individual benefits which they are able to derive from education. The two universities determine simultaneously their tuition fees. In the first stage of the game, the public university' s tuition fee is fixed by majority voting while that of the private university by profit maximization. In the second stage, the agents choose to train in the public university or in the private one or not to train at all, then the tax per head adjusts in order to balance the State budget.

Related literature A number of papers have analyzed the determination of the public provision of education through majority voting. Using a numerical example calibrated to actual data, [START_REF] Barzel | Private schools and public school finance[END_REF] was the first to raise the issue of non-single-peaked preferences for education in mixed school systems where education is provided by both public and private institutions. He noticed that while public education is provided uniformly, regardless of households' preferences, "private schools provide an opportunity to obtain the desired quantity" (page 176). In his example, he shows that the richest of the seven income segments opts indeed for private education. [START_REF] Stiglitz | The demand for education in public and private school systems[END_REF] trying to test the Pareto-optimality of the provision of education under various institutional arrangements, considered in his Section 4 mixed public-private school systems and noted (p. 365) that "the absence of single-peaked preferences means that there may not be a majority voting equilibrium" regarding the amount of publicly provided education. Glomm and Ravikumar (1998) pioneered the analysis of endogenous provision of public education when there is a private alternative and proved that a majority voting equilibrium exists, in which the median income voter is pivotal ,under conditions on the utility function and the income's distribution which amount to assume that any utility function satisfies a singlecrossing condition 4 . More precisely, this condition implies that the marginal willingness to pay5 for a higher level of public education decreases with income. Without private alternative, this "SDI" property (slope declining in income) would entail an inverse relationship between the most preferred level of public provision of education and income. This inverse relationship is even reinforced when a private alternative exists6 . Epple and Romano (1996a) did call this property "SDI" and globally confirmed Glomm and Ravikumar (1998) results. They generalized them by considering as well the opposite assumption of "SRI" (slope rising in income), under which single crossing fails. They provided only necessary conditions for the existence in that case of a majority voting equilibrium, namely the existence of two groups of equal sizes: a coalition ("the middle") of intermediate income types which would favor a higher provision of public education and a coalition ("the ends") of both lower and higher income types which would prefer a smaller level of public education expenditures, the rich opting for private education.

Another set of papers (Epple and Romano, 1996b;[START_REF] Bearse | Endogenous public expenditures on education[END_REF] focused on private supplements to public education. Contrarily to Epple and Romano (1996b), [START_REF] Bearse | Endogenous public expenditures on education[END_REF] consider in addition a private alternative for education. Other papers as De la Croix and Deopke (2009) enlarged the analysis linking voting on public schooling expenditures to private education and fertility decisions.

In this paper, we depart from the above literature in several important respects. First, we consider here that higher education is an investment good rather than a consumption good: agents who choose to train consider the additional income they will earn once educated. Second, we suppose that the private university behaves strategically rather than assuming a perfectly com-petitive education supply. Finally, rather than studying the determination of the public provision of education, we focus on the determination, through majority voting, of the tuition fees at the public universities, assuming that training is not necessarily given there for free. In a standard vertical differentiation model, we consider agents which differ in the individual benefits which they are able to derive from education and we assume that the quality of education at the private university is exogenously higher than at the public one. Depending on the private and public tuition fee levels, there are three groups of agents ranked in order of increasing return on education: those who don't train, those who train at the public university and those who train at the private university. Going to the private university is here the way to get a higher quality education leading to better paid jobs. Given that the tax level is adjusted so as to balance the public budget, the first and the third groups favor high tuition fees while the second one prefers low tuition fees: this is the "ends against the middle" feature put forward by Epple and Romano (1996a).

Main results. In this model, (i) agents' preferences are not generally singlepeaked and (ii) no single crossing condition holds. These are features typically associated with a "middle against the ends" framework. Nevertheless, whereas Epple and Romano (1996a) only provided necessary existence conditions, we are able to prove here, for any value of the private university tuition fee, the existence of a majority voting equilibrium. Endogenizing the determination of the private tuition fee, we then prove the existence of three different types of equilibria: (i) an equilibrium in which the "ends" are a strict majority and select a high public tuition fee, higher than the cost per student, (ii) an equilibrium where the "middle" is a strict majority and selects a low public tuition fee, in many cases smaller than the cost per student, and (iii) a "mixed equilibrium" in which votes tie and the public university randomizes between the high and the low public tuition fees. The cost structure is shown to determine which type of equilibrium obtains: the "ends" are a strict majority when the marginal cost per student of the private university's higher quality is small enough to make the private university attractive and the "middle" is a strict majority when this marginal cost is high enough to make the private university rather unattractive. When the quality differential between the two universities is large, a relatively small decrease of the marginal cost of quality per student or/and a small increase of the cost per student at the public university, that is a relative decrease of the private education cost with respect to the public one, may have in our framework the effect of leading to an important rise of the public tuition fee by leading to a majority reversal, the "ends" winning a strict majority against the "middle". Finally, we show that the equilibrium public tuition fee exceeds the optimal one when the "ends" are a strict majority and falls short of it when the "middle" is a strict majority: majority voting leads either to overshoot or to undershoot the optimum.

The remainder of the paper is organized as follows. Section 2 introduces the model. In Section 3, we solve the game and provide comparative statistics . We conclude in Section 4. All proofs are given in Appendix A.

The model

We consider a model with two different universities a public and a private one and where agents choose to train in the public or private university or not to train at all. This choice is made by comparing the lifetime benefits from training (additional discounted income) and the tuition fees to be paid initially. We assume that the tuition fee of the public university is fixed by majority voting while that of the private one is determined by profit maximization.

The universities. The State university proposes a training program delivering a quality q S = 1 and the private one offers training yielding a quality q P > 1 with respective tuition fees t S and t P and unit costs of training c S = c < 1 and c P = c + m(q P -1), m > 0. We suppose that the tuition fee at the State university cannot be negative.

Agents. There is a continuum of agents, the mass of which is normalized to 1 7 . If an agent does not train, he/she will earn thereafter a fixed lifetime income, which is independent of his/her type and which we normalize to 0. When choosing to train, a type θ-agent earns a type-dependent income θq once trained, where q is the quality of training and θ is distributed on the interval [0, 1]. Notice that θ represents the additional lifetime income which the type θ-agent will derive thereafter from training at the public university (where q S = 1) or, equivalently, the lifetime benefit he/she will obtain from a marginal increase in educational quality. We suppose that a type 0-agent chooses not to train when t S = 0, namely when there is no strictly positive benefit from training.

All agents, whether skilled or not, pay during the training period a uniform tax 8 per head τ which is fixed so as to balance the public budget, i.e. to cover the difference between the costs of training at the public university and the tuition fees which are paid by the students. The tax level is fixed before the students may be able to derive any benefit from higher education and, given that the agents' types are unobservable ex ante, is independent of the future benefits they will derive from training. This provides a rationale for the uniformity of the tax 9 .

Hence the payoffs of an unskilled agent, a skilled agent in the State university and a skilled agent in the private one are respectively,

U u (τ ) = -τ, U s S (θ, t S ) = θ -t S -
( t P -t S q P -1 -t S )(t S -c S ) = -τ. ( 1
)
Since there is no natural assumption on the way in which they could be distributed between domestic agents, we assume for the sake of simplicity that the profits of the private university go to non-voters (foreigners). For many countries, assuming that the alternative to public higher education is to go to a foreign university is not a bad assumption 10 .

The game. The timing of the game is such that:

• in stage 1, universities fix their tuition fees; the public university selects the tuition fee which follows from majority voting, while the private university's tuition fee is determined by profit maximization.

• in stage 2, agents choose to train in the public university or in the private one or not to train at all. The tax per head is adjusted so as to balance the public budget.

It is assumed that voters, when deciding upon the public university's tuition fee, hold rational expectations. They perfectly anticipate how, given the tuition fees at the private and public universities, the agents divide into three groups: those who don't train, those who train at the public university and those who train at the private university. They accordingly deduce the value of the income tax which will follow from any couple of tuition fees. As usual, the game is solved backward.

The equilibrium

In this section we solve the game by backward-induction. At the second stage, agents choose to train in the public university or in the private one or not to train at all. At the first stage, the private university fixes its tuition fee t P in order to maximize its profit while the State university fixes the tuition fee t S by majority voting. Private university and voters rationally anticipate the second-period outcomes.

Second-stage training decisions

The type θ-agent indifferent between training at the State university and not training at all is given by θ = t S , whereas the one indifferent between training at the State university and training at the private one is given by θ = t P -t S q P -1 . This implies that agents with low revenues (θ) prefer not to train while those with high revenues prefer to train in the private university. The middle class of agents prefer to train in the State university. Thus the respective demands for training at the private and the State universities are respectively

D P = 1 - t P -t S q P -1 , (2)
and

D S = ( t P -t S q P -1 -t S ). (3)
For the sake of simplicity, we shall only consider the case where the RHS of ( 2) and (3) take on positive equilibrium values, i.e. the case where both universities are active at equilibrium. Extending our results to cover the case where one university is closed would be straightforward but cumbersome.

First-stage tuition fees' determination

The private university's tuition fee follows simply from profit maximization, given the public tuition fee t S . Its best reply function is continuous and increasing in t S :

t P (t S ) = 1 2 (t S + c + (m + 1)(q P -1)). ( 4
)
On the other hand, the value of t S is determined by majority voting as a best reply to t P .

In order to understand this determination, it is necessary to determine precisely what is a type θ-indirect utility function as a function of t S . Substituting in the utilities of the three types of agents the value of τ from equation (1), we obtain

U u (t S ) = ( t P -t S q P -1 -t S )(t S -c S ), (5)
for a type θ-agent who does not train,

U s S (t S ) = θ -t S + ( t P -t S q P -1 -t S )(t S -c S ) (6)
for a type θ-agent who trains at the public university, and

U s P (t S ) = θq P -t P + ( t P -t S q P -1 -t S )(t S -c S ), (7)
for a type θ-agent who trains at the private university. Notice that U u (t S ), U s P (t S ) and U s S (t S ) are concave and reach their maxima respectively at t sP S = t u S = q P c+t P 2q P and t sS S = q P (c-1)+t P +1 2q P , irrespective of the agent's type θ. Notice in addition that choosing or not to train and, in the former case, choosing where to train, is an endogenous function of the public tuition fee t S : people do not belong ex ante to one of the three possible groups.

We distinguish two cases. The first occurs when θ ∈ [0, t P q P [, or, equivalently, U s P (t S ) < U u (t S ): agents always prefer not to train rather than to train at the private university, so that their only choice is between training at the public university or not training at all. Hence the utility these agents θ ∈ [0, tp qp [ is a piece-wise continuous function given by:

U θ (t S ) = U s S (t S ), iff t S ≤ θ; U u (t S ), iff t S > θ.
This function is double-peaked, with local maxima at t sP S = t u S and t sS S , when θ ∈ t sS S , t u S and single-peaked otherwise (see Figures 3,4 and 5 in Appendix). The second case occurs when θ ∈] t P q P , 1], or, equivalently, U u (t S ) < U s P (t S ): agents always prefer to train at the private university rather than not to train at all so that their only choice is between training at the State university or at the private one. Hence the utility of agents θ in ] tp qp , 1] is a piece-wise continuous function given by:

U θ (t S ) = U s S (t S ), iff t S ≤ t P -θ(q P -1); U s P (t S ), iff t S > t P -θ(q P -1).
This function is double-peaked, with local maxima at t sP S = t u S and t sS S , when t sS S < t P -θ(q P -1) < t sP S and single-peaked otherwise (see Figures 6, 7 and 8 in Appendix).

It follows from above that we cannot rely on the median voter theorems to ascertain the existence of a Condorcet winner. On one hand, we have just seen that voters' preferences are not always single-peaked: double-peakedness obtains for large intervals of values of θ. On the other hand, the preferences do not either satisfy the single-crossing property: the most preferred value of t S is not a monotonic function of the voter's type θ. More precisely, voters with respectively low and large values of θ, who don't go to the public university11 , favor a high value t sP S = t u S of the public university's tuition fee while voters with intermediate values of θ, who go to the public university, favor a low value t sS S of the public university's tuition fee. Lemma (q P (2c + 1) -1) and it is t sS S when t P > 1 2 (q P (2c + 1) -1). When t P = 1 2 (q P (2c + 1) -1), there is a tie vote between the two. When there is a majority of votes, the public university selects the Condorcet winner. In the case of a tie vote, we suppose that the public university randomizes, choosing t u S with probability λ and t sS S with probability 1 -λ. From Lemmas 1 and 2, the voters' reaction function is discontinuous at t P = 1 2 (q P (2c + 1) -1) since t sS S < t sP S = t u S . This is illustrated in Figure 1 where it can be seen that there are three cases. The first occurs when the reaction functions intersect for a value of t P < 1 2 (q P (2c + 1) -1): the extremes, which respectively don't go to any university and go to the private university, are a majority and select a high tuition fee for the public university: this is the "high tuition fees" equilibrium. The second occurs when the reaction functions intersect for a value of t P > 1 2 (q P (2c + 1) -1): the middle class which opts for the public university is a majority and selects a low tuition fee for the public university: this is the "low tuition fees" equilibrium. The third case is the one when the reaction functions do not intersect so that no pure strategy equilibrium exists. In the latter case, we see below that a mixed strategy equilibrium exists where votes tie.

The high tuition fees equilibrium: When the ends defeat the middle

In this case, the extremes represent a majority and vote for t u S . Proposition 1 characterizes the equilibrium outcome.

t P 3 (t S )
Figure 1: The tuition fees equilibria

Proposition 1 For ∀ c ≤ 1 2
, there is a "high tuition fees equilibrium", with a majority of votes at the ends, such that both universities are open, iff m < 2c -1 4q P . At equilibrium the tuition fees are given by,

t * S = (q P -1)(1 + m) + c(2q P + 1) 4q P -1 > c , t * P = q P (2(1 + m)(q P -1) + 3c) 4q P -1 ,
Notice that, equivalently, at this equilibrium t * S = (q P -1) + 2q P c S + c P 4q P -1

At a high tuition fees equilibrium any increase in the unit costs of higher education result in a higher public tuition fee as well as in a higher private tuition fee. Interestingly,

∂t * S ∂c S = 2q P 4q P -1 ∈ ( 1 2 , 1),
i.e. an increase in the cost of education at the public university is only partially passed on to students.

From the above Proposition, the agents who don't train, together with the ones who study at the private university, form a majority when the marginal cost of providing the highest educational quality is low enough, i.e. when m is small, so that the private university is attractive enough. Indeed, a smaller value of m implies lower equilibrium tuition fees for both universities but the private university's tuition fee decreases more than the public university's one. It follows that more people choose to train at the private university and that less people choose not to train. However, the former effect dominates the latter, so that the number of people training at the public university is lower.

The low tuition fees equilibrium: When the middle defeats the ends

In this case, there is a majority of voters (the "middle class") who vote for t sS S . Proposition 2 provides the equilibrium outcome.

Proposition 2 For c < 2q P -1 4q P , there is a "low tuition fees equilibrium", with a majority in the middle, such that both universities are open, iff m ∈ (2c +

1 4q P , 1 -c 2q P -1
). At this equilibrium the tuition fees are given by

t * S = (q P -1)(m -1) + c(2q P + 1) 4q P -1 , t * P = 1 + q P (2(1 + m)(q P -1) + 3c -1) 4q P -1 ,
The condition c < 2q P -1 4q P is to ensure that the interval (2c

+ 1 4q P , 1 -c 2q P -1 ) is non-void.
Notice that, equivalently, at this equilibrium

t * S = 2q P c S + c P -(q P -1) 4q P -1 .
At a low tuition fees equilibrium any increase in the unit costs of higher education result in a higher public tuition fee as well as in a higher private tuition fee. Interestingly, as in a high tuition fee equilibrium,

∂t * S ∂c S = 2q P 4q P -1 ∈ ( 1 2 , 1),
i.e. here as well an increase in the cost of education at the public university is only partially passed on to students.

From the above Proposition, the agents who train at the public university form a majority when the marginal cost m of providing the highest educational quality is high enough, so that there are few people who train at the private university. Indeed, a higher value of m implies larger equilibrium tuition fees for both universities but the private university's tuition fee increases more than the public university's one. It follows that less people choose to train at the private university and that more people choose not to train. However, the former effect dominates the latter, so that the number of people training at the public university is larger.

The Proposition only deals with the case when there is positive demand for training at both universities. For values of m larger than 1-c 2q P -1 , it is possible to show that the public university is the only one to be open and a majority of voters choose to train there, voting accordingly for a low tuition fee. This would be lengthy and would not add very much to the intuition so it is left to the reader.

Mixed equilibria: When votes tie

When there is a tie vote the public university has been assumed to choose t u S with a probability λ and t sS S with a probability (1 -λ). Proposition 3 characterizes the equilibrium outcome in this case.

Proposition 3 For ∀c < 2q P -1 4q P , there are mixed equilibria in which both universities are open iff m ∈ [2c-1 4q P , 2c+ 1 4q P ] The private and public universities respectively select

t * P = 1 2
(q P (2c + 1) -1), with probability 1, and

t * S = c + q P -1 4q P , with probability λ * , t * S = c - q P -1 4q P , with probability 1 -λ * ,
where

λ * = 1 2 (1 + 4q P (2c -m)).

Corollary 4

The expected public tuition fee when votes tie equals

t * e S = 2q P c -(c + m(q P -1)) = 2q P c S -c P .
Interestingly, it is increasing in the unit cost of public education but decreasing in the unit cost of private education and, as far as it is positive12 , increases in the case of increases of both. This is because when the latter decreases the private university becomes more attractive, raising the probability of a high public tuition fee through a "tipping of majority" effect. The impact of an increase in c S on expected public tuition fees is all the more important as it not only directly raises the public tuition fee in all occurrences (direct cost effect) but also raises the probability of the occurrence of the high tuition fee (political effect). More precisely

∂t * e S ∂c S = 2q P > 2,
i.e. the resulting increase in the expected public tuition fee is more than twice any increase in the cost of public higher education.

In the following subsection we provide some comparative statistics relative to the three types of equilibria.

4 Optimal vs equilibrium public tuition fees Stiglitz (1994) main purpose was to determine whether majority voting leads to an underprovision or an overprovision of public education. In this paper we have focused on the determination of equilibrium public tuition fees which result from majority voting. A comparison with their optimal value is of interest.

Let us first determine the aggregate social surplus by summing the agents' utilities 13 as given by ( 5) for θ ∈ [0, t S ] , by [START_REF] Epple | Public provision of private goods[END_REF] for θ ∈ t S , t P -t S q P -1 and by [START_REF] Glomm | Opting out of publicly provided services: A majority voting result[END_REF] for θ ∈ t P -t S q P -1 , 1 . We obtain:

W = q 2 P -q P (1 + 2t P -2ct S + t 2 S ) + t P (2 -2c + t P ) 2(q P -1)
.

Let us now substitute for t P in the above equation its value for the private university's reaction function [START_REF] Desrochers | Trends in College Spending: 2003-2013[END_REF]. W is maximized for

t Opt S = q P (4c + m -1) -(m + c -1) 4q P -1 . (8)
t Opt S is increasing in c and m. It is increasing in q P if m > 1 and decreasing in q P in the opposite case.

To set the stage for a comparison between the optimal and equilibrium values of the public tuition fee, it is worthwhile to see how the latter changes with the value of m. This relationship is pictured in Figure 2 below, together with the optimal public tuition fee. The equilibrium public tuition fee 14 is an increasing function of m both low values of m (for which the "high tuition fees equilibrium" obtains) and for large ones (for which the "low tuition fees equilibrium obtains"). This follows from a strategic effect: an increase in m raises directly the private tuition fee and, given strategic complementary between the tuition fees, this results in a higher equilibrium value of the public tuition fee. For values in between, the expected public tuition fee is decreasing in m, simply because an increase of m raises the probability that the public university will choose the lowest tuition fee (t sS S ). This is a "tipping of majority effect". The range 2c -1 4q P , 2c + 1 4q P , over which one obtains a decreasing relationship, is itself the smaller the largest is the private educational quality q P .

It is now easy to compare the optimal value from ( 8) with the equilibrium ones from the three Propositions above. Straightforwardly the equilibrium public tuition fee exceeds the optimal one by c 2q P when the ends are a strict majority and falls short of it by 1-c 2q P when the middle gains a strict majority 15 . So the gap between equilibrium and optimal values is larger in the majority at the ends 13 Remember that the private university's profits are not included in the aggregate social surplus.

14 In the mixed equilibrium case, we consider the expected equilibrium public tuition fee. 15 Notice that the slopes of t * S and t Opt S with respect to m are identical (= q P -1 4q P -1 ) in the intervals 0, 2c -1 4q P and 2c + 1 4q P , 1 -

c 2q P -1 .
case when c < 1 2 and in the majority at the middle case when c > 1 2 . Moreover, there is a critical value of m, in the range for which a mixed equilibrium prevails, such that the expected equilibrium tuition fee equals the optimal one. This value obtains as:

m = 2c + 1 -2c 4q P .
It is is easy to check that m ∈ 2c -1 4q P , 2c + We can conclude that the (expected) equilibrium public tuition fee is greater that the optimum one when m < m and smaller in the opposite case. The tipping of majority which occurs when the marginal cost of educational quality at the private university decreases entails a switch from a too small to a too high public tuition fee.

Conclusion

In this paper, we have studied the joint determination of tuition fees at a public university, through majority voting, and a private university, through profit maximization. We have proved the existence of an equilibrium which is one of three types, according to the cost structure. In the first one, when the marginal cost per student m of the private university's superior quality is low, the "ends" are a strict majority and the equilibrium level of the public tuition fee exceeds its optimal level. In the second, when m is high, the "middle is a strict majority the equilibrium level of the public tuition fee falls short of its optimal level. In the third, for intermediate values of m, votes tie and the public university randomizes between a tuition fee above and a tuition fee below its optimal level.

Our paper may help to understand the process which has led to very large increases in public tuition fees in several countries. Martin's (2002) claim that they result from quasi-mechanical pass-on to students of increases in training costs is not confirmed by our results: under stable majorities, only a fraction of increases in public tuition expenditures is passed on to students, so that non-dramatic increases in training costs cannot result in dramatic increases in public tuition fees. On the contrary, moderate evolutions of the training costs16 may result into dramatic variations of the public tuition fee when they trigger majority reversals. More precisely, in the latter case, large increases in the public tuition fee may occur then as a consequence of moderate increases of the public training cost and/or the private training cost.

Appendix

Proof of Lemma 1. In order to prove that t sS S and t sP S = t u S are the only possible Condorcet winners we prove that all other values of t S are defeated by t sS S or t sP S = t u S , i.e. any other values of t S might be a majority. As explained above in Subsection 3.2 the agents' utility varies according to the value of θ. We distinguish two cases when θ ∈ [0, t P q P [ (Figures 3, 4 and5) and θ ∈] t P q P , 1] (Figures 6, 7 and8):

E T E ' t S U θ t sS S U s S U u 0 θ t sP S =
T 0 E t sS S t S U s S E ' E ' U u t u S = t sP S θ U θ Figure 4: Case (ii) E T t S U θ 0 U u t sS S E ' U s S t sP S = t u S d d d θ Figure 5: Case (iii) value of t S ∈ [t sS S , t sP S ]
which might be a Condorcet winner. We start first by comparing a ∈ [t sS S , t sP S ] to t sS S and determine conditions under which a is a majority.

• For the cases (i) and (iv) (Figures 3 and6), i.e. the utility function is single-peaked at t sP S , it is clear that all voters choose a. • For the cases (iii) and (vi) (Figures 5 and8), i.e. the utility function is single-peaked at t sS S , all voters choose t sS S . • For the cases (ii) and (v) (Figures 4 and7), i.e. the utility function has

E T t S U θ 0 t sS S E ' U s P U s S t P -θ(q P -1) t sP S = t u S Figure 6: Case (iv) T 0 E t S U s S E ' E ' U s P t P -θ(q P -1) t sS S U θ t sP S = t u S Figure 7: Case (v)
two peaks at t sS S and t sP S , we compare respectively U s S (t sS S ) to U u (a) and

E T t S 0 t sS S U s P U θ U s S E ' t P -θ(q P -1) t sP S = t u S
θ 1 (a) = - (1 -2c + (c -2a)
2 )q 2 P + (1 + t P ) 2 + 2q P t P (c -1 + (c -2a -1)) 4q P (q P -1) , θ 2 (a) = (4a(a -c) + (c -1) 2 + 4t P )q 2 P + (2(c -1) + 2t P (c -3 -2a))q P + (t P + 1) 2 4q P (q P -1) 2 .

Then in order to determine the choice of voters for each case we compare θ 1 (a) and θ 2 (a) to t P q P as follows:

θ 1 (a) - t P q P = - (1 -2c + (c -2a) 2 )q 2 P + (t P -1) 2 + 2q P (c -1 + (1 -2a + c)t P ) 4q P (q P -1) = P (a) 4q P (q P -1) θ 2 (a) - t P q P = (1 -2c + (c -2a) 2 )q 2 P + (t P -1) 2 + 2q P (c -1 + (1 -2a + c)t P ) 4q P (q P -1) = R(a) 4q P (q P -1) E T t S 0 U θ E ' t sS S t sP S θ 1 (a) E ' a U u U s S U s S (t sS S ) = U u (a) Figure 9: Case (ii): U s S (t sS S ) = U u (a) 20 E T t S 0 U θ E ' t sS S t sP S E ' a U s S U s P t P -θ 2 (a)(q P -1) U s S (t sS S ) = U s P (a) Figure 10: Case (v): U s S (t sS S ) = U s P (a)
Interestingly, P (a) and R(a) are symmetrical w.r.t the t S axis. P (a) reaches its maximum at t sP S and R(a) reaches its minimum at the same value of t S . Moreover P (a) and R(a) have the same determinant ∆ = 16q 2 P (q P -1)(q P (2c -1) + 1 -2t P ) which is negative if t P > tP = 1 2 (q P (2c -1) + 1) and positive otherwise. In the latter case, both polynomials have the same roots a * and a * * such as a * < a * * . Here we are only interested in a * as a * * > t sP S whereas t sS S < a * < t sP S if t P > tP = q P (c -1) + 1 and t sS S > a * otherwise. Hence, we summarize in the following Figures the three possible

t sS S E ' E ' a * * a * R(a) tP < t P < tP : ∆ > 0 and t sS S < a * E a T 0 P/R t sS S t sP S R (a)
(P (a)

E ' E ' t P > tP : ∆ < 0 • When t P < tP , θ 1 (a) > t P
q P and θ 2 (a) < t P q P , which implies that all voters in [0, 1] choose a.

• When tP < t P < tP , for values of a < a * , θ 1 (a) < t P q P and θ 2 (a) > t P q P thus voters in [0, θ 1 (a)[ and those in ]θ 2 (a), 1] choose a whereas those in ]θ 1 (a), θ 2 (a)[ choose t sS S . For values of a > a * , θ 1 (a) > t P q P and θ 2 (a) < t P q P thus all voters in [0, 1] choose a.

• When t P > tP , θ 1 (a) < t P q P and θ 2 (a) > t P q P thus voters in [0, θ 1 (a)[ and those in ]θ 2 (a), 1] choose a whereas those in ]θ 1 (a), θ 2 (a)[ choose t sS S .

After comparing t sS S , t sP S , t P -t sS S q P -1 and t P -t sP S q P -1 to t P q P we finally obtain:

• t P < tP thus t P q P < t sS S < t sP S and t P -t sP S q P -1 < t P -t sS S q P -1 < t P q P yielding 1 0

t P q P a a
Case (i) Case (iv)

• tP < t P < tP thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 , for values of a < a * we obtain • tP < t P < tP thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 , for values of a > a * we obtain 1 0

t P q P t P -t sS S q P -1 a a Case (ii) Case (i) Case (v) Case (iv) a a t sS S
• tP < t P < q P c thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t Now we compare a to t sP S in the same way.

• For the cases (i) and (iv) it is clear that all voters choose t sP S . • For the cases (iii) and (vi) it is clear that all voters choose a.

• For the cases (ii) and (v) we compare respectively U u (t sP S ) to U s S (a) and U s P (t sP S ) to U s S (a). As depicted in Figures 11 and12 we denote respectively θ 3 (a) and θ 4 (a) the values of θ at which U u (t sP S ) = U s S (a) and U s P (t sP S ) = U s S (a).

θ 3 (a) = (4a(a -c + 1) + c 2 )q 2 P + (2t P c -4a(t P + 1))q P + t 2 P ) 4q P (q P -1) , θ 4 (a) = -(4a(a -c + 1) -4t P + c 2 )q 2 P + (2t P (c + 2) -4a(t P + 1))q P + t 2 P ) 4q P (q P -1) 2 .

We compare θ 3 (a) and θ 4 (a) to t P q P , θ 3 (a) -t P q P = (4a(a -c + 1) + c 2 )q 2 P + (2t P (c -2(a + 1)) -4a)q P + 4t P (t P + 1) 4q P (q P -1) = S(a) 4q P (q P -1)

E T 0 t S U θ E ' t sP S U u U s S θ 3 (a) t sS S E ' a U s S (a) = U u (t sP S )
Figure 11: Case (ii):

U s S (a) = U s P (t sP S ) E T 0 t S U θ E ' t sP S U s S t sS S E ' a U s P θ 4 (a) U s P (t sP S ) = U s S (a)
Figure 12: Case (v): U s S (a) = U s P (t sP S ) and θ 4 (a) -t P q P = -(4a(a -c + 1) + c 2 )q 2 P + (2t P (c -2(a + 1)) -4a)q P + 4t P (t P + 1) 4q P (q P -1) = M (a) 4q P (q P -1) S(a) and M (a) are symmetrical w.r.t t S axis. S(a) is convex and reaches its minimum value at t sS S and symmetrically M (a) is concave and reaches its maximum value at t sS S . Their common determinant is given by -16q 2 P (q P -1)((2c -1)q P + 1 -2t P ) which is positive if t P > tp and negative otherwise. We denote by a + and a ++ the common roots (if they exist) such as a + < a ++ . We are only interested in a ++ as a + < t sS S whereas t sS S < a ++ < t sP S if tP < t P < q P c and a ++ > t sP S if t P > q P c. We summarize in the following figures the possible cases according to S/M t P > q P c: ∆ > 0 and a ++ > t sP S • When t P < tP , θ 3 (a) > t P q P and θ 4 (a) < t P q P , hence all voters in [0,1] choose t sP S .

• When tP < t P < q P c, for values of a < a ++ , θ 3 (a) < t P q P and θ 4 (a) > t P q P , hence all voters in [0, θ 3 (a)[ and ]θ 4 (a), 1] choose t sP S whereas those in ]θ 3 (a), θ 4 (a)[ vote for a.

For values of a > a ++ θ 3 (a) > t P q P and θ 4 (a) < t P q P , hence all voters in [0,1] choose t sP S .

• When t P > q P c, θ 3 (a) < t P q P and θ 4 (a) > t P q P , hence all voters in [0, θ 3 (a)[ and ]θ 4 (a), 1] choose t sP S whereas those in ]θ 3 (a), θ 4 (a)[ vote for a.

Comparing t sS S , t sP S , t P -t sS S q P -1 and t P -t sP S q P -1 to t P q P we finally obtain:

• t P < tP thus t P q P < t sS S < t sP S and t P -t sP S q P -1 < t P -t sS S q P -1 < t P q P yielding 1 0 • tP < t P < q P c thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 for values of a < a ++ we obtain • tP < t P < q P c thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 for values of a > a ++ we obtain 1 0 We conclude that a might be a Condorcet winner only when tP < t P < q P c for t sS S < a < a ++ and t P > q P c ∀a ∈ [t sS S , t sP S ]. For all other values of t P a is defeated by either t sS S or t sP S . Now we prove that even for these cases a cannot be a Condorcet winner. The idea is to prove that θ 1 + 1 -θ 2 , i.e. the number of voters for a against t sS S , and θ 4 -θ 3 , i.e. the number of voters for a against t sP S , are never a majority at the same time, thus a is defeated by either t sS S or t sP S .

t P q P Case (i) Case (
t P q P t P -t sS S q P -1 Case (ii) Case (i) Case (v) Case (
θ 1 -θ 2 + 1 2 = (4a(c -a) -c 2 + 2c + 1)q 2 P -(2(c + 1) + 2t P (c + 1 -2a))q P -t 2 P + 2t P + 1 4(q P -1) 2 = L(a) 4(q P -1) 2 θ 4 -θ 3 - 1 2 = (4a(c -a -1) -c 2 -
2)q 2 P + (4(a + 1)(t P + 1) -2t P c)q P -t 2 P -4t P -2 4(q P -1) 2 = H(a) 4(q P -1) 2

On the one hand the polynomial L is concave and reaches its maximum value at t sP S , his determinant is given by ∆ L = (q P -1)(q P (2c + 1) -1 -2t P ) which is positive for values of t P < 1 2 (q P (2c + 1) -1) and negative otherwise. On the other hand the polynomial H is concave and reaches its maximum value at t sS S , his determinant is given by ∆ H = -(q P -1)(q P (2c + 1) -1 -2t P ) which is negative for values of t P < 1 2 (q P (2c + 1) -1) and positive otherwise. Hence as depicted below, when t P < 1 2 (q P (2c + 1) -1), θ 4 -θ 3 < 1 2 , i.e. a is defeated by t sP S . Otherwise when t P > 1 2 (q P (2c + 1) -1), ) for the first segment then for the second to obtain finally the choice of voters in all the interval [0, 1].

θ 1 + 1 -θ 2 < 1 2 , i
• It is clear that for cases (i) and (iv) all voters choose t sP S as the utility function is single-peaked at t sP S .

• Similarly for cases (iii) and (vi) all voters choose t sS S as U θ is single-peaked at t sS S .

• For the cases (iii) and (v) we compare respectively U s S (t t P -θ 6 (q P -1)

Figure 14: Case (v): U s S (t sS S ) = U s P (t sP S )

θ 5 = q P (2c -1) + 1 + 2t P 4q P , θ 6 = 2t P (2q P -1) -q P (2c -1) + q P -1 4q P (q P -1) .

Comparing θ 5 and θ 6 to t P q P gives:

iff t P < tP , θ 5 > t P q P and θ 6 < t P q P thus all agents choose t sP S . iff t P > tP , θ 5 < t P q P and θ 6 > t P q P thus agents in [0, θ 5 [ and ]θ 6 , 1] vote for t sP S and those in [θ 5 , θ 6 ] choose t sS S .

After comparing t sS S , t sP S , t P -t sS S q P -1 and t P -t sP S q P -1 to t P q P we finally obtain:

• t P < tP thus t P q P < t sS S < t sP S and t P -t sP S q P -1 < t P -t sS S q P -1 < t P q P yielding 1 0 t P q P Case (i) Case (iv)

t sP S t sP S
• tP < t P < tP thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 , which yields 1 0 t P q P t P -t sS S q P -1

Case (ii) Case (i)

Case (v) Case (iv) • tP < t P < q P c thus t sS S < t P q P < t sP S and t P -t sP S q P -1 < t P q P < t P -t sS S q P -1 yielding 1 0 t sS S t P q P t P -t sS S q P -1 The choice of voters is summarized as follows:

t
• iff t P < tP all voters in [0, 1] choose t sP S , • iff t P > tP the number of voters who choose t sP S is given by: V u/sP (t P ) = θ 1 + 1 -θ 2 = -3 + (3 + 2c)q P -2t P 4(q P -1) .

V u/sP is a majority when t P < 1 2 (q P (2c + 1) -1), thus t sP S is a majority for all values of t P < 1 2 (q P (2c + 1) -1) as 1 2 (q P (2c + 1) -1) > tP . Hence the best response function of voters is given by: t S (t P ) = t sP S = q P c+t P 2q P , iff t P < 1 2 (q P (2c + 1) -1); t sS S = q P (c-1)+t P +1 2q P , iff t P > 1 2 (q P (2c + 1) -1). [START_REF] Stiglitz | The demand for education in public and private school systems[END_REF] From ( 10) and (4) we distinguish three types of equilibria as depicted in Figure 1; two equilibria in pure strategies given by the intersections of t sP S (t P ) and t P 1 , i.e t sP S is a majority, and t sS S (t P ) and t P 3 , i.e. t sS S is a majority. The third equilibrium is in mixed strategies where t P 2 intersects neither t sP S (t P ) or t sS S (t P ), i.e. there is a tie vote. Proof of Proposition 1. From ( 10) and (4), when there is a majority voting for t sP S , at equilibrium tuition fees are obtained as, t * S = (q P -1)(1+m)+c(2q P +1) 4q P -1 , t * P = q P (2(1+m)(q P -1)+3c) We check under what condition 1 -c 2q P -1 > 2c + 1 4q P : this inequality holds whenever c < 2q P -1 4q P . It is easy to see that this inequality holds always true if c < 1 3 .

Proof of Proposition 3.

There is a tie vote between the extremes and the middle class when V u/sP = 1 2 yielding t P crit = 1 2 ((1 + 2c)q P -1). Hence at equilibrium, 1 2 (c + (m + 1)(q P -1) + λt sP S + (1 -λ)t sS S ) = t crit P which gives λ * = 1 2 (1+4q P (2c-m)). There is equilibrium only if 0 < λ * < 1 which means, only for values of 2c -1 4q P < m < 2c + 1 4q P . We substitute the value of t crit P respectively in t sP S and t sS S given in [START_REF] Stiglitz | The demand for education in public and private school systems[END_REF] to obtain the equilibrium values of t * S respectively with probability λ * and 1 -λ * .

Figure 8 :

 8 Figure 8: Case (vi)

  cases according to the value of t P . t P < tP : ∆ > 0 and a * < t sS S

  tP < t P < q P c: ∆ > 0 and a ++ < t sP S

4q P - 1 ,•

 1 Then we prove that for low enough values of c, there always exists a nonvoid interval of values of m, [2c + 1 4q P , 1 -c 2q P -1 ], for which the private university is open. P -1) + c + 1 -2q P 4q P -1 , D * P ≥ 0 iff m ≤ 1 -c 2q P -1 .

  1 nevertheless shows that, for every value of t P , there is a Condorcet winner which is either t sP S = t u S or t sS S . A straightforward corollary is that one only needs to compare t sP S = t u S and t sS S to determine which of the two is the Condorcet winner.Lemma 1 For a given t P , there is a Condorcet winner which is eitherThe proof of Lemma 1 is very long but it makes a very simple point. Suppose that some a = t sP S and = t sS S is a Condorcet winner, i.e. that there is a majority for a versus t sP S and a majority for a versus t sS S . We show in the Proof that, if there is a majority for a versus t sP S , there is an even larger majority for t sS S versus t sP S . We also show that if there is a majority for a versus t sS S , there is an even larger majority for t sP S versus t sS S . Accordingly, assuming that some a = t sP S and = t sS S is a Condorcet winner leads to a contradiction. Now, the same type of argument shows that if there is a majority for t sP For a given t P , the Condorcet winner is t sP

		t sP S = t u S =	q P c + t P 2q P	,
	or	t sS S =	q P (c -1) + t P + 1 2q P	.
	S S ) is the Condorcet winner. S (resp. t sS S ), then t sP S (resp.t sP S ) versus t sS (resp. t sS For suppose this not true and there is some a which defeats t sP S (resp. t sS S ): by the above argument t sS S (resp.t sP S ) would defeat t sP S (resp. t sS S ), a contradiction. Accordingly, we only need to compare t sS S and t sP S .
	Lemma 2

S

= t u S when t P < 1 2

  • t P > q P c thus t sS S < t sP S < t P q P and t P q P <

		t sP S		t sP S	t sS S	t sS S		t sP S		t sP S
					θ 5		θ 6
	Case (i)			Case (ii)	Case (v)	Case (iv)
						t P -t sP S q P -1 <	t P -t sS S q P -1 yielding
	t sP S	t sP S		t sS S	t sS S	t sS S		t sS S		t sP S	t sP S
	0	t sS S	θ 5	t sP S	t P q P	t P -t sS S q P -1	θ 6	t P -t sP S q P -1	1
	Case (i)	Case (ii)	Case (iii)	Case (vi)	Case (v) Case (iv)

Notice that, following devolution, tuition fees have been abolished in Scotland.

He also pointed to the "reduced public subsidies for higher education", i.e. a political factor.

This was pointed out byEpple and Romano (1996a). The use of single-crossing conditions in voting models go back to Roberts (1977).

through a proportional tax rate increase.

An illuminating proof is inEpple and Romano (1996a, p.308). The intuitive argument is (i) SDI entails that agents' marginal willingness to pay for public education decreases with income, absent any private alternative and (ii) the possibility of switching to private education reduces even more richer people benefits from public provision.

We may consider that a typical "agent" is a family in which the parents are purely altruistic in the sense that when voting they fully share the interest of their children.

Subsidy if negative.

More precisely the important assumption is that the amount of the tax paid by the agent (family) must be independent of the student's type. The amount of the tax may be dependent on exogenous initial family income y provided, there is no correlation between initial family income y and the student's type θ.

France, for instance, has no really important private university.

The former because they don't study at any university, the latter because they go to the private university.

A sufficient positiveness condition is c > 1 4 (1 -1 q P).

The cost incidence of the "increasing time given to professors for scholarship and other activities" seems to have been mitigated by the increased use of nonpermanent, part-time teachers (see[START_REF] Desrochers | Trends in College Spending: 2003-2013[END_REF] for the US case).

which yields the number of voters V u/sP * = 4q P (2(c+1)-m)-3 4(4q P -1)

. V u/sP * > 1 2 implies q P (2c -m) -1 4 > 0. Hence, there is a majority "ends against the middle" only for values of m < 2c -1 4q P , equivalently q P > 1 4(2c-m) . We show that c ≤ 1 2 is sufficient to ensure that both demands are positive, i.e. both universities are open.

• We first prove that for values of c ≤ 1 2 , D * P ≥ 0,

It is easy to see that

which is negative when c ≤ 1 2 , thus for all values of q P > 1 4(2c-m) , 2q P -c 2q P -1 > 2c which implies 2q P (1 -2c) + c ≥ 0. QED.

• Then we prove that when c ≤ 1 2 , D * S ≥ 0,

It is easy to see that D * s ≥ 0 iff m ≥ 2c -1. This condition holds always true since m > 2c -1 4q P > 2c -1.

Proof of Proposition 2. From ( 10) and ( 4) when there is a majority voting for t sS S , at equilibrium tuition fees are obtained as, t * S = (q P -1)(m-1)+c(2q P +1) 4q P -1 , t * P = 1+q P (2(1+m)(q P -1)+3c-1) 4q P -1 , which leads to the number of voters V u/sP * = -1+4q P (2(1+c)-m) 4(4q P -1) . V u/sP * < 1 2 implies q P (2c-m)+ 1 4 < 0. Hence there is a majority "middle against the ends" for values of m > 2c + 1 4q P , equivalently q P > 1 4(m-2c) . We determine conditions under which at equilibrium both universities are open.

• We first prove that D S is always positive,

t * P -t * S q P -1 -t * S = mq P + 3q P -2cq P -1 4q P -1 .

D S ≥ 0 iff m > -3 + 2c + 1 q P which is always true.


Albert Banal-Estañol

Jeremy Eckhause

Olivier Massol
email: olivier.massol@ifpen.fr

Incentives for early adoption of carbon capture technology: further considerations from a European perspective

Keywords: Tax incentives, Carbon capture, Learning effects, Levelized cost

This note details two comments on a recent policy proposal in Comello

Introduction

The prohibitively high cost of CC technology for first-of-a-kind plants is recurrently cited as a major barrier to its large-scale deployment. To overcome this problem, [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF] recently articulate an innovative policy proposal to enable substantial cost reductions by leveraging the sizeable deployment of thermal-based power generation projected in the U.S. during the period 2017-2027. The proposal combines two ingredients: a binding and inflexible emission standard; and the "Accelerated Carbon Capture Deployment" (ACCD) -a preannounced schedule of Investment Tax Credits (ITC) and Production Tax Credits (PTC) -aimed at providing an incentive for newly built power plants in the U.S. to adopt CC immediately. This brief note extends the analysis by considering two issues. In a first section, we apply the framework detailed in the original article1 to generate a schedule of tax-credits that is robust to alternative scenarios for CC deployments outside the U.S. In a second section, we reflect on the possible emergence of a coordination game capable of hampering the desired early deployment of that technology and propose a modified schedule of tax-credits that is sufficient to overcome that problem.

-The role of early CC deployments outside the U.S.

Using a list of proposed but still undecided projects [START_REF] Gccsi | Global CCS Institute Status of CCS Project Database[END_REF], the authors assume the installation of nearly 3 GW of foreign CC capabilities between 2014 and 2020. However, in Europe, the funding of large CC projects has recently proven to be difficult causing delays and several project cancelations [START_REF] Lupion | NER300: Lessons learnt in attempting to secure CCS projects in Europe[END_REF]. As early foreign projects are posited to engender international spillovers, one may wonder whether these withdrawals could undermine the proposal's success.

To render the proposal robust to the vicissitudes impacting foreign projects, we consider a 'worstcase' scenario whereby foreign deployments are restricted to the unique Canadian 130MW power plant finalized in 2014. To compensate for the absence of foreign early investments, augmented ITC and PTC schedules are needed (cf., Figure 1) but our evaluations confirm that this robust version is almost as attractive as the initial version:2

• The Levelized Cost of Electricity (LCOE) obtained with a facility that becomes operational by the end of 2027 is approximately 7.9 ¢/kWh if CC technology is consistently adopted by all the newly built U.S. thermal power plants. 3• The magnitude of the tax-credit levels remain politically acceptable (cf., Figure 1).

• Overall, the cumulated (undiscounted) foregone tax revenue to the U.S. Treasury reaches about $8.2 billion. 4 This robust schedule of incentives thus represents a cost-effective solution for achieving a large scale deployment of this innovative technology.

Figure 1. The modified ACCD tax credits schedule under a robust scenario [ INSERT FIGURE 1 ABOUT HERE ]

-Strategic interactions among CC adopters

Recent European literature on CC and storage has highlighted the interactions that exist among CC adopters connected to a common infrastructure system [START_REF] Mendelevitch | The role of CO2-EOR for the development of a CCTS infrastructure in the North Sea Region: A techno-economic model and applications[END_REF][START_REF] Massol | Joining the CCS Club! The economics of CO 2 pipeline projects[END_REF]. In the present paper, infrastructure issues are neglected but the use of an experience curve de facto generates some interactions. It is instructive to examine these interactions further.

Our discussion is structured as follows. A first subsection reviews the evaluation of the schedule of tax-credits used in the original article. The second subsection focuses on CC adoption in a given year by several players and proves that the incentives offered that year may not be sufficient to rule out the possible occurrence of a coordination game with multiple equilibria. As uniqueness is not achieved, these players may select an equilibrium where some emitters rationally prefer to delay CC adoption.

As that phenomenon may jeopardize the desired policy outcomes, the third subsection identifies an appropriately augmented minimum level of incentives that is sufficient to overcome that issue. The last subsection reports the numerical results obtained using the associated ITC and PTC schedules. For the sake of brevity, all the formal proofs are provided in a Supporting Document.

A -The subsidy scheme in Comello and Reichelstein (2014)

We consider a given year t in { } 2017,..., 2027 and let: t K denote the total planned capacity of all the power plants to be installed during that year; and t CK denote the state variable that gives the cumulated CC capacity already installed during the preceding years. 5For an investor that considers installing a power plant during that year, we let: In year t , we do not model the tax-credits but simply assume that their effect is to lower the LCOE measured on a power plant that early adopts CC capabilities. We let t S be the levelized subsidy and ( ) ( )

: N N t t t c x c x S = - ɶ denote the subsidized LCOE function.
Recall that the ACCD tax-credits are set so that, for a facility to be installed in a given year, it becomes advantageous to adopt CC capabilities immediately compared to retrofitting that plant by the end of 2027. The evaluation of the schedule of tax-credits presented in the original paper is detailed in an associated spreadsheet model: the "NGCC + CC Calculator" [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF]. In this model, CC adoption at the maximum level is assumed in each year t so that the cumulated CC capacity available at the beginning of the next year is ( )

: N R t t t t t S c CK K c = + -, (1)
The authors underline that, by construction, the tax-credits prevent possible 'deviation' from what they name an 'equilibrium path' of early CC adoption. Indeed, if one models CC adoption as a one-shot game involving 11 players (one for each year) where each player faces a binary choice with respect to early CC adoption for the entire capacity t K , the proposed subsidy makes 'early adoption' the best response of every player when the other players also jointly decide 'early adoption'. In this game, the proposed schedule of tax-credits thus makes the decision vector stating 'early adoption' for every player a pure strategy Nash Equilibrium (NE).

B -Is that proposed subsidy sufficient?

In reality, the problem at hand involves a sequence of annual decisions and the cumulated impact of installed CC capacity on future LCOE figures generates a 'path dependency' to past CC adoption decisions. The capacity forecasts and the standard plant size used by the authors together suggest that

CK K τ τ = + ∑
is the cumulated CC capacity in operation at that time.

several power stations will be installed in some years (particularly during the period 2023 -2027). As these plants are likely to be owned by independent companies, one may wonder whether, in each year t , the threshold level t S is sufficient to induce the joint early adoption of CC capability by all the plants to be installed that year.

To address this issue, one has to examine the strategic interactions among the plants to be installed in a given year. To clarify the presentation, we consider a simplified case and assume that two identical plants of size 2 t K are installed in year t . These plants are controlled by two independent players labeled '1' and '2' and we posit that each player faces a binary choice with respect to the immediate installation of CC capabilities. We also assume that early CC adoption has already been decided by all predecessors so that the cumulated CC capacity available at the beginning of the year is

1 2017 2017 t t CK CK K τ τ - = = + ∑ . The 'unsubsidized' game
The objective of each player is to minimize its LCOE. Absent any tax-credits, their strategic interactions can be summarized with the 'unsubsidized' normal form game in Table 1.

Table 1. The 'unsubsidized' normal form game [ INSERT TABLE 1 HERE ]

Following the authors, we posit that it pays to delay CC adoption when the rival adopts the CC technology early:

( )

R N t t t t c c CK K < + . ( 2
)
As N t c is assumed to be a strictly decreasing function, it also pays to delay CC adoption when the rival does not early adopt the capture technology. 8 Hence, the condition (2) indicates that 'late adoption' is a strictly dominant strategy for a player and therefore the pure strategy where both players decide 'late adoption' is the unique NE. Hence, absent any tax-credits, the two players rationally prefer to delay CC adoption.

Emergence of a coordination game 8 Because ( )

2 R N N t t t t t t t K c c CK K c CK   < + < +     .
We now examine the case where tax-credits are implemented and consider the levelized subsidy level, t S . For a player that early adopts CC capabilities, the payoff is now given by the subsidized LCOE.

The 'subsidized' normal form game is thus derived from the 'unsubsidized' one by using the subsidized LCOE function N t c ɶ in lieu of the unsubsidized one. Two results can be highlighted for the 'subsidized' normal form game.

Finding 1 -For any levelized subsidy that verifies t t S S ≥ , the strategy vector stating 'early adoption' for each player is a NE of the subsidized game.

Finding 2 -Implementing a levelized subsidy t S that verifies t t

S S ≥

may not be large enough for the strategy ('early adoption', 'early adoption') to be the unique NE. In particular, a subsidy level

t t S S =
is such that ('late adoption', 'late adoption') remains a NE.

Together, these two findings suggest that implementing a levelized subsidy that solely verifies t t

S S ≥

could lead to a coordination game with possibly several NEs.

However, the existence of a NE other than the one that provides early CC adoption at level t K in year t is a source of concern. The following finding proves that a lower-than-expected level of CC adoption in year t may undermine the ability of the proposed schedule of levelized subsidies to also achieve the early adoption of CC capabilities in subsequent years.

Finding 3 -Possible existence of a snowball effect: If delayed adoption were to be decided by some players in year t , a levelized subsidy 1 t S + that solely verifies the condition 1 1 t t S S + + ≥ can be too small to make 'early adoption' of CC capabilities the best response of an investor in year 1 t + .

C -A remedy

Because of this possible snowball effect, one may desire that the schedule of levelized subsidies rules out any possibility for the investors in any given year t to pick up a NE that does not lead to generalized early CC adoption. As there can be between-year variations in the number of players and in the sizes of their plants, we believe that it is preferable to opt for a threshold level that is large enough to be independent from these considerations. The following proposition offers a sufficient condition for generalized CC adoption to occur.

Proposition -

D -Application

This subsection reports the results obtained using the minimum threshold above. The numerical evaluations have been conducted assuming the robust capacity deployment scenario discussed in Section 1.9

Compared to the values in the original article, our evaluations indicate that the magnitude of the ITC levels remains similar; however, substantially augmented PTC are needed (cf., Figure 2).

Figure 2. The tax credit schedule needed to obtain a unique NE [ INSERT FIGURE 2 ABOUT HERE ]

These new levels generate a substantial increase in the cumulative (undiscounted) foregone tax revenue to the U.S. Treasury: about $14.1 billion. This is a 110% increase over the figure obtained in the original article. Nevertheless, we believe that this cost figure remains tolerable for such an ambitious policy, especially one that would now be rendered robust to both foreign adverse events and domestic gaming issues.

Conclusions

This note discusses the feasibility of the policy proposal in [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF]. Two lines of arguments have been considered. First, we have examined the effects of early CC deployments outside the U.S. Second, we have determined that the initially proposed ACCD schedule can be insufficient to engender the desired generalized early adoption of CC capabilities because of the possible co-existence of multiple Nash equilibria. In both cases, a modified version of the policy has been proposed using the detailed cost structure developed in [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF]. Though higher incentive levels have been obtained, our findings confirm that the cost of the proposed ACCD policy to the U.S. Treasury is not out of reach. This modified policy thus represents an interesting instrument to break the 'vicious circle' that currently hampers the deployment of CC technologies.

A.2 Methodology

We proceed as in [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF] and identify the minimum incentives required to bridge the gap between: (i) the Levelized Cost of Electricity (LCOE) of a facility that is retrofitted in 2027, and (ii) the LCOE of a plant that immediately adopts CC capabilities. For each year, the minimum investment tax credit (respectively production tax credits) is calibrated so that the levelized capacity (respectively variable) cost in case of immediate CC adoption 1 is equal to those incurred in case of a retrofit in 2027. 2

A.3 Results

In Table A.2, we detail the "Accelerated Carbon Capture Deployment" (ACCD) tax credits obtained in the robust scenario. These results have been generated using an adapted version of the original NGCC+CC Calculator. 3

1 For the sake of comparability, this levelized cost is determined using the fiscal depreciation schedules and the 10% first-ofa-kind premium defined in [START_REF] Comello | Incentives for early adoption of carbon capture technology[END_REF].

2 Note that these minimum tax credits schedule are not necessarily tiered. The use of different fiscal depreciation schedules between two successive years can generate a situation whereby the investment tax credit needed in a given year is larger than that needed during the previous year. To overcome that issue, we proceed as in the NGCC+CC Calculator and generate tiered schedules. In this paper, a linear programming approach is implemented to obtain these tiered schedules. The linear program is aimed at determining the tiered tax credit schedule that minimizes the cumulative foregone tax revenue to the U.S. Treasury and verifies two types of constraints: (i) the tiered tax credit in a given year must not be lower than that of the subsequent year, and (ii) the tiered tax credits must not be lower than the minimum values required for the LCOE of a new plant to be lower than the LCOE of a plant retrofitted in 2027. Note: MACRS indicates that all capital expenditures will be eligible for the five-year accelerated depreciation schedule according to MACRS; 150% DB indicates that all capital expenditures will be subject to the current 150% balance, 20-year depreciation schedule.

Appendix B -Mathematical proofs

This Appendix presents the formal proofs of the results stated in Section 2. Unless otherwise specified, the notation is based on the one introduced in the paper.

Finding 1 -For any levelized subsidy that verifies t t

S S

≥ , the strategy vector stating 'early adoption' for each player is a Nash equilibrium of the subsidized game.

Proof: Suppose both players decide 'early adoption'. Each player incurs the subsidized LCOE:

( ) ( ) N N t t t t t t t c CK K c CK K S + = + - ɶ . As t t S S ≥ , the condition ( ) N R t t t t c CK K c + ≤ ɶ
is de facto verified which indicates that 'early adoption' is the best response of a player whenever its rival also picks that strategy.

Q.E.D.

Finding 2 -Implementing a levelized subsidy t S that verifies t t

S S ≥

may not be large enough for the strategy ('early adoption', 'early adoption') to be the unique Nash equilibrium. In particular, a subsidy level t t

S S

= is such that ('late adoption', 'late adoption') remains a Nash equilibrium.

Proof: We have to check the conditions for the vector ('late adoption', 'late adoption') to no longer be a Nash equilibrium. We examine the best response of a player when the rival decides 'late adoption'. The condition for that player to systematically prefer 'early adoption' when the rival decides 'late adoption' is Note: MACRS indicates that all capital expenditures will be eligible for the five-year accelerated depreciation schedule according to MACRS; 150% DB indicates that all capital expenditures will be subject to the current 150% balance, 20-year depreciation schedule.

2 t t N R t t K CK c c +   <     ɶ
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Figure 1 .

 1 Figure 1. The modified ACCD tax credits schedule under a robust scenario (a) Modified ITC schedule (b) Modified PTC schedule

Figure 2 .

 2 Figure 2. The tax credit schedule needed to obtain a unique NE (a) Modified ITC schedule (b) Modified PTC schedule

  In each year t , any tax-credits yielding a levelized subsidy t S that verifies

	capacity available at the beginning of year t , systematically induces the installation of a CC
	capacity t K .
	t ≥ , with S S t	S	t	( ) t c CK : N t =	-	R t c	where	: CK CK t =	2017	+ ∑	t τ	1 2017 -=	K τ	is the cumulated CC

Table 1 . The 'unsubsidized' normal form game

 1

					Player 2
			Late adoption	Early adoption
	Player 1	Late adoption Early adoption	R t c ; R t c 2 t t K c CK + N t   	  	; R t c	R t c ; t c CK K N t c CK + t    ( ) N t t + ; ( N t c CK K 2 t K    t +	t	)
	Note: In this paper, the objective of each player is to minimize its LCOE.

Table A .1. Projected U.S. Capacity Deployments of NGCC Facilities and International Capacity Deployments of Carbon Capture Technology in the Robust Scenario

 A

	Year	Projected Annual Int'l Additions of Capture Unit Capacity (GCCSI) [MW]	Projected Annual Capacity Additions 2017-2027 (EIA) [MW]	Total Annual Capacity Additions [MW]	Total Cumulative Capacity Additions [MW]	Projected Net NGCC Capacity 2017-2027 (EIA) [MW]
	2014	130		130	130	189 939
	2015	0		0	130	192 122
	2016	0		0	130	192 447
	2017	0	0	0	130	192 447
	2018	0	0	0	130	192 220
	2019	0	690	690	820	192 910
	2020	0	432	432	1 252	193 342
	2021		272	272	1 524	193 615
	2022		450	450	1 974	194 064
	2023		5350	5 350	7 324	199 414
	2024		2389	2 389	9 712	201 803
	2025		5907	5 907	15 619	207 710
	2026		4307	4 307	19 926	212 017
	2027		6828	6 828	26 754	218 845

Table A .2. The ACCD tax credits in the Robust Scenario

 A

	Year	Fiscal Depreciation Schedule	Schedule of Investment Tax Credits	Schedule of Production Tax Credits [¢/kWh]
	2017	MACRS	22.4%	2.41
	2018	MACRS	20.9%	2.41
	2019	MACRS	19.1%	2.41
	2020	MACRS	17.4%	2.41
	2021	MACRS	15.2%	2.41
	2022	MACRS	13.1%	2.41
	2023	MACRS	10.0%	0.94
	2024	150% DB	10.0%	0.84
	2025	150% DB	4.4%	0.48
	2026	150% DB	0.2%	0.21
	2027	150% DB	0.0%	0.00

3

This modified spreadsheet model named "NGCC+CC Cost Calculator_Section 1.xlsx" can be downloaded from Olivier Massol's webpage: https://sites.google.com/site/oliviermassolshomepage/working-papers

  In each year t , any tax credits yielding a levelized subsidy t S that verifies

				subsidy that verifies the condition:	t S S >	t	where	S	t	: =	N t c	  	CK	t	+	2 t K	  	-	R t c	. As N t c is a
				strictly decreasing function, the threshold t S verifies t S S > . t	Q.E.D.
	Finding 3 -Possible existence of a snowball effect: If delayed adoption were to be decided by some
	players in year t , a levelized subsidy	1 S + that solely verifies the condition t	S	t	1 + ≥	1 S + t	can be too
	small to make 'early adoption' of CC capabilities the best response of an investor in year	1 t + .
				Proof: Because of non-adoption, we assume that the CC capacity constructed in year t attains
				a level t k with t k K < . If the levelized subsidy scheduled during the next year is set such that t
				S	t	1 + =	1 S + t	, the subsidized LCOE obtained in case of adoption verifies
				c ɶ	1 + N t	(	t CK k k t + +	t	+	1	)	>	1 + R t c	for any level of CC adoption	k	t	1 +	∈	[	0,	K	t	1 +	]	in year	1 t + (because
				1 c + is a strictly decreasing function). N t	Q.E.D.
	Proposition -t t S S ≥ ,
	with	S	t	( ) t c CK : N t =	R t c -where	: CK CK t =	2017	+ ∑	t τ	1 2017 -=	K τ	is the cumulated CC capacity available
	at the beginning of year t , systematically induces the installation of a CC capacity t K .
				Proof: We assume that a player has to build a unique power plant of capacity	t x	∈	[ ] 0, t K	in
				year t and let		y	t	∈	[	0,	t K x t -	]	be the aggregate amount of CC capacities decided by its
				rivals. If the levelized subsidy t S is such that	t ≥ , the LCOE obtained in case of early CC S S t
				adoption systematically verifies	c ɶ	N t	(	t CK x t + +	t y	)	≤	( c CK x y N t t t t + +	)	-	( ) t c CK N t	+	R t c	. As
				N t c is a decreasing function and	t CK x y CK t t + + ≥	t	, we have	c ɶ	N t	(	t CK x + + t	y	t	)	≤	R t c	for
				any t y in [	0, t t K x -		]	which indicates that this player's best response is systematically to
				implement CC immediately. Applying that reasoning to any other player in year t indicates
				that early CC adoption is a dominant strategy for that player which indicates that the
				generalized early adoption of CC capabilities by all the investors is the unique Nash
				equilibrium.													Q.E.D.
																								which calls for a levelized
																								4

Table C .A. The ACCD tax credits in the Robust Scenario

 C

	Year	Fiscal Depreciation Schedule	Schedule of Investment Tax Credits	Schedule of Production Tax Credits [¢/kWh]
	2017	MACRS	22.4%	2.78
	2018	MACRS	20.9%	2.78
	2019	MACRS	19.1%	2.78
	2020	MACRS	17.4%	2.78
	2021	MACRS	15.2%	2.78
	2022	MACRS	13.1%	2.78
	2023	MACRS	11.2%	2.78
	2024	150% DB	11.2%	1.11
	2025	150% DB	6.7%	1.05
	2026	150% DB	1.4%	0.62
	2027	150% DB	0.0%	0.00

The two authors must be praised for having made their data and spreadsheet model readily available to readers.

For the sake of brevity, this note solely summarizes our main conclusions. Further details on the methods used to generate the results are provided in a Supporting Document to be disseminated as a companion file to this paper.

This figure remains close to the 7.8 ¢/kWh obtained in the original article(Comello and Reichelstein, 2014 -Finding 3).

This 25% increase over the base-case scenario reveals the positive externality provided by foreign early investments in firstof-a-kind CC plants.

 5 Hence, t CK is the sum of the capacities of all the CC plants that are already in operations when year t begins.

As all power plants installed between 2017 and 2027 are forced to adopt CC by the end of that year, this LCOE figure is systematically evaluated assuming that 2027 2017 2017

As discussed in the original paper, this LCOE figure takes into consideration a 10% first-of-a-kind premium if the cumulated capacity x is lower than the first 3 GW.

For the sake of brevity, this note solely summarizes our main conclusions. Further details on the methods used to generate the results are provided in a Supporting Document to be disseminated as a companion file to this paper.

This modified spreadsheet model named "NGCC+CC Cost Calculator_Section 2.xlsx" can be downloaded from Olivier Massol's webpage: https://sites.google.com/site/oliviermassolshomepage/working-papers

Acknowledgements

The views expressed here and any remaining errors are our sole responsibility.

Supporting Document:

Incentives for early adoption of carbon capture technology: Further considerations from a European perspective

This document constitutes a technical appendix to the aforementioned paper.

• Appendix A (respectively Appendix C) clarifies the assumptions and methodology used to generate the numerical results commented in Section 1 (respectively Section 2) of the paper.

• Appendix B presents the mathematical proofs of the Finding 1, 2 and 3 and the Proposition

A.1 Assumptions

Table A.1 details the projected annual capacity deployments by year retained in the 'worst case' scenario. This conservative projection has been derived from Comello and Reichelstein (2014, Supporting document -Table A.1.) by restraining the amount of international capacity to the unique 130 MW Carbon Capture (CC) power plant that is currently in operation in Canada. Capacity deployment excepted, our assumptions are the ones discussed in the original article.

Appendix C -A robust schedule of ITC and PTC that is immune to strategic gaming considerations

This Appendix details the assumptions and results commented in Section 2.

C.1 Assumptions and methodology

The simulation is based on the projected capacity deployments retained in Appendix A (Table A.1).

We follow the methodology in Appendix A.2 except that the tax credits are now calibrated so as to provide a levelized subsidy that is at least as large as the threshold level mentioned in the Proposition in Section 2. These tax credits thus prevent the existence of Nash equilibriums where some emitters could rationally prefer to delay CC adoption.

C.2 Results

In Table C.1, we detail the "Accelerated Carbon Capture Deployment" (ACCD) tax credits obtained in the robust scenario when the incentives are derived from the levelized subsidy discussed in the Proposition in Section 2. These results have been generated using an adapted version of the original NGCC+CC Calculator.