Antoine Bonleu
email: antoine.bonleu@hotmail.fr.

Gilbert Cette
email: gilbert.cette@banque-france.fr

Guillaume Horny
email: guillaume.horny@banque-france.fr.

Document de Travail n°2011-47 Capital Utilisation and Retirement

Keywords: Capital, Capital measure, Capital retirement, Capital utilisation JEL codes: E22, D24, O16

This empirical analysis aims at assessing the effect of the economic climate and the intensity of capital utilisation on companies' capital retirement behaviour. It is conducted using individual company data, as well as original data on the degree of utilisation of production factors. The sample includes 6,998 observations over the period 1996-2008. This database is, to our knowledge, unique for the empirical analysis of the intensity of capital utilisation on firms' capital retirement behaviour. We adjust for endogeneity biases by means of instrumental variables.

, the contribution of these results is to show, through the use of unique survey data, that the effect of the intensity of capital utilisation on capital retirement is structurally positive, via a wear and tear effect, and cyclically negative, via a decelerator effect which completes that already taken into account via the effect of changes in value added.

Introduction

The macroeconomic literature underlines the decisive role of the capital stock in the potential output evaluation, and also in determining the position of the economy in the business cycle and the inflationary pressures. Not surprisingly, investment plays an important role in this literature which, nevertheless, ignores generally the capital retirement firm behaviour.

To our knowledge, the assessments of the fixed capital stock conducted by national statistics institutes always assume invariant distributions of the life span of fixed assets (law of obsolescence) over time for each one of the different products that make up capital (for a summary, see OECD, 2009). The retirement rate of capital measures that result from these approaches may then vary over time due to: i) changes in the product structure of the capital if the laws of obsolescence differ for each product; ii) changes in the age structure of the capital if the laws of obsolescence suppose that the retirement rate varies according to the age of equipment. Given how difficult it is to model and estimate in all its complexity companies' capital retirement behaviour, the assumption of invariant obsolescence laws by product adopted by national accountants seems prudent, especially as its divergence with economic reality probably has no significant consequence for the analysis of growth factors over relatively long periods or between periods characterized by similar cyclical pressures. However, it may have more significant consequences in short-term analyses or in the event of large cyclical shocks. For example, the sharp fall in capacity utilisation rate indicators during the current crisis was probably accompanied by acceleration in capital retirement. Assessments drawing on capital series based on the assumption of time-invariant obsolescence laws could show a bias of an uncertain scale. Thus, measures of potential GDP based on such assessments of capital could overestimate the potential GDP and therefore the output gap (in absolute terms) and, consequently, the disinflationary pressures caused by the crisis.

The aim of our study is to analyse empirically the capital retirement behaviour, using for this individual French firm data. The estimated model characterises the companies' capital retirement behaviour, and devotes a particular attention to the distinction of the specific impacts, on the retirement rate, of cyclical and structural effects.

Intuition suggests that the intensity of capital utilisation must probably have an impact on the retirement rate through two channels: i) the first, which is structural, corresponds to a simple wear and tear effect: the retirement rate increases with the intensity of capital utilisation; ii) the second, more cyclical, corresponds to the lags in the adjustment of capital to its targeted level through investment alone, which would result in an adjustment, at least in part, via capital retirement. In this second channel, the retirement rate would decrease with the rise in pressures on the utilisation of capital. The capital retirement behaviour would add its effects to the investment behaviour to adjust the capital to its targeted level. To the investment accelerator mechanism would corespond a capital retirement one. Beyond the mere structural and cyclical effects of capital utilisation, intuition also suggests that the financial situation of companies probably has an impact on their capital retirement behaviour: if a firm has problems financing its targeted investment level, it would lower its retirement rate.

To our knowledge, the only analysis of companies' capital retirement behaviour based on individual firm data is that conducted by [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF]. This analysis is based on panels of several hundred French, German and US companies, the data used being drawn from balance sheets and profit and loss accounts. It shows that companies' capital retirement behaviour is, to a large extent, influenced by cyclical developments. Variations in value added have an adverse effect on the retirement rate, which suggests the existence of a countercyclical effect on capital retirement, while the profit rate is found to have a generally negligible effect.

Our analysis builds on that of [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF], using a large sample of French firms. It is based on unique data on the degree of utilisation of production factors. More precisely, our empirical analysis is based on a database which merges two sorts of individual company data collected by the Banque de France: those from the FIBEN database and those obtained from a specific survey on the utilisation of production factors. FIBEN is a very large individual company database that draws on annual tax statements including balance sheets and profit and loss accounts. This database can be used to calculate retirement rates, changes in output and corporate profitability. The survey on the degree of utilisation of production factors has been carried out every year since 1989 by the Banque de France at plant level. Companies are questioned on the changes in the workweek of capital and their use of shift work. This information gives us an indication of the intensity of capital utilisation and its cyclical variations. The merger of these two databases results in an unbalanced sample of 35,679 observations over the period 1989-2008. Given the lag structure used for estimations, these may be carried out on 6,998 observations corresponding to 1,443 companies over the period 1996-2008. To our knowledge, this individual company database is unique for conducting an empirical analysis of the effects of the intensity of capital utilisation on companies' capital retirement behaviour. The empirical analysis of companies' capital retirement behaviour is conducted with these data and adjusted for endogeneity biases by means of instrumental variables.

The main results of the analysis may be summarized as follows: i) The retirement rate decreases with the variations in cyclical pressures measured by the changes in output and the workweek of capital; this relation corresponds to a countercyclical decelerator effect on capital retirement; ii) The capital retirement rate increases with the structural intensity of capital utilisation; this effect, which corresponds to a wear and tear one, is nevertheless small compared to the decelerator one; iii) The profit rate does not have a significant impact on the retirement rate. Compared with the existing literature, here mainly [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF], the contribution of these results is to show, through the use of unique survey data, that the effect of the intensity of capital utilisation on capital retirement is structurally positive, via a wear and tear effect, and cyclically negative, via a decelerator effect which completes that already taken into account via the effect of changes in value added.

These empirical results give some perspectives to the theoretical studies on growth or those based on calibrated models which often assume that the retirement rate increases with the intensity of capital utilisation (see, for example, among others, [START_REF] Greenwood | Investment, Capacity Utilisation, and the Real Business Cycle[END_REF][START_REF] Greenwood | Investment, Capacity Utilisation, and the Real Business Cycle[END_REF][START_REF] Burnside | Factor-Hoarding and the Propagation of Business-Cycle Shocks[END_REF][START_REF] Burnside | Factor-Hoarding and the Propagation of Business-Cycle Shocks[END_REF]. To use the same terminology as above, these studies assume usually that the structural wear and tear effects outweigh any opposite decelerator effects on capital retirement. This assumption is, however, not based on any empirical confirmation. It could have an important impact on the evaluation of short term impact of shocks (technological or others) on growth, since the capital retirement influences the investment expected rate of return (Cf. [START_REF] Veracierto | Plant-Level Irreversible Investment and Equilibrium Business Cycles[END_REF].

The literature on capital retirement is mainly conducted at the macroeconomic level. Several papers underline the complexity of a measurement of the capital retirement at this macroeconomic level, pointing that this measurement should be linked to assumptions concerning the technological progress, incorporated or not (see among others [START_REF] Diewert | Depreciation, Deterioration and Obsolescence when there is Embodied or Disembodied Technical Change[END_REF]Wykoff, 2006, or Bitros et al., 2007), and that it could use available data concerning the capital second-hand market (for a survey, see [START_REF] Jorgenson | Empirical Studies of Depreciation[END_REF]. Our study is, whith [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF], one of the rare ones which link the business cycle to the capital retirement behaviour.

Section 2 presents the data used in the empirical analysis. Sections 3 and 4 present the results of model estimations first by taking no account of the degree of capital utilisation, then by factoring it in. Section 5 concludes.

Data

The empirical analysis is based on a database that results from the merger of two types of individual company data collected by the Banque de France: those contained in the FIBEN database and those obtained from a specific survey on the utilisation of production factors.

FIBEN is a very large individual company database that draws on annual tax statements including balance sheets and profit and loss accounts. It covers all French companies with an annual turnover of over EUR 0.75 million or with a loan of at least EUR 0.38 million. It contains annual account data for roughly 200,000 companies. It may be used to calculate for every company and for every year the real value added (Q), the stock of capital (K), the volume of investment (I), the retirement rate (RR) and the profit rate (PR) 1 :

-The real value added (Q) is calculated by dividing the nominal value added by a sectoral value added price index supplied by the national accounts. The nominal value added is the maximum of both the value added (output minus intermediate consumption) and the compensation of employees.

-The volume of capital stock (K) is the sum of the volume of capital for two products: buildings and equipment. For each of these two products, the volume of capital is obtained by dividing the gross capital stock at historic cost obtained from company accounts by an investment price index supplied by national accounts, this index being lagged by the average age of capital. The average age of capital is calculated for each of these two products using the share of depreciated capital in the gross capital stock at historic cost.

-The volume of investment (I) is calculated by dividing productive investment in nominal terms (investment in tangible fixed assets minus the variation in gross land stock when it is positive) by an investment price index obtained from the national accounts.

-The retirement rate (RR) is obtained by dividing capital retirement in real terms (R) of the year under consideration by the stock of capital at the end of the previous year (RR t = R t / K t-1 ). Capital retirement in real terms (R) is calculated as the variation in the volume of capital minus the volume of investment (R = ∆K -I).

-The profit rate (PR) is calculated by dividing the gross operating surplus by the stock of capital in nominal terms, i.e. by capital stock in real terms multiplied by an investment price index supplied by the national accounts. The gross operating surplus is the difference between value added and the compensation of employees.

The survey on the degree of utilisation of production factors has been conducted every year since 1989 by the Banque de France on 1,500 to 2,000 companies. These companies are those usually questioned in the framework of the Banque de France's monthly survey on business activity. In the survey on the degree of utilisation of production factors, companies are questioned about the changes in the workweek of capital, with respect to the previous year, and their use of shift work. These variables give us an indication of the intensity of capital utilisation. Two variables from this database are used in the following empirical analysis: the change in the workweek of capital from one period to the next and the proportion of years during which the firm is using shift work :

-The change in the workweek of capital ݓܿ∆( ௧ ) is obtained by the answer to the question: "What is (in %) the change in the workweek of capital between the reference week of year t-1 and that of year t (no decimals)?". The explanatory note attached to the questionnaire provides a definition of the workweek of capital: it is defined as the average number of hours during which the capital is used over the reference period. It does not include maintenance time, but covers all the time during which the machine is running including the time required to prepare production.

-The proportion of years during which the firm is using shift work (ܹܵ ) is obtained by dividing the number of years during which the firm is using shift work by the total number of years during which the firm appears in our estimation period (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008). The use of shift work is known from the answer ("yes" or "no") to the question: "Have you practice shift work during the reference week?". Figure A1-3 in Appendix 1 is an histogram of this variable, showing that it is highly concentrated around 0 and 1, which confirms that the use of shifwork is mainly a structural choice.

The survey on the workweek of capital is conducted at the plant level, while the FIBEN data concern companies that may be composed of several plants. When several plants of a same company are covered by the survey on the degree of utilisation of production factors, the replies of the different plants are aggregated. The survey variables are then calculated at the company level by means of a sum weighted by the share of each plant in the company's total employment. The database only includes the companies for which the sum of the staff employed by the different plants is at least equal to 50% of employees contained in the FIBEN database. In total, only 64 observations of the database used here correspond to multi-plant companies.

The merger of these two databases results in an unbalanced database of 35,679 observations over the 1989-2008 period. Given the lag structure used for the estimations, the latter may be performed on 6,998 observations corresponding to 1,443 companies over the 1996 -2008 time period. To our knowledge, this individual company database is unique for conducting an empirical analysis of the effects of the intensity of capital utilisation on capital retirement. Descriptive statistics of the variables obtained are presented in Table A1-1 of Appendix 1. Figure A1-1 presents kernel density estimates of the retirement rate. The distribution is highly concentrated around its mode and looks like a lognormal distribution, with almost half of the observations ranging between 1% and 6% (Table A1-1). The right-hand tail is longer than the left-hand tail given the lower bound of 0. [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF] report similar average retirement rates, of around 3% to 6% in the 1970s in France.

The average retirement rates calculated for each company separately range between 0% and 41%, with a standard deviation of 5%. Slightly larger variations are observed over time for a given company. Indeed, the retirement rate shows a within standard deviation of 6%. The within firm variations in the retirement rate are thus slightly larger than the between firm variations, an observation that represents a further incentive to study what lies at the root of retirement rate dynamics at the firm level.

A reduced-form model of capital retirement

The first part of our approach builds on an accelerator -profit model, in line with [START_REF] Eisner | Factor in business investment, General series n° 102[END_REF] and [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF]. It is reduced form model mostly used for the analysis of corporate investment (see [START_REF] Bond | Microeconometric models of investment and employment[END_REF], for a survey). We extend later the specification to account for business cycles. Finally, we focus on the permanent level of capital retirement at the firm level over the whole time period, that we model using firms' characteristics.

The model

We denote by i the firm index and by t the year index. The first relation we study is a reduced form of the accelerator -profit type. It relates the capital retirement rate with the production and profits, both current and past, so that the adjustments can occur with lags owing to rigidities. It has the following general form:

(1)

ܴܴ ௧ ൌ ߚ ߚ ଵ . ݍ∆ ௧ ߚ ଵଵ . ∆ ݍ ௧ିଵ ߚ ଵଶ . ∆ ݍ ௧ିଶ ߚ ଶ . ܴܲ ௧ ߚ ଶଵ . ܴܲ ௧ିଵ ߚ ଶଶ . ܴܲ ௧ିଶ ݀ ௧ ߤ ߝ ௧ .
This specification relates the capital retirement rate ܴܴ ௧ with the value added growth rate ݍ∆ ௧ and its values lagged up to two periods, the profit rate ܴܲ ௧ and its values lagged up to two periods, yearly indicators ݀ ௧ , a fixed effect ߤ specific to each company and capturing the unobserved heterogeneity at the level of the firms, and lastly i.i.d gaussian error terms ߝ ௧ . The number of lags is arbitrarily set to two, as will be discussed in Section 4.1. The model can obviously be adjusted to a different number of lagged variables. Equation ( 1) is formulated in rates rather than in levels, to reduce the variability resulting from the size differences among firms, and so heteroskedasticity problems.

Equation ( 1) captures the role of the demand, through the value added, and of the financial state of the firm, through its' profit rate. Indeed, the firm permanently adjusts to its' optimal level of active capital, knowing that insufficient capabilities can prevent it to satisfy its' demand, whereas an excess of capabilities raises maintenance and opportunity costs. [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF] find that output variations have a negative effect on the retirement rate. This decelerator effect of the activity level on the capital retirement rate is related to the willing to keep a substantial capital stock to face a positive demand shock. Conversely, companies use phases of slow activity to reduce the costs related to overcapabilities, and enhance their productivity in scrapping unused equipments. The intuition suggests that the capital retirement behaviour can also depend on the financial state of the firm. Indeed, the most profitable firms can more easily finance new productive capital, and thus easily scrap the less performing equipments.

Conversely, the less profitable firms can hardly fund new equipments and reduce the capital retirement.

We thus consider in the model a measure of the firms' profitability. [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF] do not find a clear effect of profitability on capital retirement.

We then extend the accelerator-profit model to account for business cycles. We first want to assess if the effect of the activity on the retirement varies along the cycle. A decelerator effect of activity can be explained by the choice to adjust capital to a demand shock, either positive or negative. We want to assess whether this effect is strengthened during the ascending or descending phase of the cycle.We allow for the possibility that both effects differ in magnitude by separating the valued added increases and decreases in the model. Subsequently, we add variables measuring the workweek of capital. Indeed, work shifts may be arranged in different ways due to their market peculiarities (daily interruption, weekly interruption, etc.). It translates in different capital workweek (hereafter labelled CW), that can vary both structurally and cyclically. We focus here on cyclical variations, and thus include in the specification the variations over the last 12 months of the workweek of productive capital. Variations in capital utilisation also do not necessarily produce symmetrical effects on capital retirement, and we therefore distinguish between increases and decreases in the capital workweek :

(

2) ܴܴ ௧ ൌ ߚ ߚ ଵ . ݍ∆_ܫ ௧ ߚ ଵ . ݍ∆_ܦ ௧ ߚ ଶ . ܴܲ ௧ ߚ ଷ . ݓܿ∆_ܫ ௧ ߚ ଷ . ݓܿ∆_ܦ ௧ ݀ ௧ ߤ ߝ ௧ ,
where ݍ∆_ܫ ௧ and ݍ∆_ܦ ௧ measure increases and decreases of the value added ݍ∆_ܫ( ௧ is always positive and ݍ∆_ܦ ௧ negative), and ݓܿ∆_ܫ ௧ and ݓܿ∆_ܦ ௧ denote increases and decreases over the last 12 months of the workweek of productive capital. The expected signs of coefficients ߚ ଷ and ߚ ଷ is at first unknown. Indeed, variations in the workweek of capital can enable to adjust the production capacities to their desired levels in accordance to the retirement rate. A negative ߚ ଷ (ߚ ଷ respectively) is related to, when the capital workweek is increasing (decreasing), an increased (decreased) intensity in the use of the equipments and to a lesser (higher) retirement, that is, everything else equal, to an acceleration (a deceleration) in capital accumulation. These two mechanisms lead to an increase (decrease) in production capacities, and are complementary from this perspective. On the opposite, when ߚ ଷ or ߚ ଷ are positive, the evolutions of both the capital workweek and the stock of capital are conflicting. In this case, variations in the capital workweek allow softening the impact of the retirement on production capacities.

In a last stage, we are studying the impact of the structural workweek of capital on capital retirement. Theoretical works on growth models assume a wear and tear effect, implying that a high activity leads to a high level of retirement. To assess this point, we are investigating the variations between firms of a measure of retirement net of cyclical effects. This measure is obtained after estimation of equation ( 2), and consists in the estimated firm fixed effects ߤ̂୧ 2 . They can be regressed on the average over the time period of the firm characteristics:

(3) μ ො ൌ ߛ ߛ ଵ ܹܵ ߛ ଶ ܦܰܫ ߛ ଷ ܼܫܵ ݑ ,
where ܹܵ is the proportion of years some part of workforce of firm i works in shifts. It does vary only with the number of years in which work is organized in successive shifts. Therefore, it is only related to the duration of period of tense capital workweek. Variables ܦܰܫ and ܼܫܵ are sets of industry and size dummies. This simple specification enables us to avoid any cyclicality in the measure of the structural workweek of capital. Furthermore, it provides estimated standard-errors adjusted for any correlation of the error terms within firms.

Selection of instrumental variables

Numerous sources of endogeneity are likely to bias the estimators. On one side, it is very likely that capital retirement, value added, profitability are determined simultaneously. Indeed, the retirement rate and the profit rate are both normalized by the capital in the previous year, the value added can vary due to change in intermediate consumptions resulting from the retirement of obsolete capital, and the profit rate is a function of the value added. On the other side, the variable measuring changes in the capital workweek displays values concentrated on a few points. Figure A1-2, in Appendix 1, is a histogram of the changes in the capital workweek over the previous year. Most variations are concentrated around -20%, -10%, -5%, 0%, 5%, 10% and 20%, whereas the variable should at the first glance be continuous. It is therefore likely there are measurement errors in the observed workweek of capital.

We choose to correct the endogeneity biases with instrumental variables, using essentially lagged values of the explanatory variables, in the spirit of [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. Lagged variables can be used as instruments under the weak exogeneity assumption, meaning the absence of correlation between ߝ ௧ and the previous values of the explanatory variables. This assumption holds here if the current level of capital retirement does not depend on lags older than the ones included in the model ( 1) and (2).

Our set of instruments consists in the lagged values and two more variables. The first one is the industry yearly average of the changes in the workweek of capital, 3 and the second one is a variable indicating limits to extensions of the capital workweek. More precisely, it is a dummy equal to one when the firm answers during the survey that it encounters obstacles to extend the workweek of capital while the demand allows for it. Estimates of fixed effects Logit models, not reported in the paper, let us think that companies just raising the workweek of capital during the preceeding years are most likely to declare themselves as restricted.

Our instruments must satisfy two conditions to be valid: They must be free of any correlation with the error terms and be reasonably correlated with the potentially endogenous variables. Internal instruments are generally used in great numbers, whereas each one is only weakly correlated with the endogenous explanatory variables. This can lead to substantial biases. Angrist and Pischke (2009, pp. 205-216) describe the problems raised by the use of weak instruments. To restrict the number of instruments and include only the ones containing most of the information on the variables, we follow a pragmatic strategy. We first estimate model (1) including as instruments the lagged values of all the explanatory variables until t-6. Hence, based on the first stage results, we dropped the weaker instruments. We are enabled by Sargan's statistic to test the exogeneity of the instruments, and by [START_REF] Cragg | Testing identifiability and specification in instrumental variable models[END_REF] statistic associated to [START_REF] Stock | Testing for weak instruments in liner IV regression[END_REF] critical values, to test their correlation with the endogenous explanatory variables. We hence obtain a set of instruments for equation ( 1), whose properties are discussed in Section 4.1. We apply after the appropriate decomposition in increases and decreases to obtain model (2) instrumental variables, and then added the lagged values related to workweek of capital.

As the lagged values of the workweek of capital are susceptible to measurement errors, we prefer to use as instruments dummies of their increases and decreases. It is indeed unlikely that such broad variations are missmeasured by the answering firm. As previously, we include lags up to t-6 before removing some instruments based on the first stage results. Properties of the instruments are presented in Section 4.2.

We use the LIML estimator. This choice is motivated by several reasons. First, it is far less susceptible to biases than the usual two-stage GMM estimator in presence of weak instruments and finite samples, as shown by the values tabulated in [START_REF] Stock | Testing for weak instruments in liner IV regression[END_REF] for the LIML and two-stage GMM, as well as by [START_REF] Hahn | Estimation with weak instruments : accuracy of higherorder bias and MSE approximations[END_REF]. Several other alternative estimators are less biased than the two-stage GMM estimator when the model is overidentified, as it is the case here. The LIML estimator neverthess seems superior with respect to several criteria [START_REF] Flores-Lagunes | Finite sample evidence of IV estimators under weak instruments[END_REF]. Furthermore, [START_REF] Stock | Testing for weak instruments in liner IV regression[END_REF] provides tables allowing the directly test the assumption of weak instruments with Cragg-Donald statistic. Those tables have been obtained under the assumption of i.i.d and homoscedastik errors. We are not aware of similar tables under the assumption of heteroscedastik errors, that could be used for example to validate instruments used by the CUE-GMM estimator of Hansen et al. (2006). As it seems of importance to us to reduce as far as possible potential biases, as well as being able to discuss the relevance of the instruments, we decide to use the LIML estimator. Estimates are performed using Stata and the ivreg2 (Baum et al., 2010) procedure. Firms fixed effects are eliminated using the within transformation.

Results

In a first section, we present the results of an accelerator -profit model similar to the one used by [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF]. We then introduce variables which capture the intensity of capital utilisation, first from a cyclical viewpoint is Section 4.2, and second from a structural viewpoint in Section 4.3.

The accelerator -profit model

Table 1 shows several estimates derived from equation (1). A first set of results, denoted by W, are obtained using the within estimator that assumes all the explanatory variables to be exogenous. We examine in particular the lags structure that are pertinent under this assumption. A second set of results, obtained using the LIML estimator, assumes that the explanatory variables are endogenous and the error terms homoscedastik. The choice of instruments used in the tables below is described in Section 3.2. -the coefficients in bold characters are significant at the 5% threshold.

All the within estimates indicate lower value-added change coefficients for lagged values than for the current value, which is coherent with a model where the capital adjustments are slowed by rigidities. Only the coefficients of contemporary values are significant. Conversely, the profit rate is not significant, even under different lags structure. Similar results were obtained by [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF].

The LIML estimator here used the first set of instruments presented in Section 3.2. The validity of the instruments is tested in a variety of ways. A Sargan test concludes at the threshold of 77% to the absence of a correlation between the instruments and the error term. We test the assumption of weak instruments using [START_REF] Stock | Testing for weak instruments in liner IV regression[END_REF] procedure. The [START_REF] Cragg | Testing identifiability and specification in instrumental variable models[END_REF] statistic is evaluated 17.78, for the model in the penultimate column, and 25.51 for the one in the last column. The critical value for the LIML estimator when there are two endogeneous variables and six excluded instruments is of 4.06 (Stock and Yogo, 2002, Table 4, p. 61). We can thus reject the assumption of weak instruments in both specifications. First stage results are show in Table A2.1 in Appendix 2.

The results obtained with the LIML estimator indicate that the coefficient of ݍ∆ ௧ remains significant when the endogeneity is taken into account and the current profit rate is included among the covariates.

Compared to the within estimates, the LIML ones react and decrease when we use instrumental variables.

This let us think there is here a simulataneity that must be handled. As we did with the within estimator, we test different specifications including several lags and therefore several sets of instruments. Their results indicate that the contemporary values of the explanatory variables capture most of the dynamics and that no lagged variable has any significant effect on the capital retirement rate, nor increases the model fit. The adjustment periods do not play an important role either in the results of the within estimator or in the results of the LIML one. We thus focus hereafter on a specification that only introduces explanatory variables at time t.

Taking all the specifications together, we observe that a growth in value-added has a negative impact on the capital retirement rate. This decelerator effect, already found by [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF], means that in the event of a cyclical positive demand shock, companies adjust partly their production capacity by reducing the retirements. Conversly, in the event of weak business, firms increases the retirement of useless equipments.

The effects of cyclical variations of the capital workweek

This Section presents the estimates of model ( 2). We first assess whether the decelerator effect pointed in Section 4.1 is strengthened during the ascending or descending phase of the cycle; with a specification that allows growth in value added to have asymmetrical effects on capital retirement. Then, we broaden this specification to assess how the decelerator effect reacts to variations in the capital workweek. All the results are obtained using the LIML estimator in order to correct the estimates for endogeneity problems. -the coefficients in bold characters are significant at the 5% threshold.

We study in several ways the properties of the instruments used to obtain the results displayed in Table 2. Firstly, a valid instrument has to be exogeneous. For all models reported in Table 2, the absence of correlation between the residual and the instruments is accepted to at least the 40% level. Secondly, the instruments must not be weak. We document this property using Fisher and [START_REF] Cragg | Testing identifiability and specification in instrumental variable models[END_REF] statistics. Indeed, [START_REF] Staiger | Instrumental variables regression with weak instruments[END_REF] suggest as a rule of thumb, in model with a single endogenous variable, to consider instruments as weak when their first stage Fisher statistic does not excess the threshold of 10. Our model comprises multiple endogeneous variables, and this rule does not apply here. We however report in Table A2-2 in Appendix 2, for information, the results of the first stage and the Fisher statistics computed with the number of excluded instruments of the model in the last column of Table 2. An approach dedicated to model with multiple endogeneous variables is described in [START_REF] Stock | Testing for weak instruments in liner IV regression[END_REF]. They propose to test the assumption of weak instruments in comparing the Cragg -Donald statistic with values they tabulate. However, critical values are only tabulated for models with less than two endogeneous variables, where they are respectively of 3.31 and 3.54 for the LIML estimator of a model with 15 excluded instruments (Stock and Yogo, 2002, Table 4, p. 61). Those values are far less than [START_REF] Staiger | Instrumental variables regression with weak instruments[END_REF] threshold of 10. Our model comprises four endogeneous variables. Cragg -Donald statistics of more than 9 for the two first models and of 4.99 for the last one let us think that the instruments are reasonably strong and that the biases are at most small. Results of the third model, where the test statistic is greater than the availables critical values, have to be considered more carefully.

Rises and falls in value added have different effects. Unlike falls, rises have virtually no impact on capital retirement. A 1% fall in value added is associated with a rise from 8 to 12 precentage points in retirement.

The adjustement of production capacities through the retirement thus does seem to occur only in a weak activity setting, companies thus retiring useless overcapacities. This result is not impacted by the inclusion of variables measuring the capital workweek.

The effect of increases and decreases in the workweek of capital are also differenciated. Whereas the magnitude of the coefficients is similar, their significancy is not. A decrease in the workweek of capital does not have a clear significant effect. On the opposite, an increase of 1% in the capital workweek is related to a decrease in retirement of -0.29 percentage points, showing a complementarity in the increase of production capacity between capital retirement and more intensives capital workweeks.

The effect of the structural intensity in capital usage

Model (3) is now estimated, so as to characterize the impact of the structural intensity of usage on the retirement. In this model, the permanent level of retirement is measured by the firms fixed effects ߤ̂୧, estimated with the last model of the preceeding Section ( Once the heterogeneity at the industry and size levels is accounted for, the coefficient of the structural intensity variable is positive and significant. The retirement rate is hence increased when some durable pressure is put on the equipement. This result is in line with the intuition of a wear and tear effect. This is however a mechanism of low magnitude. A company managing its' workforce in successive shifts during all the 9 years of the period under study has a ߤ̂୧ greater of 1 percentage point, and thus its' retirement rate, than a similar firm having never used work in successive shifts. Compared to the influence of the value added, whom a decrease of 1% induces an increase of 12 percentage points, the impact of the wear and tear effect seems of minor importance.

Contrary to short term variations in the capital workweek, which have a decelerator impact on the capital retirement, long term levels of intensity of use have an accelerator effect. The capital workweek thus has two distinct and opposite effects: cyclically, the capital workweek contributes through the retirement to adjust production capacities, and structurally, the capital retirement is increasing with the intensity of capital usage.

Conclusion

The empirical analysis of individual company data proposed in this study uses not just the data from the company accounts to calculate the retirement rate, variations in value added and the profitability rate, but also other original data from a survey on the utilisation of production factors. The latter tells us to what extent companies resorted to shift work and to what degree the workweeks of capital varied. The extent to which a company resorts to shift work is used to explain the structural intensity of capital utilisation. Variations in the workweek of capital tell us more about cyclical variations in the intensity of capital utilisation. The sample used for the estimates includes 6,998 observations over the 1996-2008 period. This individual database is, as far as we know, unique for the empirical analysis of the effects of the intensity of capital utilisation on capital retirement. The empirical analysis of capital retirement behaviour is conducted with this data and adjusted for endogeneity biases by means of instrumental variables.

The principal results obtained from the estimation of capital retirement models can be summarised as follows: i) The retirement rate decreases with the variations in cyclical pressures measured by the changes in output and the workweek of capital; this relation corresponds to a countercyclical decelerator effect on capital retirement; ii) The capital retirement rate increases with the structural intensity of capital utilisation; this effect, which corresponds to a wear and tear one, is nevertheless small compared to the decelerator one; iii) The profit rate does not have a significant impact on the retirement rate. Compared to previous studies in this field, notably [START_REF] Mairesse | Labor and Investment Demand at the Firm Level[END_REF], the contribution of these results is to show, using original survey data, that the intensity of capital utilisation influences capital retirement in a structurally positive way, via a wear and tear effect, and in a cyclically negative way, via a decelerator effect that completes the already-accounted-for effect of variations in value added.

The approach proposed here is still very partial. A more in-depth representation of capital retirement behaviour would need to be based on a more complete model of the dynamics of capital, analysing the relationship between capital accumulation and capital retirement. Nevertheless, the results obtained appear fairly robust and they comply with basic economic intuition. They suggest that during the current crisis -which notably involved a sharp cyclical contraction of business activity and the intensity of capital utilisation -the retirement of capital must have experienced a marked acceleration.

An impact of this nature is not negligible in assessing the potential level of production, and consequently, in determining the position of the economy in the business cycle, for example, using output gap indicators. -Fisher statistics are evaluated with the number of excluded instruments.

Figure A1- 2

 2 Figure A1-2 Histogram of the variations of the capital workweek over the last 12 months

Table 1 Estimates of relation (1) with different lags and exogeneity assumptions

 1

	Estimateur	W	W	W	W	LIML	LIML
	∆	-3.25 *	-3.43 *	-2.94 *	-3.26 *	-4.74 *	-2.06
		(-5.31)	(-5.72)	(-5.49)	(-6.28)	(-2.17)	(-0.68)
	∆ ି	0.25	-0.22
		(0.38)	(-0.38)
	∆ ି	0.59
		(1.02)
		0.14	0.23	-0.18		-3.24
		(0.29)	(0.49)	(-0.43)		(-1.61)
	ି	-1.03 *	-0.76		-0.71		0.20
		(-2.09)	(-1.73)		(-1.85)		(0.15)
	ି	0.37
		(0.86)
	Year dummies	Yes	Yes	Yes	Yes	Yes	Yes
	Firm fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
	Instrumental variables	No	No	No	No	Yes	Yes
	Critical prob. of Sargan's	-	-	-	-	0.77	0.84
	test
	N	6998	6998	6998	6998	6998	6998
	Note: -the Student statistics are in brackets;

  Table2, rightmost column). Those fixed effects are regressed on the proportion of years during which company i organized its' work in successive shifts This variable captures the structural intensity of the equipments usage over the time period, and FigureA1-3 in Appendix 1 displays its' histogram. It thus measures the impact of an intensive and durable use of capital over several years. We also include as control variables industry dummies (5 different industries are considered) and size dummies (4 different sizes, corresponding to the quartiles of the sizes). Results are provided in Table3below.

		(1.35)		(1.26)
	-	capital goods	0.01 *		0.01 *
		(3.29)		(3.16)
	Workforce :
	-	first quartile		-0.01 * -0.01 *
				(-2.94) (-3.01)
	-	second quartile		-0.01 *	-0.01
				(-2.15) (-1.76)
	-	third quartile		-0.01 * -0.01 *
				(-2.23) (-2.11)
	N		1443	1443	1443
	Note : -the Student statistics are in brackets;
	-the coefficients in bold characters are significant at the 5% threshold;
	-the industry reference category is the intermediate goods ;
	-the workforce reference category is the fourth quartile.
		Tableau 3
		Estimates of the structural effects
	Estimator	OLS	OLS	OLS
			0.01 *	0.00	0.01 *
		(1.98) (1.15) (2.16)
	Industries :
	-	food	0.00		0.00
		(0.22)		(0.34)
	-	consommer goods	0.02 *		0.02 *
		(6.44)		(6.47)
	-	automotive	0.01		0.01

Table 2 .2 Results of the first stage regressions of Table 2

 2

	Variable	_ࢤ		_ _
	_ࢤ ି	-0.30 *	0.10 *	-1.33	0.50
		(-20.42)	(3.23) (-1.58)	(0.73)
	_ࢤ ି	-0.23 *	0.10 * -2.78 *	-0.64
		(-14.72)	(3.07) (-3.22) (-0.91)
	_ࢤ ି	-0.22 *	0.07 * -2.83 *	-0.46
		(-14.09)	(2.19) (-3.26) (-0.65)
	_ࢤ ି	-0.16 *	0.06 * -1.67 *	-0.13
		(-12.10)	(2.16) (-2.19) (-0.21)
	_ࢤ ି	-0.12 *	0.04	-0.86	-0.95
		(-8.82)	(1.32) (-1.14) (-1.55)
	ି	-0.04 *	0.27 *	-0.44	-0.45
		(-6.57) (21.23) (-1.25) (-1.58)
	ܟ܋ തതതത ܜܒ	0.01 *	0.01	0.40 *	0.56 *
		(4.52)	(1.65) (3.09)	(5.33)
	_ࢤ ି	0.00 *	0.00 * -0.01 *	-0.00 *
		(3.83)	(2.64) (-2.44) (-2.00)
	_ࢤ ି	0.00	0.00 -0.01 *	-0.00
		(0.56)	(0.47) (-4.97) (-0.71)
	_ࢤ ି	0.00	0.00 -0.01 *	-0.00
		(1.92)	(1.21) (-4.30) (-0.47)
	_ࢤ ି	-0.00	-0.00 -0.01 *	-0.00 *
		(-0.26) (-1.64) (-4.17) (-2.04)
	_ࢤ ି	0.00	0.00 * -0.01 *	0.00
		(1.57)	(2.48) (-4.65)	(0.50)
	_ࢤ ି	-0.00	-0.00	0.00	0.01 *
		(-0.65) (-1.91) (0.47)	(5.90)
	_ࢤ ି	0.00	-0.00	-0.00	0.01 *
		(1.15) (-1.48) (-1.79)	(4.76)
	_	0.00 *	0.00	0.01	0.01
		(3.64)	(1.05) (6.50)	(6.89)
	F	61.26	44.96	11.29	10.60
	N	6998	6998	6998	6998
	Note : -The Student t are in brackets ;

In this study, the variables in upper or lower case indicate respectively their level or their natural (napieren) logarithm, and ∆ before a variable corresponds to its change compared with the previous period.

A method to estimate the firm fixed effects in a linear model for panel data is provided inWooldridge (2001, p. 273).

Sixteen industries are defined, according to the Nomenclature Economique de Synthèse of the INSEE.


Olivier Sterck

Geoengineering as an alternative to mitigation: specification and dynamic implications

Keywords: Geoengineering, Abatement, Climate change, Global warming, R&D, Intergenerational issues JEL Classification: Q01, Q52, Q54, Q55, O44

Geoengineering, i.e. the use of artificial techniques aiming at cooling the planet, is increasingly considered as a realistic alternative to emission mitigation. Several methods are promising for their capacity to quickly halt global warming at a moderate cost. Such cheap technologies might be very beneficial to countries profoundly affected by global warming. In this paper, I propose a dynamic model in which geoengineering is introduced as an alternative to mitigation. Contrary to abatement, geoengineering is fast and cheap, but requires a large initial investment in research and development. Within this framework, I confirm the fear which is common among geoengineering opponents: abatement is reduced if geoengineering is expected to be available in the future. The long-run implications of the model are also alarming as geoengineering will not be undertaken progressively. The sudden implementation of geoengineering, together with the sharp jump in temperature induced, may disturb climate equilibrium and fragile ecosystems. Furthermore, the availability of geoengineering will exacerbate intergenerational issues: while current generations will anticipate the use of geoengineering by increasing their emissions, future generations will have to reduce their emissions, to bear the cost of sustaining geoengineering for centuries and to suffer from its negative side-effects.

Introduction

The Earth system is continuously heated by the shortwave light coming from the Sun. Before reaching the Earth surface, a fraction of this short-wave radiation, the albedo, is reflected by clouds and white surfaces. The rest, about two-thirds of the incident solar radiation, is absorbed by the Earth system and converted into heat. This heat is then re-emitted from the Earth surface as long-wave radiation. In equilibrium, the energy coming from the sun and absorbed by the Earth surface is 1 IRES, Université catholique de Louvain, Place Montesquieu 3, B-1348 Louvain-la-Neuve, Belgium (e-mail: olivier.sterck@uclouvain.be).

The author would like to thank the ARC project 09/14-018 on "sustainability" (French speaking community of Belgium) for its financial support. This paper was partially written while the author was visiting the GREQAM at the University of Aix-Marseille. The author is thankful to Raouf Boucekkine and Henry Tulkens for their helpful comments. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the author. exactly compensated by the heat that escapes the Earth system. This delicate energy balance determines Earth temperature: if the radiation stream is perturbed by 1%, the surface temperature would change by about 1.8 • C.

The increase in the atmospheric concentration of greenhouse gases (GHGs), principally carbon dioxide (CO 2 ), methane and water vapor, affects this fragile equilibrium. Indeed, while GHGs allow the passage of the short-wave radiation (the sunlight and the albedo), they absorb and re-emit a fraction of the long-wave radiation (heat) escaping from the Earth. Part of this energy is sent back in direction of the Earth surface and again converted into heat. This back-and-forth of long-wave radiation, commonly called "greenhouse effect", is seen as the main responsible for global warming. Indeed, because of the industrial revolution, the atmospheric concentration of CO 2 increased over the past two centuries from 280 ppmv (parts per million by volume) in preindustrial time to 379 ppmv in 2005 (+40%) October 17, 2011 (IPCC Working Group I, 2007). Annual emission of carbon dioxide was increased by 80% between 1970 and 2004. With such a trend, atmospheric CO 2 concentration is projected to range from 540 to 970 ppmv by the end of the 21st century. Similarly, methane concentration is nowadays over 150% above its preindustrial level. Changes in atmospheric concentration of greenhouse gases have already risen the surface Earth's temperature of 0.8 • C. Even if the concentration of greenhouse gases remains stable, the average temperature is still expected to rise of one more degree (IPCC Working Group I, 2007). This is due to the time lag in global warming caused by the large heat capacity of the oceans. The IPCC anticipates a global warming ranging between 1.8 and 4 • C, leading to the collapse of the major ice sheets and a sea level rise of tens of meters (IPCC Working Group II, 2007). Even more frightening is the likelihood of unpredictable non-linearities in climate change, or "tipping points", at which temperature, or other factors, may rapidly generate irreversible and potentially very destructive changes (IPCC Working Group II, 2007;[START_REF] Swart | Policy options in a worst case climate change world[END_REF][START_REF] Kousky | Responding to threats of climate change mega-catastrophes[END_REF].

In order to prevent such dangerous and irreversible consequences, the most popular approach, mitigation, seeks to reduce the emission of greenhouse gases in the atmosphere. Despite the last twenty years of proactive diplomatic talks, this strategy seems limited because subject to free-riding. The stabilization of CO 2 concentration would require a 60-80% worldwide reduction in current anthropogenic CO 2 emissions. Nevertheless, fossil fuels provide over 80% of the world's energy, and emissions of CO 2 are actually increasing by around 2% each year [START_REF] Bickel | Albedo enhancement by stratospheric sulfur injections: A contribution to resolve a policy dilemma?[END_REF]. Many governments are reluctant to engage in unpopular binding commitments whose cost is born in the short-run, while benefits mainly arise in the long-run.

In response to this coordination failure, geoengineering has been put forward in the scientific debate as a credible but controversial alternative to mitigation2 . Geoengineering is defined by the National Academy of Sciences (1992) as the "options that would involve large-scale engineering of our environment in order to combat or counteract the effects of changes in atmospheric chemistry". Geoengineering techniques may be classified in two categories, depending on whether they aim at reducing the greenhouse effect, or at diminishing the share of sunlight that reaches and warms the Earth system. On the one hand, carbon dioxide removal (CDR) seek to remove CO 2 from the atmosphere in order to reduce the amount of long-wave radiation trapped in the Earth system through the greenhouse effect. For example, the literature proposes to extend the surface of the Earth covered by trees, to increase the oceanic uptake of CO 2 through enhanced algae and plankton growth, to accelerate artificially the mineral sequestration of CO 2 , or to create "artificial trees" that capture CO 2 thanks to a chemical sorbent.

On the other hand, solar radiation management (SRM) aims at repelling short-wave sunlight before the radiation hits the Earth surface and turns into heat. Offsetting the warming caused by a doubling of the preindustrial revolution atmospheric concentration of CO 2 would require shielding or reflecting approximately 1.8% of the incident solar radiation. The two most effective methods are the injection of reflective particles, principally sulfate aerosols, in the high-atmosphere [START_REF] Bickel | Albedo enhancement by stratospheric sulfur injections: A contribution to resolve a policy dilemma?[END_REF], and the increase of the albedo of maritime areas by seeding and whitening stratocumulus clouds over the oceans [START_REF] Salter | Sea-going hardware for the cloud albedo method of reversing global warming[END_REF][START_REF] Latham | Global temperature stabilization via controlled albedo enhancement of low-level maritime clouds[END_REF]. Recently, these two geoengineering techniques were evaluated using modified DICE models [START_REF] Nordhaus | The Challenge of Global Warming: Economic Models and Environmental Policy[END_REF][START_REF] Bickel | An analysis of climate engineering as a response to climate change[END_REF][START_REF] Goes | The economics (or lack thereof) of aerosol geoengineering[END_REF][START_REF] Bickel | Albedo enhancement by stratospheric sulfur injections: A contribution to resolve a policy dilemma?[END_REF]. All but [START_REF] Goes | The economics (or lack thereof) of aerosol geoengineering[END_REF] conclude that geoengineering is highly cost-effective. For example, [START_REF] Bickel | An analysis of climate engineering as a response to climate change[END_REF] estimates benefit-cost ratios of around 25 to 1 for aerosols injection in the stratosphere and around 5000 to 1 for cloud albedo enhancement. The study of [START_REF] Goes | The economics (or lack thereof) of aerosol geoengineering[END_REF] is at odds with these conclusions. Rather, they argue that aerosol geoengineering is not cost-effective because too risky in case of failure to sustain aerosol injections. Another SRM method would be to launch space sunshields between the sun and the Earth to block incoming radiations. Proposals have also been made to increase Earth albedo by brightening land or ocean surfaces, for example by painting roofs or covering deserts with reflective sheets. These last two options appear to be less cost-effective than the injection of sulfate aerosols in the high-atmosphere or the enhancement of cloud albedo (see [START_REF] Shepherd | Geoengineering the climate: Science, governance and uncertainty[END_REF] for a detailed review of all geoengineering methods).

This paper aims at comparing mitigation (abatement) and geoengineering. More specifically, I will construct a theoretical model assessing under which conditions geoengineering may be used as cheap substitute for abatement. As both mitigation and carbon management seek to reduce the quantity of CO 2 in the atmosphere, these two methods would be formalized in the same way. CDR can be regarded as special kind of mitigation method. Therefore, I focus only on SRM whose formalization and properties are sharply different from abatement. In what follows, I use the term geoengineering to refer to SRM. I distinguish seven differences between geoengineering and abatement. First, while it would be technically feasible to drastically decrease emissions nowadays, geoengineering is not yet implementable and requires a sustained effort in research and development. Second, geoengineering is cheaper than abatement, but requires large investments in research and development. Third, geoengineering may act much faster than abatement. Indeed, abatement requires a sustained effort to mitigate emissions. Furthermore, because GHGs is a stock accumulating over time, and because the large heat capacity of the oceans induces a time lag in global warming, the positive impact of mitigating emissions is only expected in the long-run. Conversely, geoengineering is fast at decreasing the average temperature when it is implemented. Fourth, contrary to abatement, geoengineering does not fight the cause of global warming. Rather, it aims at artificially offsetting the greenhouse effect in order to prevent its detrimental consequences. Fifth, geoengineering is imperfect as it may have large side-effects on the environment. For example, stratospheric sulfate injection may have harmful consequences on stratospheric ozone and biological productivity. Cloud albedo enhancement may affect weather patterns and ocean currents. Both these methods have non-uniform effects, and induce significant regional climate changes [START_REF] Jones | A comparison of the climate impacts of geoengineering by stratospheric so2 injection and by brightening of marine stratocumulus cloud[END_REF]. The sixth difference between geoengineering and abatement is related to their surrounding uncertainties. For geoengineering, the uncertainty is related to the cost, the effectiveness and the presence of possible side-effects. Conversely, for mitigation, the amount of abatement needed to avoid harmful consequences and "tipping points" is unknown. Finally, while countries implementing geoengineering may be clearly identified, it is much harder to assess which country has to mitigate its emissions, and by how much. Consequently, a country deciding unilaterally to use geoengineering may be punished by others. This is not the case for countries emitting more GHGs.

The literature on the economics of geoengineering is scarce. [START_REF] Barrett | The incredible economics of geoengineering[END_REF] was the first economist to discuss governance challenges brought about by geoengineering. He argues that any countries for which implementation costs are lower than benefits will be willing to try it. As [START_REF] Barrett | The incredible economics of geoengineering[END_REF] does not propose a formal model, he can not predict which country or group of countries will bear the cost. Only four papers introduced geoengineering in economic models as a cheap alternative to abatement. First, using a one-country model with climate damages, Moreno-Cruz and Smulders (2010) conclude that geoengineering is an imperfect substitute for abatement as it generates negative side-effects (fourth and fifth differences). Second, by using numerical estimations, [START_REF] Moreno-Cruz | Climate change under uncertainty: A case for geoengineering[END_REF] expand this framework to allow for uncertainty on the effectiveness and the consequences of geoengineering (sixth difference). They show that geoengineering is an effective means to approach the uncertainty on climate damages as it can be implemented quickly if the bad outcome of the uncertainty is revealed. Finally, [START_REF] Moreno-Cruz | The long and short of climate change: Abatement versus geoengineering[END_REF][START_REF] Millard-Ball | The tuvalu syndrome[END_REF] show that countries act strategically when geoengineering is made available in a multi-country framework (Seventh difference). For example, [START_REF] Moreno-Cruz | The long and short of climate change: Abatement versus geoengineering[END_REF] shows that a country may substitute away from abatement to induce higher levels of geoengineering in other countries. Conversely, if a country fears the side-effects of geoengineering, it may increase its level of abatement to deter the use of geoengineering by other countries. Similarly, [START_REF] Millard-Ball | The tuvalu syndrome[END_REF] shows that countries not much affected by climate change may nevertheless strengthen their level of abatement to avoid the threat of unilateral geoengineering use coming from a country highly affected by global warming.

For now, the theoretical literature on geoengineering disregarded the three first differences between abatement and geoengineering. Introducing these in a theoretical framework is the main objective of the paper. I will construct a dynamic model that takes into account both the fastness and the specific cost structure of geoengineering. More specifically, geoengineering will be introduced besides abatement in a two-period model with climate damages. Geoengineering will be characterized by a fixed cost which includes both research and development investments as well as indirect costs related to governance issues or conflict mitigation. While geoengineering may quickly reduce temperature and climate damages, abatement is much slower as emissions accumulate as a stock. Within this framework, I will study the short-and the long-run implications of geoengineering availability, with a special focus on intergenerational issues. Some crucial differences with Moreno-Cruz and Smulders (2010) will be highlighted along the way.

In the next section, I introduce the two-period model, in which both abatement and geoengineering may be undertaken to prevent climate change damages. In section 3, I assess the long-run implications of the model and I discuss the main findings of the paper. Section 4 concludes.

A model of climate change and geoengineering

Set-up

We consider the maximization problem of a unique country that seeks to maximize consumption over two periods. In order to produce one unique good, the country uses energy e t . One unit of energy costs π, and generates one unit of GHG emission (later, e t will refer for energy use as well as emissions at time t).

The accumulation of GHG emissions in the atmosphere damages the environment through global warming. In order to avoid dramatic loss of production due to the temperature increase, the country may choose to reduce its emissions (mitigation or abatement) or to implement geoengineering. Both strategies are costly. On the one hand, by limiting emissions, abatement constrains production. On the other hand, implementing geoengineering requires a large initial investment in the research and the development of the cooling technology as well as a sustained effort for maintaining temperature at the desired level.

At each period, the consumption is equal to the production f (e t ), minus the energy cost πe t , the damages due to global warming Ω(T t ), and the geoengineering cost Γ(G t ).

c t = f (e t ) -πe t -Ω(T t ) -Γ(G t ).
Formally, we assume that the production function of the country is quadratic and given by:

f (e t ) = e t (α -βe t ).
We assume that α > π to ensure the existence of a positive solution for e t in the "no damage case" (

f ′ > 0, f ′′ < 0, f (0) = 0).
The evolution of the stock of GHG in the atmosphere, P t+1 , is determined by the sum of current emissions and the remaining stock of pollution:

P t+1 = P t (1 -δ) + e t .
(1)

The factor δ is constant and represents the length of life of CO 2 in the atmosphere (0 ≤ δ < 1). Current emissions only have an impact in the next period. This specification reflects the time lag in global warming.

The world temperature T t at time t is positively related to the stock of greenhouse gases in the atmosphere P t and negatively related to the total quantity of geoengineering used G t . We assume a linear specification:

T t = Φ(P t , G t ) = P t -κG t .
(2)

It is worth noting that pollution P t is a stock. Hence, emissions have a long-lasting impact on temperature.

Conversely, the quantity of geoengineering G t is a flow. At each period, geoengineering has to be reimplemented if it is optimal to cool the planet.

The global temperature T t affects the economy through the damage function Ω(T t ). We assume a quadratic form for the damage function:

Ω(T t ) = (T t -γ) 2 .
The damage function is positive, strictly convex and Ushaped with a minimum in γ. This minimum is called the optimal temperature.

Geoengineering is costly. Contrary to the existing literature, we assume that the geoengineering cost function Γ(G t ) is not continuous near Γ(0) = 0 because of the existence of a fixed cost p. This fixed cost represents the large investment in R&D and infrastructure needed to put geoengineering technologies into service. For G t > 0, we assume that the cost function of geoengineering is linear:

Γ(G t ) = 0 if G t = 0 mG t + p if G t > 0
In summary, the objective of the country is to maximize the discounted sum of consumption flows:

max e 1 ,e 2 ,G 1 ,G 2 V(e 1 , e 2 , G 1 , G 2 ) = c 1 (e 1 , G 1 ) + ρc 2 (e 1 , e 2 , G 2 ) = t=1,2 ρ t-1 [e t (α -βe t ) -πe t -(P t -κG t -γ) 2 -Γ(G t )].
The decision variables are the energy consumed and the geoengineering levels at each period of time (e 1 , e 2 , G 1 , G 2 ). The stock of greenhouse gases in the atmosphere at time t = 1, P 1 ≥ 0 is given.

Solution without climate damages

In order to define abatement, we consider the benchmark case of an economy which does not suffer from global warming (Ω(T t ) = 0). The problem is then static. The consumption of energy is chosen such that the marginal benefit of energy equals its marginal cost:

f ′ (e t ) = α -2βe t = π ⇔ e * = α -π 2β . (3)
The solution, denoted e * , is constant, unique and positive as α > π. Figure 1 shows the graphical solution of the maximization. In the lower-part of the figure, the green line represents the production function, and the red line the energy cost. The marginal counterparts of these functions are drawn in the upper-part. The optimal use of energy is defined by the equality between the marginal productivity of energy (the green line) and its marginal cost (the red line).

Contrary to the existing literature [START_REF] Moreno-Cruz | Climate change under uncertainty: A case for geoengineering[END_REF][START_REF] Moreno-Cruz | The long and short of climate change: Abatement versus geoengineering[END_REF][START_REF] Moreno-Cruz | Revisiting the economics of climate change: The role of geoengineering[END_REF], emissions are explictly considered in this model, and abatement is only an indirect decision variable. This difference makes possible the dynamic analysis of geoengineering and climate change. We define abatement A t , as the optimal level of emission without climate damages, e * , minus the emission level at time t, e t : A t = e * -e t .

With geoengineering

The problem is dynamic as present emissions have an impact on both present production and future damages. In order to prevent damages from climate change, the country may mitigate its emissions or invest in geoengineering. The country chooses e 1 , e 2 , G 1 and G 2 to maximize the discounted sum of consumption flows:

max e 1 ,e 2 ,G 1 ,G 2 V(e 1 , e 2 ) = f (e 1 ) -Ω(T 1 ) -πe 1 -Γ(G 1 ) +ρ[ f (e 2 ) -Ω(T 2 ) -πe 2 -Γ(G 2 )],
Subject to:

                   T t = Φ(P t , G t ) P 2 = P 1 (1 -δ) + e 1 G 1 , G 2 ≥ 0 P 1 > 0 given.
Because of the fixed cost, the objective function is not continuous. Therefore, we cannot apply the Mangasarian Lemma: the first-order conditions are not sufficient to have a maximum. Because of the discontinuity near G t = 0, we will compare the solution given by the resolution of the first-order conditions with the solution for G 1 = 0 and/or G 2 = 0 (section 2.4). As both G 1 and G 2 should be positive, we form the Generalized Lagrangean:

max e 1 ,e 2 L(e 1 , e 2 , G 1 , G 2 ) = t=1,2 ρ t-1 [ f (e t ) -Ω(T t ) -πe t -Γ(G t )] + λ[P 2 -P 1 (1 -δ) -e 1 ] + µ 1 G 1 + µ 2 G 2 .
Because of the inequality constraints G 1 ≥ 0 and G 2 ≥ 0, two complementary slackness conditions should hold simultaneously:

µ i ≥ 0 , G i ≥ 0 and µ i G i = 0 , i = 1, 2. ( 4
)
The two slackness conditions imply four cases.

CASE NN: G 1 = G 2 = 0.
Geoengineering is never used to counteract global warming, either because it is too expensive or because the pollution stock is low. The first-order conditions of the maximization program give the optimal levels of emission and geoengineering:

⇔                            e NN = α -π -2ρ[P 1 (1 -δ) -γ] 2(β + ρ) = e * -ρ e * + P 1 (1 -δ) -γ β + ρ e NN = α -π 2β = e * .
(

) (6) 5
The marginal return of emissions at time t = 1 equals its marginal cost π plus the discounted value of the marginal damage due to these emissions. Emissions in t = 1 are positively related to the discount rate ρ, the dissipation of CO 2 δ and the optimal temperature γ. They are negatively related to the energy price π and the initial pollution stock P 1 . Abatement in the first period is positive if emitting e * would induce a temperature in t = 2 which is above the optimal temperature level γ, that is, if emitting more than e * would increase climate damages. As our framework is limited to two periods, emissions at time t = 2 will always be equal to e * as they do not induce damage in the future.

Slackness conditions imply that both µ NN 1 and µ NN 2 are strictly positive:

           µ NN 1 = m -2κ(P 1 -γ) > 0 µ NN 2 = 2βκρ β + ρ m 2κ -[P 1 (1 -δ) -γ + e * - mρ 2βκ ] > 0.
The complementary slackness conditions are sufficient conditions for geoengineering to be suboptimal. However, these conditions are not necessary: if the fixed cost of geoengineering is high, geoengineering may not be implemented even if slackness conditions hold. For sufficient conditions, we will need to compare the discounted sum of consumption flows for the four cases (section 2.4). By reversing the sign of the slackness conditions, we obtain necessary conditions for geoengineering to be optimal.

Graphically, the damage function and its marginal counterpart are represented by the blue lines in figures 1 and 2. The optimal level of emission is determined by the intersection between the marginal productivity of energy (the green line in the upper-part) and the sum of its marginal cost plus the marginal damages at time t = 2 due to emissions at time t = 1 (the dotted blue line in the upper-part). In figure 1, abatement is positive as increasing emissions would increase climate damages. In figure 2

CASE GG: G 1 > 0, G 2 > 0.
If geoengineering is cheap and the initial pollution stock is high, geoengineering is used in both periods to counteract climate change.

The first-order conditions of the maximization program give the optimal levels of emission and geoengineering:

                                 e GG 1 = e * - mρ 2βκ e GG 2 = e * G GG 1 = 1 κ P 1 -γ - m 2κ > 0 G GG 2 = 1 κ P 1 (1 -δ) -γ + e * - mρ 2βκ - m 2κ > 0.
It is worth noting that abatement in the first period is positive even if geoengineering is used. Abatement is positively related to the marginal cost of geoengineering π and to the discount rate ρ, and negatively related to geoengineering efficiency κ. Slackness conditions imply µ 1 = µ 2 = 0, G GG 1 > 0 and G GG 2 > 0. Because of the fixed cost of geoengineering, G GG 1 > 0 and G GG 2 > 0 are necessary, but not sufficient for geoengineering to be optimal. Therefore, if G GG 1 < 0 or G GG 2 < 0, the case GG is not optimal.

CASE NG: G 1 = 0, G 2 > 0. Geoengineering is only used at time t = 2. In the first period, the pollution stock is low and geoengineering is not optimal. Emissions at time t = 1 rise the pollution stock such that geoengineering becomes optimal at time t = 2. The first-order conditions of the maximization program give the optimal levels of emission and geoengineering:

                             e NG 1 = e * - mρ 2βκ e NG 2 = e * G NG 1 = 0 G NG 2 = 1 κ P 1 (1 -δ) -γ + e * - mρ 2βκ - m 2κ > 0.
Again, abatement is positive in the first period even if emissions will be compensated by geoengineering in the second period. The complementary slackness condition at time t = 1 imposes µ NG 1 > 0 (necessary condition), that is: µ NG 1 = m -2κ(P 1 -γ) > 0. At time t = 2, the slackness condition requires G NG 2 > 0. Therefore, if µ NG 1 < 0 or G NG 2 < 0, the case NG is not optimal.

CASE GN: G 1 > 0, G 2 = 0. Geoengineering is only used at time t = 1. In the first period, the pollution stock is high and geoengineering is optimal. Emissions at time t = 1 are low and the dissipation rate of CO 2 is high. Consequently, the pollution stock sharply decreases, and geoengineering is not optimal anymore in the second period. This case is of course not realistic, at least in the short-run. The first-order conditions of the maximization program give the optimal levels of emission and geoengineering:

                                 e GN 1 = e * -ρ e * + P 1 (1 -δ) -γ β + ρ e GN 2 = α -π 2β G GN 1 = 1 κ P 1 -γ - m 2κ G GN 2 = 0.
Abatement in period 1 is high as the country anticipates that geoengineering will not be used in the second period. The two slackness necessary conditions are

G GN 1 > 0 and µ GN 2 = 2βκρ β+ρ m 2κ -[P 1 (1-δ)-γ+e * -mρ 2βκ ] > 0.
If one of these two conditions are not satisfied, the case NG is not optimal.

The optimal case

Until now, we computed the optimal levels of emission and geoengineering for each case. We still have to compare the discounted sum of consumption flows for each case, taking into account that the country has to pay a fixed cost to undertake geoengineering. Let us compare the four cases by plugging the optimal emission and geoengineering levels obtained for each case into V(e 1 , e 2 , G 1 , G 2 ). We find the following results.

Proposition 2.1. Geoengineering at time 1 is optimal if the two following conditions are satisfied:

               p < P 1 -γ - m 2κ 2 P 1 -γ > m 2κ . (7) (8)
Proof Simple algebra gives V GG -V NG = V GN -V NN . Inequality ( 7) is derived from the following equivalences:

V GG > V NG ⇔ V GN > V NN ⇔ p < P 1 -γ -m 2κ 2 .
The inequality ( 8) is derived from the complementary slackness conditions: µ NN 1 < 0 and µ NG 1 < 0 are necessary conditions for geoengineering to be optimal in the first period.

Taken together, conditions ( 7) and ( 8) imply that geoengineering is optimal at time t = 1 if the temperature in the first period, P 1 -γ would be high without the use of geoengineering, and if the price of geoengineering for a one-unit reduction in the temperature, m κ , is low compared to the fixed cost of geoengineering p. Proposition 2.2. Geoengineering at time 2 is optimal if the two following conditions are satisfied:

               β β + ρ P 1 (1 -δ) -γ + e * - mρ 2βκ - m 2κ 2 > p P 1 (1 -δ) -γ + e * - mρ 2βκ > m 2κ . (9)
(10)

Proof The proof is similar. Simple algebra gives V GG -V GN = V NG -V NN . Inequality ( 7) is derived from the following equivalences:

V GG > V GN ⇔ V NG > V NN ⇔ Inequality (9) holds.
The condition ( 10) is derived from the complementary slackness conditions: µ NN 2 < 0 and µ GN 2 < 0 are necessary conditions for geoengineering to be optimal in the second period.

As e GG 1 = e NG 1 = e * -mρ/2βκ, proposition 2.2 may be interpreted as follows. Conditions ( 9) and ( 10) imply that geoengineering is optimal at time t = 2 if the temperature at time t = 2 without using geoengineering would be high and if the price of geoengineering for a one-unit reduction in the temperature, m κ , is low compared to the fixed cost of geoengineering p. Strong preferences for the present (low ρ) increase the relative utility of geoengineering use at time t = 2. These propositions should be contrasted with the results of [START_REF] Moreno-Cruz | Revisiting the economics of climate change: The role of geoengineering[END_REF]. In the present model, inequalities ( 7) and ( 9) induce that geoengineering may not be implemented even if the GHG stock generates climate damages. In Moreno-Cruz and Smulders (2010), geoengineering is always used in the presence of climate damages as the fixed cost is not considered.

Everything else being equal, we derive from these propositions that the initial concentration of GHGs in the atmosphere is higher when geoengineering is implemented.

Corollary 2.3. Geoengineering is implemented in the short-run if the initial concentration of GHGs is high:

P GG 1 > P GN 1 > P NN 1 . (11)
Similarly, geoengineering is implemented in the longrun if the initial concentration of GHGs is high and if the dissipation of GHGs is low:

P GG 1 > P NG 1 > P NN 1 . (12)
In the present analysis of the optimal case, I assumed that the fixed cost is paid each time geoengineering is used. Under this assumption, the fixed cost encompasses investments needed to maintain geoengineering capacities as well as indirect costs related to governance issues or conflict mitigation. If we assume that the fixed cost is paid only once, for example if the fixed cost is related to R&D investments needed to develop geoengineering techniques, then inequalities ( 7) and ( 9) are slightly modified for the case GG (the right-hand side of inequalities ( 7) and ( 9) is divided by (1 + ρ)). Indeed, as the fixed cost is paid only once, using geoengineering in both periods becomes more attractive. The case GN becomes even irrelevant if G GG 2 is positive. This is always the case when the GHG stock is high enough to generate climate damages.

Comparing abatement, geoengineering and temperature

Let us first compare the abatement levels across the four cases. Regarding the level of emission in the first period, we have that:

e NN 1 -e NG 1 = ρ β + ρ m 2κ -[P 1 (1 -δ) -γ + e * - mρ 2βκ ] = µ NN 2 2βκ .
As µ NN 2 < 0 is a necessary condition for geoengineering to be optimal in the second period, we conclude that emissions in the first period are higher when geoengineering is used to compensate the global warming in the second period. Furthermore, as P GN 1 > P NN 1 (corollary 2.3), and as emissions are negatively related to the initial pollution stock for cases GN and NN, we conclude that the following relations hold: e GG 1 = e NG 1 > e NN 1 and e NN 1 > e GN 1 . Hence, the country increases its emissions at time t = 1 if it is anticipated that emissions will be compensated by the use of geoengineering in period 2. Conversely, the country reduces its emissions in the first period if geoengineering is not expected to be implemented in the second period. The following proposition summarizes this reasoning.

Proposition 2.4. Abatement is reduced in the first period if the country anticipates that geoengineering will be used in the second period. This proposition confirms a fear which is common among opponents of geoengineering R&D. They argue that promoting research to develop these artificial techniques may give the false impression that environmental issues are resolved, and therefore discourage efforts to reduce emissions.

Regarding the level of emission in the second period, it is straightforward that: e GG 2 = e NG 2 = e NN 2 = e GN 2 . This set of equalities should be interpreted as an "end of life" effect: emissions at time 2 have no impact on damages as our model is limited to two periods. The long-run evolution of abatement is discussed in section 3.

Second, let us analyze the use of geoengineering over time. When geoengineering is optimal at time t, it is straightforward to see that the amount of geoengineering used is a linear function of the pollution stock:

G t = 1 κ (P t -γ - m 2κ ). ( 13
)
As the 1/κ is positive, we have the following property.

Proposition 2.5. If geoengineering is optimal, the use of geoengineering increases as long as the pollution stock grows, that is, as long as emissions e t are higher than the dissipation of the pollution stock δP t .

Finally, let us assess the impact of geoengineering availability on temperature. For a given initial pollution stock P 1 , it is direct to see that the climate is colder in the first period when geoengineering is used as T 1 = P 1 -κG 1 . In the second period, the availability of geoengineering affects the temperature in two opposite directions. On the one hand, the direct effect of geoengineering is to decrease the temperature. On the other hand, the pollution stock is higher as abatement is reduced in the first period (proposition 2.4). For a given GHGs initial stock P 1 , we have that:

T GG 2 -T NN 2 = T NG 2 -T NN 2 = µ NN 2 /2κρ
. As µ NN 2 < 0 is a necessary condition for geoengineering to be used, we conclude that the net impact of the two opposite effects on temperature is negative. The following proposition summarizes this statement.

Proposition 2.6. Even if the availability of geoengineering reduces abatement, the temperature is lower when geoengineering is used.

By plugging equation (13) into equation (2), we find that the temperature with geoengineering is constant and equal to:

T Geo ss = γ + m 2κ (14)
This constant temperature is positively related to the optimal temperature, γ, and to the marginal cost of geoengineering, m. Conversely, this temperature is lower when geoengineering is highly effective, that is, if κ is high.

Graphical solution

The damage function and its marginal counterpart are represented by the blue lines in figures 3 and 4. Without geoengineering, the optimal level of emission is determined by the intersection between the marginal productivity of energy (the green line in the upper-part) and the sum of the marginal cost of energy and the second period marginal damages induced by emissions at time t = 1 (the dotted blue line in the upper-part). The amount available for consumption is equal to the blue area A. The brown area B represents the loss due to climate damages. In figure 4, the fixed cost of geoengineering is represented by the area in red. The hatched red area is its variable cost: for a marginal increase of geoengineering, the marginal damages curve moves to the right such that climate damages decrease. Each unit of geoengineering costs m. The country will increase its use of geoengineering as long as the avoided damages are higher than the variable cost m. It is worth noting that first period abatement is reduced as climate damages are lower (proposition 2.4).

Finally, the country will implement geoengineering if and only if the brown area A in figure 3 is bigger than the brown area A'+A" in figure 4, or, similarly, if the brown area A" is bigger than the red area p. This condition is the graphical equivalent of inequality (9).

Discussion: A long-term scenario

Even without an infinite-horizon model, a realistic analysis of the long-term use of abatement and geoengineering is possible thanks to the 2-periods model. Figure 5 represents the long-run evolution of emissions, pollution stock, geoengineering and temperature, depending on whether the country undertakes geoengineering (plain lines) or not (dotted lines). If the initial GHG stock is low (i.e. at the preindustrial level), conditions (7) to (10) are not satisfied and geoengineering is not implemented. Emissions are high because future damages due to current emissions are low. Abatement may even be negative if the initial stock of pollution is such that the initial temperature is below its optimal level γ. The pollution stock increases over time. Consequently, emissions diminish because their impact on damages becomes more and more harmful.

If geoengineering is cheap, geoengineering becomes optimal when the pollution stock exceeds a certain threshold. Because the country anticipates the implementation of geoengineering, emissions will be increased even before the threshold is reached (proposition 2.4). Hence, the long-term level of emission and the long-term stock of pollution will be higher when geoengineering is undertaken. However, the temperature is lower when geoengineering is used, even if abatement is reduced (proposition 2.6).

Because of the fixed cost (accounted for in inequalities ( 7) to ( 10)), geoengineering is not always implemented, even if the GHG stock generates climate damages. When the pollution stock and the climate damages are high enough, geoengineering will be suddenly used in large quantities. Indeed, by plugging equation (13) into a generalized formula for inequalities ( 7) and ( 9), we have 0 < p ≤ κ 2 G 2 t . Consequently, the temperature jumps abruptly from a high level to a low level, which may disrupt sensible climate equilibrium or fragile ecosystems. Once geoengineering is implemented, the temperature remains constant at a level slightly above the optimal temperature γ (taking into account that the variable price of geoengineering, m, is low). Emissions remain constant, and geoengineering increases proportionally to the pollution stock so as to maintain the temperature constant. Abatement is also constant, and positive. This long-term analysis should be contrasted with the results of Moreno-Cruz and Smulders (2010). As there is no fixed cost in their model, geoengineering is always used in presence of climate damages. Moreover, as geoengineering may be used in very small quantities, [START_REF] Moreno-Cruz | Revisiting the economics of climate change: The role of geoengineering[END_REF] would not be able to find a jump in geoengineering and temperature. However, this jump is important to characterize as rapid changes may have dramatic effects if the environment and ecosystems are not able to adapt as fast.

Our long-run analysis also underlines disconcerting intergenerational issues. Indeed, while current generations will anticipate the use geoengineering by increasing their emissions, future generations will have to reduce their emissions, to bear the cost of sustaining geoengineering and will suffer from its negative sideeffects. If emissions are not mitigated, future generations may even not have the choice to use geoengineering, especially in presence of "tipping points". Furthermore, geoengineering will have to be sustained for centuries in order to avoid dramatic consequences induced by a sharp increase in temperature [START_REF] Goes | The economics (or lack thereof) of aerosol geoengineering[END_REF]. Hence, the forthcoming availability of geoengineering methods increases the intergenerational negative transfer from now to the future. This is particularly objectionable as current generations from developed countries already live beyond the means of Earth system.

Conclusion

Geoengineering is troubling. Indeed, while it is expected to have a high cost-effectiveness at preventing damages from global warming, it will nevertheless generate worrying side-effects. For example, the two most popular SRM methods, clouds albedo enhancement and sulfate aerosols injections are expected to have nonuniform effects, and to induce significant regional climate changes. Furthermore, Stratospheric sulfate injection may have adverse consequences on stratospheric ozone and biological productivity, and cloud albedo enhancement will affect weather patterns and ocean currents.

In this paper, I showed that geoengineering availability will also induce indirect side-effects, both in the short-and the long-run. Using a two-period dynamic model with climate change, in which geoengineering may be used as an alterative to abatement, I deduced the conditions under which geoengineering is undertaken. I confirmed the fear of geoengineering opponents who argue that the anticipation of geoengineering availability will dangerously decrease abatement in the short, and in the long-run. Furthermore, I showed that the optimal path of geoengineering use is characterized by a jump, which will in turn induce a sharp and sudden decrease in temperature. This abrupt temperature drop may prove to be particularly damaging for fragile ecosystems and vulnerable regions. These indirect consequences raise disturbing intergenerational issues. The net benefit from geoengineering for current generations is positive as they may increase their emissions without engaging in costly abatement. For future generations, the situation appears to be much worse: as they inherit a higher stock of pollution, they may be compelled to implement geoengineering in order to prevent catastrophic climate damages and to avoid reaching "tipping points". Consequently, future generation will have to support the negative side-effects of geoengineering, as well as to bear the full cost of geoengineering, which includes R&D investments as well as the capacities required for maintaining high levels of geoengineering for centuries.

These conclusions underline the need of an urgent research agenda paired with large-scale experiments of promising geoengineering methods, in order to assess the effectiveness and the dangerousness of each methods, and to share more equally the whole cost of geoengineering across generations.

Figure 1 :

 1 Figure 1: Emissions are lower with the damage function

Figure 3 :

 3 Figure 3: Potential gain from geoengineering

  Figure 4: Optimal geoengineering

Figure 5 :

 5 Figure 5: Long-run scenario: emissions, temperature, pollution stock and geoengineering

A third approach, adaptation, proposes to render the society and the environment more robust to the changes in climate that are occurring. The evaluation of this third solution is beyond the scope of this paper.