Clément Bosquet
email: c.bosquet@lse.ac.uk.

Pierre-Philippe Combes
email: ppcombes@univmed.fr.

Are academics who

Keywords: J24, O31, J45 economics of science, productivity determinants, knowledge diffusion, publication scores, citation indexes

Thanks to a unique individual dataset of French academics in economics, we explain individual publication and citation records by gender and age, coauthorship patterns (average number of authors per article and size of the co-author network) and specialisation choices (percentage of output in each JEL code). The analysis is performed on both Econ-Lit publication scores (adjusted for journal quality) and Google Scholar citation indexes, which allows us to present a broad picture of knowledge diffusion in economics.

Introduction

Up until recently, assessing the quality of an article through the average quality of the journal in which it was published was the only possible way to go. However more and more citation indexes exist for individual publications, as for example those of the Journal of Citations Report, Scopus, or Google Scholar. Currently, the two ways of assessing publication quality, through the average quality of the journal or individual citations, co-exist. There has been very few attempts to assess whether the two approaches really measure the same phenomenon. [START_REF] Abramo | Citations versus journal impact factor as proxy of quality: could the latter ever be preferable?[END_REF] for instance compare Italian university rankings obtained when using Web of Science citations and journal impact factors over 2004-2007 but they do not control for any of the characteristics of the universities, nor of their academics.

Using an individual dataset of French academics and their publications over 1969-2008, we provide the first econometric assessment of the determinants of the two types of measures and we evaluate the extent to which they are substitutes or not. Importantly, we control for a large number of co-variates at the individual level as age, gender, field of specialisation, team size, and network.

We propose to answer three sets of questions. First, are some of the standard determinants of productivity in market activities (age and gender) also determinants of productivity in academic research, or do other variables (as typically the size of the author's team and network) also play an important role? Second, to what extent are publications and citations records driven by specialisation fields? Third, do individual citations and publication scores adjusted for journal quality measure the same dimension of academic productivity, and which variables (age/gender, specialisation, or team size/network) drive the gap between the two?

We use an exhaustive dataset of French academic economists in 2008, their publication records in EconLit and their Google Scholar citation indexes. Alongside the academic's age, age-squared and gender, we introduce a first variable specific to the organisation of labour in academic research, which is the average number of authors per publication (we refer to this variable as the author's team size). We find that larger author teams have a more robust impact on publications adjusted for journal quality and citations than standard Mincerian determinants like age or gender. We also introduce the number of published articles and the size of the co-author network (the author's total number of different co-authors) as determinants of the average quality of publications. We demonstrate increasing returns to scale with respect to both variables. Academics who have published more articles and who have had more different co-authors reach a higher average quality of publications.

Then, we introduce specialisation patterns measured by the share of each academic's articles in each JEL (Journal of Economic Literature) classification code. It turns out that, even if these specialisation choices do not have much impact on the overall results described above, some evidence of disparities between fields are observed, both in terms of publication and citation patterns. For instance, French academics specialised in the fields of "Microeconomics" and "Labor and Demographic Economics" publish more articles, of a higher average quality, and they are more cited than the average. Finally, regressing citation indexes on publication scores and on the variables mentioned above, we find that the largest part of the variance in citations is explained by publication scores. This allows us to conclude that on the whole, publications adjusted for journal quality and citations measure the same dimension of publication productivity. Nevertheless, we observe some non-random deviations between the two measures for specific over-or undercited fields and due to strong team size and network effects related to the organisation of the research and publication activity. For a given publication record, larger team size and a larger co-author network generate more citations. Since the publication volume and journal quality are controlled for, we interpret this result as a pure impact of team size and network on knowledge diffusion at identical levels of academic activity. This can emerge from two types of effects. First, different co-authors of an article present their study in different conferences, seminars, informal talks, etc. The more numerous the co-authors are, the stronger the diffusion. Second, because academics talk about their new papers to their former co-authors, the more numerous these latter are, the larger the knowledge diffusion to their colleagues and their other co-authors. While these effects are fairly intuitive, our study, using both individual publications and citations simultaneously, is the first, to the best of our knowledge, to provide systematic evidence on them and to quantify their overall magnitude.

In the 1970s, the quantity of academics' publications was shown to be an important determinant of academic wages. [START_REF] Katz | Faculty salaries, rates of promotion and productivity at a large university[END_REF] evaluates the return to an article between $18 and $102 in 1969 in one large US public university. In a multi-equation system modelling jobquality, research productivity and earnings of a sample of 863 economists in 1966, [START_REF] Hansen | Modeling the earnings and research productivity of academic economists[END_REF] find that experience, measured by the number of years since Ph.D., has a significant impact on research productivity. They also evaluate the return to an additional published article or book at an almost 8% increase in annual earnings. From the 1980s, the quality of publications, measured by citations, has been shown to be a more important determinant of salaries than their quantity. On a sample of 148 full professors of economics at seven large public universities, [START_REF] Hamermesh | Scholarship, citations and salaries: Economic rewards in economics[END_REF] show that citations, which they define as indirect contributions to knowledge, have a superior effect on academic earnings than the number of publications, which they define as direct contributions to knowledge. [START_REF] Diamond | What is a citation worth?[END_REF] goes further in the analysis and estimates that the marginal value of a citation lies between $50 and $1,300, depending on the discipline.

With a panel of 140 academic economists, [START_REF] Sauer | Estimates of the returns to quality and coauthorship in economic academia[END_REF] confirms this existence of incentives to knowledge growth and estimates that an individual's return from a co-authored paper with n authors is approximately 1/n times that of a single-authored paper. Finally, [START_REF] Kenny | Economists' salaries and lifetime productivity[END_REF] show that economists' salaries are best characterised by implicit long-term contracts (predicted publications and citations) than by current productivity (presumably because of mobility costs) and find an insignificant effect of field choice on academic wages. This paper takes a similar perspective, assessing the respective roles on research productivity of age, gender, the number of articles published, and specialisation, to which we add the role of team size and networks.

Adopting a macro perspective, [START_REF] Lovell | The production of economic literature: An interpretation[END_REF] estimates production functions of publications and citations, which he defines as contributions to economic knowledge. At the aggregate level, considering previously published articles as the stock of capital and the number of PhDs granted in the USA as the labour input in a Cobb-Douglas production function, he explains the tendency of scientific literature to grow exponentially between 1895 and 1965 by the exponential growth of the labour input. Here, we take another road and study the micro determinants of publications and citations, which have been proved to be important determinants of academic wages and promotions and are therefore considered to be evidence of research productivity. Hence, this allows us to shed some light on the impact on knowledge creation and diffusion of the behaviour of academics in terms of co-authorship, specialisation choices and research strategy in general. Recently, some studies have studied the role of seniority [START_REF] Mishra | Are more senior academics really more research productive than junior academics? Evidence from australian law schools[END_REF], gender (van Arensbergen, van der Weijden and van den Besselaar, 2012) and networks [START_REF] Badar | Examining the relationship of co-authorship network centrality and gender on academic research performance: The case of chemistry researchers in pakistan[END_REF][START_REF] Egghe | Measuring co-authors' contribution to an article's visibility[END_REF] on different measures of academic productivity. However the impact of these variables has been never studied simultaneously, with the further possible role of specialisation, which we do here. From a policy point of view, understanding and quantifying such mechanisms is an important step in designing a more efficient academic system. Finally, [START_REF] Stigler | The citation practices of doctorates in economics[END_REF] is the study the most closely related to ours. They examine the citation of articles published between 1950 and 1968 in two economic sub-fields, cited in doctorates in economics from six major US universities, to identify patterns of intellectual debtors and creditors. Regressing the number of citations -considered as a measure of intellectual influence -on the number of published articles, they find some evidence of weak but significant increasing returns to quantity. However, their specification does not include the age, gender, team size, and network variables we consider here.

Another contribution of this paper is the use of a new tool to measure the impact of academics: their Google Scholar citations. Standard studies on citations use the Journal of Citation Reports but even its latest version only refers to about 300 journals in economics, and many less were used in most of the studies mentioned above, in which citations of and by articles in non-referenced journals are excluded. Using Google Scholar citations presents the decisive advantage of taking into account a much wider range of supports for both cited and citing articles and it also includes books, working papers, and policy reports, since all supports on academic websites are concerned. As a consequence, Google Scholar is sometimes considered as a better tool for comparisons across disciplines [START_REF] Landry | Counting citations in the field of business and management: why use google scholar rather than the web of science[END_REF][START_REF] Harzing | A preliminary test of google scholar as a source for citation data: A longitudinal study of nobel prize winners[END_REF] and we also contribute to this emerging literature that assesses the properties of Google Scholar citations. As regards our measures of publications adjusted for journal quality, we use all 1206 Econlit journals, and their relative quality, which is also a large extension with respect to previous studies. Finally, we do not restrict the academic sample used, which can induce selection biases when, for instance, only the best universities are kept;

we consider the full sample of 2782 French academics in 2008, whether they publish a lot or not at all. Data and the econometric strategy are presented in sections 2 and 3, respectively. The individual determinants of publication scores and citation indexes are analysed in section 4.

Section 5 tests the robustness of the findings when specialisation choices are taken into account. Comparing citation indexes and publication scores, section 6 analyses the patterns of knowledge diffusion and section 7 concludes.

Data

Measure of output

We measure the research output of the academic i in two ways: her number of publications adjusted for journal quality and her number of citations in Google Scholar.

Publication records

Publication records are measured as weighted sums of articles referenced in EconLit, which lists more than 560,000 publications in more than 1200 journals between 1969 and 2008. Three dimensions enter the weighting of publications: the relative number of pages, the number of authors and the quality of the journal.

We take into account the number of pages to capture the idea that longer articles contain more ideas (normal vs short papers in the American Economic Review, for instance). However, since the layout can be very different from one journal to another and since we do not want to favorise some journals for that reason, the weighting is made within each journal. The weight is the ratio of the number of pages of article a over the average number of pages of the articles published in that journal in the same year, which assumes the consistency of editorial policy over one year only. By contrast, differences in the length of articles between journals (Economic Letters vs American Economic Review, for instance) are considered to be captured directly by the journal quality index (presented below).

We also take into account the number of authors of each article. As is standard, we assume an equal split of the article between its authors.

Finally, we take the quality of publications into account using the Combes and Linnemer (2010) journal weighting scheme. For robustness purposes, we compare our results using two of their indexes assuming different degrees of convexity in the distribution of journals' weights.

To sum up, the output of academic i is a weighted sum of her articles a: We refer to these two schemes as the "Quality" and "Top quality" publication measures respectively. They are illustrated for the top 50 journals in Appendix A. E is referred to as "Quantity" (which is corrected for the number of authors only).

y i = a W (

Citation records

We assess citations through the Google Scholar citations of articles, books and working papers (which we refer to as entries) written by the academics of our database. These citations were extracted on January 2010, around two years after the date at which we want to measure research productivity, which seems quite reasonable given the time needed for studies to be cited.

In order to avoid problems of homonyms involving academics with identical names in fields other than economics, we restrict the fields on Google Scholar to the "subject areas" "Business, Administration, Finance, and Economics" and "Social Sciences, Arts, and Humanities". To have a period of time comparable to that used for EconLit, we only keep entries dated between 1969 and 2008.

As we did for publications, we take into account the number of authors for each entry.

However, it is no longer necessary to take into account the quality of the support or the relative length of publications, since we assume that the number of citations directly reflects the entry quality.

We first build an index of total citations, T C i , which is the total number of citations received by all academic i's entries, each divided by its number of authors. Then, to combine this (quality-adjusted) measure of quantity with something closer to the average quality of publications, we use a synthetic index. We do not use the famous H-index, H i , proposed by [START_REF] Hirsh | An index to quantify an individual's scientific research output[END_REF], because typically two academics can have the same H-index when one of them has some very highly-cited entries and the other not. In other words, the H-index ignores the internal distribution of citations received by the articles used to calculate it, and we consider that as one of its strong limits. Therefore we prefer to use the G-index, proposed by [START_REF] Egghe | Theory and practise of the g-index[END_REF] which states that academic i has a G i -index equal to g, which is unique, if her g most-cited articles have received g2 citations in total, or g citations on average. It can be shown that G i ≥ H i . The difference between the two indexes relates to the number of citations received on average by the most-cited articles.

To take the number of co-authors of each article into account, we follow [START_REF] Schreiber | To share the fame in a fair way, hm modifies h for multi-authored manuscripts[END_REF] who proposes to attribute all its citations to any entry but simply a fraction of the entry to each author. Then the H and G indexes are not necessary integers anymore, but they keep the same signification. For instance, a G-index of 7.5 means that the academic has had published at least 7.5 "single-author equivalent" articles with 7.5 citations each on average.

Population and descriptive statistics

The EconLit database, which enables us to compute our different scores of publication, is matched with a list of academics in economics provided by the French Ministry of Education and Research.1 In 2008, year for which the analysis was conducted, 2782 academics were considered.

2

Table 1 presents descriptive statistics about our main dependent and independent variables. The average academic was around 47 years old, had published 3.5 (single-author equivalent) articles referenced in the EconLit database between 1969 and 2008, had entries cited 107 times with a G-index of 7.25 (meaning that her 7.25 most-cited entries had been cited 7.25 times on average). In 2008, 30% of French academic economists were women and 73% had published at least one article. We refer to them as the "Published". Also, 85%

have at least one entry with at least one citation in Google Scholar; we refer to them as the "Cited".

If we restrict our sample to the academics who have published at least one article referenced in the EconLit database (Panel (b)), the average published author is slightly younger (around 46 years old) and more likely to be a man (27% of women, compared with 30% for all academics). She has published 4.8 single-author equivalent articles, has been cited 137 times and has a G-index of 9 on average. 95% of published academics have been cited at least once on Google Scholar. The average number of authors per article is around 2 (1.85 in EconLit, 2.04 in Google Scholar) while the average network size (total number of different co-authors) is 4.12.

Alternatively, if we restrict our sample to academics who have been cited at least once on Google Scholar (Panel (c)), 81% have published at least one article and 28% are women. The average cited academic has published 4.05 single-author equivalent articles, has been cited

126 times and has a G-index of 8.5. The average network size of cited academics is slightly smaller (3.49 vs 4.12) than that of published academics.

Finally, Table 2 provides some simple correlations between EconLit publication scores and Google Scholar citation indexes. 3 The following observations are noteworthy. As expected, the academics who have published more articles are more cited. The correlation between citation indexes and publication scores is higher when the quality of journals is taken into account with a medium degree of convexity rather than a high degree of convexity. However, the average quality of publications is more correlated with citation indexes when there is a high degree of convexity in the journal weighting scheme. Quantity: number of single-author equivalent articles in EconLit (E). Quality and top quality: publication scores with a low and high degree of convexity in the journal weighting scheme total scores (CLm and CLh), respectively. Average quality and top quality: CLm/E and CLh/E. Correlations are computed on logarithms to match the econometric specification presented in section 3.

Econometric specification

For our different measures of publication scores or citation indexes, we estimate the following specification using ordinary least squares:

log y i = β 0 + β 1 Gender i + β 2 Age i + β 3 Age 2 i + β 4 log nau i + β 5 log(1 + N et i ) + 18 j=1 γ j y ij y i + i
where y i is the publication score or citation index of academic i, Gender i is a dummy variable taking 1 for women, nau i is the average number of authors per publication by academic i and N et i is the size of her co-authorship network, i.e., her total number of different co-authors.4

Finally, y ij y i is the share of academic i's output in JEL code j at the first letter level (calculated in terms of E, the number of single-author equivalent articles published in EconLit). 56 Since some academics have never published an article in EconLit or have no citation on Google Scholar, we also take selection into account using a Heckman 2-step procedure. The first step is the probit selection equation and the second step is the main equation augmented by the inverse of Mills' ratio. In the present case, this corresponds to a model where academics who have not published or are not cited are those who do not reach a sufficient quality threshold in their research activity, and this can be explained by the same type of variables as those that determine the level of publication or citations. Unfortunately, for this very reason, it is pretty difficult to find exclusion restrictions -variables that would explain the probability of being published or cited, but not the levels. Therefore the selection effect is identified on non-linearities only.

Individual determinants of publication and citation records

In this section, we analyse the individual determinants of publication and citation records by regressing total publication scores (E -the number of single-author equivalent articles, and CLm and CLh, the quality and top quality publication scores using a medium and a high degree of convexity in the journal weighting scheme, respectively) and citation indexes (T C -Google Scholar total citations discounted by the number of authors per paper and G -Google Scholar G-index) on gender, age and its square and the average number of authors per article. This is presented in Table 3. We also regress CLm/E and CLh/E, the average quality and top quality of publications on the same explicative variables augmented by E (quantity) and the network size to identify possible increasing returns to quantity and co-authorship. Since network size is by construction highly correlated with quantity, it is included in the regressions only when quantity is also included to identify the network effect on top of its natural quantity effect. Except for age and gender, all variables are in logs.

Women are less productive, whatever the measure of research output. Older academics publish more articles, which are on average of lower quality. These two effects cancel out when the dependent variable is the total publication score taking quality of publications into account. Older academics are also more cited (with a slightly concave effect), which could be due to the fact that their articles were published longer ago and have therefore had more time to be cited. Moreover, we find increasing returns to the average number of co-authors own network in this case.

per article for all research output measures, except for the number of single-author equivalent articles. A published academic who has on average two co-authors instead of only one (per article) has 10.1% less publications but their average quality is 8.4% higher and their average top quality is 36.8% higher.7 Her total quality and top quality publication scores are 11.4% and 96.9% higher, respectively, while she is cited 53.4% more and has a G-index 41.8% higher. 8We also find increasing returns to the individual quantity and to the size of the co-authors' network on the average quality of publications, and they are stronger when the journal's quality assessment is more selective. The more articles academics have published, the more different co-authors they have had, the higher the average quality of their publications. Researchers who have published five single-author equivalent articles instead of four have an average quality of publications 3% higher and an average top quality of publications 15.5% higher.9 Having a stock of five different co-authors instead of four, meaning a total network size of 6 instead of 5, increases average publication quality by 4% and average publication top quality by 15.5%.10 Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. E = number of single-author equivalent articles in EconLit. CLm and CLh are publication scores with a low and high degree of convexity in the journal weighting scheme, respectively. CLm/E and CLh/E = average quality and top quality. T C = Google Scholar total citations. G = Google Scholar G-index.

In Table 4, we repeat the same exercise as in Table 3 except that we take selection into account using the Heckman 2-step procedure. Hence, in the first column of Table 4, we run a probit equation of the probability of having been published, which allows us to calculate the inverse of Mills' ratio ("Selection") that we include in the columns 2 to 6. The same exercise is done for Google Scholar citation indexes in columns 7 to 9. Without clear exclusion restrictions, the inverse of Mills' ratio should possibly control for the presence of non-linearities in the model. Comparing the results from Table 3 and Table 4, we find that if women have published less articles in EconLit on average, it is because of the women who have not published at all.

Once we have controlled for this selection process, published women have had as many papers published as men, and cited women are cited as often as cited men.

Taking selection into account, older academics do not produce lower quality articles but they are still more cited. Results on the increasing returns to the average number of authors per article and to the quantity of publications and the size of the co-authorship network for the average quality of publications are not impacted by the Heckman procedure.

The impact of specialisation choices

In this second step, we test the robustness of section 4's results and analyse the effect of specialisation choices by including the shares of research output published in each field (JEL codes at the first letter level) as a control variable. This tests whether academics specialised in certain fields are published more or cited more. 11

Specialisation only

We start by regressing publication scores and citation indexes on specialisation shares alone.

As seen in Table 5, French economists specialised in the fields of "General Economics and

Teaching" (A), "Macroeconomics and Monetary Economics" (E), "Microeconomics" (D) and "History of Economic Thought, Methodology, and Heterodox Approaches" (B) have published more articles than the average (by order of magnitude of the coefficients). At the other extreme, academics specialised in "Business Administration and Business Economics; Marketing; Accounting" (M), "Economic History" (N) and "Health, Education, and Welfare" (I)

have published less single-author equivalent articles.

The French economists' average quality of publications is higher in the fields of "Mathematical and Quantitative Methods" (C), "Microeconomics" (D), "Public Economics" (H), "Labor and Demographic Economics" (J) and "Macroeconomics and Monetary Economics" (E) (by order of magnitude of the coefficients). Total publication scores are higher than average in the same fields (except for Public Economics (H) in CLm), meaning that fields which are over-represented in terms of quantity as "General Economics and Teaching" (A) or "History of Economic Thought, Methodology, and Heterodox Approaches" (B) are published in low-quality journals.

In terms of Google Scholar citation indexes, French economists specialised in the fields of "Microeconomics" (D), "Labor and Demographic Economics" (J), "Agricultural and Natural

Resource Economics; Environmental and Ecological Economics" (Q) and "Mathematical and Quantitative Methods" (C) (only considering total citations for the latter) are more cited than the average. At the other extreme, French economists specialised in the fields of "History of Economic Thought, Methodology, and Heterodox Approaches" (B), "Financial Economics" (G) and "Economic History" (N) are less cited than the average.

Interestingly, comparing Table 5 with Table 3 in section 4, we observe that the R 2 of the model is higher with specialisation shares alone than it is with individual characteristics alone to explain total publication scores. The total explained variance by specialisation is 13.8% for CLm and 19.5% for CLh, whereas the total explained variance by individual characteristics (Table 3) is 3.4% for CLm and 6.6% for CLh. By contrast, the R 2 of the model explaining citation indexes is higher when we introduce individual characteristics alone (8.2% and 10.1%

for Google Scholar total citations and G-index, respectively) than it is when we introduce specialisation choices alone (2.9% and 3.4% for total citations and G-index, respectively). In 11 The number of observations in the regressions falls slightly because some academics have only published articles for which JEL codes are not recorded in EconLit. Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. E = number of single-author equivalent articles in EconLit. CLm and CLh are publication scores with a low and high degree of convexity in the journal weighting scheme, respectively. CLm/E and CLh/E = average quality and top quality. T C = Google Scholar total citations. G = Google Scholar G-index.

any case, the variance explained by individual characteristics is weak compared with what is found in the literature on wage equations for instance (in which individual characteristics often explain between 30 and 40% of wages, a measure of labour productivity). In academic research, specialisation choices appear to be a stronger determinant of publication outcome than the individual variables we considered in section 4, while the reverse is true for citations. Now we need to assess whether the conclusions we presented in section 4 are driven by the omitted specialisation variables or if the two sets of variables correspond to different effects.

Specialisation and individual characteristics

We now introduce both specialisation patterns and individual characteristics. We conclude from Table 6 that the average number of authors per article is the only individual characteristic that matters for explaining total publication scores and citation indexes. As found in section 4, it has a negative impact on the quantity of published articles and a positive impact on average quality of publications, total publication scores (except in CLm) and citation indexes. Coefficients associated with age, its square and gender are never significant anymore, underlining the fact that the demographic impact on total publication scores observed in section 4 was in fact due to specialisation choices.

This contrasts with results on the average quality of publications, which is still impacted, in the same order of magnitude, by increasing returns to quantity and size of the co-authorship network. Then we also observe that French academics in economics specialised in the fields of "Microeconomics" (D) and "Labor and Demographic Economics" (J) have more articles of a higher average quality published, hence reach higher total publication scores and are more cited. French economists specialised in the fields of "Macroeconomics and Monetary

Economics" (E), "Mathematical and Quantitative Methods" (C) and "Public Economics" (H) also have higher publication scores but are not more cited than the average. At the other extreme, academics specialised in the fields of "History of Economic Thought, Methodology, and Heterodox Approaches" (B) and "Economic History" (N) publish lower-quality articles and are less cited than the average, whereas academics specialised in "Business Administration and Business Economics; Marketing; Accounting" (M) and "Urban, Rural, and Regional

Economics" (R) also publish lower-quality articles but are not less cited. 6 The patterns of knowledge diffusion

Finally, we compare citation indexes and publication scores to assess whether they measure the same dimensions of research productivity. The above analysis shows that this is not fully the case, since the effects of certain variables differ in significance and magnitude for the two types of variables. To go further into the analysis, we regress Google Scholar total citation indexes and G-indexes on EconLit publication scores, age, gender, specialisation fields, team size and co-author networks. This allows us to analyse the patterns and determinants of knowledge diffusion, that is the number of citations received everything else being equal, including the number and quality of publications. In other words, publication scores and individual citations could differ only because the former are more noisy (because quality is assessed at the journal level and not at the individual level), but not on average. If we show that some variables explain the gap between the two, this means that citations and publication scores do not measure exactly the same dimensions of productivity. And this also allows us to quantify the determinants of the non-random divergence of citations with respect to publication scores, i.e., the driving forces behind citations, reflecting knowledge diffusion, at identical levels of publication.

Citations regressed on publications and specialisation

We start by regressing citation indexes on publication scores (quantity and average quality of publications) and specialisation shares to assess whether citation patterns differ from one field to another. Unsurprisingly, academics who publish more and in journals of higher quality are more cited. This confirms that Google Scholar citation indexes capture both the quantity and quality aspects of publication records. In Table 7, the total R 2 of the model is slightly higher when the average quality of publications is measured with a high degree of convexity in the journal weighting scheme. With this measure, having published five single-author equivalent articles instead of four increases Google Scholar total citations by 22.4% and the Google Scholar G-index by 11.9%.12 On top of that, increasing the average publication top quality by 10% increases Google Scholar total citations by 2.1% and the Google Scholar G-index by 1.2%.13

Controlling for publication scores, academics specialised in the fields of "Business Administration and Business Economics; Marketing; Accounting" (M), "Industrial Organization" (L), "Agricultural and Natural Resource Economics; Environmental and Ecological Economics" (Q) and "Urban, Rural, and Regional Economics" (R) ( by order of magnitude of the coefficients) are relatively more cited. At the other extreme, academics specialised in the fields of Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. Quantity: number of single-author equivalent articles in EconLit (E). Average quality and top quality: CLm/E and CLh/E. T C: Google Scholar total citations. G: Google Scholar G-index.

"History of Economic Thought, Methodology, and Heterodox Approaches" (B), "Macroeconomics and Monetary Economics" (E), "Microeconomics" (D), "Public Economics" (H) and "Financial Economics" (G) are relatively less cited, given their level of publications.

6.2

The determinants of citations when publications have been controlled

for Finally, we regress citation indexes on publication scores (quantity and average quality of publications), specialisation, and individual characteristics including co-authorship variables.

The results of section 6.1 are robust to the introduction of individual characteristics, except that academics specialised in "Agricultural and Natural Resource Economics; Environmental and Ecological Economics" (Q) are no longer over-cited and academics specialised in "Financial Economics" (G) are no longer under-cited. Controlling for the quantity of published papers and the average quality of these publications, older academics have slightly higher total Google Scholar citation scores, which could be due to the fact that their articles were published longer ago and hence have had more time to be cited. But they do not have higher G-indexes.

Importantly, at given publication records, academics with a higher average number of authors per article and with a larger network size (total number of different co-authors) are significantly more cited. Our interpretation of this result is that larger team sizes generate more knowledge diffusion through conferences, seminars, informal talks, etc. Also, academics may generate knowledge diffusion of their new publications through their former co-authors, which is stronger when they are more numerous. When the average quality (and not the top quality) is controlled for, a published academic who has on average two co-authors instead of only one (per article) is cited 9.7% more and has a G-index 20.5% higher.14 Moreover, a stock of five different co-authors instead of four, meaning a total network size of 6 instead of 5, increases Google Scholar total citations by 5.7% and the Google Scholar G-index by 4.2%.15

We perform a variance analysis to infer the relative explanatory power of each variable (or group of variables). To do so, we first calculate the effect of each variable as the product of the variable and its coefficient. Then, we calculate the standard error of this effect on all observations and the correlation coefficients between the calculated effects and the dependent variable. To have an important explanatory power of the dependent variable, the effect of an explanatory variable should first have a high standard error by comparison with the standard error of the dependent variable. Most importantly, a variable -or a group of variables -has a large explanatory power when its effect is largely correlated with the dependent variable.

In Table 9, we perform the variance analysis of the estimations reported in columns 3 and 5 of Table 8, where the log of Google Scholar total citations is the dependent variable. In Table 10, we perform the variance analysis of columns 4 and 6 of Table 8, where the log of the Google Scholar G-index is the dependent variable. In both cases, with a standard error equal to around 55 -60% of the dependent variable and a correlation coefficient of around 0.65, EconLit publication scores are the most important determinants of citation indexes. 16 In this group of variables, the quantity of publications has a much higher explanatory power than the average quality of these publications with a standard error between 1.5 times and twice as large and a correlation coefficient 1.5 times as large, whatever the journal weighting scheme.

Then, we observe that other explanatory variables have much weaker explanatory powers of citation indexes than publication scores. For total Google Scholar citations, even if individual demographic characteristics (gender and age) have a higher standard error than co-authorship effects, the correlation coefficient of the latter with the dependent variables are 1.8 times as large as the correlation coefficients of individual demographic characteristics. 17 For Google Scholar G-indexes, it is even clearer that co-authorship patterns matter more than individual demographic characteristics, since both standard errors and correlation coefficients with the dependent variables are higher for co-authorship effects than for individual demographic characteristics (importantly, correlation coefficients are more than twice as large). 18 Hence, the determinants of publication productivity seems to differ from those in market activities and encompass network effects that are typical of this activity.

In the "individual demographic characteristics" group of variables, age has the largest explanatory power. In the "co-authorship patterns" group of variables, network size has a larger explanatory power than the average number of authors per article. Knowledge diffusion is driven more by total co-authorship network size than by team size. Finally, even if we observe some disparities between fields in the citation practices described above, we observe in Tables 9 and10

Conclusion

We study the individual determinants of EconLit publication scores and Google Scholar citation indexes of French academic economists. We show that when co-author patterns have been controlled for (the average number of authors per article and the total co-author network size), gender and age do not matter anymore, except for the probabilities of being published and of being cited at least once. Moreover, we carefully analyse the role of the specialisation patterns of the academics. Those specialised in "Microeconomics" and "Labor and Demographic

Economics" publish more articles of a higher average quality and are more cited.

Importantly, we exhibit increasing returns to quantity and to co-author network size for the average quality of publications, whatever the chosen weighting scheme of journals, and increasing returns to the average number of authors per article for all research output taking quality into account, including citation indexes. Finally, by looking at the pattern of knowledge diffusion, we find that publication scores are the most important determinant of citation indexes. Nevertheless, we also show that team size and co-author networks constitute its second largest explanatory group of variables. Academics who publish more papers and of higher quality are more cited, which was expected. Now, we also show that academics working in larger co-author teams and who have a larger total co-author network are also more cited. Our interpretation is the following. Co-authored articles are presented in conferences, seminars and workshops by their several authors. Moreover, academics discuss their new findings with their peers, including those with whom they have already worked. Both attitudes generate more knowledge diffusion, which we measure through citations. Therefore we confirm the widespread intuition that networks do matter for citations.

It has already been shown that citations are an important determinant of academic wages and that network effects matter for academic promotions (see [START_REF] Mcdowell | The Effect of Gender-sorting on Propensity to coauthor: Implication for academic Promotion[END_REF], [START_REF] Combes | Publish or Peer-Rich? The Role of Skills and Networks in Hiring Economics Professors[END_REF] or [START_REF] Zinovyevay | It's Not What You Know, But Who You Know: The Role of Connections in Academic Promotions[END_REF] for instance). We prove here that network effects matter for citations. Therefore, data on wage and position would allow us to disentangle the direct effects of networks and their indirect effects through citations, in the explanation of wages. Also, studying the sources of citations (the citing academics/articles) would allow us to build a more precise picture of the patterns of knowledge diffusion. Unfortunately, Google Scholar does not provide such information yet. Finally, it would be interesting to perform some more structural estimations or to benefit from richer datasets (with a panel dimension possibly and some instruments) to deal with endogeneity concerns and to infer the directions of causalities, as [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF] propose, for instance, to identify peer effects in recreational services. This would improve on the correlations established here and move us to more causal interpretations. This is a difficult exercise though, which awaits further research.

Appendix

A Top 50 journals

  a) is the number of pages of the article, p the annual average number of pages of articles in the journal, n(a) the number of authors of the article and W (a) the journal weighting scheme. We consider the medium and high degree of convexity of journals' weights, noted CLm and CLh respectively. Scores using neither journal weights nor the correction for the relative number of pages are noted E. CLm ranges from a weight equal to 100 for the Quarterly Journal of Economics to a weight of 4 for the last journal, passing by 55.1 for the Journal of Labor Economics for instance. CLh ranges from 100 for the Quarterly Journal of Economics to 0.0007 for the last journal, passing by 16.7 for the Journal of Labor Economics.

Table 1 :

 1 Individual descriptive statistics

		Observations	Minimum	Maximum	Mean	Stand. err.
	Panel (a): All academics
	Women	2782	0	1	0.30	0.46
	Age	2782	26	85	46.87	10.79
	Publisher	2782	0	1	0.73	0.44
	Quantity	2782	0	78.5	3.53	6.07
	Quality total score	2782	0	5866.6	59.99	209.57
	Top quality total score	2782	0	3867.7	16.46	112.90
	Cited	2782	0	1	0.85	0.35
	Total citations	2782	0	30069.0	107.17	652.19
	G-index	2782	0	165.8	7.25	10.31
	GS Authors number	2782	1	6	1.96	0.65
	Panel (b): Published
	Women	2040	0	1	0.27	0.44
	Age	2040	27	85	46.09	10.58
	Quantity	2040	0.17	78.5	4.81	6.64
	Quality total score	2040	0.40	5866.6	81.81	241.08
	Top quality total score	2040	0	3867.7	22.45	131.34
	Average quality	2040	0.65	114.2	12.03	11.20
	Average top quality	2040	0	73.0	1.84	5.69
	EL Authors number	2040	1	6	1.85	0.63
	Network size	2040	0	53	4.12	5.35
	Cited	2040	0	1	0.95	0.23
	Total citations	2040	0	30069.0	137.48	758.03
	G-index	2040	0	165.8	8.89	11.29
	GS Authors number	2040	1	5	2.04	0.58
	Panel (c): Cited
	Women	2374	0	1	0.28	0.45
	Age	2374	27	85	46.31	10.70
	Publisher	2374	0	1	0.81	0.39
	Quantity	2374	0	78.5	4.05	6.41
	Quality total score	2374	0	5866.6	69.75	225.38
	Top quality total score	2374	0	3867.7	19.27	122.00
	Average quality	1929	0.65	114.2	12.35	11.41
	Average top quality	1929	0	73.0	1.94	5.84
	EL Authors number	1929	1	5	1.87	0.61
	Network size	2374	0	53	3.49	5.17
	Total citations	2374	0.20	30069.0	125.59	704.39
	G-index	2374	0.20	165.8	8.50	10.68
	GS Authors number	2374	1	5	2.05	0.58

Quantity: number of single-author equivalent articles in EconLit (E). Quality and top quality total scores: publication scores with a low and high degree of convexity in the journal weighting scheme (CLm and CLh), respectively. Authors number: average number of authors by article. Network size: total number of different co-authors.

Table 2 :

 2 Correlations of EconLit and Google Scholar indexes

		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	Quantity (1)	1	0.89	0.70	0.32	0.43	0.61	0.60
	Quality total score (2)		1	0.93	0.72	0.77	0.64	0.64
	Top quality total score (3)			1	0.85	0.94	0.59	0.60
	Average quality (4)				1	0.93	0.41	0.40
	Average top quality (5)					1	0.47	0.48
	Total citations (6)						1	0.95
	G-index (7)							1

Table 3 :

 3 Determinants of publication scores and citation indexes

	Women	E -0.357 a	CLm -0.556 a	CLh -1.126 a	CLm/E -0.106 a	CLh/E -0.372 a	TCit -0.517 a	G -0.298 a
		(0.05)	(0.07)	(0.13)	(0.03)	(0.09)	(0.08)	(0.04)
	Age	0.059 b	0.029	-0.037	-0.048 a	-0.171 a	0.104 a	0.055 b
		(0.03)	(0.04)	(0.07)	(0.01)	(0.04)	(0.04)	(0.02)
	Age 2	-0.000 c	-0.000	0.000	0.000 a	0.001 a	-0.001 c	-0.000 c
		(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)
	Authors number	-0.262 a	0.266 a	1.671 a	0.200 b	0.773 a	1.056 a	0.861 a
	Quantity	(0.07)	(0.10)	(0.18)	(0.09) 0.132 a	(0.28) 0.644 a	(0.12)	(0.08)
	Network size				(0.03) 0.216 a	(0.08) 0.789 a
	Constant	-0.386	2.501 a	-0.504	(0.04) 3.134 a	(0.14) 0.848	-0.535	-0.486
		(0.59)	(0.82)	(1.65)	(0.31)	(0.95)	(0.87)	(0.48)
	R 2	0.059	0.034	0.066	0.251	0.355	0.082	0.101
	Observations	2040	2040	2040	2040	2040	2374	2374

Table 4 :

 4 Determinants of publication scores and citation indexes, with selection

		Prob.Pub	E	CLm CLh CLm/E CLh/E Prob.Cit TCit	G
	Women	-0.405 a -0.781 -1.385 -3.580 b -0.477 c -2.215 a -0.325 a -0.408 -0.083
		(0.06)	(0.65) (0.87) (1.63) (0.27) (0.79) (0.07) (0.71) (0.42)
	Age	0.028	0.096 c 0.103 0.181 -0.015 -0.008 0.006 0.100 b 0.048 b
		(0.03)	(0.05) (0.07) (0.14) (0.03) (0.08) (0.03) (0.04) (0.02)
	Age 2	-0.000 c -0.001 -0.001 -0.003 -0.000 -0.001 -0.000 -0.001 -0.000
		(0.00)	(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
	Authors number		-0.262 a 0.267 a 1.673 a 0.195 b 0.747 a		1.056 a 0.861 a
	Quantity		(0.07) (0.10) (0.18) (0.09) (0.28) 0.130 a 0.634 a		(0.12) (0.08)
	Network size				(0.03) (0.08) 0.219 a 0.805 a
					(0.04) (0.14)
	Selection		2.204 4.300 12.736 1.925 9.565 b		-1.251 -2.465
			(3.38) (4.51) (8.42) (1.39) (3.95)		(8.31) (4.93)
	Constant	0.571	-1.625 0.084 -7.662 c 2.054 b -4.517 c 1.628 b -0.367 -0.155
		(0.59)	(1.76) (2.36) (4.49) (0.85) (2.44) (0.77) (1.27) (0.75)
	R 2		0.060 0.035 0.068 0.251 0.356		0.082 0.101
	Observations	2782	2040 2040 2040	2040	2040	2782	2374 2374

Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. E: Number of single-author equivalent articles in EconLit. CLm and CLh: Publication scores with a low and high degree of convexity in the journal weighting scheme, respectively. CLm/E and CLh/E: average quality and top quality. T C: Google Scholar total citations. G: Google Scholar G-index.

Table 5 :

 5 Determinants of publication scores and citation indexes: specialisation only

	% A:General	E 0.736 c	CLm 0.418	CLh -0.116	CLm/E -0.318	CLh/E -0.852	TCit 0.990	G 0.455
	% B:Thought	(0.43) 0.401 a	(0.57) -0.127	(1.04) -1.751 a	(0.27) -0.528 a	(0.78) -2.152 a	(0.73) -0.770 a	(0.41) -0.537 a
	% C:Maths.	(0.13) 0.104	(0.13) 1.172 a	(0.21) 3.548 a	(0.07) 1.068 a	(0.18) 3.444 a	(0.21) 0.747 c	(0.12) 0.351
	% D:Micro.	(0.24) 0.506 a	(0.35) 1.344 a	(0.70) 3.466 a	(0.16) 0.838 a	(0.52) 2.959 a	(0.43) 0.645 a	(0.23) 0.358 a
		(0.11)	(0.16)	(0.31)	(0.07)	(0.23)	(0.19)	(0.10)
	% E:Macro.	0.642 a	0.873 a	1.427 a	0.231 b	0.785 a	-0.028	0.023
		(0.16)	(0.20)	(0.39)	(0.09)	(0.30)	(0.24)	(0.13)
	% F:Inter.	0.030	0.039	0.097	0.009	0.067	-0.240	-0.106
		(0.14)	(0.19)	(0.36)	(0.09)	(0.28)	(0.26)	(0.14)
	% G:Finance	-0.112	-0.062	-0.084	0.050	0.028	-0.401 b	-0.228 b
	% H:Public	(0.11) 0.070	(0.14) 0.398	(0.26) 1.513 a	(0.07) 0.329 a	(0.21) 1.443 a	(0.19) -0.027	(0.11) 0.044
		(0.21)	(0.27)	(0.55)	(0.13)	(0.44)	(0.34)	(0.19)
	% I:Health	-0.467 b	-0.428 c	-0.450	0.039	0.017	-0.084	-0.034
		(0.21)	(0.26)	(0.47)	(0.10)	(0.35)	(0.32)	(0.20)
	% J:Labor	0.146	0.429 a	1.301 a	0.283 a	1.155 a	0.431 b	0.251 b
	% K:Law	(0.12) -0.304	(0.16) -0.616	(0.33) -1.065	(0.08) -0.312 c	(0.27) -0.761	(0.20) -0.722	(0.12) -0.308
	% L:I.O.	(0.33) -0.156	(0.41) -0.275 c	(0.72) -0.467	(0.17) -0.119 c	(0.53) -0.311	(0.63) 0.205	(0.32) 0.080
	% M:Business	(0.12) -1.059 a	(0.16) -1.641 a	(0.31) -3.304 a	(0.07) -0.582 a	(0.23) -2.245 a	(0.21) -0.021	(0.12) -0.015
		(0.13)	(0.16)	(0.27)	(0.07)	(0.21)	(0.34)	(0.19)
	% N:History	-0.513 c	-1.075 b	-2.314 b	-0.562 a	-1.801 a	-1.128 c	-0.693 c
		(0.31)	(0.44)	(0.90)	(0.20)	(0.70)	(0.63)	(0.39)
	% O:Develop.	-0.133	-0.245	-0.720 b	-0.112	-0.587 b	-0.005	0.042
		(0.13)	(0.17)	(0.32)	(0.08)	(0.24)	(0.25)	(0.14)
	% P:Systems	0.163	-0.134	-1.094 b	-0.297 b	-1.257 a	-0.306	-0.205
		(0.27)	(0.31)	(0.53)	(0.13)	(0.39)	(0.48)	(0.26)
	% Q:Agr.	-0.024	0.090	0.595 c	0.114	0.619 a	0.404 b	0.421 a
		(0.12)	(0.16)	(0.31)	(0.07)	(0.24)	(0.19)	(0.11)
	% R:Urban	-0.031	-0.161	-0.582 c	-0.130 c	-0.551 b	0.309	0.102
	Constant	(0.13) 0.986 a	(0.16) 3.113 a	(0.33) -1.423 a	(0.08) 2.127 a	(0.27) -2.409 a	(0.23) 3.262 a	(0.14) 1.727 a
		(0.04)	(0.05)	(0.10)	(0.02)	(0.08)	(0.08)	(0.04)
	R 2	0.058	0.138	0.195	0.085	0.160	0.029	0.034
	Observations	1923	1923	1923	1923	1923	1835	1835

Table 6 :

 6 Determinants of publication scores and citation indexes: specialisation and individual

	characteristics Women	E -0.390	CLm -0.714	CLh -2.239	CLm/E -0.278	CLh/E -1.597 c	TCit 0.236	G 0.094
		(0.62)	(0.84)	(1.68)	(0.29)	(0.89)	(0.84)	(0.46)
	Age	0.050	0.039	0.079	-0.022	-0.018	0.051	0.027
		(0.05)	(0.07)	(0.14)	(0.03)	(0.09)	(0.05)	(0.03)
	Age 2	-0.000	-0.000	-0.001	0.000	-0.001	0.000	0.000
		(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)
	Authors number	-0.338 a	0.094	1.232 a	0.203 b	0.880 a	1.025 a	0.954 a
	Quantity	(0.08)	(0.10)	(0.19)	(0.09) 0.129 a	(0.27) 0.685 a	(0.15)	(0.09)
	Network size				(0.03) 0.167 a	(0.08) 0.565 a
					(0.04)	(0.14)
	% A:General	0.273	0.099	-0.168	-0.189	-0.559	0.400	0.222
	% B:Thought	(0.36) 0.216 c	(0.53) -0.146	(1.06) -1.355 a	(0.28) -0.374 a	(0.83) -1.665 a	(0.73) -0.631 a	(0.39) -0.346 a
	% C:Maths.	(0.12) 0.080	(0.13) 0.978 a	(0.23) 2.865 a	(0.07) 0.871 a	(0.21) 2.675 a	(0.22) 0.318	(0.12) 0.043
	% D:Micro.	(0.22) 0.581 a	(0.34) 1.354 a	(0.69) 3.326 a	(0.14) 0.640 a	(0.41) 2.149 a	(0.39) 0.731 a	(0.20) 0.377 a
	% E:Macro.	(0.11) 0.602 a	(0.16) 0.796 a	(0.31) 1.244 a	(0.06) 0.081	(0.19) 0.109	(0.17) -0.016	(0.10) 0.050
		(0.16)	(0.20)	(0.38)	(0.08)	(0.25)	(0.23)	(0.13)
	% F:Inter.	0.022	0.059	0.200	0.027	0.137	-0.200	-0.069
		(0.14)	(0.19)	(0.36)	(0.08)	(0.23)	(0.25)	(0.13)
	% G:Finance	-0.004	0.040	0.030	0.041	0.025	-0.215	-0.141
		(0.10)	(0.13)	(0.26)	(0.06)	(0.20)	(0.17)	(0.09)
	% H:Public	0.106	0.379	1.356 b	0.238 b	1.106 a	-0.006	0.028
	% I:Health	(0.21) -0.352 c	(0.27) -0.365	(0.55) -0.513	(0.11) 0.076	(0.37) 0.225	(0.32) -0.165	(0.18) -0.142
		(0.19)	(0.25)	(0.49)	(0.09)	(0.30)	(0.30)	(0.18)
	% J:Labor	0.332 a	0.592 a	1.452 a	0.172 b	0.737 a	0.634 a	0.309 a
		(0.11)	(0.16)	(0.32)	(0.08)	(0.23)	(0.19)	(0.11)
	% K:Law	-0.140	-0.427	-0.753	-0.257	-0.475	-0.284	-0.055
		(0.33)	(0.40)	(0.72)	(0.18)	(0.58)	(0.51)	(0.26)
	% L:I.O.	-0.083	-0.247	-0.539 c	-0.151 b	-0.394 b	0.204	0.037
	% M:Business	(0.12) -1.102 a	(0.16) -1.657 a	(0.30) -3.249 a	(0.06) -0.325 a	(0.19) -1.097 a	(0.19) -0.137	(0.11) -0.111
		(0.12)	(0.16)	(0.30)	(0.08)	(0.22)	(0.31)	(0.18)
	% N:History	-0.621 c	-1.123 b	-2.215 b	-0.370 b	-0.990 c	-1.065 b	-0.550 b
		(0.35)	(0.47)	(0.94)	(0.18)	(0.58)	(0.48)	(0.26)
	% O:Develop.	-0.081	-0.190	-0.643 b	-0.092	-0.483 b	0.060	0.078
		(0.13)	(0.17)	(0.32)	(0.07)	(0.22)	(0.25)	(0.14)
	% P:Systems	0.051	-0.093	-0.679	-0.171	-0.834 b	-0.126	-0.022
	% Q:Agr.	(0.27) 0.094	(0.32) 0.073	(0.55) 0.229	(0.12) -0.059	(0.37) -0.019	(0.49) 0.171	(0.25) 0.194 c
		(0.13)	(0.17)	(0.32)	(0.07)	(0.21)	(0.20)	(0.11)
	% R:Urban	0.025	-0.124	-0.588 c	-0.157 b	-0.646 a	0.328	0.100
		(0.13)	(0.16)	(0.32)	(0.07)	(0.24)	(0.22)	(0.14)
	Selection	0.374	1.318	6.954	1.088	6.972	-7.935	-3.857
		(3.26)	(4.42)	(8.77)	(1.49)	(4.56)	(9.93)	(5.48)
	Constant	-0.454	1.722	-4.994	2.278 b	-4.078	0.934	0.190
		(1.74)	(2.36)	(4.69)	(0.90)	(2.70)	(1.51)	(0.84)
	R 2	0.143	0.109	0.239	0.457	0.570	0.147	0.176
	Observations	1923	1923	1923	1923	1923	1835	1835

Table 7 :

 7 Citation indexes regressed on publication scores and specialisation

	Quantity	TCit 1.086 a	G 0.605 a	TCit 0.970 a	G 0.543 a	TCit 0.907 a	G 0.503 a
	Average quality	(0.03)	(0.02)	(0.03) 0.697 a	(0.02) 0.377 a	(0.03)	(0.02)
	Average top quality			(0.05)	(0.03)	0.221 a	0.126 a
						(0.02)	(0.01)
	% A:General	0.103	-0.040	0.498	0.174	0.487	0.179
	% B:Thought	(0.72) -1.156 a	(0.39) -0.752 a	(0.76) -0.732 a	(0.41) -0.523 a	(0.76) -0.597 a	(0.42) -0.434 a
		(0.19)	(0.11)	(0.18)	(0.11)	(0.18)	(0.11)
	% C:Maths.	0.712 b	0.332 b	-0.029	-0.069	-0.032	-0.092
	% D:Micro.	(0.30) 0.101	(0.15) 0.054	(0.30) -0.464 a	(0.16) -0.251 a	(0.30) -0.504 a	(0.15) -0.291 a
	% E:Macro.	(0.14) -0.699 a	(0.08) -0.351 a	(0.15) -0.780 a	(0.08) -0.395 a	(0.15) -0.752 a	(0.08) -0.381 a
		(0.20)	(0.11)	(0.19)	(0.10)	(0.19)	(0.11)
	% F:Inter.	-0.265	-0.119	-0.276	-0.126	-0.293	-0.136
	% G:Finance	(0.21) -0.225	(0.12) -0.130	(0.22) -0.273 c	(0.12) -0.156 c	(0.21) -0.250 c	(0.11) -0.144 c
		(0.16)	(0.09)	(0.15)	(0.08)	(0.15)	(0.08)
	% H:Public	-0.223	-0.065	-0.482 c	-0.205	-0.570 b	-0.263 c
		(0.28)	(0.16)	(0.27)	(0.15)	(0.27)	(0.15)
	% I:Health	0.332	0.198	0.283	0.171	0.272	0.163
		(0.22)	(0.13)	(0.23)	(0.14)	(0.22)	(0.14)
	% J:Labor	0.349 b	0.205 b	0.178	0.112	0.130	0.080
		(0.15)	(0.09)	(0.15)	(0.09)	(0.15)	(0.09)
	% K:Law	-0.405	-0.132	-0.238	-0.041	-0.313	-0.079
		(0.56)	(0.29)	(0.53)	(0.28)	(0.51)	(0.27)
	% L:I.O.	0.418 b	0.198 c	0.493 a	0.239 b	0.467 a	0.226 b
	% M:Business	(0.18) 1.203 a	(0.10) 0.667 a	(0.18) 1.504 a	(0.10) 0.831 a	(0.18) 1.525 a	(0.10) 0.851 a
		(0.34)	(0.19)	(0.34)	(0.19)	(0.33)	(0.19)
	% N:History	-0.854	-0.540	-0.576	-0.389	-0.570	-0.378
	% O:Develop.	(0.60) 0.166	(0.39) 0.138	(0.57) 0.227	(0.37) 0.170	(0.58) 0.278	(0.37) 0.201 c
		(0.21)	(0.12)	(0.21)	(0.12)	(0.20)	(0.11)
	% P:Systems	-0.393	-0.254	-0.172	-0.134	-0.087	-0.079
		(0.31)	(0.17)	(0.31)	(0.17)	(0.30)	(0.16)
	% Q:Agr.	0.422 b	0.431 a	0.329 b	0.380 a	0.273	0.346 a
		(0.17)	(0.09)	(0.17)	(0.09)	(0.17)	(0.09)
	% R:Urban	0.413 b	0.160	0.509 b	0.212 c	0.536 a	0.230 c
	Constant	(0.21) 2.126 a	(0.13) 1.094 a	(0.20) 0.752 a	(0.12) 0.349 a	(0.20) 2.827 a	(0.12) 1.494 a
		(0.08)	(0.04)	(0.14)	(0.08)	(0.09)	(0.05)
	R 2	0.426	0.423	0.481	0.474	0.482	0.481
	Observations	1835	1835	1835	1835	1835	1835

  that the measured effects of specialisation shares are very weakly correlated with the dependent variables, meaning that specialisation patterns have a weak explanatory power of citation indexes like Google Scholar total citations and G-indexes, when all individual effects are controlled for. In both Tables, using CLm or CLh to measure journal quality leads to very similar results.

	16 1.021 1.756 = 0.581, 1.037 1.756 = 0.591, 0.532 0.988 = 0.538, 0.544 0.988 = 0.551 17 0.514 0.284 = 1.810, 0.516 0.283 = 1.823 18 0.539 0.256 = 2.105, 0.537 0.255 = 2.106

Table 8 :

 8 Determinants of citation indexes controlled for publication scores

	Quantity	TCit 0.804 a	G 0.434 a	TCit 0.744 a	G 0.407 a	TCit 0.693 a	G 0.381 a
	Average quality	(0.05)	(0.03)	(0.04) 0.681 a	(0.02) 0.314 a	(0.05)	(0.02)
	Average top quality			(0.05)	(0.03)	0.216 a	0.104 a
	Women	-0.119 c	-0.052	-0.070	-0.029	(0.02) -0.063	(0.01) -0.025
	Age	(0.07) 0.019	(0.04) 0.007	(0.07) 0.048 c	(0.04) 0.020	(0.07) 0.050 c	(0.04) 0.022
		(0.03)	(0.02)	(0.03)	(0.01)	(0.03)	(0.01)
	Age 2	0.000	0.000	-0.000	-0.000	-0.000	-0.000
		(0.00)	(0.00)	(0.00)	(0.00)	(0.00)	(0.00)
	Authors number	0.342 b	0.512 a	0.229	0.460 a	0.199	0.443 a
	Network size	(0.16) 0.452 a	(0.09) 0.294 a	(0.16) 0.303 a	(0.09) 0.225 a	(0.16) 0.280 a	(0.09) 0.211 a
		(0.06)	(0.04)	(0.06)	(0.03)	(0.06)	(0.03)
	% A:General	0.213	0.135	0.404	0.223	0.368	0.210
		(0.68)	(0.35)	(0.74)	(0.37)	(0.74)	(0.38)
	% B:Thought	-0.747 a	-0.396 a	-0.503 a	-0.283 a	-0.399 b	-0.229 b
	% C:Maths.	(0.19) 0.274	(0.10) 0.006	(0.19) -0.338	(0.10) -0.277 c	(0.19) -0.320	(0.10) -0.280 c
	% D:Micro.	(0.28) 0.096	(0.14) 0.014	(0.29) -0.376 a	(0.15) -0.204 a	(0.28) -0.406 a	(0.14) -0.227 a
	% E:Macro.	(0.14) -0.610 a	(0.07) -0.284 a	(0.14) -0.677 a	(0.07) -0.315 a	(0.14) -0.650 a	(0.08) -0.304 a
		(0.19)	(0.11)	(0.18)	(0.10)	(0.18)	(0.10)
	% F:Inter.	-0.267	-0.112	-0.285	-0.120	-0.302	-0.129
	% G:Finance	(0.20) -0.171	(0.10) -0.116	(0.21) -0.200	(0.10) -0.129 c	(0.20) -0.177	(0.10) -0.118
		(0.14)	(0.07)	(0.13)	(0.07)	(0.13)	(0.07)
	% H:Public	-0.293	-0.138	-0.475 c	-0.223	-0.552 b	-0.263 c
		(0.27)	(0.15)	(0.25)	(0.14)	(0.25)	(0.14)
	% I:Health	0.200	0.066	0.177	0.056	0.173	0.053
		(0.23)	(0.14)	(0.22)	(0.14)	(0.22)	(0.14)
	% J:Labor	0.362 b	0.155 c	0.261 c	0.108	0.224	0.088
		(0.15)	(0.08)	(0.14)	(0.08)	(0.14)	(0.08)
	% K:Law	-0.161	0.016	0.004	0.092	-0.073	0.058
		(0.50)	(0.25)	(0.46)	(0.23)	(0.44)	(0.23)
	% L:I.O.	0.342 b	0.114	0.462 a	0.169 c	0.444 a	0.163 c
	% M:Business	(0.17) 1.189 a	(0.09) 0.649 a	(0.17) 1.451 a	(0.10) 0.770 a	(0.17) 1.469 a	(0.10) 0.783 a
	% N:History	(0.32) -0.759 c	(0.18) -0.385	(0.32) -0.563	(0.18) -0.294	(0.31) -0.571	(0.18) -0.294
	% O:Develop.	(0.41) 0.196	(0.26) 0.155	(0.41) 0.250	(0.25) 0.179 c	(0.42) 0.299	(0.26) 0.204 c
		(0.20)	(0.11)	(0.20)	(0.11)	(0.20)	(0.11)
	% P:Systems	-0.252	-0.094	-0.135	-0.040	-0.069	-0.006
		(0.31)	(0.16)	(0.32)	(0.16)	(0.31)	(0.16)
	% Q:Agr.	0.035	0.105	0.077	0.124	0.047	0.110
		(0.17)	(0.09)	(0.16)	(0.09)	(0.16)	(0.09)
	% R:Urban	0.353 c	0.111	0.466 b	0.163	0.495 b	0.179
		(0.20)	(0.12)	(0.19)	(0.11)	(0.19)	(0.11)
	Constant	0.670	0.153	-1.316 b	-0.765 b	0.682	0.159
		(0.65)	(0.35)	(0.61)	(0.34)	(0.59)	(0.32)
	R 2	0.482	0.517	0.528	0.548	0.527	0.550
	Observations	1835	1835	1835	1835	1835	1835

Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. Quantity: number of single-author equivalent articles in EconLit (E). Average quality and top quality: CLm/E and CLh/E. T C: Google Scholar total citations. G: Google Scholar G-index.

Table 9 :

 9 Variance analysis, Google Scholar total citationsThe table presents the variance analysis of the estimation reported in Table8 columns (3) and (5). 'Stand. error' columns report the standard error of the effect of a variable or a group of variables. For the first line, it reports the standard error of the dependent variable. 'Correlation' columns report the correlation between the effect of a variable or of a group of variables and the dependent variable.

		CLm as quality	CLh as top quality
		Stand. error	Correlation	Stand. error	Correlation
	Explained: GS Total citations	1.756	1.000	1.756	1.000
	EconLit Publications	1.021	0.659	1.037	0.662
	Quantity	0.782	0.618	0.728	0.618
	Average quality	0.465	0.410	.	.
	Average top quality	.	.	0.501	0.472
	Individual effects	0.328	0.284	0.332	0.283
	Gender	0.031	0.171	0.028	0.171
	Age	0.320	0.275	0.325	0.275
	Coauthorship	0.287	0.514	0.262	0.516
	Authors number	0.062	0.157	0.054	0.157
	Network size	0.251	0.547	0.232	0.547
	Specialisation	0.333	0.040	0.330	0.031
	Residuals	1.206	0.687	1.207	0.688

Table 10 :

 10 Variance analysis, Google Scholar G-indexThe table presents the variance analysis of the estimation reported in Table8columns (4) and (6). 'Stand.

		CLm as quality	CLh as top quality
		Stand. error	Correlation	Stand. error	Correlation
	Explained: G-index	0.988	1.000	0.988	1.000
	EconLit Publications	0.532	0.653	0.544	0.660
	Quantity	0.427	0.611	0.400	0.611
	Average quality	0.215	0.404	.	.
	Average top quality	.	.	0.241	0.476
	Individual effects	0.160	0.256	0.163	0.255
	Gender	0.013	0.159	0.011	0.159
	Age	0.157	0.249	0.160	0.249
	Coauthorship	0.270	0.539	0.256	0.537
	Authors number	0.125	0.258	0.120	0.258
	Network size	0.187	0.607	0.175	0.607
	Specialisation	0.169	0.043	0.169	0.031
	Residuals	0.664	0.672	0.663	0.671

error' columns report the standard error of the effect of a variable or a group of variables. For the first line, it reports the standard error of the dependent variable. Columns 'Correlation' reports the correlation between the effect of a variable or a group of variables and the dependent variable.

Table 11

 11

	: Top 50 journals

Ministère de l'Enseignement Supérieur et de la Recherche -Direction Générale de la Recherche et de l'Innovation.

We matched data from the Universities, from the CNRS (Centre National de la Recherche Scientifique) and from the INRA (Institut National de la Recherche Agronomique).

In logarithms to match the econometric analysis presented in section 3.

The academics who have never co-authored an article have a network size of size 0, so we add 1 to N eti to take its log. 1 + N eti can be seen as the total network size, including academic i who would belong to her

We ignore the fields "Miscellaneous Categories" (Y) and "Other Special Topics" (Z). We also slightly modify the codes C and D by merging code C7 (Game Theory and Bargaining Theory) and C9 (Design of Experiments) with Microeconomics (D), which seems to us more consistent.

We estimate the γj coefficients under the constraint 18 j=1 γj = 0. Therefore significance tests reported for the γj are with respect to the average γj.

1.5 -0.262 -1, 1.5 0.200 -1 and 1.5 0.773 -1, respectively.

1.5 0.266 -1, 1.5 1.671 -1, 1.5 1.056 -1 and 1.5 0.861 -1, respectively.

1.25 0.132 -1 and 1.25 0.644 -1, respectively.

1.2 0.216 -1 and 1.2 0.789 -1, respectively.

Standard error between brackets. a , b , c significant at the 1%, 5% and 10% level, respectively. E = number of single-author equivalent articles in EconLit. CLm and CLh are publication scores with a low and high degree of convexity in the journal weighting scheme, respectively. CLm/E and CLh/E = average quality and top quality. T C = Google Scholar total citations. G = Google Scholar G-index. "Selection" is the inverse of Mill's ratio from columns 1 (for publications) and 7 (for citations) of Table4.

1.25 0.907 -1 and 1.25 0.503 -1, respectively.

1.1 0.221 -1 and 1.1 0.126 -1, respectively.

1.5 0.229 -1 and 1.5 0.460 -1, respectively.

1.2 0.303 -1 and 1.2 0.225 -1, respectively.


Patrice Loisel
email: patrice.loisel@supagro.inra.fr

Pierre Cartigny
email: pierre.cartigny@supagro.inra.fr

How to model marine reserves ?

Keywords: dynamical system, calculus of variation, infinite horizon, marine reserve, bioeconomic model

The safeguarding of resources is one of the principal subjects of halieutics studies. Among the solutions proposed to avert the disappearance of species, the setting in place of no take reserves is often mentioned. Most work on this subject, theoretical as well as applied, was undertaken in recent years. In this paper, we seek to compare two different models presented in existing literature by highlighting their underlying assumptions. Both models were derived from what is often referred to as the "model of Schaefer-Clark" (reference to the work of the last author on Mathematical Bioeconomics : Clark [7]). We show that various variations of this model lead to properties that can be very different.

Introduction

It is now well known and largely accepted within the scientific community that the exploitation of halieutics resources has reached a critical threshold and solutions must be found urgently to conserve marine biodiversityand indeed the existence of certain species. Beyond the application of quota policies, other strategies have been proposed such as the creation of marine reserves. The study of the role of reserves in fishery management has been the subject of renewed interest in recent years [START_REF] Arnason | Marine Reserves: Is there an Economic Justification?[END_REF][START_REF] Carr | Conceptual Issues Relevant to Marine harvest Refuges: Examples from Temperate reef Fishes[END_REF][START_REF] Conrad | The Bioeconomics of Marine Sanctuaries[END_REF][START_REF] Hannesson | Marine Reserves: What would thet Accomplish[END_REF][START_REF] Houde | Testinomy on Use of Marine protected Areas (MPAs) as a Fisheries Management Tool, as a Means to Protect and restore Marine Ecosystem, and as a Research Tool[END_REF][START_REF] Pezzey | A simple bioeconomic model of a marine reserve[END_REF]. The International Conference on the Economics of Marine Protected Areas (MPA) held in July 2000 in Vancouver was one of the starting points in the development of this new paradigm: the use of MPAs as an instrument in the management of fisheries.

In economics literature, Sanchirico and Wilen [START_REF] Sanchirico | A Bioeconomic Model of Marine Reserve Creation[END_REF] seem to have been the first to suggest that MPAs could be beneficial not only from an ecological but also from an economic point of view. In their dynamic and spatial model of a Marine Reserve Creation, they analyze whether the transfer of biomass from the reserve to areas where catch is allowed could create economics profit from the MPA since its creation could help to improve a depleted biomass and increase catch outside of the reserve. They call this a double-payoff because in this case the MPA would increase both biomass and economic profits from the fishery.

Both theoretical and applied aspects of the subject are well documented in the literature. From a theoretical point of view, numerous types of models have been proposed based on differential equations, mixing an inter temporal dynamic that corresponds to the population growth under consideration with a spatial distribution of this dynamic over diverse zones [START_REF] Dubey | A model for fishery resource with reserve area[END_REF][START_REF] Sanchirico | Additivity properties in metapopulation models : application for the assessment of marine reserve[END_REF][START_REF] Sanchirico | A Bioeconomic Model of Marine Reserve Creation[END_REF].

Sanchirico and Wilen construct a model where "the population structure is characterized in a manner consistent with modern biological ideas that stress patchiness, heterogeneity and interconnections among and between patches" [START_REF] Sanchirico | Bioeconomics of Spacial Exploitation in a Patchy Environment[END_REF]. In this model, independent growth dynamics thus are associated with different patches.

Another type of model has been used in the literature to account for an analogue structure [START_REF] Srinivasu | Dynamics and Bio-Economics of a Renewable Resource in the Vicinity[END_REF] where a population develops different characteristics in sub-zones. In these models, the population of the entire zone under consideration follows a given dynamic evolution and the diverse sub-zones have dynamics such that by aggregating them together one may rediscover the global dynamic.

The question that one then must ask is whether these two approaches may be used interchangeably. Few studies have focused on comparing these different model types. We would like to demonstrate that the choice between a patch model and a global model is not a neutral one, and highlight char-acteristics of these two models that often are not specified in the literature.

The two models that we will compare are both Clark type (Schaefer, Gordon,...) [START_REF] Clark | Mathematical Bioecomics: The Optimal Management of Renewable Resources[END_REF] whose dynamic is logistic (Verhulst) and therefore widely used in halieutics dynamics. We use them to study the consequences of setting up a marine reserve from both an economic (inter temporal revenue) and a biologic point of view (population stocks).

The first, the patch model, assumes a relative autonomy between reserve and non-reserve zones. This type of model is fairly widely used. The second model, known as the global model, assumes for its part a greater interaction between zones.

The rest of the paper is organized as follows. In Section 3, we introduce and examine the two models we wish to study. We then compare results obtained, particularly using numerical simulation, in Section 4. Section 5 concludes and is followed by a series of annexes that present the demonstration of various results.

Two variations on the Clark model

We present two modelizations for a protected area in a given zone. These two modelizations derive from the well known fishery model studied for instance by Clark [START_REF] Clark | Mathematical Bioecomics: The Optimal Management of Renewable Resources[END_REF] among others.

2.1

The first variation: patches model

The model

We consider a fish population that lives in a zone caracterised by a carrying capacity K = 1. We assume that this zone splits in two sub-zones, with capacity respectively equal to α and 1 -α; we denote the stock of the corresponding sub-populations by x 1 and x 2 . These two populations follow two independent evolutions laws and this is the reason why we use the "patches-concept". These evolutions are given by:

dx 1 dt (t) := ẋ1 (t) = F 1 (x 1 (t)) dx 2 dt (t) := ẋ2 (t) = F 2 (x 2 (t)).
The standard reference for the evolutions law is the logistic law:

F 1 (x 1 ) := r 1 x 1 (1 - x 1 α ) F 2 (x 2 ) := r 2 x 2 (1 - x 2 1 -α ).
But for our purpose it is enough to assume that the F i are strictly concave, C 1 functions defined on [0, α] respectively on [0, 1-α] and the F i satisfy

F i (0) = 0, F 1 (α) = F 2 (1 -α) = 0.
We now assume that some exchange exists between these two patches and that this can be represented in terms of the density of these populations. More precisely we assume that the existence of some diffusion between these two patches can be captured by the following:

λ( x 2 1 -α - x 1 α )
where λ ≥ 0 represents a diffusion coefficient ( x 2 1-α , x 1 α being the density of the populations). The value of the diffusion coefficient depends on the location of the protected area.

From now on, we decide that the first zone with capacity α is a protected area where no catch is allowed, whereas in the second zone fishing is allowed. The "normal" situation, i.e. the protected area acts like a source of biomass, corresponds to the case where the density inside the protected area is bigger than outside, i.e.

x 2

1 -α ≤ x 1 α .
The growth of the two sub-populations are governed respectively by the following dynamics:

ẋ1 (t) = F 1 (x 1 (t)) + λ(t)( x 2 (t) 1-α -x 1 (t) α ) ẋ2 (t) = F 2 (x 2 (t)) -λ(t)( x 2 (t) 1-α -x 1 (t) α ) -h(t) (1)
where h(t) is the capture rate at time t. We note, from the positiveness of λ and of the functions F i (.), that if the sytem (1) possesses an equilibrium, it has to be necessarily normal.

As it is generally assumed the catch is proportional to the fishing effort E, and to the density of the population [START_REF] Boncoeur | Fish, Fishers, Seals and Tourists: Economic Consequences of Creating a marine reserve in a Multi-Species, multi-activity Context[END_REF], therefore given by:

h(t) = qE(t) x 2 (t) 1 -α
where q, the catchability coefficient, represents the fishing death rate when the density of the population is equal to one. We assume

0 ≤ E(t) ≤ E M q > 0
The catch is sold on a market. In order to simplify we assume a constant price, p, over time and a constant cost, c, proportional to the effort. Therefore the revenue at t time is

ph(t) -cE(t) = (pq x 2 (t) 1 -α -c)E(t)
We then consider the discounted total revenue on an infinite horizon is given by

J(E(.), λ(.)) := ∞ 0 e -δt (pq x 2 (t) 1 -α -c)E(t) dt (2)
where δ > 0 is an actualisation factor. We assume the existence of a manager whose goal is the maximisation of this total revenue. Moreover we assume that this manager can act on the fishing effort E and on some caracteristics of the reserve (closure, location) that are captured by λ. Then the manager faces the following control problem: max E(.), λ(.)

J(E(.), λ(.))

s.t. (1) 0 ≤ E(t) ≤ E M (3)
Remark In many papers the states variables stand for the densities of the populations and not for the amount of the biomass. The link with the present model is obtained in setting:

X 1 = x 1 α , X 2 = x 2 1 -α
The two dynamical equations that give the evolution of the populations in the logistic case, then become:

Ẋ1 = r 1 X 1 (1 -X 1 ) + λ α (X 2 -X 1 ) Ẋ2 = r 2 X 2 (1 -X 2 ) -λ 1-α (X 2 -X 1 ) -QEX 2 (4)
with Q = q 1-α . This model with patches could be considered as a more general one than the [START_REF] Ami | Can marine protected areas enhance both economic and biological situations ?[END_REF] paper which corresponds to α = 1 2 .

Analysis of the solutions

We will study the previous optimal control problem by the help of the calculus of variations theory .

From the dynamic (1), we deduce the expression of the effort in terms of the state variables:

E(t) = 1 -α qx 2 (t) (F 1 (x 1 (t)) + F 2 (x 2 (t)) -ẋ1 (t) -ẋ2 (t))
and then we obtain the new form of the objective. Thus the optimisation problem becomes:

max X∈C ∞ 0 e -δt (p - c(1 -α) qx 2 )[F 1 (x 1 ) + F 2 (x 2 ) -ẋ1 -ẋ2 ] dt
C being the set of admissible curves:

C = {x(.) = (x 1 (.), x 2 (.)) x i (.) ∈ BC 1 ([0, ∞[), x i (0) given , G(x 1 , x 2 ) -qE M x 2 1-α ≤ ẋ1 + ẋ2 ≤ G(x 1 , x 2 )} (5) with G(x 1 , x 2 ) := F 1 (x 1 ) + F 2 (x 2 )
and BC 1 stands for the bounded with bounded derivative functions defined on the interval [0, ∞[.

It is known that on BC 1 , the first order optimality conditions given by the Euler-Lagrange equations, apply (see [START_REF] Blot | Bounded Solutions and Oscillations of Concave Lagrangian Systems in Presence of a Discount Rate[END_REF]). We suppose that x(.) stands for an interior solution and then x(.) has to satisfy:

l x i (x 1 (t), x 2 (t)) - d dt l ẋi (x 1 (t), x 2 (t)) + δl ẋi (x 1 (t), x 2 (t)) = 0 l(., .
) being the non actualised Lagrangian of the calculus of variations problem, l x i (., .) stands for the derivative with respect to x i and l x i (., .) stands for the derivative with respect to x i .. The Euler-Lagrange equations becomes then

ẋ1 = x 2 ( pqx 2 c(1-α) -1)(F ′ 2 (x 2 ) -δ) + F 1 (x 1 ) + F 2 (x 2 ) ẋ2 = x 2 ( pqx 2 c(1-α) -1)(δ -F ′ 1 (x 1 )). (6)
We first are interested by the non trivial equilibria , (x * 1 , x * 2 ), of ( 6) i.e. such that F i (x * i ) = 0, i.e. x * 1 = 0, α and x * 2 = 0, 1 -α. It is easy to establish that such equilibria have to satisfy (Appendix 1)

F ′ 1 (x * 1 ) = δ.
We now assume that r 1 > δ. Then in the logistic case, from the strict concavity of F 1 (.), we immediatly deduce the existence of a unique x *

1 ∈]0, α/2[. Therefore we obtain the following result whose proof is postponed in Appendix 1.

Lemma 1 In the logistic case with r 1 > δ, there is a unique positive non trivial solution, (x * 1 , x * 2 ), of the Euler-Lagrange equations ( 6). This solution is caracterised by

x * 1 = α(r 1 -δ) 2r 1 and x * 2 given by x 2 [ 2r 2 pq c(1 -α) 2 x 2 2 -( pq c(1 -α) (r 2 -δ) + r 2 1 -α )x 2 -δ] = α (r 1 -δ)(r 1 + δ) 4r 1 . (7)
Clearly

x * 1 ∈]0, α[. It remains to show that x * 2 ∈ [0, 1 -α].
This can be done straightforwordly but we prefer to use the following approach. We recall that if (1) possesses a non-zero equilibrium (x * 1 , x * 2 ) then it is necessarily normal that is to say

x * 2 1-α ≤ x * 1
α and we first observe that if this last condition holds then x * 2 ≤

x * 1 α (1 -α) ≤ 1 -α. We prove now that this condition holds under conditions on the ratio pq c and on the size of the different zones given by α. Let us introduce T (.) defined from the left hand member in ( 7)

T (z) = z[2r 2 θz 2 -(θ(r 2 -δ) + r 2 )z -δ]
where θ := pq c . Then:

T ( x * 2 1 -α ) = α 1 -α (r 1 -δ)(r 1 + δ) 4r 1 .
As we assume r 1 > δ, then T (

x * 2 1-α ) > 0.
From the graph of T (.) we deduce that the inequality

x * 2 1 -α ≤ x * 1 α holds if and only if T ( x * 2 1 -α ) ≤ T ( x * 1 α ).
Let us introduce when r 1 = r 2

θ 0 := ( 2r 1 r 1 -δ + r 2 δ ) r 1 r 1 -r 2 . ( 8
)
The following result is detailled in Appendix 2.

Lemma 2

In the logistic case,

1. if r 1 > r 2 , for each θ > θ 0 if α satisfies α(r 1 + δ + r 1 -δ r 1 (θδ r 1 -r 2 r 1 -r 2 ) -2δ) ≤ r 1 -δ r 1 (θδ r 1 -r 2 r 1 -r 2 ) -2δ (9)
then the solution (x * 1 , x * 2 ), of the Euler Lagrange equation [START_REF] Cartigny | The optimal spatial distribution of small and large scale fisheries Gomez, to appear in Ecological Modelling[END_REF] given in Lemma 1, is normal i.e. satisfies:

x * 2 1 -α ≤ x * 1 α 2. if r 1 ≤ r 2 , (x * 1 , x * 2 ) is never normal i.e. x * 2 1 -α > x * 1 α .
Then it remains to prove that (x * 1 , x * 2 ) is a candidate to the optimisation problem (3), thus that the constraints are satisfied.

If the conditions in the Lemma 1 and Lemma 2.1 hold, then

F 1 (x * 1 ) > 0 and x * 2 1 -α < x * 1 α * . Therefore to this corresponds a unique coefficient of diffusion λ * > 0.
From (1) we deduce that

E * = 1 -α qx * 2 (F 1 (x * 1 ) + F 2 (x * 2 )) > 0. ( 10
)
Moreover the expression of the total revenue is given by

J * = (pq x * 2 1 -α -c) E * δ . ( 11
)
This revenue is positive if the fishery profit is positive at this equilibrium, that is to say if

x * 2 1 -α > c pq = 1 θ .
But this inequality holds because we have

T ( x * 2 1 -α ) > T ( 1 θ ) = (1 -θ) r 2 θ 2
and due to the fact that θ ≥ θ 0 > 1, this last term is nonpositive .

Proposition 1

In the logistic case if r 1 > δ the problem (3) possesses at most a non trivial and positive optimal stationary solution caracterised by

x * 1 = α(r 1 -δ) 2r 1
and x * 2 given by [START_REF] Clark | Mathematical Bioecomics: The Optimal Management of Renewable Resources[END_REF]. The corresponding effort, diffusion coefficient and total revenue are given respectively by ( 10), ( 1), [START_REF] Houde | Testinomy on Use of Marine protected Areas (MPAs) as a Fisheries Management Tool, as a Means to Protect and restore Marine Ecosystem, and as a Research Tool[END_REF].

Remarks 1) When α = 1 2 the value of θ 0 coincides with the value pm given in [START_REF] Ami | Can marine protected areas enhance both economic and biological situations ?[END_REF]. 2) From the expression of θ 0 in (8), we observe that r 1 can't be closed to r 2 . If this is not the case, then the value of the dimensionless ratio θ has to be very high. But this can be unrealistic because the value of θ is given by the economic environment.

3) From ( 8) with a given value for θ we can precise a bound for r 2 expressed in terms of r 1 , δ:

r 2 ≤ θ -2r 1 r 1 -δ θ + r 1 δ r 1 .

The second variation: the splitting of a unique zone

In this second model we start with an unique zone with capacity K that we normalise to one, K = 1. Let us assume that the population follows a standard evolution law:

ż(t) = φ(z(t)) (for instance = rz(t)(1 -z))
φ(.) being a C 1 concave function defined on [0, 1], with φ(0) = φ(1) = 0 and φ ′ (1) < 0. We assume that this zone splits first in a part that is a reserved area where no fishing is allowed and a complementary part that is open to harvest. We assume that these two parts have respectively α and 1 -α as a carying capacity. The main difference with the previous model is that the two populations, which stocks are respectively x 1 and x 2 follow the evolution laws given by:

ẋ1 (t) = F (x 1 (t), x 1 (t) + x 2 (t)) ẋ2 (t) = F (x 2 (t), x 1 (t) + x 2 (t))
where F (., .) satisfies the standard assumption of regularity with F (x, z) = 0 if and only if x = 0 or z = 1 and where F (.) and φ(.) satisfy

F (x 1 , x 1 + x 2 ) + F (x 2 , x 1 + x 2 ) = φ(x 1 + x 2 ).
For instance F can be a logistic function

F (x, z) = rx(1 -z) = rx(1 -(x 1 + x 2 )).
As in the previous model, there is some diffusion between the two zones which can be represented by: λ(

x 2 1 -α - x 1 α ).
Then the two populations evolve following the dynamics:

ẋ1 (t) = F (x 1 (t), x 1 (t) + x 2 (t)) + λ( x 2 (t) 1 -α - x 1 (t) α ) ẋ2 (t) = F (x 2 (t), x 1 (t) + x 2 (t)) -λ( x 2 (t) 1 -α - x 1 (t) α ).
We want to stress on the fact that this new model is consistant in the sense that the sum of the two dynamics is exactly the evolution law of the total population. Now taking into account the catch in the zone where fishing is allowed, we derive the final dynamics of the populations

ẋ1 (t) = F (x 1 (t), x 1 (t) + x 2 (t)) + λ( x 2 (t) 1-α -x 1 (t) α ) ẋ2 (t) = F (x 2 (t), x 1 (t) + x 2 (t)) -λ( x 2 (t) 1-α -x 1 (t) α ) -qE x 2 (t) 1-α (12)
where E ∈ [0, E M ] stands for the fishing effort and q > 0 is the catchability coefficient.

In order to compare with the previous model, we assume that a manager has as an objective to maximise the actualised total revenue as presented before. To do so, he has to act on two controls, the fishing effort and the location of the reserve area given by λ. Therefore the manager faces the following program of optimisation max E(.), λ(.)

∞ 0 e -δt (pq x 2 (t) 1-α -c)E(t) dt s.c. [START_REF] Lauck | Implementing the Precautionary Principles in Fisheries Management through Marine Reserves[END_REF] 0 ≤ E(t) ≤ E M (13)

Analysis of the solutions

From the dynamic equations ( 12) we can derive the expression of the effort

E(t) = 1 -α qx 2 (t) (φ(x 1 (t) + x 2 (t)) -ẋ1 (t) -ẋ2 (t)) ( 14
)
and then we obtain the equivalent problem to (13) as a calculus of variations problem:

max X∈C ∞ 0 e -δt (p - c(1 -α) qx 2 )[φ(x 1 + x 2 ) -ẋ1 -ẋ2 ] dt
where C stands for the set of feasible curves defined by:

C = {x(.) = (x 1 (.), x 2 (.)) x i (.) ∈ BC 1 ([0, ∞[), x i (0) given , φ(x 1 + x 2 ) -qE M x 2 1-α ≤ ẋ1 + ẋ2 ≤ φ(x 1 + x 2 )}.
We know that in this framework a necessary optimality condition for an interior solution x(.) is given by the Euler-Lagrange equations that are:

ẋ1 = x 2 ( pqx 2 c(1-α) -1)(φ ′ (x 1 + x 2 ) -δ) + φ(x 1 + x 2 ) ẋ2 = x 2 ( pqx 2 c(1-α) -1)(δ -φ ′ (x 1 + x 2 )).
For now, we will stick to the logistic case. The Euler-Lagrange equations are then:

ẋ1 = x 2 ( pqx 2 c(1-α) -1)(r -2r(x 1 + x 2 ) -δ) + r(x 1 + x 2 )(1 -(x 1 + x 2 )) ẋ2 = x 2 ( pqx 2 c(1-α) -1)(δ -r + 2r(x 1 + x 2 )). ( 15
)
In order to derive the non trivial positive equilibria, denoted by (x * 1 , x * 2 ), we first consider the second equation in [START_REF] Sanchirico | Additivity properties in metapopulation models : application for the assessment of marine reserve[END_REF] with the condition

x * 1 + x * 2 = r -δ 2r .
This implies as a result

0 = r(x * 1 + x * 2 )(1 -(x * 1 + x * 2 )) = (r -δ)(r + δ) 4r .
A contradiction if r > δ. In the case where r = δ, then we obtain the trivial solution x * 1 = x * 2 = 0. Then we deduce that an equilibrium has to necessarily satisfy

x * 2 = c(1 -α) pq .
With the help of first equation in [START_REF] Sanchirico | Additivity properties in metapopulation models : application for the assessment of marine reserve[END_REF] we find that either

x * 1 + x * 2 = 0 or x * 1 + x * 2 = 1.
Finding a non trivial equilibrium implies to exclude the first condition. Therefore we have proved the following result Lemma 3 In the logistic case there is a unique non trivial and positive solution for the Euler-Lagrange solutions [START_REF] Sanchirico | Additivity properties in metapopulation models : application for the assessment of marine reserve[END_REF] given by:

(x * 1 , x * 2 ) = (1 - c(1 -α) pq , c(1 -α) pq ).
In order to examine whether this candidate solution of the problem (13) can be optimal or not, we have to derive the corresponding effort and diffusion coefficient. From the expression of the effort [START_REF] Pezzey | A simple bioeconomic model of a marine reserve[END_REF], we obtain φ(x

* 1 + x * 2 ) = φ(1) = 0 that is to say E * = 0.
We also deduce that λ * = 0 except if θ = 1. Finally at this equilibrium the intertemporal revenue is null too. We have established the following proposition

Proposition 2

In the logistic case the problem (13) possesses at most a non trivial and positive stationary solution given by

(x * 1 , x * 2 ) = (1 - c(1 -α) pq , c(1 -α) pq ).
The corresponding effort, diffusion coefficient and total revenue are null. Remarks

1) It is easy to obtain that this equilibrium is normal, i.e.

x *

1 α > x * 2 1 -α if the fishery is profitable, that is to say if pq -c > 0.
2) An adaptation of the model given in Gomez et al. [START_REF] Cartigny | The optimal spatial distribution of small and large scale fisheries Gomez, to appear in Ecological Modelling[END_REF] to our case of a no take zone is:

ẋ1 = αr(x 1 + x 2 )(1 -(x 1 + x 2 )) + λ( x 2 1-α -x 1 α ) ẋ2 = (1 -α)r(x 1 + x 2 )(1 -(x 1 + x 2 )) -λ( x 2 1-α -x 1 α ) -q 2 E 2 x 2 1-α . (16)
We can get the same results as those given earlier. In Gomez et al. [START_REF] Cartigny | The optimal spatial distribution of small and large scale fisheries Gomez, to appear in Ecological Modelling[END_REF], fishing is allowed in the so called artisanal zone (corresponding to the protected area in our case) and the objective to be maximised is somewhat different (it takes into account the revenues of the artisanal and industrial fisheries). Here also it has been proved that a unique solution exists but with a non null effort and a non null revenue.

areas. It should be the same for a wide class of models. For instance for models that correspond to a situation where fishing is allowed in the the two areas ( [START_REF] Mchich | The dynamics of a fishstock exploited in two fishing zones[END_REF]). We consider thus:

ẋ1 = F 1 (x 1 ) + λ( x 2 1-α -x 1 α ) -qEx 1 ẋ2 = F 2 (x 2 ) -λ( x 2 1-α -x 1 α ) -qEx 2 (17)
and

ẋ1 = F (x 1 , x 1 + x 2 ) + λ( x 2 1-α -x 1 α ) -qEx 1 ẋ2 = F (x 2 , x 1 + x 2 ) -λ( x 2 1-α -x 1 α ) -qEx 2 (18)
with the same assumptions as before. In order to compare numerically ( 17) and ( 18) we will face the same issue to determine significant growth rates.

We propose to use this new situation in order to fix values for r 1 , r 2 , r. The new problem we consider now is to maximize the same objective as before

J(E) = ∞ 0 (pq(x 1 + x 2 ) -c)Ee -δt dt
but with ( 17) and [START_REF] Sanchirico | A Bioeconomic Model of Marine Reserve Creation[END_REF]. We observe that [START_REF] Sanchirico | A Bioeconomic Model of Marine Reserve Creation[END_REF] corresponds to the classic Clark model, it is enough to let z = x 1 + x 2 to obtain that the dynamic is ż = φ(z) -qEz.

We can assume that a manager has no reason to use one model rather than another. Then the two models can be considered as equivalent in the sense that they provide the same optimal effort. Hence, we propose the following procedure to determine a system of growth rates: Let's set an arbitrary choice of values for r 1 and r 2 . From the first order optimality conditions, given here by the Pontryagin principle, we can derive the optimal value of the corresponding effort for problem [START_REF] Sanchirico | Bioeconomics of Spacial Exploitation in a Patchy Environment[END_REF]. We hand-over this value in the first order optimality conditions of the second problem [START_REF] Sanchirico | A Bioeconomic Model of Marine Reserve Creation[END_REF] from which we derive the value of the growth rate r.

To follow this procedure we set: α = .5, δ = .05, c = .15, q = 2., λ = 20 and we obtained for r 1 = 0.4, r 2 = 0.05 that Ē = .0566 and r = 0.28739.

Let's now go back to our models with protected areas from where we take the previous values for the parameters and where we set r 1 = .4, r 2 = .05 and r = 0.28739 . Then for the model with patches we found that:

• the optimal effort is E * = .0457, and the biomass values are respectively

x * 1 = .21875, x * 2 = .0302

and for the global model

• E * = 0 et x * 1 = .875, x * 2 = .125 .
We observe that the optimal efforts corresponding to the patches case, E * = .0457, and the Clark model ( 18), Ē = 0.0566, have similar sizes. We know that the optimal value of the effort in this last model can't be considered as null. Therefore we can deduce that in the first model with patches the effort is not null. Then the two models have different qualitative behaviour:total revenues and optimal effort are totally different.

Conclusion

In this article, we have shown that different models have been proposed and used in the literature for the same MPA problematic. We focused on demonstrating the importance of the hypotheses underlying two types of models -the patch model and sub-zone model -particularly the crucial role played by the growth functions (rate and form), and on studying the different results produced by them.

A manager who wishes to study the role of an MPA in a given zone must first know if the entire zone is artificially divided or if it can be broken down into patches (entities with their own dynamics). Without taking this precaution, and in obtaining the very different results that we have seen, the manager risks taking erroneous decisions.

x 2 ( pqx 2 c(1 -α) -1)(F ′ 2 (x 2 ) -δ) + F 2 (x 2 ) = -F 1 (x * 1 )
which is [START_REF] Clark | Mathematical Bioecomics: The Optimal Management of Renewable Resources[END_REF] in the logistic case

x 2 [ 2r 2 pq c(1 -α) 2 x 2 2 -( pq c(1 -α) (r 2 -δ) + r 2 1 -α )x 2 -δ] = α (r 1 -δ)(r 1 + δ) 4r 1 .
The graphes of the functions defined by the left and right hand members are curves that crosse in a single x * 2 if r 1 ≥ δ. But this last inequality holds because from our assumption we have

F 1 (x * 1 ) = α (r 1 -δ)(r 1 + δ) 4r 1 ≥ 0.
This ends the proof of Lemma 1.

Appendix 2

In order to find conditions for the inequality

x * 2 1-α ≤ x * 1
α to be true, we know that it is equivalent to consider the inequality T (

x * 2 1-α ) ≤ T ( x * 1 α ). This last inequality becomes α 1-α (r 1 -δ)(r 1 +δ) 4r 1 ≤ T ( x * 1 α ) = r 1 -δ 2r 1 [ 2r 2 pq c ( r 1 -δ 2r 1 ) 2 -( pq c (r 2 -δ) + r 2 ) r 1 -δ 2r 1 -δ] that is equivalent to α(r 1 + δ) ≤ (1 -α)( r 1 -δ r 1 (θδ r 1 -r 2 r 1 -r 2 ) -2δ).
1) If r 1 > r 2 the right hand member has to be positive, this implies the following condition on θ:

θ > θ 0 := ( 2r 1 r 1 -δ + r 2 δ ) r 1 r 1 -r 2 .
Now if this condition on θ holds, from the previous inequality we should deduce [START_REF] Dubey | A model for fishery resource with reserve area[END_REF].

2) If r 1 ≤ r 2 , the right hand member is always negative and therefore (x * 1 , x * 2 ) can't be normal. This ends the proof of Lemma 2.

Appendix 3

We consider the case where the manager has the size of the preserved area as control. We will first prove that the diffusion coefficient has to depend on this size. We start with the dynamics and the objective given in the second variation ( § 3.2). We suppose that the manager has to maximise his objective by using the fishing effort E and the size of the preserved area that is captured by α. We always denote by z the stock of the total population and the two subpopulations stocks by x 1 and x 2 respectively. Then the densities in the two regions are d 1 = x 1 α and d 2 = x 2 1-α . When α = 0 we can only find a single zone and thus x 1 = 0 and x 2 = z. In this case, it is natural to set for the densities: d 1 = 0 and d 2 = z respectively. Now if α = 1, it is natural to set: d 1 = z d 2 = 0. As we have done before, we assume that some diffusion exists between the two zones and that it is proportional to the difference of the two densities. Therefore in order to respect our previous remark, we have to set

λ(α)( x 2 1 -α - x 1 α ) (20)
where the diffusion coefficient depends on α. Indeed if α = 0, from d 1 = 0 we deduce that λ(0) x 2 1-α = λ(0)z = 0 because in this case we can only find a single zone, and thus λ(0) = 0. From a similar argument, we deduce that for α = 1, we have λ(1) = 0. Now for α ∈]0, 1[ the coefficient λ(α) is certainly not null. For instance we can let

λ(α) = λ 0 α(1 -α) (21)
and in this case the diffusion is modelised by

λ 0 α(1 -α)( x 2 1 -α - x 1 α ). ( 22
)
This expression is the one proposed by Boncoeur in [START_REF] Boncoeur | Fish, Fishers, Seals and Tourists: Economic Consequences of Creating a marine reserve in a Multi-Species, multi-activity Context[END_REF].

Then the problem of the manager becomes in this setting max E(.), α

∞ 0 e -δt (pq x 2 (t) 1-α -c)E(t) dt s.c. [START_REF] Lauck | Implementing the Precautionary Principles in Fisheries Management through Marine Reserves[END_REF] (23)

Comparison, Numerical application

In this section we underline the differences between the results we obtained in the previous sections for both the patches case and the global model. From their expressions given in the Propositions 1 and Proposition 2, we can make the following remarks for the equilibria (x * 1 , x * 2 ):

• In the model with patches the first component x * 1 doesn't depend explicitely on the ratio c pq , whereas it does in the global model.

• In the global model the second component is given by x * 2 = c(1-α) pq , whilst the patches model doesn't possess any equilibrium with such a component as the second, cf. Appendix 1.

Thus the expressions of the equilibria are different in the two models. Now we established that the optimal effort and the corresponding total revenu at (x * 1 , x * 2 ) was null for the global model. This doesn't seem to be the case for the patches model, we will later show with simulations that optimal effort and total revenue are not significantly close to zero.

In order to continue the comparison, let's arbitrarily fix the parameters p, q, c, α, δ. Thus the models depend only on the instantaneous growth rates r 1 , r 2 and r respectively. If we let r 1 = r 2 = r, in Lemma 2 we established that the equilibrium (x * 1 , x * 2 ) was never normal in the patches case, while it is always normal for the global model (Remark 1 in section 3.2). Now to compare our models with r 1 and r 2 only near r, we noted in the Remark 2 of section 3.1 that this situation wasn't a realistic one.

Then comparing these two models is not an easy task. Our first conclusion is: the role of the instantaneous growth rates of the biomasses are crucial to choose such or such model. An assumption that is not underlined in general. Now let's come back to the comparison of the optimal efforts and revenues by using simulations. The main issue is to determine significant growth rates that are not equal. But this choice shouldn't depend on our particular models with preserving

The two preceding models of resource dynamics are adapted to the case where control instruments are independent of the size of the no take reserve. If the manager must take size into account in his decisions, the modelling of the dynamic has to be changed. For instance, it is necessary to consider a depending of α diffusion coefficient. A justification is given in Appendix 3. In this Appendix, we also underline that this coefficient can be given by formula

which is the expression considered by Boncoeur (cf. [START_REF] Boncoeur | Fish, Fishers, Seals and Tourists: Economic Consequences of Creating a marine reserve in a Multi-Species, multi-activity Context[END_REF]).

5 Appendix

Appendix 1

We determine the non trivial equilibria of the Euler-Lagrange equations ( 6)

If we assume that x * 2 = c(1-α) pq , from the first equation:

From the assumption of the non triviality of the equilibria, we have that c = 0, α = 1, c = pq and then F 2 (x * 2 ) > 0. Therefore we can't find any x 1 ∈ [0, α] such that (19) holds. Thus there is no non trivial equilibrium with x * 2 = c(1-α) pq . Therefore in order for a non trivial equilibrium to exist it is necessary that

)) = δ. In the logistic case it is easy to compute that

where the diffusion coefficient in ( 12) is given by (21). The solution of this problem is straightforward using the Pontryagin maximum principle. We won't mention it in this paper.