Bruno Amable
email: bruno.amable@univ-paris1.fr

Ivan Ledezma
email: ivan.ledezma@dauphine.fr

Export performance and product market regulation *

Keywords: exports, product market regulation, competitiveness

This paper analyses the impact of product market regulation on the propensity to export at the industry level for 13 OECD countries and 13 industries over the 1977-2007 period. Recent economic policy and academic literature insists on the negative effects of product market regulation on productivity or innovation, and hence on "competitiveness", a term that we interpret as the ability to export. Similar to the conclusions of some contributions to a recent literature on competition and growth, the "common sense" is that product market regulation should be detrimental to competitiveness. Testing through a two-step estimation the impact of upstream pressures of product market regulation on productivity and the effect of the latter on the propensity to export, this paper shows that upstream regulatory pressures have a significantly positive impact on productivity and thereby on the capability of an industry to attract resources and to sell its production in international markets.

Performance à l'exportation et réglementation sur les marchés de biens et services

Résumé

Cet article analyse l'impact de la réglementation sur les marchés de biens et services sur la propension à exporter au niveau sectoriel pour 13 pays de l'OCDE, 13 secteurs sur la période 1977-2007. La littérature académique récente comme celle portant sur la politique économique insiste sur les effets négatifs de la réglementation sur la productivité ou l'innovation et par conséquent sur la « compétitivité », que nous définissons ici comme la propension à exporter. Suivant les conclusions d'une certaine littérature sur la croissance et la concurrence, le « sens commun » est que la réglementation sur les marchés de biens et services nuit à la compétitivité. En testant en deux étapes les conséquences de la réglementation en amont sur la productivité et les effets de celle-ci sur la propension à exporter, ce papier montre que la réglementation a un effet significativement positif sur la productivité et par là-même sur la capacité d'un secteur à attirer des ressources et vendre sa production sur les marchés internationaux.

Mots-clés : exportations, réglementation sur les marchés de biens et services, compétitivité JEL: F14

Introduction

The issue of competitiveness has gone back to the forefront of policy debates on the importance of industry for Europe since the Great Depression (European Commission, 2010). A competitive manufacturing industry is seen as indispensable for a sustainable growth and the achievement of full employment. Facing 'intensified global challenges' such as the competition of new industrialised countries or the pressure on resources, European countries would need to put competitiveness and sustainability at the centre-stage of industrial policy. In this respect, competition policy would have to play a major role because functioning competitive markets would contribute to improve the international market position of European industries. Competition would be a driver of innovation and efficiency gains and create strong incentives for firms to increase productivity. Therefore the implementation of a competition-promoting legal environment would be a central element of an economic policy for the promotion of competitiveness.

The positive influence that product market competition would play on innovation, productivity, or growth has been a central theme of the recent literature on growth (Aghion et al. 2005) and the conclusion that less product market regulation (PMR) would imply more innovation/productivity/growth has been promoted as a central policy recommendation by the OECD. 2 Although the link between PMR and export performance has received little attention in empirical academic research, there is now a quite substantial literature on the influence of product market regulation on innovation and technical progress. The mainstream view whereby higher levels of PMR should hinder innovation, slow down productivity gains and hamper macroeconomic growth have actually dominated a rapidly growing policy-oriented empirical research that links measures of product market regulation to measures of productivity at the industry level (Nicoletti and Scarpetta, 2003; Conway et al., 2006; Arnold et al., 2008; Bourlès et al., 2012). Particular attention in this literature has been paid to the extent to which the (usually expected) negative effect of PMR is especially pronounced at the leading edge in technology, the so called "world technology frontier". This is motivated, in fact, by a rough translation of the results in Aghion et al. (2005) who suggest that the relationship between product market competition (PMC) and innovation is hump-shaped and that the peak of this curve is 'larger and occurs at a higher degree of competition in more neck & neck industries'3 , that is to say in industries where firms compete at the same technological level. Some sort of conflation between product-market liberalisation and low profitability (and so competition), and between the "world technology frontier" (WTF) and "neck & neck industries" has implicitly been assumed to give a rationale and to structure the above mentioned empirical literature. 4 Elsewhere (Amable, Demmou and Ledezma, 2010, 2013; Amable, Ledezma and Robin, 2011; Ledezma, 2013) we have critically examined this line of argument, both theoretically and empirically. Knowledge standardisation consequences of PMR (Ledezma, 2013) and leader R&D pre-emption in an otherwise standard step-by-step model without innovative leaders à l 'Aghion et al. (2005) (Amable, Demmou and Ledezma, 2010) have been put forward to highlight, and more generally to recall, the ambiguities in the theoretical relationship between competition, regulation and innovation. 5 We have also argued therein that our empirical results, pointing out a positive interaction between PMR and the proximity to the technology frontier in determining innovation and productivity, are by no means an isolated finding. Although the claims differ in interpretation, Nicoletti and Scarpetta (2003) and Conway et al. (2006) find parameters estimates of similar sign with other economy-wide and country level time-varying indicators of regulation. 6 In this paper we go further on these issues and test whether countries where industries are more regulated suffer from a technological handicap on their ability to export and therefore perform a lower level of export activity. We use industry-level information for a sample of 13 OECD manufacturing industries in 13 countries from 1977 to 2007 and rely on a two-step estimation strategy that seeks to explain the export share of production of an industry in a given country by traditional determinants of comparative advantage such as international differences in factor endowments as well as international differences in productivity at the industry level. These latter are in turn explained by product market regulatory pressures. Hence, in our modelling framework, PMR affects export activity through its influence on technical progress.

The above mentioned debate of competitiveness cannot be treated in an underlying theoretical context of comparative advantage without some embarrassment. 7 However no empirically implementable theory of competitiveness is available and we need to put some theoretical structure into the analysis. Hence, we interpret the word "competitiveness" in its fairly popular signification of "ability to export", which still has sense when the analysis is grounded on relative productivity measures, as we do. We rely on sophisticated estimations of multifactor productivity levels provided by the Groningen and Growth Development Centre, the construction of which is relative to the United States. We focus on the export share of production, which can be interpreted as both a measure of export specialisation as well as a measure of export performance in an international comparative analysis, once national scale effects are taken into account. This variable has also the highest coverage, with information starting in 1979. 8 The particularities of the indicator used to capture PMR give further precisions on the scope of the analysis. We use the regulation impact indicator of the OECD. It measures the impact of regulatory pressures in key input sectors on the rest of the economy accordingly to their use, which leads to a panel data structure that explains its wide use in the PMR empirical literature. As such, its most popular interpretation is that of the knock-on effect of upstream regulation on downstream industries.9 But it also presents high correlation with other more direct economy-wide product market regulation indicators so that the regulation impact indicator also remains a good proxy of business constraints (See Amable, Demmou and Ledezma, 2010).

Estimations obtained with these data do not support the postulate associating lessregulated industries with higher levels of export orientation. Rather the contrary appears: the pressures generated in manufacturing industries from upstream provisions appear to positively influence their productivity level, which in turn translates in higher export specialisation. Such a result recalls the ambiguities in the relationship between competition and technical progress and is consistent with our previously mentioned findings.

To the best of our knowledge this is the first attempt to make a link from PMR to technology driven comparative advantage at the industry level. Since at least Leontief (1953) or Balassa (1963), there is an important literature testing classical and neoclassical predictions about international trade specialisation. Recent empirical attempts (e.g. Harrigan, 1997; Redding, 2002; Nickell, Redding and Swaffield, 2008) have implemented a theoretically grounded empirical framework based on a GDP function derived from duality to incorporate both the technology and the endowment explanations of trade within the same estimation of GDP shares.10 This approach has also proved to be compatible for new trade theories (Helpman and Krugman 1985, Feenstra and Klee 2008). The link to market institutions, however, has received scarce attention. An exception is Nickell, Redding and Swaffield (2008) who, among other analysis, establish a relationship between deindustrialisation in the OECD and labour market institutions by relating estimates of specialisation adjustments with proxies of labour market protection. In our empirical framework we integrate the role of PMR directly in the identification strategy as an excluded variable. Moreover, GDP-shares dualitybased estimates usually require industry-by-industry identification, which yields directly interpretable coefficients, namely on Rybzcinsky elasticities, but does not exploit the full sample variance for other explanatory variables. Here, in this paper we have preferred to follow Romalis (2004) for an empirical strategy that considers factor endowments scaled by their respective factor intensity. Hence, it is possible to run full sample regressions and expect a positive coefficient on the composed terms of factor inputs, as the national availability of a production factor should positively influence trade specialisation when it is intensely used in the production process. 11 The other way around (the link from trade to technical progress) has also been largely tracked in empirical studies at several levels of aggregation. There is a large body of macroeconomic literature, recently surveyed by Harrison and Rodriguez-Clare (2010), analysing how trade openness affects economic growth at the country level. After the methodological criticism of Rodriguez and Rodrik (2001) that had tempered initial claims of sizeable positive effects of trade openness, recent panel data studies have provided new evidence on a positive correlation between trade volumes and economic growth, but generally after other reforms than trade liberalisation have taken place. In any case the usefulness of using trade volumes as proxies of trade liberalisation policies still lacks of general agreement, even if the problem is (partially) addressed through instrumenting strategies. At the firm level, the empirical research is also abundant on country specific case studies testing the effects of trade on firm productivity.12 Recent works on this research line emphasise the heterogeneous responses stemming from heterogeneous firms, with a general consensus on positive effects at least for most performing firms.

Although substantial, these macro and micro empirical literatures on the relationship between trade openness and productivity have not provided a consistent discussion on the role of national product market provisions. Recent papers in the PMR literature rely on firm-level data to explore the link between trade, national regulatory reforms and productivity. Ben Yahmed and Dougherty (2012) do so using micro-level data for several OECD countries. They test whether import penetration improves firm productivity accordingly to different national regulatory environments. They find negative interactions between import penetration and PMR at the leading edge with economy-wide indicators of barriers to entrepreneurship and administrative burdens. The overall effect of PMR, however, cannot be identified as the specification considers only an interaction term for PMR, without including it alone in linear form, which reflects the authors' focus on the ability of national industries to face foreign competition. With our modelling framework, we allow for a larger role of PMR on the internationalisation of countries since it is considered at the root of technology driven comparative advantage, while still controlling from reverse channels.

The rest of the paper gives the details of what we have announced so far. Next section presents the data and methodology used; the following section discusses descriptive data statistics and explores reduced forms relationships. We then turn to the main estimations and briefly conclude in a final section.

Methodology a. Empirical strategy

Following standards theories of comparative advantage, we are interested in testing the following system

ln(𝑀𝐹𝑃 𝑖𝑐𝑡 ) = 𝛼 0 + � 𝜃 𝑗 𝑗∈𝐽 𝐼 𝑗𝑡 ln�𝐹 𝑗 � + 𝜌 ln(𝑃𝑀𝑅 𝑖𝑐𝑡 ) + 𝜖 𝑖 + 𝜖 𝑐 + 𝜖 𝑡 + 𝑢 (1) ln(𝑋𝑃 𝑖𝑐𝑡 ) = 𝛽 0 + � 𝛾 𝑗 𝑗∈𝐽 𝐼 𝑗𝑡 ln�𝐹 𝑗𝑡 � + 𝛿 ln(𝑀𝐹𝑃 𝑖𝑐𝑡 ) + 𝜀 𝑖 + 𝜀 𝑐 + 𝜀 𝑡 + 𝜇 (2)
Equation ( 2) explains an export performance measure 𝑋𝑃 𝑖𝑐𝑡 of country 𝑐 in industry 𝑖 at time period 𝑡, by its multifactor productivity level 𝑀𝐹𝑃 𝑖𝑐𝑡 and a set of country's factor endowments 𝐹 𝑗𝑡 that are scaled by their factor intensity in production. This latter transformation seeks to estimate the extent to which the increase in factor availability reinforces export performance in sectors that intensively use them. 13 Equation (1) expresses that multifactor productivity 𝑀𝐹𝑃 𝑖𝑐𝑡 is in turn explained by factor endowments and product market regulation 𝑃𝑀𝑅 𝑖𝑐𝑡 , which does not directly participate in the explanation of export performance. As a consequence, the identification will follow an instrumental variable approach where we treat 𝑀𝐹𝑃 𝑖𝑐𝑡 as an endogenous variable and 𝑃𝑀𝑅 𝑖𝑐𝑡 as the instrument within a two-stage ordinary least square estimation. Equation (2) will thus be exactly identified, 𝑃𝑀𝑅 𝑖𝑐𝑡 will be the excluded instrument and factor endowments 𝐹 𝑗𝑡 and other dummy variables will be the included instruments. We rely in fact on the hypothesis that the policy measure is exogenous to technical progress, which should be a good approximation at least in the short run of our static specification. Moreover, in the estimations that follow the null hypothesis of weak instruments is systematically rejected.

With this specification we can estimate the parameters without splitting the sample by industry, as it would be the case if factor inputs were not scaled by their relative utilisation so that different Rybczynski effects would be expected to depend on the industry. In this manner we exploit the full sample variability of regulation environments and that of technological performances while controlling for factor inputs. From standard international trade theories we expect then 𝛾 𝑗 and 𝛿 to be positive. Usual arguments postulating a positive impact of market liberalisation policies would suggest 𝜌 < 0, which we are about to test.

Unobserved heterogeneity is supposed to come from the three dimensions of the data: country, industry and time. We consider then fixed effects for each of them. In the robustness checks we shall also consider individual fixed effects (one for each country-industry couple) in a model where national industries present specific characteristics explaining their export share.

b. Data and sources

We have collected data on capital assets, hours worked and value added from the EU KLEMS database, constructed by the Groningen Growth and Development Centre (GGDC). Details on labour inputs are only available in the 2008 release whereas the rest of series have been drawn from the 2009 release, updated on March 2011. 14 The sample considered in the main regressions potentially contains information on 13 manufacturing industries in 13 OECD countries for the 1977-2007 period, with main estimations varying from 3534 to 3781 observations. The latest two years are dropped when estimations include a higher level of detail for labour inputs. A superior coverage can yet be obtained in pairwise descriptive regressions with consistent results.

As EUKLEMS is consistent with national accounts, we have aggregated data on total hours worked by persons engaged (L) and data on gross fixed capital formation (K) for the whole economy and used them as measures of factor endowments. In alternative specifications, we also consider more detailed data on hours worked by high-medium-and low-skilled workers (resp. HS, MS and LS). Factor intensities are proxied via the share of the factor compensation on value added, also available in EUKLEMS.

EU KLEMS also provides measures of multifactor productivity (MFP) growth, but not of MFP levels. In order to obtain the latter, we use the GGDC Productivity Level Database, which gives a benchmark of MFP in levels relative to the United States for 1997. Since these measures need to be comparable across time, countries and industries, a specific deflation procedure is performed with heavy data details, especially to construct purchasing power parities at the industry level. For this reason MFP measures in levels are available only for the benchmark year 1997. Combining this benchmark with MFP growth of EU KLEMS it is possible, however, to reproduce MFP series in levels for our sample period.15 These data have been completed with other information from the OECD. We use the OECD indicator of regulation impact (Henceforth REGIMP). It informs about the "knock-on" effects of PMR in key input sectors (energy, transport and communication, retail and distribution, finance and professional services) on the rest of the economy. For each industry, regulation in these sectors is weighted accordingly to their use thanks to harmonised inputoutput tables constructed by the OECD. Hence, a large value of the regulation impact indicator may be the effect of both a restrictive regulation in a particular input sector or a heavy use of the latter as supplier. This proxy is useful as it gives a measure of the strength of regulatory provisions in business operations and at the same time has the advantage of being available in a panel format (i.e. time-varying for each country-industry couple). 16 Finally, the export share of production has been obtained from the OECD STAN indicators database (v. 2009).

Preliminary analysis a. Data patterns

We start with an overview of the data. Box plots in Figure 1 show the central tendency and the dispersion of our main variables, available on a yearly basis for each industry in each country. 17 In order to better assess the specificity of our sample heterogeneity, the graph on each variable is displayed twice, by industry and by country, which are the most structuring dimensions of the data. Panels (a) and (b) present the export share of production (XSP). The hierarchy of countries is in conformity with expectations, large countries (the US or Japan) have lower and less dispersed export shares than smaller countries.

For several countries, the export shares of textiles (17t19), electrical and optical equipment (30t33) as well as chemical products (24), are substantially higher in the respective country sample distribution. However, if one considers the country by country representation, there is in general one specific outlier industry exhibiting an exceptional export performance. In an analogous fashion, for a given industry there is in general one specific country outperforming the rest in relative terms, with the smaller highly dispersed countries (notably, the Netherland, Belgium and Denmark) being remarkable in this respect Panels (c) and (d) give the median and dispersion of the regulation indicator (regimpact), which measures the strength of downstream restrictiveness caused by upstream regulation, the so-called knock-on effect of upstream regulation. Clearly, considering the country specificity rather than the industry specificity leads to clearer data patterns, although scarcely any observable association between the regulation indicator and the size of the country appears. Interestingly, however, the top and the bottom of the country hierarchy of the export share of manufacturing and the knock-on effect of upstream regulation are inverted. Whereas Belgium appears as having the strongest average knock-on effects of upstream regulation it is the country featuring highest export orientation, which is also consistent with its small size. The opposite is true for the US. A similar graphical analysis is displayed for the levels of multifactor productivity (relative to the US) in panels (c) and (d). As can be seen, there is substantial dispersion in the levels of productivity according to both country and industry dimensions. The US is not on average the most productive country. Some industries seem to be characterised by extreme values of the productivity level, in particular industries belonging to the chemical, rubber, plastics and fuel products (23 to 25). Some countries exhibit that pattern too, notably Ireland and UK. The highest multifactor productivity level in the sample belongs actually to Ireland in the chemical industry (24), however, excluding Ireland, this industry presents substantially less extreme values.

This first descriptive analysis suggests that data patterns are heavily structured by country and industry specificities that should be kept in mind at the moment of explaining different export 29 30t33 29 34t35 29 29 30t33 30t33 30t33 29 30t33 29 30t33 29 30t33 30t33 29 30t33 30t33 30t33 30t33 30t33 30t33 30t33 30t33 30t33 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 34t35 30t33 30t33 industry-level orientation in our sample. The next section goes further on this by proposing an exploratory regression on reduced-form relationships.

U S A D N K N L D S W E I R L A U S U K F I N F R A G E R H U N A U T J P N C Z E E S P I T A B E L Panel (c)

b. Exploratory regressions

Before starting the main econometric analysis it is instructive to examine the empirical relationship between the regulatory indicator and the performance measures of export orientation and productivity. This analysis is done through a set of exploratory regressions shown in Table 2. Regression in column ( 1) is a simple pooled OLS regression of the (log of) export market share on the (log of) the product market regulation indicator. Such a bivariate estimation gives a significantly negative correlation, which would be compatible with the dominant "common sense" (Aghion and Griffith, 2005) interpretation of the negative influence of PMR on productivity and hence competitiveness. However, when this model is extended to include year dummies (column (2)), industry dummies (regression (3)) and country dummies (column (4)), the consequence is to turn progressively the significantly negative elasticity of REGIMP into a significantly positive one. Therefore, correcting for common shocks and heterogeneity across countries and industries eliminates the "common sense" negative correlation between product market regulation and export orientation. The same conclusion arises in a fixed-effect model assuming the unobserved heterogeneity at the country-industry level, that is to say, in a model where national industries present specific unobserved time-invariant characteristics explaining its export propensity.

Regressions reported in columns ( 6)-( 8) describe the empirical relationship between multifactor productivity and the product market regulation indicator. Consistent with the previous findings, the popular view portraying highly regulated environment as intrinsically inefficient is not found. With all the set of dummies included (column (6)) and even within a more exigent fixed-effect specification (column (7)) the estimated elasticities are again significantly positive. The last regression (column (7)) is a robust one in the sense that it seeks to minimise the effect of outliers, namely of those regarding the dependent variable. 18 This additional robustness check confirms the sign of the previous estimates. Note: Standard errors in parentheses. POLS stands for pooled OLS regression, WG for within group estimates and RR for robust regression to outliers in the sample.

In the same spirit of tracking the effect of possible outliers in the previous estimates, the upper panels of Figure 2 plots the partial elasticity relating the regulation indicator to the export market share as specified by regression in column (4), which controls for country, industry and time fixed effects. 19 This regression is run twice, including (Panel (a)) and excluding (Panel (b)) the Coke, refined petroleum products and nuclear fuel industry (23), the rubber and plastic industry (25) as well as Ireland. As previously mentioned, there we observe the most extreme values of multifactor productivity, performing in general more than five times the efficiency level of the US. Clearly, such a filter implies a significantly higher and more precise elasticity of REGIMP. Panel (c) presents the estimates of the elasticity of the product market regulation indicator after performing quantile regressions at different percentiles of the conditional distribution of the dependent variable (see Koenker and Hallock, 2001). 19 This is done in a two dimensional space by plotting conditional residuals, which is an application of the Frisch-Waugh theorem. Finally, given the important heterogeneity in the sample, one may ask if the fit performed at the mean is actually representative of the relationship in other location of the distribution. Panel (c) in Figure 2 plots the coefficients and confidence intervals stemming from quantile regressions having the same specification as regression in column ( 4) and performed at different percentile of the conditional distribution of (the log) of XSP. Basically, these estimations seek to fit the model at different quantiles of the conditional distribution of the export share of production, instead of fitting at its expected conditional mean (as in a standard OLS). The graph also presents the OLS elasticity with its confidence intervals (the horizontal lines depicted in the figure). This exercise reveals that the estimated OLS elasticity provides a fairly representative picture of the relationship between the export share of production and the regulation indicator in most of the conditional distribution.

To sum up, the "common sense" notion that product market regulation should lead to lower productivity and lower export orientation is not supported by this preliminary inspection of the data. In order to assess the robustness of these findings, we now turn to the estimation of the system of equation discussed above, which includes, in a more parsimoniously way, traditional determinants of comparative advantage.

Estimation results

Table 3 presents the estimation results for the first stage equation (Table 3.a) linking product market regulation to productivity and for the second stage (Table 3.b) where the export share of production is determined by the productivity level. Six different specifications are considered, depending on whether all variables are considered relative to the US (regressions in columns (3), ( 4) and ( 6)) or not, whether the labour input is detailed according to the skill level (high, medium, low, presented in columns (2) and (4)) or not, and whether the model consider individual (country-industry couples) fixed effect (columns ( 5) and ( 6)) or not.

The simplest specification is an instrumental variable (IV) regression distinguishing only aggregate capital and labour input (column (1)). In this estimation, the first stage results show a negative influence of factor inputs variables on multifactor productivity, which may reflect a smaller than unity scale effect or an inaccurate accounting of factor inputs, but a significantly positive influence of product market regulation. Recall that factor input variables are multiplicative terms that interact the national availability of the factor with the intensity of their use in production. Hence, as expected, the availability of factors intensely used in an industry has a positive impact on its export share of production at the second stage. Moreover, multifactor productivity also significantly boosts export orientation of the industry. Therefore, this IV estimation suggests that product market regulation has a positive influence on productivity, which in turns favourably impacts exports. Distinguishing several skill levels gives different estimations of the contributions of factor inputs (column (2)). In the first stage only the medium skilled labour input obtains a significantly negative coefficient, but high skill labour input is significantly positively contributing to productivity. As in the previous estimation, the coefficient of the regulatory indicator is substantially higher and still significantly positive. In the second stage, can be noted the positive contributions of capital and high-skill labour inputs, which means that export share are higher when these factors are proportionally heavily used in production. Here also, productivity favourably influences exports, but the coefficient is somewhat lower than in the regression reported in column (1).

The estimates presented in column (3) take into account all variables relative to the US. This is done as a robustness check on the fact that the measures of productivity levels that we can obtain are only available relative to the US, which means that equations are more meaningful if expressed in relative terms. These estimates can be compared with those presented in column (1). Results are not substantially different. The same conclusion applies to regression (4), which can be compared to regression (2). One may note however that the positive impact of product market regulation is larger in specifications relative to the US.

Regressions reported in columns ( 5) and ( 6) take a more precise account of heterogeneity. Besides time fixed effects they include individual country-industry fixed effects. Remarks regarding the impact of factor inputs at each stage are similar to those made with the other specifications. The positive impact of product market regulation is somewhat higher in fixedeffect specifications than in the equivalent IV regressions and the impact of productivity on exports slightly lower.

Overall, the conclusions drawn from these estimations are that, contrary to the "common sense" regarding the consequences of product market regulation, the impact of the latter on productivity is significantly positive, which translates in better export performance at the industry level.

Conclusion

This paper has analysed the impact of product market regulation on the propensity to export at the industry level for 13 OECD countries and 13 industries over the 1977-2007 period. Recent economic policy and academic literature insists on the negative effects of product market regulation on productivity or innovation. Since the latter can be held to be positive factors influencing a broadly defined "competitiveness", the conclusion from the dominant opinion on that matter is that product market regulation should be detrimental to exports. The results of the tests performed in this paper lead to a different conclusion. Using various estimation specifications, it is shown that the knock-on effect of product market regulation in key input sectors (energy, transport and communication, retail and distribution, finance and professional services) has a positive influence on the productivity of industries. In turn, productivity has, as expected, a positive effect on the propensity to export. Therefore, the common sense opinion that product market regulation harms competitiveness is not supported. These results confirm previous findings on the influence of PMR on innovation (Amable, Demmou and Ledezma, 2010, 2013; Amable, Ledezma and Robin, 2010; Ledezma, 2013).

Figure 1 .

 1 Figure 1. Note: The figure presents box-plot graphical analysis of central tendency and dispersion of main variables by country and industry. See footnote (4) for further details.

  ITA ITA ITA ITA ITA ITA ITA ITA ITA ITA UK BEL IRL IRL UK AUS BEL AUS IRL AUS UK BEL UK UK UK UK UK FRA UK AUS GER IRL IRL IRL IRL IRL IRL IRL IRL IRL UK UK DNK UK UK DNK UK UK UK DNK DNK DNK DNK DNK DNK UK UK DNK DNK DNK UK DNK UK UK DNK UK UK UK UK UK UK UK DNK UK DNK UK DNK UK UK UK UK UK UK FRA ITA FRA ITA ESP ITA BEL ITA ITA ITA BEL BEL BEL BEL BEL ESP FRA ESP FIN SWE FRA FIN FIN FIN FIN ESP FIN FRA FRA FIN FIN FRA ESP FRA FIN FRA ESP ESP FRA ESP FRA FRA FRA HUN HUN HUN CZE ITA ITA ITA ITA ITA ITA ITA SVN ITA FRA BEL BEL FRA FRA FRA BEL BEL BEL FRA BEL BEL FRA BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL BEL FRA FRA BEL Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

Figure 2 .

 2 Figure 2. Note: Panel (a) and (b) presents the partial elasticity of the regression of the export share of production on the product market regulation indicator (in log form) after controlling for time, country and industry fixed effects. Panel (a) considers the full sample whereas panel (b) exclude potential outliers (industry 23, 25 and Ireland).Panel (c) presents the estimates of the elasticity of the product market regulation indicator after performing quantile regressions at different percentiles of the conditional distribution of the dependent variable (seeKoenker and Hallock, 2001).

  Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

Table 1 .

 1 Main variables

	Variable Description	Panel structure	Source
	K	Gross fixed capital formation	country, year	EU-KLEMS
	L	Total hours worked	country , year	EU-KLEMS
	HS	Total hours worked by high skilled	country, year	EU-KLEMS
		labour
	MS	Total hours worked by medium	country, year	EU-KLEMS
		skilled workers
	LS	Total hours worked by medium	country, year	EU-KLEMS
		skilled workers
	I j	Factor rewards over value added j ∈	country, industry,	EU-KLEMS
		{ K, L, H S , MS, LS}	year
	MFP	Multifactor productivity relative to	country, industry,	EU-KLEMS &
		the US	year	Productivity
				Levels
				Database
				(GGDC)
	XSP	Export share of production	country, industry,	OECD STAN
			year
	REGIMP Knock-on effect of upstream	country, industry,	OECD
		regulation	year	PMR

Table 2

 2

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
		𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑀𝐹𝑃) 𝑙𝑛(𝑀𝐹𝑃) 𝑙𝑛(𝑀𝐹𝑃)
	Estimation	POLS	POLS	POLS	POLS	WG	POLS	WG	RR
	𝑙𝑛(𝑅𝐸𝐺𝐼𝑀𝑃) -0.153***	0.052	0.070** 0.219*** 0.125*** 0.335*** 0.510*** 0.453***
		(0.035)	(0.037)	(0.033)	(0.041)	(0.04)	(0.055)	(0.067)	(0.045)
	Fixed-effects	No	Year	Industry,	Country,	Individual	Country,	Individual	Country,
				Year	Industry,	(country-	Industry,	(country-	Industry,
					Year	industry),	Year	industry),	Year
						Year		Year
	Obs.	4977	4977	4977	4977	4977	5518	5518	5518

Table 3 .

 3 a. First-stage estimations

		(1)	(2)	(3)	(4)	(5)	(6)
		𝑙𝑛(𝑋𝑆𝑃)	𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃)	𝑙𝑛(𝑋𝑆𝑃)	𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃)
				-𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )	-𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )		-𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )
	Estimation	IV	IV	IV	IV	FE-IV	FE-IV
	𝐼 𝐾 𝑙𝑛(𝐾)	-0.056*** 0.019**			-0.027***
		(0.009)	(0.009)			(0.006)
	𝐼 𝐿 𝑙𝑛(𝐿)	-0.163***				-0.130***
		(0.012)				(0.007)
	𝑙𝑛(𝑅𝐸𝐺𝐼𝑀𝑃)	0.126**	0.398***			0.160***
		(0.060)	(0.059)			(0.060)
	𝐼 𝐻𝑆 𝑙𝑛(𝐻𝑆)		0.167***
			(0.051)
	𝐼 𝑀𝑆 𝑙𝑛(𝑀𝑆)		-0.138***
			(0.028)
	𝐼 𝐿𝑆 𝑙𝑛(𝐿𝑆)		-0.008
			(0.034)
	𝐼 𝐾 𝑙𝑛(𝐾)			-0.056***	0.021**		-0.033***
	-𝐼 𝐾 𝑈𝑆 𝑙𝑛(𝐾 𝑈𝑆 )		(0.009)	(0.009)		(0.006)
	𝐼 𝐿 𝑙𝑛(𝐿)			-0.155***			-0.120***
	-𝐼 𝐿 𝑈𝑆 𝑙𝑛(𝐿 𝑈𝑆 )		(0.011)			(0.006)
	𝑙𝑛(𝑅𝐸𝐺𝐼𝑀𝑃)			0.153**	0.415***		0.226***
	-𝑙𝑛 �𝑅𝐸𝐺𝐼𝑀𝑃	𝑈𝑆 �		(0.060)	(0.059)		(0.062)
	𝐼 𝐻𝑆 𝑙𝑛(𝐻𝑆)				0.167***
	-𝐼 𝐻𝑆 𝑈𝑆 𝑙𝑛(𝐻𝑆 𝑈𝑆 )			(0.045)
	𝐼 𝑀𝑆 𝑙𝑛(𝑀𝑆)				-0.147***
	-𝐼 𝑀𝑆 𝑈𝑆 𝑙𝑛(𝑀𝑆 𝑈𝑆 )			(0.027)
	𝐼 𝐿𝑆 𝑙𝑛(𝐿𝑆)				-0.012
	-𝐼 𝐿𝑆 𝑈𝑆 𝑙𝑛(𝐿𝑆 𝑈𝑆 )			(0.035)
	Fixed Effects	Country,	Country,	Country,	Country,	Individual	Individual
		Industry,	Industry,	Industry,	Industry,	(country-	(country-
		Year	Year	Year	Year	industry),	industry),
						Year	Year
	Obs.	3781	3534	3781	3534	3781	3781

Table 3 .

 3 b. Second-stage estimations.

		(1)	(2)	(3)	(4)	(5)	(6)
		𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) 𝑙𝑛(𝑋𝑆𝑃) -𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )	𝑙𝑛(𝑋𝑆𝑃) -𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )	𝑙𝑛(𝑋𝑆𝑃)	𝑙𝑛(𝑋𝑆𝑃) -𝑙𝑛(𝑋𝑆𝑃 𝑈𝑆 )
	Estimation	IV	IV	IV	IV	FE-IV	FE-IV
	𝑙𝑛(𝑀𝐹𝑃)	1.778* 0.518*** 1.447**	0.479***	1.310**	0.423*
		(0.909) (0.145)	(0.627)	(0.137)	(0.581)	(0.252)
	𝐼 𝐾 𝑙𝑛(𝐾)	0.115**	0.009			0.035*
		(0.057) (0.009)			(0.018)
	𝐼 𝐿 𝑙𝑛(𝐿)	0.303*				0.181**
		(0.158)				(0.078)
	𝐼 𝐻𝑆 𝑙𝑛(𝐻𝑆)		0.283***
			(0.055)
	𝐼 𝑀𝑆 𝑙𝑛(𝑀𝑆)		0.052
			(0.033)
	𝐼 𝐿𝑆 𝑙𝑛(𝐿𝑆)		0.039
			(0.033)
	𝐼 𝐾 𝑙𝑛(𝐾)			0.102**	0.021**		0.027***
	-𝐼 𝐾 𝑈𝑆 𝑙𝑛(𝐾 𝑈𝑆 )			(0.041)	(0.009)		(0.010)
	𝐼 𝐿 𝑙𝑛(𝐿)			0.242**			0.070**
	-𝐼 𝐿 𝑈𝑆 𝑙𝑛(𝐿 𝑈𝑆 )			(0.105)			(0.032)
	𝐼 𝐻𝑆 𝑙𝑛(𝐻𝑆)				0.247***
	-𝐼 𝐻𝑆 𝑈𝑆 𝑙𝑛(𝐻𝑆 𝑈𝑆 )				(0.049)
	𝐼 𝑀𝑆 𝑙𝑛(𝑀𝑆)				0.097***
	-𝐼 𝑀𝑆 𝑈𝑆 𝑙𝑛(𝑀𝑆 𝑈𝑆 )				(0.032)
	𝐼 𝐿𝑆 𝑙𝑛(𝐿𝑆)				0.036
	-𝐼 𝐿𝑆 𝑈𝑆 𝑙𝑛(𝐿𝑆 𝑈𝑆 )				(0.033)
	Fixed Effects	Country,	Country,	Country,	Country,	Individual	Individual
		Industry,	Industry,	Industry,	Industry,	(country-	(country-
		Year	Year	Year	Year	industry),	industry),
						Year	Year
	Number of Obs 3781	3534	3781	3534	3781	3781
	Note: Standard errors in parentheses. p<0.10, ** p<0.05, *** p<0.01

Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

Table A2 .

 A2 Documents de Travail du Centre d'Economie de la Sorbonne -2013.10 Chemicals and chemical products List of countries

	25	Rubber and plastics
	26	OTHER NON-METALLIC MINERAL
	27t28 BASIC METALS AND FABRICATED METAL
	29	Machinery,
	30t33 ELECTRICAL AND OPTICAL EQUIPMENT
	34t35 TRANSPORT EQUIPMENT
	36t37 MANUFACTURING NEC; RECYCLING
	Note: capital letters indicate 2-digit industry aggregation
		Code	Description
		AUS *	Australia
		AUT	Austria
		BEL	Belgium
		CZE *	Czech Republic
		DNK	Denmark
		ESP	Spain
		FIN	Finland
		FRA	France
		GER	Germany
		HUN *	Hungary
		IRL *	Ireland
		ITA	Italy
		JPN	Japan
		NLD	Netherland
		SVN *	Slovenia
		SWE	Sweden
		UK	United Kingdom
		USA	United States
		Note : * indicates countries without information on
		export share, but considered in the exploratory
		regression of multifactor productivity.

Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

See the various issues of the yearly publication of the OECD: Economic Policy Reforms: Going for Growth

[START_REF] Aghion | Competition and Innovation: An Inverted-U Relationship[END_REF] Proposition 5

We use the term conflation in light of a recent wave of innovation models that focus on endogenous entry (See Etro, 2007 for a systematic treatment). This possibility generally leads to equilibrium market structures with a monopolist featuring persistently positive profits, so that low profitability and intense competition are by no means two faces of the same coin. Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

The traditional Schumpeterian insight whereby market power provides incentives to innovate actually appears in most theoretical models of the endogenous growth and patent-races literatures. For[START_REF] Aghion | Competition and Innovation: An Inverted-U Relationship[END_REF], product market competition encourages innovation in a step-by-step innovation process since in a neck-and-neck technological competition firms will try to escape competition by innovating. However, at the same time, laggards' innovation will be discouraged by competition as they anticipate lower post innovation profits. As we argue in the above-mentioned papers, the absence of innovative activity of leaders and more generally the lack of analysis of entry deterrence are important neglected elements.

In an error correction model, the authors claim a slowing-down effect of PMR in the natural catching-up process of laggard industries.

See[START_REF] Krugman | What Do Undergrads Need to Know About Trade?[END_REF] for an assertive point on this. 8 A revealed comparative advantage indicator considering world markets is available in the STAN Indicators database, but only from 2000 on. Using it would imply losing information on key decades of the current wave of globalisation.Documents de Travail du Centre d'Economie de laSorbonne -2013.10

Indeed, recent debates on manufacturing competitiveness, in particular with respect to the France-Germany comparisons, have focused on the role of services (e.g.[START_REF] Gallois | Pacte pour la compétitivité de l'industrie française[END_REF].

See Dixit and Norman (1980) for a formal presentation.

[START_REF] Nunn | Relationship-Specificity, Incomplete Contracts, and the Pattern of Trade[END_REF] also follows a similar strategy to test prediction from the application of incomplete contracts to trade theory. Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

See the trade related works surveyed by[START_REF] Syverson | What Determines Productivity?[END_REF]

See Nunn (2007) or[START_REF] Romalis | Factor Proportions and the Structure of Commodity Trade[END_REF] for similar identifications strategies. Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

http://www.ggdc.net/databases/euklems.htm. A complete description of EU KLEMS can be found in O'Mahony and Timmer (2009)

See Inklaar and Timmer (2008 ) for details. Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

www.oecd.org/eco/pmr. SeeConway and Nicoletti (2006)

Each box displays the interval between the 25 th percentile and the 75 th percentile (i.e. the interquartile range), with the horizontal line inside the box showing the median. The length of the vertical lines (portrayed bellow the lower quartile and above the upper one) are given by the so-called adjacent values. These values are computed as the most extreme values within an interval equal to one and a half times the length of the interquartile range. Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

The method used performs an iterative process of assessment of outliers based on residuals which are in turn weighted accordingly to their magnitude (See Hamilton,

1991). Documents de Travail du Centre d'Economie de la Sorbonne -2013.10

* The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2011) under grant agreement n°225349 (ICaTSEM project).

Appendix


Gérald Gurtner

Marc Durand
email: marc.durand@univ-paris-diderot.fr

Stiffest Elastic Networks

Keywords: Hashin-Shtrikman bounds, elastic networks, frameworks, truss, foams, Maxwell's criterion, beams

The rigidity of a network of elastic beams crucially depends on the specific details of its structure. We show both numerically and theoretically that there is a class of isotropic networks which are stiffer than any other isotropic network with same density. The elastic moduli of these stiffest elastic networks are explicitly given. They constitute upper-bounds which compete or improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks, and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions.

Introduction

Various elastic systems can be understood as networks of interconnected beams. Examples include polymer gels, protein networks and cytoskeletal structures [1][2][3][4][5], or wood and bones [START_REF] Gibson | Cellular Solids -Structure and properties[END_REF][START_REF] Ashby | [END_REF][8][9][10]. On a length scale much larger than the typical beam length ("macroscopic" scale), such a network can be viewed as a continuous and homogeneous medium characterized by spatially constant elastic moduli. When this medium is subjected to uniform stresses at its boundaries, it will undergo a homogeneous deformation with a constant strain [START_REF] Landau | Theory of elasticity[END_REF]. Such homogeneous deformations are called affine. This picture of affine strain is generally valid at length scales large compared to any characteristic inhomogeneities: the macroscopic displacement field ū(r), defined as the spatial average of displacements over a sufficiently large domain surrounding every point r, is affine [START_REF] Didonna | [END_REF]. On a microscopic level however, the displacement field is generally not affine, and beams can deform by a combination of stretching, bending and twisting mechanisms. In addition, the nature of the junctions between beams plays an important role in the stability and mechanical properties of such systems: networks with junctions that either fix the relative orientation of the beams ("rigid junctions") or allow free rotation ("free hinges") can have very different mechanical responses [3]. Actually, junctions in most of the real systems have an intermediate behaviour: a small but finite energy cost is associated with the change of their geometry.

The relationship between the mechanical properties of elastic networks on a macroscopic level and the details of their structures is still poorly understood. The stiffness of such a system clearly depends on its density φ, defined as the volume of beams per unit volume of material. But for a given value of φ, it is also dramatically affected by the specific spatial arrangement of the elastic phase within the material. On dimensional grounds [START_REF] Ashby | [END_REF], the volumetric density of strain energy ε associated with a stretch-dominated deformation varies linearly with φ, while it scales as φ 2 for the deformation of a three-dimensional network dominated by the beam bending mode. Thus, for the low-density materials considered here (φ ≪ 1), a structure deforming primarily through the beam stretching mode is usually much stiffer. However, the constant of proportionality between ε and φ still varies significantly among stretched-dominated networks.

In this paper, we reveal both numerically and theoretically the existence of a class of elastic networks which are stiffer than any other ones having the same symmetry, same density and same elastic phase. These stiffest networks deform through the stretching mode exclusively, and their displacement field is affine down to microscopic scale for any loading conditions. We derive and analyse the necessary and sufficient conditions under which a network belongs to this class. We will limit our study to isotropic structures, though the reasoning can be transposed without difficulty to materials with lower symmetries.

In section 2 we investigate numerically the role of node connectivity and structural disorder on the affinity of the displacement field and the stiffness of two-dimensional elastic networks. Our simulations show that stretch-dominated networks with same node connectivity can present significantly different elastic moduli and displacement fields. In section 3 we propose a theoretical framework to rationalize these observations: we derive bounds on the elastic moduli of isotropic networks using a variational approach, and establish simple rules on the geometrical and topological arrangement of beams in a network to make it stiffer for the same amount of material. In section 4, we analyse the restrictions imposed by these structural conditions, and show that they rationalize our numerical findings. In particular, they prove that stiffest networks deform affinely down to the microscopic scale. Finally, in section 5, we provide examples of both two-dimensional (2d) and three-dimensional (3d) regular structures with highest elastic moduli.

Simulations

method

In order to investigate the interplay between node connectivity, affinity of the strain, and stiffness of the network, we simulated the mechanical behaviour of different 2d regular networks, made of straight and uniform beams (same cross-section and material). Five different networks have been simulated in order to inspect the effect of both connectivity z and disorder on the mechanical properties. Three regular ones: hexagonal (connectivity z = 3), kagome (z = 4), triangular (z = 6), and two disordered ones: Voronoi (z = 3) and Delaunay (z = 6) networks, which are generated from uniformly distributed points on the plane.

The hamiltonian of a network of beams has three different contributions:

H = (i,j) H s ij + (i,j) H b ij + i H n i . (1)
The first two sums are carried over all beams (i, j) of the network, and represent the contributions of the stretching and bending energy of the beams, respectively:

H s ij = lij 0 (κ s /2)(∂u s /∂l) 2 dl, and H b ij = lij 0 (κ b /2)(∂ 2 u n /∂l 2 ) 2
dl, where u s and u n are the tangential and normal components of the displacement u, respectively. κ s and κ b are the stretching and bending moduli, related to the Young's modulus of the material E 0 , the section of the beam s and the second moment of area I by: κ s = E 0 s and κ b = E 0 I. In the simulations each network is represented by the set of mobile nodes {x i } consisting of all beam junctions and midpoints between junctions (the latter so as to include the first bending mode of the beams), with each contribution linearized with respect to changes in the {x i }. The discretized version of the stretching and bending contributions are:

H s ij = κ s (u ik • e ij ) 2 + (u jk • e ij ) 2 /l ij , and H b ij = 4κ b ((u ik + u jk ) × e ij ) 2 /l 3 ij , where u ij = u j -u i .
The third sum in Eq. ( 1) is carried over all junctions i, and H n i denotes the energy cost associated with the change of geometry of junction i. In absence of this energy cost (free hinges), torques cannot be sustained and beams deform through the stretching mode exclusively and act as hookean springs. Conversely, if some "node rigidity" is included, other modes of deformation are solicited. Different expressions can be considered for H n i [2,3]. In the simulations presented here, we use

H n i = κ n k,k ′ ∈N (i) (∆θ ikk ′ ) 2 /(l ik + l ik ′ )
, where N (i) denotes the set of nodes that are connected to node i and ∆θ ikk ′ is the change of the angle between beam bending beam stretching node bending i j k k'

Figure 1: sketch of discretized beams. Each beam is composed of two segments of equal length. A beam bending energy is associated with a change of angle between the two segments of a beam, a node bending energy is associated with a change of angle between adjoining beams, and a stretching energy is associated with a change of length of the beam segments.

the adjoining beams (i, k) and (i, k ′ ) (see Fig. 2.1). Because of the analogy of this term with the bending energy of a beam, we will refer this mode of deformation to as "node bending" in the following, and κ n as the "bending modulus of a junction". The size of the box L is approximately 40 times larger than the average beam length l. Range of values used in our simulations for κ s , κ b , and κ n are such that κ s l 2 ≫ κ b ≫ κ n (typically, κ s l 2 ∼ 10 3 κ b , and

κ b ∼ 10 3 κ n ).
Either a shear or uniaxial strain γ is applied across the top and bottom boundaries, while periodic boundary conditions are used in the other direction. Within our linearized scheme, H({x i }) is a high dimensional paraboloid with a unique global minimum, corresponding to the state of mechanical equilibrium. Hence, the linear system can be solved directly using a LU decomposition method included in the UMFPACK routines [13]. Once the equilibrium positions are obtained, the associate strain energy can be calculated. Finally the shear modulus µ and longitudinal modulus M are obtained by equating the energy with µγ 2 L 2 /2 for shear strain, and M γ 2 L 2 /2 for uniaxial strain. Any other elastic moduli can then be calculated. For instance, the Young's modulus E is related to µ and M as:

E = 2µ(dM -2(d -1)µ)/((d -1)M -2(d -2)µ) [14].

Results

For each simulated network, we represent the repartition of energy between beam stretching, beam bending, and node bending modes, and measure the affinity of the displacement field. Different parameters have been proposed in literature [2,5,15] to quantify the degree of (non)affinity of a displacement field, but most of them are global measures of the affinity. We use instead the local affinity measure m i defined at every node (junction and midpoint) i as m 2 i = (ū iu i )) 2 /(γL) 2 , where ūi is the macroscopic, affine, displacement field of the node i.

Figure 2 shows the repartition of energy between beam stretching, beam bending and node bending when the networks are subjected to shear strain (similar results, not shown here, are obtained for uniaxial strain). Hexagonal and Voronoi networks deform mainly through the node bending mode, while kagome, triangular and Delaunay networks deform mainly through the stretching mode. These results are consistent with Maxwell criterion on rigidity of 2d frameworks [8,[16][17][18]: networks with connectivity above 4 are rigid; they deform primarily through the stretching of beams, while those with connectivity below 4 are soft; that is, they deform through floppy modes if κ n = 0 [19] (free hinges), or through the bending of beams and nodes if κ n = 0. The repartition of energy between the beam and node bending modes depends on the relative importance of κ b and κ n (rigid junctions with fixed angles correspond to the limit κ n ≫ κ b ). In our simulations the bending of beams is quasi-absent (see Fig. 2), because κ n ≪ κ b .

Figure 3 shows the affinity of the displacement field for the five networks under shear strain. Unsurprisingly, all networks with connectivity < 4 deform through the bending modes and thus present non-affine displacement fields. On the other hand, the measure of the affinity reveals major differences between networks with connectivity ≥ 4. Comparison of triangular and Delaunay lattices specially is intriguing: despite the fact that these two networks share similar topological features (they have same mean connectivity and both are fully triangulated), the triangular network has a pure affine strain, while the Delaunay lattice presents a non affine strain. Evaluation of the elastic moduli also shows differences between these two networks: we found µ/(E 0 φ) ≃ 0.091 and M/(E 0 φ) ≃ 0.277 for Delaunay lattice, while µ/(E 0 φ) ≃ 0.125 and M/(E 0 φ) ≃ 0.374 for the triangular lattice (where φ denotes the network density). Therefore, while both networks are stretchdominated, the triangular lattice is significantly (≃ 27%) stiffer than the Delaunay lattice. We checked that the numerical values we obtained were independent of κ n , consistent with our choice for the range of parameter values (κ n ≪ κ s l 2 ). Quite unexpectedly, the kagome and triangular lattices share similar mechanical properties, although these two lattices are structurally very different (kagome has connectivity z = 4 and is only partially triangulated): both present an affine displacement field, and the values obtained for their elastic moduli are very close (we obtained for kagome lattice: µ/(E 0 φ) ≃ 0.125 and M/(E 0 φ) ≃ 0.376).

(a) (b) (c) (d) (e) (f)
These results show that no clear connexion can be established between the rigidity of a network and the (mean) connectivity of its nodes, aside from Maxwell criterion. On the other hand, affinity of the displacement field and network stiffness seems correlated: among the five structures studied, the two networks that deform in an affine way (triangular and kagome lattices) are those with highest elastic moduli. The Delaunay network presents a piecewise affine displacement field [20], and has significantly lower elastic moduli. Finally the two soft, bending-dominated, networks (hexagonal and Voronoi) have highly non-affine displacement fields. In the next section we provide a theoretical framework to rationalize the relationships between node connectivity, network stiffness, and affinity of the displacement field.

Stiffest networks: existence, structure, elastic moduli

In the following we show that there exists a class of networks which are stiffer than any other with same symmetry and density (and beam Young's modulus). We restrict our analysis to isotropic structures, although our theoretical framework can be transposed to non-isotropic structures as well. We first establish upper bounds on the elastic moduli of an isotropic network of beams. These bounds coincide with the numerical values obtained for the triangular and kagome lattices. We then derive the structural properties (actually, a set of necessary and sufficient conditions) of optimal networks, i.e. networks that have maximal elastic moduli for a given value of E 0 and φ. These conditions involve both geometry and topology, which explain why connectivity alone is not enough to predict the macroscopic behaviour of an elastic network.

Isotropy conditions and bounds on the elastic moduli of isotropic networks

Assuming that on the macroscopic scale, the network is a continuous, homogeneous and isotropic medium, the strain energy per unit volume is (for small strains [START_REF] Landau | Theory of elasticity[END_REF]):

ε = λ 2 ( α ūαα ) 2 + µ α,β ū2 αβ , ( 2
)
where λ is the Lamé's first parameter of the network, µ the shear modulus (or Lamé's second parameter), ū the macroscopic displacement field (displacement averaged over a domain large compared to any characteristic homogeneities in the network), and ūαβ = 1 2

∂ ūα ∂x β + ∂ ūβ ∂xα
are the components of the strain tensor (ū α are the components of ū (r)). Under uniform loading conditions, ū varies linearly with the position r: ū(r) = A r, where A is a matrix that characterizes the strain. For instance, A αβ = γδ αx δ βy for a uniform shear strain γ in the xy plan, and A αβ = γδ αβ for a uniform radial strain γ. According to Eq. ( 2), the density of strain energy corresponding to such homogeneous strain is:

ε = λ 2 ( α a αα ) 2 + µ α,β a 2 αβ ( 3
)
with

a αβ = (A αβ + A βα ) /2.
On the microscopic scale, we suppose that the network is made of interconnected Euler-Bernoulli beams that can have inhomogeneous natural curvatures and cross-sections, thus extending our previous analysis on stiff networks [21]. It is worth mentioning that for 3d structures, the Hamiltonian has more terms than in equation (1), because of the existence of two bending modes in different planes and the presence of twisting. But since their expressions are not going to appear explicitly in the following, we do not need their precise forms.

The equations of mechanical equilibrium derived from the theory of linear elasticity can be equivalently expressed in terms of a variational principle, sometimes known as the Principle of Minimum Potential Energy (PMPE): consider a body with volume V and prescribed displacements on its boundaries; the PMPE states that, among all kinematically admissible displacement fields (i.e. all continuous displacement fields satisfying the displacement constraints on the boundary), the actual displacement (i.e. the one satisfying the equations of mechanical equilibrium) is the one that makes the energy functional E = V εdV an absolute minimum, where ε is defined through Eq. ( 2). This principle is commonly used to find approximate solutions to boundary valued problems (Rayleigh-Ritz method). Here we use it to derive rigorous upper bounds on the elastic moduli by choosing an ad-hoc trial displacement field, and then look for the network architecture for which the actual displacement field matches the trial displacement field.

Let us note r ij (l) and s ij (l) the position vector and cross-sectional area along the beam (i, j), respectively, where l refers to the arc-length starting at node i (see Fig. 3.1). We also note l ij the length of beam (i, j), L ij the distance between nodes i and j, and e ij the unit vector along this straight line. For any network structure, one can always define a continuous displacement field such that every infinitesimal piece of the beam (i, j) undergoes the same rotation ω ij and elongation ǫ ij . This trial displacement field is (for small strains):

u ij (l) = u i + ǫ ij r ij (l) + ω ij × r ij (l) ∀ l ∈ [0, l ij ] , (4)
with

ǫ ij = e ij • (u j -u i ) /L ij and ω ij = e ij × (u j -u i ) /L ij .
Thus, every beam (i, j) is subject to a translation u i , a homothety with ratio ǫ ij and a global rotation ω ij . To complete our definition of the trial displacement field, we impose that each node follows the macroscopic affine displacement: u i = A r i , where r i denotes the position vector of node i. Clearly, the trial displacement field defined this way is kinematically admissible. Since each elementary piece of beam undergoes the same rotation ω ij , there is no bending or twisting deformations, and the only contribution to the strain energy of a piece of beam with infinitesimal length dl is the stretching term (E 0 /2) s ij (l)ǫ 2 ij dl. Moreover, for low-density structures (s 1/2 ij ≪ l ij ), the node contribution to the strain energy is negligible compared to the typical stretching energy of a beam [22]. In this limit, the total energy associated with the trial displacement field defined above reduces to

E trial = E 0 /2 (i,j) v ij ǫ 2 ij
, where the summation is over all the beams that constitute the network, and v ij = lij 0 s ij (l)dl is the volume of beam (i, j). Introducing the density φ as the total beam volume per unit area (2d) or unit volume (3d) of the network, the volumetric density of strain energy ε trial = E trial /V associated with the trial displacement field reads:

ε trial = E 0 2 φ ǫ 2 ij , (5)
where the brackets denote the average defined, for any quantity q ij , as:

q = (i,j) v ij q ij / (i,j) v ij .
The elongation ǫ ij can be rewritten in terms of the elements of A:

ǫ ij = e ij • A • e ij = α,β A αβ e α ij e β ij , (6)
where e α ij = e α • e ij is the cosine of the angle between the beam (i, j) and the α axis (α ∈ {x, y} for 2d materials, and α ∈ {x, y, z} for 3d materials). Similarly, the components of the rotation vector ω ij = e ij × (A • e ij ) can be expressed in terms of the elements of A.

The expression of ε trial involves averaged quantities which can be evaluated using symmetry arguments; for isotropic structures, the strain energy must be identical for any orientation of the applied strain. Thus, the bound expressions must remain invariant under rotation around the perpendicular axes or under permutation of the axis labels. After simple but lengthy calculations that we detail in Appendix A, these invariance properties lead to a set of relations on the global structural properties of optimal networks, which can be summarized as:

       e α ij 2 = 1 d e α ij 4 = 3 d(d+2) e α ij 2 e β ij e γ ij = 0 (β = γ) (7)
with α, β, γ ∈ {x, y} (resp. {x, y, z}), and d = 2 (resp. d = 3) for 2d (resp. 3d) structures. These relations will be referred to as isotropy conditions and their consequences will be analysed in the next section. Equations [START_REF] Ashby | [END_REF] imply in particular that (e β ij e γ ij ) 2 = 1/ (d (d + 2)) and e β ij e γ ij = 0 (for β = γ).

Using these relations together with Eqs ( 5) and ( 6), the density of strain energy simplifies to (see Appendix A):

ε trial = E 0 φ d (d + 2)   1 2 ( α a αα ) 2 + α,β a 2 αβ   . (8)
According to the PMPE, comparison of Eqs. ( 3) and ( 8) yields:

λ 2 ( α a αα ) 2 + µ α,β a 2 αβ ≤ E 0 φ d (d + 2)   1 2 ( α a αα ) 2 + α,β a 2 αβ   . (9)
Bounds on the elastic moduli can then be deduced from this inequality. Consider first the affine displacement field A αβ = γδ αx δ βy , corresponding to a shear strain in the xy-plane. Inequality (9) reduces to: 1: Elastic moduli of stiffest isotropic structures normalized by the Young's modulus of the beam material E 0 : (µ), Young's modulus (E), bulk modulus (κ), longitudinal modulus (M ). We also report the Lamé's first parameter (λ) and Poisson's ratio (ν) of such optimal structures; note however that these two quantities are not elastic moduli, so λ ⋆ and ν ⋆ do not necessarily correspond to their highest possible values.

µ ≤ E 0 φ d (d + 2) = µ ⋆ . ( 10
) µ ⋆ /E 0 E ⋆ /E 0 κ ⋆ /E 0 M ⋆ /E 0 λ ⋆ /E 0 ν ⋆ 2d φ/8 φ/
µ (HS) /E 0 E (HS) /E 0 κ (HS) /E 0 M (HS) /E 0 2d φ/8 φ/3 φ/4 3φ/8 3d φ 15 (1+ φ 6 1 + 3 49κ 0 + 8µ 0 69κ 0 + 8µ 0 φ 9 1 + 9κ 0 3κ 0 + 4µ 0 φ 5 (1+ 18κ 2 0 + 8µ 2 0 + 21κ 0 µ 0 3κ 0 (3κ 0 + 4µ 0 ) 141κ 2 0 + 4µ 2 0 + 116κ 0 µ 0 9κ 0 (3κ 0 + 4µ 0 )
Table 2: Hashin-Shtrikman upper-bounds on the elastic moduli of low-density isotropic materials with one single elastic phase [23][24][25]: shear modulus(µ (HS) ), Young's modulus (E (HS) ), bulk modulus (κ (HS) ), longitudinal modulus (M (HS) ). The bounds are normalized by E 0 . The bounds derived in the present paper and reported in Table 1 coincide with the HS bounds for low-density 2d structures, and are strictly lower (i.e. tighter) for 3d structures.

Consider then the affine displacement field A αβ = γδ αβ , corresponding to a uniform compression. Inequality (9) yields:

κ ≤ E 0 φ d 2 = κ ⋆ , (11)
where κ = λ + 2µ/d denotes the bulk modulus of a d-dimensional body. Any other elastic modulus of an isotropic body is related to κ and µ. In particular, Young's modulus reads E = 2d 2 κµ/(2µ + d(d -1)κ). Since E is an increasing function of κ and µ, it comes that:

E ≤ 2E 0 φ 2 + (d -1) (d + 2) = E ⋆ . ( 12
)
Values of µ, E, M , and κ for 2d and 3d stiffest networks are reported in table 1. We also report the values of the Lamé's parameter λ and Poisson's ratio ν for this class of networks. However, it must be pointed out that they generally do not correspond to the highest possible values of λ and ν. Indeed these two quantities are not elastic moduli, as they are not a direct measure of some strain-stress relationship. It can be noticed that the Poisson's ratio of such structures is below those of incompressible bodies (1 for 2d bodies, 1/2 for 3d bodies), and is independent of the elastic properties of the beam material (E 0 ).

Values of the elastic moduli reported in Table 1 coincide with the Hashin-Shtrikman (HS) bounds for low-density 2d structures [23,24], but are strictly lower (i.e. tighter) for 3d structures [25] (see Table 2). However, it must be pointed out that this improvement of the HS upper-bounds is restricted to materials whose elastic phase is organized into slender objects (beams), while the HS bounds apply to any diphasic structures, including for instance those where the continuous phase in 3d structures is assembled into 2d sheets.

Mechanical conditions

We now show that inequalities ( 9)-( 12) become strict equalities for some specific network geometries, determined by the following set of rules (along with the isotropy conditions (7)):

(a) All the beams are straight.

(b) Every beam (i, j) has uniform cross-sectional area: s ij (l) = s ij .

(c) At every junction i of the 2d (resp. 3d) network, and for all e α , e β , e γ ∈ {e x , e y } (resp. {e x , e y , e z }), the following equality is satisfied:

j∈N (i) s ij e α ij e β ij e γ ij = 0,
where N (i) denotes the set of neighbouring nodes that are connected to node i. Unlike the isotropy conditions [START_REF] Ashby | [END_REF], these mechanical conditions are local rules on the geometry and topology of the network. The demonstration is straightforward: according to the PMPE, the inequalities ( 9)-( 12) become strict equalities if and only if the respective trial displacement fields coincide with the displacement field that satisfies the equations of mechanical equilibrium. Inspection of force and moment balances along each beam and at each junction leads to the three necessary and sufficient conditions stated above: with the trial displacement field (4), each infinitesimal piece of beam is subjected to an axial deformation only. Therefore, the force and local moment deriving from the trial displacement fields are F ij (l) = -E 0 s ij (l) ǫ ij t ij (l) and M ij (l) = 0, respectively, where t ij (l) = dr ij /dl is the local tangent unit vector (F ij (l) and M ij (l) are defined as the force and local torque exerted at position l by the i-side to the j-side of beam (i, j)). The force balance equation dF ij (l) /dl = 0 along the beam (i, j) yields:

ds ij dl (l)t ij (l) + s ij (l) dt ij dl = 0. ( 13
)
Since t ij (l) and dt ij /dl are orthogonal vectors, it comes that ds ij (l)/dl = 0 and dt ij /dl = 0 for all l ∈ [0,

l ij ],
what immediately leads to condition (a) and (b). The moment balance equation dM ij (l) /dl + t ij (l) × F ij (l) = 0 is automatically satisfied for such geometry. Mechanical equilibrium must be satisfied at every junction i as well. When condition (a) and (b) are fulfilled, the force exerted by the straight and uniform beam (i, j) on node i is E 0 s ij ǫ ij e ij . Thus, no torque is exerted on the junction, while the balance of forces yields:

j∈N (i) s ij ǫ ij e ij = 0. This relation must hold for any orientation of the strain field. Replacing ǫ ij by its expression (6) and using the same rotational invariance arguments than in (4.1) eventually yields condition (c).

Analysis of the isotropy and mechanical conditions

Isotropy conditions

The isotropy conditions (7) constitute a set or 4 (resp. 15) conditions for 2d (resp. 3d) structures. We first check that these conditions are satisfied with a continuous and uniform angular distribution of beams (such a distribution is rather unrealistic, but it must have isotropic properties by construction). For that purpose, we reformulate the average defined in Eq. ( 5) for the case of a continuous distribution: let us note f (v, n) the probability density function for a beam to have a volume v and an orientation along the unit vector n. Then, the average of any quantity X(v, n) is:

X = (i,j) v ij V X ij = N V V Ω vf (v, n)X(v, n)dvdΩ, ( 14
)
where N is the total number of beams within the network, V their total volume, and dΩ denotes the elementary angle (2d) or solid angle (3d) of orientation n. If the distribution f is isotropic (i.e.: f is independent of n), and the quantity X depends only on the orientation n (e.g.: X ij = cos 2 θ ij ), Eq. ( 14) simplifies to

X = c Ω X(n)dΩ. (15)
The constant c is obtained by normalization: c = v/π (resp. c = v/(2π)) for 2d (resp. 3d) networks, where v is the mean beam volume in the network. Using Eq. ( 15), it is straightforward to see that a uniform distribution of beams satisfies the isotropy conditions [START_REF] Ashby | [END_REF]. More realistic, discrete distributions of beam orientation can also satisfy the isotropy conditions. In two dimensions, conditions (7) can be rewritten as:

cos 2θ ij = 0, sin 2θ ij = 0, cos 4θ ij = 0, sin 4θ ij = 0, (16)
where θ ij is the angle between the beam (i, j) and the x-axis. We can draw some general properties for networks with uniform angular distribution of beams, i.e. such that the total volume of beams with same (discrete) orientation θ m is independent of θ m (special cases of such networks are the periodic frameworks with identical beams and with junctions which are all similarly situated [8]). Using the complex variables y m = cos θ m + ı sin θ m , the equations ( 16) reduce to:

{m} y 2 m = 0, {m} y 4 m = 0, (17)
where the summation is over all the distinct beam orientations. From this set of equations, it is clear that a distribution with two distinct beam orientations cannot have isotropic properties: square or rectangular lattices, for examples, are known to have anisotropic elastic properties [START_REF] Landau | Theory of elasticity[END_REF]. On the other hand, one can build isotropic networks with three sets of parallel beams. A simple analysis of Eqs (17) shows that the three sets of beams must be tilted from each other with equal angles (modulo π) of π/3. Hexagonal, triangular, and kagome lattices are examples of such isotropic structures. The analysis of the mechanical conditions below allows to determine which ones of them have highest stiffness.

Mechanical conditions

The condition (c) produces a set of 4 (resp. 10) equations per node for 2d (resp. 3d) structures, imposing severe restrictions on the geometry and topology of a junction. In this section we inspect the solutions to this set of equations for 2d networks. We suppose that conditions (a) and (b) are satisfied (straight and uniform beams). Moreover, we assume that beams have the same cross-section. Let z i be the connectivity of node i, and θ ij the angle between the beam (i, j) and the x axis. The vectorial condition (c) can be rewritten as 4 trigonometric relations:

zi j=1 cos θ ij = 0, zi j=1 sin θ ij = 0, zi j=1 cos 3θ ij = 0, zi j=1 sin 3θ ij = 0. (18)
Clearly, there is no solution to this set of equations for a monovalent node (z i =1): all the beams of an optimal network are connected at their both ends and can contribute to the storage of elastic energy. For a divalent node (z i =2), the only solutions are the trivial configurations θ i2 = θ i1 + π. To inspect the possible configurations of nodes with higher connectivity, we rewrite Eqs (18) as:

zi j=1 y ij = 0, zi j=1 y 3 ij = 0, (19)
where we introduced the complex variables y ij = cos θ ij +ı sin θ ij . One can now show that there is no solution to Eqs (19) for a trivalent node (z i =3): eliminating y i3 from these equations yields y i1 y i2 (y i1 +y i2 ) = 0; either y i1 , y i2 or y i3 is zero, what is not compatible with the normalization constraint |y ij | = 1. For quadrivalent nodes (z i =4), eliminating y i4 from Eqs (19) yields: (y i1 + y i2 )(y i2 + y i3 )(y i1 + y i3 ) = 0. Thus, the only possible configurations for such nodes are θ i3 = θ i1 + π, θ i4 = θ i2 + π, and all other subscript permutations, i.e. beams must be collinear in pairs. In agreement with Maxwell's criterion [8,[16][17][18], it comes that 2d stiff networks must have a node connectivity ≥ 4. But in addition, our analysis specifies what must be the geometry of the junctions (see Fig. 5). In the special case where nodes are all similarly situated, it can then be shown that the minimal node valency is 6 [8]. It is possible, in principle, to find the solutions to Eqs (18) for nodes with higher valencies. The solutions are certainly more tedious to find. However, it can be mentioned that for junctions with an even number of adjoining beams, the geometry with beams parallel in pairs is always a solution to Eqs (18). As a consequence of our analysis, it comes that 2d optimal networks must contain triangular cells (but the converse is generally not true, e.g. see Delaunay network in Fig. 3(e)); the number of cells C, the number of sides (beams) N , and the number of junctions J of a 2d network are related by the Euler's formula: C -N + J = 1 [8]. Therefore, the mean coordination number z = 2N/J and the mean number of sides per cell n = 2N/C satisfy: 1/n + 1/z = 1/2. Thus, in optimal networks, z ≥ 4 and n ≤ 4. Clearly, optimal 16) and (18). These are the only solutions when z i ≤ 4.

networks contain triangles when z > 4. The marginal case z = 4 (n = 4) must be inspected separately: either each cell has exactly 4 sides; in order to satisfy the mechanical condition, every vertex must be a fourfold junction with beams collinear in pairs, but we have seen that a structure with only two distinct beam orientations does not satisfy the isotropy conditions. Or cells do not have all the same number of sides; then the constraint n = 4 imposes that such a network must contain triangles.

A striking feature of the stiffest networks is that, under uniform loading, the displacement field is affine down to the microscopic scale and thus coincides with the macroscopic displacement field ū: since every beam must be straight, one has r ij (l) = le ij . Thus, according to Eq. ( 4), the displacement field at location r = r i + r ij (l) simplifies to u(r) = A r i + A le ij = A r = ū(r).

Finally, it is worth mentioning that isotropic networks with optimal elastic properties also have optimal transport properties. Indeed, the isotropy conditions (7) encompass the isotropy conditions e α ij e β ij = δ α,β /d for transport properties. Furthermore, using the identity d α=1 e α ij 2 = 1, it is easy to show that condition (c) implies j s ij e ij = 0, which is the condition required at every junction of a network (together with conditions (a) and (b)) to have optimal transport properties [START_REF] Durand | Proc. R. Soc. Lond. A[END_REF][START_REF] Durand | [END_REF].

Comparison with numerical results

The analysis of the isotropy and mechanical conditions made above sheds light on the numerical results reported in section 2: the five simulated networks satisfy the isotropy conditions ( 16), but only the kagome and triangular networks satisfy the mechanical conditions. It is noteworthy that the kagome lattice is one of the stiffest networks, in spite of its large number of floppy modes [START_REF] Hutchinson | Microarchitectured cellular solidsthe hunt for statically determinate periodic trusses[END_REF][START_REF] Hutchinson | The structural performance of the periodic truss[END_REF][START_REF] Sun | Surface phonons, elastic responsen and conformal invariance in twisted kagome lattices[END_REF]. However, small defects in its structure will dramatically affect its mechanical response [31].

The mechanical conditions involve both geometry (orientation and cross-sectional areas of the beams) and topology (connectivity) of each junction. That explains why two elastic networks with same connectivity, like triangular and Delaunay lattices, can have very different macroscopic responses. Moreover, our analysis has revealed the strong correlation between the stiffness of a network and the affinity of its displacement field. This is also what we observe in the simulations: the two networks with highest elastic moduli -kagome and triangular lattices -are also the only ones with pure affine deformations. It must be emphasized that all stretch-dominated networks do not deform in an affine way. This is illustrated in our simulations with the Delaunay lattice: this structure deforms exclusively through the stretching of its members (see Fig. 2(e)), but does not deform affinely [20] (see Fig. 3(e)). Indeed, the condition (c) is not satisfied in a Delaunay lattice.

Examples of optimal structures

In sections 3 and 4, we have derived the structural conditions that an isotropic network must fulfilled to have highest elastic moduli for a given density and Young's modulus of beam material. Nevertheless, one might wonder whether such networks do exist, that is, whether one can effectively build networks that satisfy all these conditions simultaneously. The answer is obviously yes in two dimensions, as we already provided two examples of optimal networks (kagome and triangular lattices). In this section we show how our theoretical analysis can be used to identify other optimal networks in both two and three dimensions. Since isotropy and mechanical conditions are explicit, it is easy to use them to find new optimal networks. Figure 5 shows three additional optimal networks with 2d periodic arrangements, here made of non-identical beams. The first one, often referred to as the Union Jack lattice, is made with two different kinds of beams. Beams are collinear in pairs at every junction, so the mechanical conditions are satisfied. The ratio of beam cross-sections is then adjusted to fulfil the 2d isotropy conditions ( 16): the only restrictive condition is cos 4θ = 0 [32], yielding 2(l 1 s 1l 2 s 2 ) = 0 (see notations on Fig. 6(a)). Since, from geometry, l 2 = l 1 / √ 2, it comes that s 2 /s 1 = √ 2. The second network (Fig. 6(b)) is built with three different kinds of beams. Here beams are not collinear in pairs at every junction. The isotropy conditions sin 2θ = cos 2θ = 0 are satisfied. The angle α and the two cross-section ratios r 2 = s 2 /s 1 and r 3 = s 3 /s 1 (see notations in Fig. 6(b)) are then adjusted to satisfy the remaining isotropy and mechanical conditions. We obtain: This set of equations can be solved numerically; we obtain: α ≃ 0.927 rad ≃ 53.15 • , r 2 ≃ 0.506, r 3 = 0.805. It can be noticed that, like the kagome lattice, this optimal structure is not fully triangulated.

r 3 = r 2 √ 2 -sin α 2 + cos α 2 , r 2 = √ 2
The third network (Fig. 6(c)), based on the kisrhombille tiling, is made of three different kinds of beams. From geometry, l 2 = √ 3l 1 and l 3 = 2l 1 . As before, the isotropy conditions are satisfied by construction, and the cross-section ratios r 2 and r 3 are adjusted to satisfy the mechanical conditions. One obtains that the optimality is reached when r 3 = 1, whatever the value of r 2 .

We also performed simulations of these networks and checked that the numerical values of their elastic moduli are in very good agreement with the expected values: µ/(E 0 φ) ≃ 0.125, M/(E 0 φ) ≃ 0.375 for the network of Fig. 6 The three-dimensional case is certainly more complex. As a matter of fact, we were not able to identify any periodic and isotropic optimal structure made with one single kind of beam. However, stiffest structures built with two (or more) types of beams do exist. For instance, the structure depicted on Fig. 7 satisfies the mechanical conditions and can be repeated periodically: the bemas join the centres of the faces of a Kelvin cell (which is known for tiling the space [START_REF] Weaire | The Kelvin Problem: Foam Structures of Minimal Surface Area[END_REF]). The length ratio of the two kinds of beams is imposed by the Kelvin cell geometry: l 2 /l 1 = √ 3/2. We then adjust the section ratio for the isotropy conditions (7) to be satisfied. We obtain: s 2 /s 1 = 3 √ 3/4. To our knowledge, this optimal (and isotropic) network with periodic arrangement has never been published before and represents one of the main result of the present study.

Conclusions

In summary, we showed the existence of a class of isotropic networks having highest possible values of elastic moduli for a given density. We established a convenient set of conditions that allow to identify these optimal networks. The elastic moduli of these networks are also derived, and can be simply expressed in terms of the Young's modulus of the beam material and the relative density of the structure. Examples of two-and three-dimensional optimal structures with periodic arrangements are also given. These results may be of interest for structural applications as well as for our understanding of biological systems. Moreover, they can be used to study the mechanical properties of any (isotropic) structure which can be viewed as a slightly perturbed optimal network [31].

Figure 2 :

 2 Figure 2: Repartition of the strain energy in various 2d isotropic networks subject to horizontal shear deformation: (a) hexagonal network; (b) kagome network; (c) triangular network; (d) Voronoi network; (e) Delaunay network; (f) the calibration triangleshows what proportion of the total energy is due to beam stretching, beam bending and node bending. Beams are coloured black if relatively undeformed, red if the deformation energy is predominantly beam stretching, green if the deformation energy is predominantly beam bending, and blue if the deformation energy is predominantly node bending of the two associated junctions.

Figure 3 :

 3 Figure 3: Affinity measure m i = ūiu i ) /(γL) of the displacement field for the same networks as in Figure 2 subject to horizontal shear strain. The (logarithmic) calibration bar (f) shows the coloration scheme used for the nodes (junctions and midpoints): they are coloured in red if their displacements match the affine displacement field, and blue in the opposite case.

Figure 4 :

 4 Figure4: Geometry of a typical beam (i, j). In the initial configuration, the beam can have a non-zero natural curvature and an inhomogeneous cross-section. Under the trial displacement field (4), the beam is subjected to a translation u i , a uniform compression/extension ǫ ij and a uniform rotation ω ij .

Figure 5 :

 5 Figure5: Possible geometries for a junction with connectivity z i in an optimal (and isotropic) network, with beams having the same-cross-section. The allowed geometries are checked in red. Configurations where beams are parallel in pairs are always solutions to Eqs (16) and(18). These are the only solutions when z i ≤ 4.

Figure 6 :

 6 Figure 6: Optimal networks with beams of different sizes (lengths l i and cross-sections s i ); (a): Union Jack lattice, made with two different kinds of beams. The optimality is obtained for s 2 /s 1 = √ 2; (b): network with three different kinds of beams. The optimality is obtained for α ≃ 53.15 • , s 2 /s 1 ≃ 0.506 and s 3 /s 1 ≃ 0.805; (c): network based on the kisrhombille tiling and containing three different kinds of beams. The optimality is reached when s 3 = s 1 , whatever the value of s 2 .

1 -sin α cos α 2 -sin α 2 .

 122 sin αcos α -2 cos 2α 3 sin (α/2)cos (α/2) , r 2 = √ 2(20)

Figure 7 :

 7 Figure7: Example of 3d optimal structure with periodic arrangement. Left: a single mesh is inscribed in a Kelvin cell, and thus tiles the space. It has two different sorts of beams, with respective length and cross-section l 1 , s 1 (in red) and l 2 , s 2 (in yellow). The mechanical conditions are satisfied by construction, and the isotropy conditions (7) are fulfilled for s 2 /s 1 = 3 √ 3/4. Right: illustration of the structure obtained from tiling with this unit cell. One can note that it forms two entangled cubic networks (in red), linked to each other by transverse beams (in yellow).

  (a), µ/(E 0 φ) ≃ 0.124, M/(E 0 φ) ≃ 0.374 for the network of Fig. 6(b), and µ/(E 0 φ) ≃ 0.125, M/(E 0 φ) ≃ 0.375 for the network of Fig. 6(c).

  Table

			3	φ/4	3φ/8	φ/8	1/3
	3d	φ/15	φ/6	φ/9	φ/5	φ/15	1/4

Appendix A. Isotropy conditions

Introducing the expression [START_REF] Gibson | Cellular Solids -Structure and properties[END_REF] of the elongation ǫ ij in Eq. ( 5) yields where a αβ = (A αβ +A βα )/2. For an isotropic network, the expression (A.1) must be invariant by permutation or inversion of axes, for any applied strain. These invariance properties lead to restrictions on the structure of the network, and simplify the expression of ε trial . First we note that the last sum in Eq. (A.1) vanishes for 2d networks, as α, β and γ cannot be all distinct. We show that this sum also vanishes for 3d isotropic networks, by analysing the strain field defined as: a xx = a 0 , a yz = a zy = b 0 , and the other components a ij = 0. Thus Eq. (A.1) reduces

. Inverting the y or z axes changes the sign of the last term only. But as ε trial must remain unchanged under such an operation, one necessarily has e x ij 2 e y ij e z ij = 0, and more generally, by permutation of the axes: e α ij 2 e β ij e γ ij = 0 with α, β and γ all distinct. We then choose the strain field defined as a xx = a 0 , a xy = a yx = b 0 , and the other components a ij = 0 to show that the third sum in Eq. (A.1) also cancels out: Eq. (A.1) reduces to

. Inverting the x axis transforms e x ij into -e x ij . As the energy must remain unchanged, one must have e x ij 3 e y ij = 0. Performing the same procedure for any pair of axes α and β = α eventually leads to the conditions e α ij 3 e β ij = 0. We now choose the strain field defined as a αβ = a 0 δ αx δ βx (this is a particular case of the field defined above). Eq. (A.