Exploring the effect of crisis on cooperatives: a Bayesian performance analysis of French craftsmen cooperatives

ABSTRACT The aim of this paper is to understand the economic performance of craftsmen cooperatives during the crisis period. These cooperatives have the distinctive feature of being supply cooperatives. We use an exhaustive dataset for the French craftsmen cooperatives (2004–2014). We estimate Bayesian Translog econometric models in order to underline the impact of the 2008 crisis on these cooperatives. On the one hand, cooperatives’ turnover and economies of scales decrease during the crisis, the effect is lower for elder cooperatives and varies across sectors. On the other hand, there is a convergence towards the mean for the various generations of cooperatives. These findings are robust to alternative econometric specifications.


I. Introduction
According to different authors, cooperatives play an important role in building a more balanced economy (Stiglitz 2009;Birchall 2013). Hannan (2014) explains that cooperatives 'are part of the market economy but possess a multidimensionality that enables them to perform in market economies while providing members with a range of tangible and intangible benefits that have the potential to enhance their socio-economic position and voice'. The resilience of cooperatives has been challenged during the recent economic crisis (Carini and Costa 2013;Vieta 2010).
Identifying the comparative advantages and disadvantages of member-owned businesses, Birchall (2013) states that the cooperative model may potentially be stronger than other businesses during the economic recession. Several studies show better social and economic performances of cooperatives relative to other businesses (Cheney et al. 2014;Lambru and Petrescu 2014;Bentivogli and Viviano 2012;Zamagni 2012;Costa and Carini 2016;Carini and Carpita 2014), but there are counter-examples and the crisis effect may vary by sector and with the market context (Birchall 2013). According to Nunez-Nickel and Moyano-Fuentes (2004); Ingram (2003, 2004) and Rousselière (2019), agricultural cooperatives and kibbutzim are more sensitive to changes in the regulatory environment, but have a greater ability to adapt to macroeconomic fluctuations. Staber (1992) pointed out that agricultural marketing cooperatives are highly resistant to recessions. For workers cooperatives, Navarra and Tortia (2014) highlighted a greater wage fluctuation but at the same time a greater employment stability in worker cooperatives, because at the theoretical level the workers' objectives are internalized into the firm's objective function.
This literature globally shows therefore that cooperatives demonstrate a greater resilience than other types of enterprises. However, cooperatives may be less profitable than investor-owned firms, but operate more efficiently, present a stronger financial position van Dijk 2012, 2014), and have a stabilizing effect on employment with respect to shocks (Alves, Burdin, and Dean 2016;Delbono and Reggiani 2013). The main features of cooperatives' organization are the source of their resilience, since cooperative members join their (non-monetary) resources to build collective networks and skills, improve their capability to innovate, and attract government funding (Borda-Rodriguez et al. 2016). However, the cooperatives' resilience capacity differs across the sector of activity, the size and the geographic location of the cooperative (Costa and Carini 2016;Fakhfakh, Perotin, and Gago 2013;Soboh, Lansink, and van Dijk 2014;Borda-Rodriguez et al. 2016).
While most of the literature focuses on the agricultural and financial sectors (Rousselière 2019), as well as worker cooperatives (Navarra and Tortia 2014), we study here the case of French craftsmen cooperatives, which have the distinctive feature of being supply cooperatives. This paper aims at understanding the economic performance of these cooperatives during crise periods. Particular attention is paid to the age of the cooperative and the differences between sectors of activity. We are particularly interested in the resilience of French craftsmen cooperatives, namely how effective they are at surviving economic recession. We use exhaustive accounting data from the AMADEUS database over the years [2004][2005][2006][2007][2008][2009][2010][2011][2012][2013][2014]. We show that the negative effect of the crisis depends on the sector and the date of creation of the cooperative. Economies of scale, estimated with elasticities, also present differences depending on the sector, the experience, and the size of craftsmen cooperatives. These analyses are important for public policy design since they indicate whether cooperatives need support on investment or labour expenses to improve their resilience. We use Bayesian analysis to compare different models underlying the impact of cooperatives characteristics, such as their location. State-of-the-art Bayesian regression techniques allow us to perform sensitivity analyses, robustness checks (following Leamer (1983Leamer ( , 1985), and to ensure the transparency of results. This is the first study addressing this issue for craftsmen cooperatives, a type of cooperatives largely understudied in the literature.
The remainder of the article is organized as follows. The next section describes the context of the study. Section 3 presents the empirical analysis. In section 4, we report our main results, that show a low impact of the crisis on the economic performance of cooperatives, which diminishes with age and is smaller in some sectors, and a convergence towards the mean of effects specific to various generations of cooperatives. Furthermore, the economies of scales decreased dramatically after the crisis. Alternative estimations and robustness checks are reported in Appendix A. Finally, in section 5 we discuss the implications of our empirical findings.

II. Context and data: cooperatives in a craft industry under pressure
Since the 2008 crisis, the French craft industry faces a sharp slowdown. This business sector is directly dependent on public and private consumption, the consequences of the economic breakdown came into effect quickly and dramatically. Figure 1 shows the evolution of craft activity since 2009 (in Figure 1. Activity growth rate, relative to the past quarter. Source: CAPEB quarterly dynamics). Figure 2 shows the evolution of the number of employees of small businesses (less than 20 employees) in two crafts sectors (carpentry and plumbing) that we analyse in this paper.
The impact of the economic crisis appears dramatically in Figures 1 and 2. The business activity of craftsmen contracted, similarly to sector employment. In this context, we can ask if a craftsmen supply cooperative may be a reliable and efficient backstop for the members, as it was the case in transition economies (Surubaru 2012). If craftsmen cooperatives are effective at surviving economic recession, it appears necessary to understand how this resilience effect is working.
It is interesting to note that the literature on craftsmen cooperatives in developed economies is sparse, while at the same time there is an extensive literature on the benefits of horizontal cooperation between SME (Small and Medium Enterprises) (Villa and Bruno 2012). Ohnemus (1994) provides some empirical evidence on plumbing cooperatives in the USA; Lee and Mulford (1990) andRawwas and Iyer (2013) document the activity of small business cooperatives in the Japanese retail and wholesale sector. In the context of transitional economies, cooperation may be a convenient strategy for small businesses to survive economy transformation and shocks (Cordell 1993;Surubaru 2012). Richomme (2001) and Rymeyko (2016, 2017) study various cases of French craftsmen cooperatives in the construction sector. Auvolat (2008) provides a comprehensive overview of the development of craftsmen cooperatives in France. Supply cooperatives, that provide their members with intermediate inputs, emerged when investorowned firms had a substantial monopoly power over small business enterprises (including farmers) in the supply of these inputs (Mikami 2003). Therefore, the craftsmen cooperatives share common features with other purchasing cooperatives (e.g. agricultural cooperatives, fishermen cooperatives). Grouping together as a cooperative thus enabled the craftsmen, who are self-employed, to pool the management of their purchases. This helps them to be no longer been dependent on the wholesalers, who have great strength in an increasingly concentrated market (Lapayre, Pierson, and Rymeyko 2017). Cooperatives supply members with input for their own individual economic activity: for example, negotiating lower costs for plumbing supplies, or purchasing material and goods in the construction sector on behalf of their members. Contrary to marketing cooperatives or bargaining cooperatives, the craftsmen supply cooperatives do not, respectively, sell or negotiate with final customers. According to Rawwas and Iyer (2013) and Richomme (2001), these cooperatives have established trust among members, which has improved their overall performance.
Building on a mixed methodological approach (Starr 2014), we used an exhaustive dataset covering the 49 French craftsmen supply cooperatives (2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014), based on a matching of the directory of craftsmen cooperatives provided by the French Federation of Craftsmen Cooperatives and accounting data from the Amadeus/Orbis database. Orbis is often viewed as an inaccurate, imprecise, unreliable source because of missing data. To address this issue, some authors use listwise deletion methods (complete case analysis) van Dijk 2011, 2012;Hirsch and Hartmann 2014), that may produce biased estimates when there are informative drop-outs or missing not at random data (Seiler and Heumann 2013).
Our analysis is not affected by these problems, although no comparison can be made with for-profit enterprises, for which no exhaustive directory is available. Note that all cooperatives have survived over the entire period, suggesting the absence of informative drop-outs and of survivor bias.
Following data quality assessment procedures described in Hazen et al. (2014), the data were also checked for accuracy, reliability and consistency using qualitative data coming from interviews with directors of cooperatives (10). The interviews took place face-to-face at the cooperatives' premises in April and May 2015. The sampled cooperatives are all members of the French network ORCAB. A judgement sampling (Teddlie and Yu 2007) was carried out, thanks to the expertise of a former cooperative director. This specialist in the sector has accompanied us in this research work on several occasions. It made it possible, on the one hand, to set-up the sampling for the survey and to obtain interviews through its network, but its ex-post role was also important. Indeed, he attended all the interviews and his specialist insight into some of the answers was invaluable. Finally, all the cooperatives surveyed are located in the Pays-de-la-Loire Region. The tables in Appendix 3 show the characteristics and the main findings of the interviews carried out (description of the cooperative, impact of the crisis, cooperative reactions) and will help us discuss the empirical finding of our econometric models (Starr 2014).
The population of interest are the 49 craftsmen supply cooperatives. To increase the homogeneity of the analysed population of cooperatives, we do not take into account the marketing cooperatives that exist also in this industry 1 As shown in Figure 3, cooperatives in our panel were created between 1968 and 2012. 2 Accordingly, we have a unbalanced longitudinal database of our population between 2004 and 2014.
The average size of cooperatives is 8 to 10 millions € of turnover, with an average number of 100 members. These cooperatives are under the umbrella of ORCsAB, the French Union of Craftsmen Cooperatives, an association created in 1990. Since its transformation into a consortium of cooperatives (a second-level cooperative) in 1998, ORCAB plays an active role in the development of the network: creation of a collective brand, promotion of collective intangible investment, development of human capital (training of elected members), providing (by purchasing) the cooperatives with commodities to sell to their members and, furthermore, support the creation of new cooperatives in a more 'top-down' approach (Billaudeau, Poutier, and Martineau 2016). The success of this cooperation among cooperatives (Fici 2015) leads to the creation of new cooperatives (half of cooperatives were created since 2000).

III. Empirical strategy: a bayesian econometric approach to production function modelling
We estimate a classical translog model, which is a more flexible production function than its special case (cobb-douglas). This production function has already been used for cooperatives: see Fakhfakh, Perotin, and Gago (2013) for an application to worker Maietta and Sena (2008), Maietta and Sena (2010)) for an application to producers cooperatives, Battaglia et al. (2010) for an application to cooperative banks or Soboh, Lansink, and van Dijk (2012) for an application to dairy cooperatives.
For a cooperative i observed at a period t, a panel-data translog model is: where O is the cooperative's output, I is a vector of k input variables, μ and are errors term, with μ,Nð0; σÞ and ,Nð0; ϕÞ.
As we suspect the heterogeneous effects of crisis by age and sector, our benchmark model is: with A, C and S standing for age, crisis, and, respectively, sector.
There is no consensus in the literature regarding the objective function of a cooperative. It can be, for example, the utility or welfare of its members (as in Fulton and Giannakas (2001) or Giannakas and Fulton (2005)), or its profits with a patronage refunds paid to its members (as in Agbo, Rousselière, and Salanié (2015)). Soboh et al. (2009) provide a more comprehensive review on the objective functions of cooperatives. As highlighted by the Conceptual Framework for Statistics on Cooperatives of the ILO (International Labour Organization) (Bouchard, Le Guernic, and Rousselière 2017), value added and profits have to be used with caution for marketing and supply cooperatives. For a marketing cooperative, producers' income is part of value added (patronage refunds or interests on social shares), on the one hand, and reflect a decrease in value added (payments for raw materials), on the other hand. For a supply cooperative, producer expenses may reflect different price strategies (combinations of price and patronage refunds). In our case, the available output is the cooperative's total turnover. The logarithm of total turnover is usually used in the literature as an acceptable proxy for economic performance (Fakhfakh, Perotin, and Gago 2013;Sena 2008, 2010;Gagliardi 2009;Soboh, Lansink, and van Dijk 2012). Moreover, as highlighted by interviews with directors of the cooperatives, patronage refunds and price strategies are similar across all craftsmen cooperatives federated at ORCAB.
As discussed above, the dependent variables is the turnover (T), expressed in thousands of euros. The independent (explanatory) variables are, on one hand, the traditional inputs of the production function: intermediate consumption (IC), labour expenditure (LE) (including wages, salaries, and benefits), intangible assets (IA), and tangible assets (TA). All these variables are log-transformed. On the other hand, the following control variables are available: age (A), sector (S) and crisis (C). A is the age of the cooperative in years. The sector is a dummy variable, equal to 1 for the carpentry sector (nace4673: 'Wholesale of wood, construction materials and sanitary equipment') and 0 for the plumbing sector (nace4674: 'Wholesale of hardware, plumbing and heating equipment and supplies'). The crisis variable is defined as a dummy. It takes the value 1 for 2008 and the following years and 0 otherwise. 3 Note that we can not use the number of members as an explanatory variable, as we have this information only the first year. As revealed by responses to interviews, there is a strong correlation between the number of a cooperative's members and its turnover, as the population of craftsmen is relatively homogeneous. Finally, because of unreliable data, we can not control for sectoral competition (with for profit organizations) 4 We choose a Bayesian modelling approach that takes into account uncertainty, sparse data, and moderate-sized sample (Gelman et al. 2014a), especially with weakly informative priors (Gelman et al. 2008). In order to conduct Bayesian model selection, we use the WAIC (Widely Applicable Information Criterion) and LOO (Leave-One-Out cross-validation) implemented in Vehtari, Gelman, and Gabry (2017). WAIC and LOO (Gelman, Hwang, and Vehtari 2014b) are fully Bayesian methods for estimating pointwise out-of-sample prediction accuracy from a fitted Bayesian model, using the log-likelihood evaluated at the posterior simulations of the parameter values.
Estimations are conducted with package Brms for R (Bürkner 2017), that called Stan, a C++ programme performing Bayesian inference and optimization (Gelman, Lee, and Guo 2015).

Estimation
With weakly informative priors (Cauchy (0, 2.5) prior distributions (Gelman et al. 2008)), our benchmark model is based on four chains of 5,000 iterations, of which the first 2,500 are used as a warm-up to calibrate the sample, leading to a total of 10,000 posterior samples. The Stan algorithm is highly efficient as the autocorrelation of Markov chains disappears quickly (see Figure 4 for the estimation of the log Posterior). We have, therefore, a large effective sample size (ESS).
The estimation of the benchmark model is reported in Table 1. We estimate also alternative specifications as robustness checks. These estima-tions account for alternative functional forms, endogeneity, unobserved heterogeneity, and spatial correlations, and are reported in Appendix A2. They allow us the same interpretation. There is a negative effect of the crisis on performance. Note also that this effect decreases with cooperatives' age and is smaller in the carpentry sector. This result is revealed by the positive effect of the interaction of the crisis dummy with cooperative's age and with the carpentry sector dummy. The magnitude of the coefficient of a dummy variable cannot be interpreted directly in a regression with a log dependent variable as a semi-elasticity (Van Garderen and Shah 2002). Therefore, we choose to interpret these estimates using various prediction of the economies of scale (see section 4.2).

Economies of scale before and after the crisis
Economies of scale and elasticites may be calculated at the mean, the median, or at various representative values (see (Kumbhakar, Wang, and Horncastle 2015)). We estimate additional models on distinct subsets of the population (before and after the crisis).  In Table 2 we report the mean using a Bayesian test, which is just the posterior probability under the hypothesis against its alternative (Gelman et al. 2014a). Economies of scale are equal to 1.66, with a high contribution of intermediate consumption (1.52) and a non-significant effect of assets. This estimation is higher than in previous studies on cooperatives. For example, Maietta and Sena (2008) found constant economies of scale and an elasticity of intermediate consumption equal to 0.82 for producer cooperatives. The economies of scale are decreasing (0.62) in the case of European dairy cooperatives (Soboh, Lansink, and van Dijk 2014). Finally, we can see that the economies of scale decreased dramatically after the crisis.
In the same Table 2, we estimate the impact of the 2008 crisis on performance for cooperatives of different ages from the two sectors, with other independent variables fixed at the mean. The marginal impact of the crisis is relatively low in all cases, suggesting a high resilience of cooperatives during crisis periods. Carpentry sector cooperatives suffer less from the crisis than plumbing cooperatives.
Although the impact of the crisis is small, we note that younger cooperatives seem to be more affected than their older counterparts (Table 3). Accordingly, there is a convergence towards the mean for cooperatives of different age during the crisis, as illustrated in Figure 5, where we report the predictions by cooperatives' age and sector of activity. This result holds especially for cooperatives from the plumbing sector. As discussed in section 4.1, this empirical finding may be linked to the different dynamic of the creation of various generations of cooperatives.

V. Discussion and conclusion
Using an exhaustive database, we underline the presence of high economies of scale for the craftsmen cooperatives. Regarding agricultural cooperatives (e.g. Liu and Bailey (2013); Soboh, Lansink, and van Dijk (2014)), the magnitude of the returns to scale show that the best strategy to improve the cooperative's competitive position is to increase its size. One should note that this is one of the main accusations against cooperatives and especially worker coops, since high economies of scale may imply inefficient production level and, as per the Ward (1958) model on worker coops, low production can correspond to the condition of profit maximization in the short run in which marginal product is equal to average product 5 However, contrary to workers cooperatives, craftsmen supply cooperatives have a huge incentive to growth, as countervailing power tends to increase with the number of members. One main reason the cooperatives stay small is that they are federated and must respect a geographically delimited territory (even with loose boundaries). The high economies of scale show important avenue of expansion for cooperatives. We observe a weak growth and a strong resilience of craftsmen cooperatives, which questions the alleged inexorable trade-off between democracy and efficiency (Jones and Kalmi 2012). Previous studies on cooperatives found a decreasing relationship between size and social capital (Nilsson, Svendsen, and Svendsen 2012;Feng, Friis, and Nilsson 2016). This suggests that in large cooperatives profits from economies of scale and scope may be outweighed by weak democracy governance, widespread free-riding and loss of solidarity. Craftsmen cooperatives, as small grassroots organizations, seem to be at the first step of growth described by Nilsson, Svendsen, and Svendsen (2012), where larger size increases economic performance without the drain of social capital. According to Malikov, Zhao, and Kumbhakar (2017), the incentive to grow in size may be fuelled not only by present economies of scale, but also by economies of diversification. Economies of diversification especially concern cooperatives that are not for profit maximizers, but rather seek to maximize service provision (in terms of quantity, price, and variety) to their members. As cooperatives are able to secure the demand for a more diversified scope of services, larger cooperative naturally have a higher incentive to grow in order to capitalize on economies of diversification.
We investigate how the 2008 crisis affected the French cooperatives in the carpentry and plumbing sectors. We find that, during the crisis the decrease in cooperatives' turnover was smaller in the carpentry sector. This points to a higher resilience of cooperatives from this sector in comparison to other components of the social economy (Bouchard and Rousselière 2016;Pape et al. 2016). Bouchard and Rousselière (2016) highlights a split in the population of the Montreal social economy between a component characterized by low growth and low hazard and a component with high growth and high hazard. According to our results, craftsmen cooperatives seem to belong to the former type.
We also underline a convergence effect or a regression towards the mean, but of a different nature than the one highlighted in Hart (2000). Young craftsmen cooperatives have a higher turnover than the older ones, but during the crisis, they face a stronger decrease in income. This suggests that two complimentary dynamics are at stake. On the one hand, for the ecology of organizations perspective (Hannan 2005), older cooperatives have a higher capacity to adapt to a changing economic context. They benefit more from economies of learning, while younger cooperatives may face the liability of newness due to their immature social organization (Brüderl and Schusseler 1990). There is a negative overall relationship between firm age and closure or performance, which may result from the need for new firms to go through an initial learning and organizing phase (Pérotin 2004). On the other hand, we showed in a companion paper on the investment behaviour of craftsmen cooperatives (Musson and Rousselière 2019) that assets increase with size and age (although at a decreasing rate as investment tend to decrease with age), and interestingly after the crisis the older cooperatives invest more than the younger one. Therefore, older cooperatives may perform better and be more stable just because they are better capitalized and have more patrimonial stability than younger coops (Tortia 2018). However, there is no contradiction between these two perspectives because as shown by Freeman, Carroll, and Hannan (1983), older firms may thus have access to more economic and non-economic resources, which should make them less vulnerable to selection pressures. Furthermore as an experienced organization they may select a sustainable longrun strategy, whereas younger organization may pursue member utility maximization in a myopic way 6 This empirical finding is in line with the heterogenous responses towards the crisis suggesting by the theoretical literature on SME (Cucculelli and Peruzzi 2018).
An explanation based on the concept of commitment, coherent with the ecology of organizations perspective (Hannan and Freeman 1984;Barron, West, and Hannan 1994;Pérotin 2004), can also be provided. Commitment is defined by Fulton (1999) and Fulton and Giannakas (2001) as the preference of cooperative members to patronize a cooperative even when the cooperative's price or service is not as good as that provided by an investor-owned firm. Therefore, member commitment is 'a sort of glue that allows membership and business volume to be maintained even as trade becomes more fluid and barriers to a reorganization are broken down' (Fulton 1999, 418). Younger cooperatives were created from scratch in a 'top-down' way (with the support of the federation trying to develop the cooperative system) (Auvolat 2008;Billaudeau, Poutier, and Martineau 2016). Because of their weaker link with their members, they may suffer more from the crisis and a change in the economic environment (Pérotin 2004). The few empirical papers describing 'top-down cooperatives' provide mixed evidence on this issue (Kurakin and Visser 2017). On the other hand, older cooperatives (developed before the federation) were created in a 'bottom-up' manner at grass-root level. Their members have a deeper commitment, which may explain not only why they suffer less during the crisis, but also why they continue to growth. Commitment acts as a glue and determines members to be more supportive of their cooperative, or to get more involved in its governance during an economic crisis. Cechin et al. (2013) explain how democracy improves commitment, while control, monitoring, and increased formalization of agreements negatively affect members' commitment and the cooperative's performance. However, as we do not have an access to accurate data on organizational commitment as in Bareille, Bonnet-Beaugrand, and Duvaleix-Treguer (2017) or in Amghrous and Rousselière (2019), these interpretations should be taken with caution.
Our qualitative investigation (based on interviews with cooperative leaders) provides some support to the previous interpretations (see Appendix 3). It revealed different strategies of resilience on behalf of the cooperatives as a reaction to organizational tensions (Audebrand 2017): selective membership, annual monitoring that sometimes turns into control, coaching and loyalty. Facing the crisis, the most dynamic reaction in terms of investments (exhibitions, buildings, subsidiary), but also regarding support (training, tools), is that of the oldest cooperative, but this trend is not confirmed in the rest of our sample.
Most cooperatives react to maintain strong financial fundamentals, although this requires a high selectivity of members, or even the eviction of a few. In coherence with both Tortia (2018) and Hannan (2005) perspectives, we observe that the more finances have been consolidated over several decades, the more dynamic and inclusive the cooperative's response to the crisis is; on the contrary, a cooperative, despite 40 years of existence, finds itself in an extremely delicate situation due to doubtful provisions. In the latter case, the change in the relationship with members is radical.
Nevertheless, several cooperative leaders explain that their development plan is not considered from the point of view of the development of the catchment area. Indeed, each cooperative radiates in its geographical area, following the recommendations of the federation. They do not wish to compete fiercely with each other, but may consider mergers (a cooperative we studied is the result of a merger) in order to extend their territory. They are more interested in intrinsic development, attracting more members from their territory, and being as close as possible to members, helping them adapt to their customers (see the table related to cooperative reactions to the crisis). Many stress the importance of understanding the particularities of the market for craftsmen in a predominantly rural environment 7 We note that the older the cooperative is, if the modes of governance are effective, the easier this development is, because it simply has to adjust. On the other hand, for one of them, poor governance, rooted in time, requires a drastic change and is probably more difficult than for a young cooperative. As highlighted by the ecology of organizations, these organizational inertia may limit effective collective action and mitigate the positive effect of time on efficiency (Barron, West, and Hannan 1994). This qualitative results suggest an heterogeneous effect of time to be further investigated in future researches.
If governance's issues are almost shared by all cooperatives, craftsmen cooperatives differ from the agricultural ones from the point of view of the business model adopted by their members. Auvolat (2008) explains that craftsmen face an individual and local demand, and that their economic performance is less related to suppliers than in the case of farmers. Another distinctive feature of craftsmen is their suspicion of democracy with the delegation, the rejection of gigantism and depersonalization. If craftsmen cooperatives, especially the oldest ones, choose not to grow in spite of the important economies of scale they could achieve, it may be due to the fact that members consider commitment and democracy more important for their performance than economic efficiency. Billaudeau, Poutier, and Martineau (2016), presenting a special case of a French craftsmen cooperative, demonstrated that this type of governance, emphasizing strong relationships and trust between the members and the cooperative and among members, leads to a successful adaptation in a crisis period. Moreover, the optimal size of the cooperative may be different for craftsmen and farmers, as the competition between members is not of the same nature. Craftsmen face a local demand and the competitive advantage may consist of workers' know-how, while farmers may face a global market with a competition based on quantity and global prices (in which they may cooperate) and a local market (in which they compete) (Agbo, Rousselière, and Salanié 2015). Finally, the need of craftsmen for direct democracy and autonomy, coupled with commitment as a competitive advantage brought to the cooperative, leads to a special strategy. Craftsmen cooperatives follow the 'low growth, low hazard' model in spite of large economies of scale and of our results proving them efficient during crises.

Disclosure statement
No potential conflict of interest was reported by the authors. Various additional robustness checks have been made. A large set of flexible functional forms is available to the empirical researcher (Thompson 1988). Giannakas, Tran, and Tzouvelekas (2003) show that an inappropriate choice of the functional form could result in significantly biased efficiency estimates and misleading policy recommendations regarding efficiency improvements. Their results strongly reject the ad hoc imposition of a functional form and underline the importance of searching for the write specification. Cobb-Douglas is a special case of the Translog model. Less parsimonious function forms such as Generalized Leontieff can also be used, but with they lack parsimony in the case of our small sample.
We conduct a Bayesian model selection based on WAIC and LOO. These two information criteria reject the Cobb-Douglas function that has a slightly worse fit (WAIC ¼ À2232:62:19 and LOO ¼ À2229:19 for the Cobb-Douglas function vs. WAIC ¼ À2261:61 and LOO ¼ À2255:17 for the Translog function). We can also place a lasso prior on the populationlevel effects (Park and Casella 2008). This shrinkage prior is the Bayesian equivalent to the lasso method for performing variable selection. The model with a lasso prior selects at least some parameters included in the Translog model ( ln IC 2 , ln TA 2 , ln LE Â ln TA, ln IA Â ln TA), suggesting a better fit for this model.
Next, we test for unobserved heterogeneity. Unobserved heterogeneity is where the correlation between observables and unobservables may be expected (Arellano 2003). This issue may lead to biased estimations. We test several alternative methods that lead to the same results. A first possibility is the choice to add an inefficiency error term in the production function regression, that becomes a random effects stochastic frontier analysis (Greene 2005). A first preliminary analysis allows us to reject this approach. 8 More generally, as any production analysis based on longitudinal data, our results may suffer from endogeneity (between the dependent variable and one or several independent variables) or unobserved heterogeneity. We have a sample with a small T (short time frame) and small N (few individuals). Therefore, the GMM (Generalized Method of Moments) may produce even more biased estimates (Roodman 2009). A first simple and robust approach is to use the Mundlak-Chamberlain correction (Wooldridge 2010). In production function and efficiency analysis, this method has already been implemented by Emvalomatis (2012) and Griffiths and Hajargasht (2016). We just have to test the joint hypothesis of nullity for added parameters, namely the average of each time-varying variable, using a VAT (Variable addition test). The Bayesian test reject this hypothesis (Estimate ¼ 0:0394 with e:s: ¼ 0:0447). We have also a higher WAIC. Another possibility is to reestimate our model using Lewbel (2012)'s method. This method serves to identify structural parameters in regression models with potentially endogeneous regressors in the absence of traditional identifying information, such as external instruments or repeated measurements based on large panels (see Mishra and Smyth (2015) for an example).
Results are slightly similar to our previous results (economies of scale = 1.56), for both the amplitude and the statistical significance of parameters. We implement the unconditional quasimaximum likelihood estimator of Hsiao, Pesaran, and Tahmiscioglu (2002) for linear dynamic panel models with fixed effects. Authors provide simulation evidence indicating that their estimator performs better than the GMM estimator.
Obtained results are also similar and lead us to reject the addition of a lagged dependent variable. The economies of scale estimated are slightly higher (1.82). Finally, in order to give a broader comparison, we provide the estimations for the classical Arellano and Bond (1991) GMM method, although results may be biased for our panel, as explained above. Again, the economies of scale are similar to those estimated with our benchmark model (economies of scale = 1.49). Another point is the spatial dimension of data. The craftsmen cooperatives are unevenly distributed across France and in some regions they may be in competition. A simple first approach is to count the number of cooperatives in a given radius (50, 100 or 200 km). We can also estimate GAMM (Generalized Additive Mixed Models) (Wood 2006; Santias, Cadrso-Suarez, and Rodriguez-Alvarez 2011) using the latitude and longitude coordinates. Note that because of unreliable data, we can have control for sectoral competition (with for profit organizations). The  GAMM models control for these sectoral effects, although they can be disentangled form other local effects. Adding the number of cooperatives in a 100 km radius leads to a slightly better fit, but has no impact on the parameters of interest. The 200 km radius model and the GAMM model lead to a worst fit.
Finally, in order to account for uncertainty induced by model selection inconstancy (Piironen and Vehatari 2017), we can correct the estimates with model averaging based on models weights (Yao et al. 2018). These model weights are analogous to posterior probabilities of models, conditional on expected future data. Following Piironen and Vehatari (2017) , we choose two main weighting methods. The first one are the Aikake-type weights (McElreath 2016): Population estimates are reported in Table A12. 9 We see that there are only slightly changes with respect to the benchmark model.