**5. Summary**

The division of labor in agriculture is influenced by factors such as the innate characteristics of the crops, variations of the seasons, duration of a product's shelf life, and the interconnectedness of the production process; these are all heavily interlinked, making it difficult to completely separate the factors in searching for farmer's maximum profit. Moreover, marginal analysis in economics cannot be used to model the division of labor mathematically. Our work here is the first attempt to analyze the division of labor using infra-marginal model in agriculture by treating heterogeneous farmers as a single producer–consumer integrated unit.

One of our major contributions in this study is to apply the corner equilibrium analysis in studying farmers' selection logic. When we impose reasonable budget constraints, positive utility, and comparative advantage, the number of possible production–consumption decision modes can be reduced from the maximum of 64 to an optimal of 3. If we assume that at least one of the farmers selects a specialized mode and each farmer prefers a different production–consumption mode, then four division of labor structures can be derived. Solving the nonlinear programming problem of the utility function within each respective labor structure leads to a corner equilibrium. We discovered the ranges for transaction efficiency coefficients, *k*, and preference parameter, *β*, under which each structure can achieve general equilibrium. Our work is concluded by showing how farmers' exogenous comparative advantage influence the way labor is divided and labor structures are selected.

The general equilibrium is determined by the relative productivity, relative preferences, and transaction efficiency levels of the two farmers. When other parameters are set, the improvement of transaction efficiency causes the general equilibrium to jump from self-sufficiency to partial division of labor and then to complete division of labor. Given the terms of the transaction and the relative preference for the two products, the greater is the comparative advantage of the farmer, the higher is the level of division of labor. Given the conditions of trade, the more balanced are the relative preferences compared with relative productivity, the higher is the equilibrium division of labor. With the improvement of the level of equilibrium division of labor, the equilibrium aggregate productivity of the economy in which the farmer is located increases. The aforementioned super-marginal comparative static analysis of general equilibrium explains the selection logic and decision path for the participation of superior farmers in the division of labor, and also provides a general equilibrium mechanism for the development of agricultural economy. In this mechanism, exogenous comparative advantage and transaction efficiency are the driving forces of agricultural economic development.

It is worth pointing out that our simplified model only takes into consideration the exogenous comparative technical advantages in understanding farmers' decision-making and selection logic. Further research to investigate the role of endogenous comparative advantages, which are obtained through one's practices and experiences, with the improvement of production and trading environment is much needed. On the other hand, the applicability of our work can be strengthened and validated with numerical studies of actual field data. Data that are currently collected from large-scale agricultural production activities in China will be extremely useful for this purpose.

In reality, there are many critical factors such as the demographic population and the factor endowment of the farmers, the level of expertise in the agricultural production, and the market transaction efficiency that can influence the selection space of farmers and the ultimate division of labor structure. A brand new set of mathematical models and accompanying analysis would most likely be needed to provide a more comprehensive result in this area.

**Author Contributions:** Conceptualization, X.J.; methodology, X.J.; formal analysis, X.J.; investigation, X.J.; writing—original draft preparation, X.J.; writing—review and editing, X.J. and J.-M.C.; and supervision, H.S.

**Funding:** X.J. was funded by National Natural Science Foundation of China gran<sup>t</sup> numbers 71742003 and 71703041, Ministry of Education Humanities and Social Sciences Youth Fund Project gran<sup>t</sup> number 15YJC790036, and China Scholarship Council gran<sup>t</sup> number 201808440675. H.S. was funded by Simons Foundation Collaborative Award gran<sup>t</sup> number 522790.

**Acknowledgments:** The authors gratefully acknowledge the valuable comments and numerous suggestions from the editor and the anonymous reviewers.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
