Abstract
This chapter introduces the mathematical foundations for models of economies in which economic wealth is generated through a social division of labour. We introduce the notion of an economic commodity, separating consumables that have use value from intermediary inputs in production processes. All wealth creation processes are performed through consumer-producers, a mathematical representation of a rational decision-maker, who produces as well as consumes economic commodities: consumer-producers form the building blocks of any mathematical theory of the social division of labour. We also consider two mathematical models of the fundamental property of Increasing Returns to Specialisation (IRSpec) in human productive abilities. The chapter concludes with surprising insights from decision-making by consumer-producers in (standard) economic trade environments.
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- 1.
The sociality or networking capabilities of an economic agent are very hard to represent through appropriately selected mathematical constructs. The chosen network representation seems to be the most plausible, although it remains less than perfect.
- 2.
This is derived from the introduction of an economic agent as the embodiment of the social brain hypothesis as well as an individualistic bearer of productive and consumptive abilities. We refer to Gilles (2018, Chapter 1) for more details.
- 3.
Yang (2001) formulates the principle of Increasing Returns to Specialisation through his mathematical conception of “inframarginal analysis”. This refers to the determination of optimal production planning through exploration of the corners of the assigned production set.
- 4.
Note here that the definition of an economic commodity implies that the delivery of such goods is explicitly a social task. It is through the prevailing trade infrastructure in the foundation of the socio-economic space that such commodities can be provided. It is not an individual task to make this delivery possible; rather it is communal or social. Of course, such provision can be executed by individual economic agents, but ontologically it is a task of the whole socio-economic trade infrastructure to deliver this commodity.
- 5.
A more general setup is to represent preferences as a mathematical relation on the consumption space \(\mathcal {C}\). This approach lies at the foundation of general equilibrium theory, representing the fundamental theory of the competitive market mechanism. I refer to Debreu (1959), Hildenbrand (1974), and Jehle and Reny (2000) for a complete development of this mathematical theory.
- 6.
For details on this standard analysis of utility functions and representation of preferences, I refer to, e.g., Jehle and Reny (2000, Chapters 1–3).
- 7.
- 8.
The consumer-producer representation of an economic agent describes the variables that are solely under the control of an individual economic agent. In particular, one can view the utility function as a descriptor of the objectives of the individual economic agent and the agent’s production set as a descriptor of the abilities of this agent to meet these objectives. The latter essentially represent the constraints put on the agent’s choice set.
- 9.
We emphasise that the Cobb–Douglas utility function used here does not in principle satisfy the conditions imposed in Axiom 1.5, since the Cobb–Douglas utility function is not strictly monotone on the boundary \(\partial \mathcal {C} = \{ (x,y) \mid x=0\) or y = 0} of the consumption space. The computational advantages of the Cobb–Douglas utility function are rather significant, since its specification affords us simple formulations of solutions to utility maximisation problems.
- 10.
We refer to the discussion in the concluding section of the next chapter for an elaboration of this point.
- 11.
Regarding to the quantities traded in the market we use the convention that x t > 0 is a net quantity demanded, while x t < 0 represents a net quantity supplied.
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Gilles, R.P. (2019). Commodities, Consumption and Production. In: Economic Wealth Creation and the Social Division of Labour. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-04426-8_1
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