# Simple model for the transition from local to centralized production

Guillermo Benito-Calvino, Pablo Jensen

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Guillermo Benito-Calvino, Pablo Jensen. Simple model for the transition from local to centralized production. Physical Review E, 2025, 111 (3), pp.034307. 10.1103/PhysRevE.111.034307. hal-04993734

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Submitted on 17 Mar 2025

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Distributed under a Creative Commons Attribution 4.0 International License
A simple model for the transition from local to centralized
production

Guillermo Benito-Calvino* and Pablo Jensen†

CNRS, ENS de Lyon, LPENSL, UMR5672, IXXI, Lyon, France.
Abstract

We present a simple model of a crucial phenomena of modern societies: The shift from local to centralized production, leading to economies of scale and mass production transported over long-distance networks. Agents combine two distinct (and limited) resources, time and raw materials, either to produce for self-consumption or to sell on the market. The model shows a rich landscape of diverse production regimes, including mixed regimes where agents optimize utility by combining time spent working for the market, for self-subsistence or by taking time off.

Keywords: Modeling, Socio-physics, Complex systems, Centralization, Transition

# I. INTRODUCTION

Good models are tools that enable us to think better. There is a long tradition of modeling in physics that seeks to develop intuition of complex phenomena by using very simple models, the paradigmatic example being the Ising model for ferromagnetism [1]. By simplifying a complex system of interacting spins, his model provided a clear picture of the phase transitions and critical behaviour of magnetic materials.

More recently, physicists have tried to extend this simple modeling approach to social systems [2–6]. Social physics includes seven main research directions [7]: complex networks [8, 9], econophysics [10, 11], opinion dynamics [12], evolutionary games [13], community detection [14, 15], collective motion [16] and human dynamics, including epidemics propagation [17, 18]. There are two key contribution of (statistical) physics to social modeling. First, its ability to imagine simple yet interesting models, that incorporate the essential ingredients of the phenomenon to be described, allowing to capture the main conceptual mechanisms at work. Second, its capacity to link mathematically the local and global system dynamics, the micro behavior (utility function, or social atom characteristics) to the macro properties. For example, evolutionary games [13] commonly assumes that the more successful local strategies, those that accrue the highest payoffs to individuals (micro level), are imitated and thus spread globally (macro level). Mathematical calculations show that the details of spread do depend on the structure of the local interaction networks, and on

<footnote>* Also at Centre Inria de Lyon et IXXI; Also at Università di Padova, Padua, Italy; guillermo.benito.calvino@gmail.com</footnote>

<footnote>† pablo.jensen@ens-lyon.fr</footnote>
the interaction rules (public good games with different punishment or rewarding rules, tolerance...). For instance, Hauser et al [19] suggest that when individuals are provided with information about other individual's cooperative actions, they reward them for their contributions to the community, leading to the global adoption of beneficial strategies. Another famous example is the model of racial segregation proposed by economist Thomas Schelling in 1969 [20]. This model is interesting because it shows that our intuitions about collective phenomena may prove wrong: At a conceptual level, a segregated city does not logically imply that individuals seek segregation or are racist. More generally, the model shows that the famous "invisible hand" is not that powerful, as agents pursuing their own interests may not achieve an optimal situation [6, 21]. However, simple social models should be used with caution, since, contrary to physical systems, social systems cannot be transformed in laboratories, to resemble a mathematical model (see [7] or Chapter 5 in [6]). Social systems are in general far too complex to be fully captured by such simple models.

In this paper, we present a simple model of a crucial phenomenon of modern societies: The transition from a local mode of production to a centralized one, leading to economies of scale and mass production. Market products need cheap long-distance networks to centralize raw materials and then sell their production to a large population. For example, the "industrious revolution", which started in the mid-17th century, led to "the reallocation of labor from goods and services for direct consumption to marketed goods" [22]. Later, in the 19th century, Chicago took advantage of cheap rail transport to become "Nature's Metropolis" [23]. It centralized a wide range of rural productions (meat, wheat, corn ...) to mass produce and sell manufactured goods to a vast region.

As explained above, our objective is not to achieve a detailed understanding of these transitions. We expect a simple model to enrich our understanding of them in several ways. First, mathematical models force us to make our assumptions and ingredients explicit. These hypothesis should reproduce in a consistent way some features of the phenomena under study with simple ingredients. Second, the small number of ingredients gives these models an undeniable pedagogical interest, allowing one to understand their outcomes and the causalities at work in depth, without being overwhelmed by the complexity of the real world. Finally, the model proves useful when it enables new connections to be established between different historical examples of centralization or if it shows that some intuitions were wrong or too simple.
## II. MODEL INGREDIENTS

A significant challenge in creating this kind of model is to produce interesting results starting with a few simple ingredients. This is crucial for developing models that are both illuminating and comprehensible. As we explore new territory, to avoid getting lost in the vast space of all possible models, we will start from a well established economics model proposed by Gary Becker [24]. The core of Becker's model is to consider households as productive entities, that combine time and raw materials to produce the goods they need. Since its publication in 1965, this model has been used to compute agent's decisions, in order to anticipate the impact of public policy measures. For example, in Environmental studies, the model allows to account for changes in energy consumption as a function of household budget [25]; in Transportation economics to compute "generalised" costs of trips that include, beyond the monetary price of a travel, the time loss undertaking the journey [26]. Becker's work has also been cited in Gender studies, albeit from a more critical perspective [27], to understand women's employment rates or the inequalities experienced by women both in the labour market and in the domestic sphere.

As explained above, we focus here in the industrious revolution [22], the important transformation that lead households, which used to spend most of their time self-producing (dairy products, vegetables in their own garden ...), to devote most of their time working for a salary to buy market goods from mass producing companies. Historians have already suggested that Becker's model could help understand the industrious revolution [22] but without developing any formal study.

Specifically, our model consists of *N* individuals, which can choose to be either households (H, number *N*<sub>H</sub>) or companies (C, number *N*<sub>C</sub>). Companies symbolize centralized production, whereas households represent local producers. Households interact with companies by purchasing products and offering labor in exchange for wages.

### A. Households

Households represent the typical population of consumers and workers. They aim to maximize their utility, which can be achieved in two distinct ways. Firstly, they can produce their own goods ($Z_S^H(x_S, T_S)$) by using self-production time ($T_S^H$) and raw materials ($x_S^H$).
Alternatively, they can purchase goods directly from the market ($Z_M^H$), incurring a monetary cost but without the need to invest time in local production. Note however that, in this case, they will have to spend some time working ($T_w^H$), in order to produce the goods in the companies and to obtain the money necessary to buy them.

Finally, they can also obtain utility by just having free time ($T_f^H$). The utility is thus defined as:

$$U_H = l_S Z_S^H(x_S^H, T_S^H) + l_M Z_M^H + f T_f^H, \quad (1)$$

where $l_{S/M}$ represents how much individuals like the different types of products and $f$ stands for the value of the free time. $f$ is assumed to be lower than $l_S$ and $l_M$ since basic needs must be satisfied before spending free time.

To maximize their utility, households have limited time and budget. This is taken into account by the two constraints in Eqs (2) and (3). The first states that time can be allocated to work in the market for a wage ($w$), to self-produce or enjoyed as free time:

$$T = T_S^H + T_w^H + T_f^H, T_i^H \ge 0. \quad (2)$$

The budget constraint can be written as:

$$p Z_M^H + c_S x_S^H = w T_w^H + V, \quad (3)$$

where $p$ is the price of the market product, $w$ is the monetary resource obtained per unit of time (wage), $c_S$ is the economic cost of a unit of raw material and $V$ is any other form of income<fnref n="1" />.

## B. Companies

In this model, companies are individuals that have switched to production of normalized market goods, denoted as $Z_M$. Their objective is to maximize utility by increasing production and reducing costs. To achieve this, companies can hire labor from households to increase their available time. They also obtain utility from free time.

The utility function for companies is defined as follows:

<footnote n="1">¹ This term is taken from Becker, and represents the resources taken from outside the Household / Companies system, which compensate for the external inputs (raw materials, $x_S^H$ and $x_M^C$, see below, Figure 1).</footnote>
$$U_C = l_M k Z_M^C (x_M^C, T_M^C + T_w^C) + f T_f^C, \quad (4)$$

where k is the fraction of the centralized product retained by the companies for their own consumption. This means that they sell the remaining $(1-k)Z_M^C$. The decision to define utility in this manner, rather than as pure monetary earnings, is primarily due to the need for consistency in utility units between companies and households.<fnref n="2" /> $T_w^C$ represents the amount of labour that each company wants to hire.

Companies also have monetary and time constraints: they can allocate time to produce for the market ($T_M^C$) or for free time, with:

$$T = T_M^C + T_f^C. \quad (5)$$

The income, obtained by selling $(1-k)$ normalized product at a price $p$, is equal to the expenses of buying raw materials (for market goods) at price $c_M$ and hiring people with wages $w$:

$$(1-k)pZ_M^C (x_M^C, T_M^C) = c_M x_M^C + wT_w^C. \quad (6)$$

### C. Producing goods

The production of the goods $Z_i$ (where $i = S, M$) is accomplished by combining the two resources, raw materials and time. Several production functions have been proposed by economists, with varying degrees of substitutability between inputs (i.e. the ability to replace time with raw materials or the inverse, see [28]). The two most popular are the Harrod-Domar production function (no substitutability) and the Cobb-Douglas production function (elasticity of substitution equal to one). In this first exploration of the centralization/decentralization transition, we adopt the simpler Harrod-Domar function, which allows to obtain analytical solutions.

$$Z_i(x_i, T_i) = \min\left(\frac{x_i}{\chi_i}, \frac{T_i}{\tau_i}\right) \quad (7)$$

<footnote n="2">² Secondly, it allows companies to generate utility within a circular economy. Essentially, without taking into account external elements (V and raw materials), the revenue that companies earn, which corresponds to household expenditure on market products, is equivalent to the wages paid to households. In this closed system with only two types of agents, the monetary income of companies equals their wage expenses, resulting in a null net income.</footnote>
The required quantities of time and raw materials to produce one unity of item are denoted as $\tau_i$ and $\chi_i$ respectively and may be different for self-produced or market products. Both market or self-produced goods follow the same production function (7) but with a major difference, as companies can hire households' time $T_w^C$. Note that the variables $\tau_i$ and $\chi_i$ do not need a upper index, distinguishing companies or households, since each type of agent can only produce one kind of product, so that the sub index $i$ uniquely identifies it. Since both inputs are necessary, the production function is defined as the minimum of the two. Therefore, the maximum production with the least resources is always achieved when:

$$ \frac{x_i}{\chi_i} = \frac{T_i}{\tau_i}. \qquad (8) $$

To understand this equality, consider this simple example: Constructing a cane requires both wood and time. Suppose it takes 1 day ($\tau_{cane}$) and 1 kilogram of wood ($\chi_{cane}$) to build one cane. If you only have 2 kilograms of wood, no matter how many days you allocate, you will only be able to produce two canes. The reverse is true when time is limited and raw materials are in excess.

## D. Normalizing variables

To improve clarity and facilitate calculations, we eliminate units for the utilities and reduce the number of variables in the system. We begin by defining the monetary expense per unit of goods as $\epsilon_i = c_i\chi_i$ and the productivity as $\pi_i = 1/\tau_i$. Our objective is to analyze the system in terms of the exogenous (uncontrollable) variables $\epsilon_i$, $\pi_i$, and $l_i$. To further simplify, we consider the ratios between the variables related to market and self-subsistence production: $\epsilon = \frac{\epsilon_M}{\epsilon_S}$, $\pi = \frac{\pi_M}{\pi_S}$ and $l = \frac{l_M}{l_S}$.

Next, we normalize the utilities relative to the self-production parameters<fnref n="3" />, $\pi_S$ and $l_S$, and the total available time $T$. This normalization ensures that the new quantities $U'_H$ and $U'_C$ (described below) are dimensionless.

For households:

$$ U'_{H} = \frac{U_{H}}{\pi_{S}l_{S}T} = t_{S}^{H} + \frac{l}{p'}(w't_{w}^{H} + V' - t_{S}^{H}) + f'(1 - t_{S}^{H} - t_{w}^{H}), \qquad (9) $$

<footnote n="3">³ The choice of self-production parameters is arbitrary. One could also normalize using market parameters, as long as the definitions of the ratios $\epsilon_i$, $\pi_i$, and $l_i$ remain consistent.</footnote>
where the terms in the right-hand side represent self-production, the purchase of market goods with available income and free time, respectively.

For companies:

$$U'_{C} = \frac{U_{C}}{\pi_{S}l_{S}T} = l \left[ \pi(t_{M}^{C} + t_{w}^{C}) - \frac{1}{p'}(w't_{w}^{C} + \epsilon\pi(t_{M}^{C} + t_{w}^{C})) \right] + f'(1 - t_{M}^{C}), \quad (10)$$

where the first term is the production of market goods (through self time and hired time), the second is the cost of producing and hiring and the last, free time.

Additionally, all time variables are normalized with respect to the total available time per individual, T, so that $t_i^j$ ranges between 0 and 1.

The normalization introduces new dimensionless quantities:

$$V' = \frac{V}{T\epsilon_{S}\pi_{S}}, \quad p' = \frac{p}{\epsilon_{S}}, \quad w' = \frac{w}{\epsilon_{S}\pi_{S}}, \quad f' = \frac{f}{l_{S}\pi_{S}}, \quad (f' < 1).$$

Here, $p'$, $w'$, and $f'$ are straightforward normalized versions of the respective variables. $V'$ represents the normalized ratio of external income (used to buy raw materials) to total time available. When $V' < 1$, raw materials is the limiting production factor, which means that the total available time $T$ exceeds the time necessary to produce goods with the limited amount of raw materials available $V$.

### III. SOLVING THE MODEL

We now need to add some hypothesis to obtain results from the model. First, it is crucial to define the limits of the system, to distinguish between endogenous and exogenous variables. Second, we need to define different hypothesis about the equilibrium conditions: Are household demand and companies production equal? Are the utilities of all agents the same? ...).

#### A. The role of variables

The distinction between endogenous and exogenous variables is shown in Figure 1. First, some variables are assumed to be exogenous, to change only slowly and to be beyond agent's direct control. These constants include the ratios for productivity ($\pi$), monetary costs ($\epsilon$), preferences ($l$) and the other income ($V'$).
<figure><img src="image_2.png" /><figcaption>FIG. 1. Overall view of the model, showing the three different types of variables for households (H, on the left) and companies (C, on the right): the exogenous ($V', \pi, \epsilon, l$) on top, those fixed by the market equilibrium ($p', w', n$) inside the yellow circle and, finally, the endogenous and controllable ($T_i^j, x_i^j$) inside the agents.</figcaption></figure>

Second, some variables are endogenous but they are supposed to be beyond the direct control of agents, as they are determined by the system's equilibrium. These include the price ($p'$), wage ($w'$) and the population ratio $n = N_C/N_H$.

Finally, some variables are assumed to be endogenous and controllable by the individuals to maximize their utility. These include time allocations ($T_S^H, T_w^H, T_f^H, T_M^C, T_w^C, T_f^C$) and the quantity of raw materials acquired by agents ($x_S^H, x_M^C$). Note that $x_i$ and $T_i$ are dependent through Equation (8).

## B. Equilibrium

Let's now turn to the system's equilibrium. In scenarios without companies, only local production of $Z_S^H$ exists. In this case, only $U_H$ is relevant:

$$U'_H = t_s^H + f'(1 - t_s^H), \quad t_s^H = \min(V', 1). \qquad (11)$$

Households use all available resources for self-production, limited by time or raw materials. The min function in Equation (11) reflects this constraint, with maximum values of 1 for time and $V'$ for raw materials.

When companies are present, three equilibrium conditions emerge: Companies' production has to match Households' consumption; Households' working time must match the
Companies' hiring time; both agents should enjoy the same utility, as they are supposed to be free to change roles. These conditions yield solutions for the market variables (endogenous but not directly controllable) $p'$, $w'$ and $n$.

### Time condition:

Households' working time must match the hiring time of the companies: $N_H T_w^H = N_C T_w^C$, so we define $T_w = T_w^H = nT_w^C$. The factor $n = N_C/N_H$ accounts for the distribution of total labor among all companies. Additionally, for convenience in calculations, we assume that both agent types have equal free time: $T_f^H = T_f^C = T_f$ and, consequently, $T_M^C = T_S^H + T_w$.

### Equal offer and demand, $N_C(1-k)Z_M^C = N_H Z_M^H$:

$$ \frac{N_C}{p'}[\epsilon\pi(t_S + t_w(1 + 1/n)) + w't_w/n] = \frac{N_H}{p'}(w't_w + V' - t_S) \quad (12) $$

We obtain this equation from constraints (6) and (3) and using the relation between $x_i$ and $T_i$ (8)$^4$. This equation leads to the following equilibrium expression between the concentrations:

$$ n_{eq} = \left( \frac{N_C}{N_H} \right)_{eq} = \frac{V' - \epsilon\pi t_w - t_S}{\epsilon\pi t_M} \quad (13) $$

### Equal utilities, $U'_C = U'_H = U'$:

$$ l\pi(t_S + t_w(1 + 1/n)) - \frac{l}{p'}[\epsilon\pi(t_S + t_w(1 + 1/n)) + w't_w/n] = t_S + \frac{l}{p'}(w't_w + V' - t_S) \quad (14) $$

Now we can obtain a relation between $w'$ and $p'$.

$$ p'_{eq} = \frac{(\epsilon\pi - 1)t_s + (\epsilon\pi + w')(1 + 1/n)t_w + V'}{\pi(t_s + t_w(1 + 1/n)) - \frac{1}{l}t_S} \quad (15) $$

Once we have found the solutions for the equilibrium, we can plug them inside the utility. Note that $U'(n_{eq}, p'_{eq})$ is independent from $w'$ because utilities are fixed inside the market mechanism. In other words, an increase in wages leads to a higher household income, which in turn increases their spending capacity, thereby resulting in a higher price. This results in a higher expense for companies due to $w'$ but also in a higher income due to $p'$.

<footnote>$^4$ We drop the upper index notation for $t_i^j$ since it is no more necessary due to equilibrium conditions. For example $T_s^H = T_s$ since it is uniquely identified.</footnote>
We end up with a quite simple expression for the utility and two constraints:

$$U' = l\pi t_w + \left( \frac{l\pi V'_{left} + \epsilon\pi(1 - t_f)}{V'_{left} + \epsilon\pi(1 - t_f)} \right) t_S + f't_f, \quad (16)$$

$$\begin{cases} 1 - t_S - t_w = t_f, & 0 \le t_i \le 1 \\ V' - t_S - \epsilon\pi t_w = V'_{left}, & V'_{left} \ge 0, \end{cases} \quad (17)$$

where $V'_{left}$ is the unused amount of external income $V'$. The first constraint represents the time allocation of individuals. The second ensures non-negative $n_{eq}$, which is related to the offer and demand equality, indicating that system external income ($V'$) must meet or exceed total external expenditure.

Agents can now decide how to spend their time, i.e. fix those endogenous variables that maximize utility (Equation 16). If the maximum utility value in this context is greater than the utility in the household-only setting (Equation (11)), the market setting is preferred, leading to the emergence of companies.

## IV. MODEL RESULTS

We will focus on the parameters $\pi$ and $\epsilon$ as they are the most significant for our model. Recall that $\epsilon$ is the relative cost of producing market goods (compared to self-subsistence goods) and $\pi_i = 1/\tau_i$ the relative productivity of the market. These parameters represent the two dimensions (time and materials) of the difficulty or ease of centralizing the production. In the following, the super index * denotes equilibrium values, those that achieve optimal allocation.

We have identified four distinct regimes. To provide a first intuition about the behavior of the model, we describe here the results obtained for $f' = 0$ (free time is not valued for itself) and $V' < 1$ (the limiting resource is raw materials, not time). The results for the cases $f' > 0$ and/or $V' > 1$ are described in section IV B.

### A. Simple case : $f' = 0$ and $V' < 1$

* **Self-production:** Let's start with region A in Figure 2. This regime occurs when the relative cost of market inputs is too high compared to the preference for market goods
<figure><img src="image_3.png" /><figcaption>FIG. 2. Depending on the values of the relative cost ($\epsilon$) and productivity ($\pi$), four different regimes appear: **A**: Only households, **B**: Self-production + market work, **C**: Market work + free time, **D**: Pure market work. The pink triangle shows the point where $l/\pi = V'$ (see text).</figcaption></figure>

($\epsilon > l$). Then, only households exist because allocating time solely to self-production yields greater utility than centralizing production in companies. Historically, this can correspond to the common case of high transportation costs, as market products need both transportation of raw materials to the production centers and transportation of the transformed products to the consumers. Note that this regime is observed whatever the productivity of a hypothetical company (horizontal line in Figure 2). Households also have free time ($t_S^* = V'$ and $t_f^* = 1 - V'$) since in this case ($V' < 1$) the limiting resource is raw materials, not time. In this specific case, $U' = V'$.

When the cost of producing market goods decreases (as always, relative to the cost of self-produced goods), for example when transportation becomes cheaper or technical innovations allows a more efficient extraction of raw materials, thus decreasing their cost, two options exist, depending on the relative productivity of markets $\pi$.

* **Self-production + work:** When $\pi$ is low ($\pi < V'/l$, shown by the pink triangle in Figure 2), the system settles into a mixed regime (**B** in Figure 2), where market products exist, but people share their time between self-production ($t_S^* = \frac{V'-\epsilon\pi}{1-\epsilon\pi}$) and working in companies ($t_w^* = \frac{1-V'}{1-\epsilon\pi}$). This allocation of time gives $U' = \frac{V'(1-l\pi)+\pi(l-\epsilon)}{1-\epsilon\pi}$. In this regime, the relative cost favors market production, but the relative productivity favors self-production. Consequently, allocating time to both activities maximizes utility, as it achieves the optimal trade-off between productivity and costs. The proportion
of companies is minimum ($n_{eq}^* = 1/(N-1)$), as all households work in a single large company. This allows to maximize the total available income ($N_H V'$) from outside the market system.

An unexpected outcome in this regime is that market production decreases as $\epsilon$ decreases, i.e. when the (relative) cost of market raw materials decreases. However, as the above expression of $t_w^*$ indicates, it reduces the fraction of market time. The explanation is the following. A reduction in costs would allow companies to afford more raw materials for the same economical cost, but the point is that transforming these additional raw materials into utility would require additional time from households working for the market (Eqs. 4 and 8). However, devoting more time to companies would decrease the household's (and overall) utility, because in this regime self producing has a higher productivity. This phenomenon can also be understood using Equation (16). It is simple to show that in this regime, utility is maximum when $t_f = 0$ and $V_{left}' = 0$. By combining then Equations (17), it is clear that when $\epsilon$ decreases, the only way to ensure $t_f = 0$ and $V_{left}' = 0$ is by reducing $t_w^*$ as shown by the above formula.

* **Work + free time:** Let's now examine the transition from regime **A** when the cost of producing market goods decreases (decreasing $\epsilon$), and the relative market productivity is higher ($\pi > V'/l$) than in the previous case. Now, both dimensions of production (time and raw materials cost) favor markets over self-production, which should lead to all time being devoted to production in companies. However, when the cost of market inputs are relatively high ($\epsilon > V'/\pi$), another mixed regime appears, combining working in companies with free-time (region **C** in Figure 2). Here, as the cost of market inputs are relatively high, $V'$ cannot cover the raw material expenses that would obtain if all the available time were devoted to produce in companies (which now have a high productivity). Therefore, people work until they exhaust the available raw materials ($t_w^* = V'/(\epsilon\pi)$), and spend the rest as free time ($t_f^* = 1 - t_w^*$, $t_S = 0$), even for $f = 0$. Consequently, $U' = lV'/\epsilon$.

* **Pure work:** Finally, when both parameters favor market production (high relative market productivity ($\pi > V'/l$) and low relative cost of market inputs ($\epsilon < V'/\pi$), another pure regime appears (region **D** in Figure 2). The maximum utility, which is
$U' = l\pi$, occurs for $t_w^* = 1$ (with $t_S^* = t_f^* = 0$). In this regime, $n_{eq}^* = \frac{V' - \epsilon\pi}{\epsilon\pi}$, so that, contrary to regimes **B** and **C**, the number of companies is not at a minimum. The reason is that, because costs are low and productivity high, companies do not need to sell a high fraction of their production to buy inputs (Equation (6)), and can keep more production to increase their own utility. This leads to less offer in the market, pushing households to become companies to satisfy the demand. Moreover, as costs are low, it is not important to keep a high number of households to bring a high amount of total other income $N_H V'$ into the system.

**B. General cases $f' > 0$ and/or $V' > 1$**

When $V' > 1$ and $f' = 0$, we observe the same regimes as in section IV A, with identical optimal allocations of time and $n_{eq}^*$, but displayed in different positions (see Figure 3.a. While regimes **C** and **D** (work + free time and pure work) remain unchanged, **A** and **B** (only households and self-production + work) have swapped positions. Remember that, when $V' < 1$, acquiring raw materials is more challenging than having available time, making the parameter $\epsilon$ more influential than $\pi$ (Figure 2). Consequently, regime **B** appears when $\epsilon$ favors the market. Here, when $V' > 1$, the primary challenge is the availability of time, making $\pi$ more influential and thus positioning the mixed regime symmetrically (top of the *a* diagram in Figure 3). Note that, when both resources have the same value ($V' = 1$), regime **C** disappears, merging with an extended regime **D**.

When $f' > 0$, the model's behavior remains largely unchanged ($b$ and $c$ diagrams of Figure 3). As expected, all regimes that include allocation in free time expand their boundaries. Specifically, region **A** when $V' < 1$ and region **C** for any value of $V'$ become larger. Conversely, the mixed region **C** diminishes for any value of $V'$. This affects only to the top boundary of this last region. The new boundary is more complex, extending from $\epsilon = l$ to $\epsilon = \frac{l\pi - f'}{\pi(1-f')}$.

**V. DISCUSSION AND CONCLUSION**

In the introduction, we suggested a list of increasingly demanding requirements for a simple model. First, it should make our assumptions explicit and reproduce, in a consistent
<figure><img src="image_4.png" /><figcaption>FIG. 3. The four possible regimes for allocations of time depending on the values of the relative cost ($\epsilon$) and productivity ($\pi$) for different cases of $V$ and $f$. $V' > 1$, $f' = 0$ (a), $V' < 1$, $f' > 0$ (b) and $V' > 1$, $f' > 0$ (c). **A**: Only households, **B**: Self-production + work, **C**: Work + free time, **D**: Pure work.</figcaption></figure>

way, some features of the phenomena under study. Second, it should allow us to enrich our understanding of social situations, in this case the transition from local to centralized production. Finally, it could reveal some unexpected phenomenon or help connecting apparently unrelated phenomena, such as the centralization transition of other phenomena than production.

Clearly, our model fulfills the first requirement. It reproduces some features of the transition from a local production regime to a centralized regime, when both market costs decrease and market productivity increases beyond self-sustenance ones. Furthermore, Figure 4 shows that the model is consistent, i.e. that utility $U'$ (utility with respect to the self-substance parameters) monotonically increases when the normalized production conditions improve ($\pi$ increases or $\epsilon$ decreases). Note that if we were to measure the utility normalized by the market parameters, $U'$ would increase with better self-production conditions.

What about the second point? Does the model enrich the understanding of the transition? Readers can judge by themselves, but we argue that by generating a rich diversity of regimes from simple ingredients, the model enhances our understanding of the centralization transition. This diversity emerges from *the need to combine two distinct (and limited) resources for production*, namely time and raw materials. The relative availability of these resources ($V$ and $T$) and the effectiveness of production ($\pi$ and $\epsilon$) intricately define the system regimes. These variables dictate how individuals split between households and companies
<figure><img src="image_5.png" /><figcaption>FIG. 4. Example of the variations of the optimized $U'$ for different values of $\epsilon$ and $\pi$. Here, the values of $U'$ (16) have been maximized using the best values for the time allocations and the quantity of raw materials. We use the following values for the exogenous parameters: $f' = 0.2$, $V' = 0.8$, $l = 1$. (Left) $U'$ as a function of $\epsilon$ and $\pi$. The color scale on the right shows that $U'$ increases when market parameters become more favorable (cost $\epsilon$ decreases and productivity $\pi$ increases, white region in the bottom right). The colored discontinuous lines (blue, horizontal and green, vertical) represent the cross-sections chosen for the right-hand side of the figure. (Right, bottom): Maximized utility changes with $\epsilon$ for $\pi \approx 1.6$ (vertical green line in the left figure). From left to right the system is in regimes D, C, A. (Right, top) Maximized utility for varying $\pi$ and $\epsilon \approx 0.7$ (horizontal blue line in the left figure). From left to right the system is in regimes B, D, C. The red discontinuous lines mark the transition between different regimes.</figcaption></figure>

and determine their optimal time allocations, leading to different kinds of productive systems. Instead of thinking of a simple transition between two regimes, pure self-production to pure market, the model shows a richer landscape of possibilities, with mixed production regimes, including various proportions of free time. Regime C, for example, could be thought as one compatible with a limitation in the use of raw materials imposed by Earth limits [29], producing higher utility than pure self-production but preserving free time and Earth habitability. In this perspective, it would be interesting to connect our model to an old paper that also used Becker's work to understand how "the interrelations of market and non-market domains of social life" is a "possible source of the environmental degradation which derives from consumption activities" [30]. Note also that the model leads to an unex-
pected result in regime **B**: market production decreases as relative market costs decrease, when one would expect that lowering $\epsilon$ would favor market production. It is tempting then to look for historical examples of this unexpected situation.

At a more abstract level, the model could help thinking about centralization transitions that need to combine two resources, not necessarily time and raw materials. For example, the production of scientific knowledge could be seen as the centralized production, in laboratories rather than factories, of generic knowledge (similar to $Z_M$), i.e. information that can circulate within techno-scientific networks [5]. Instead of producing local knowledge, useful for growing vegetables in home gardens or local fields, scientists in labs produce standardized knowledge about plant genomes, which is then useful for creating new plant varieties and optimizing their production in controlled fields. Under which conditions is the investment in scientific knowledge worth the effort, when resources are limited? If one wanted to make an analogy with our model, one would compare the cost of market raw materials to the difficulty of transforming the world in order to make generic knowledge useful. In the case of plant genomic research, this would correspond to the difficulty of making agricultural fields comparable to the controlled ones (by using fertilizers and pesticides). Another example from physics is the electronic band theory derived from quantum mechanics, which remained useless for creating electronic devices as long as it remained too difficult to purify semiconductors [31]. The model then prompts us to imagine a dimension analogous to the time / productivity dimension that could capture some feature of the scientific transition. It would then help differentiating various regimes combining local and global types of knowledge, as in the mixed regime (**B**) of the present model, which combines the production of $Z_M$ and $Z_S$. This suggestion and further refinements of the model, such as introducing different free times for households and companies, allowing a fixed number of companies to optimize their own utility at the expense of households utilities or using the Cobb-Douglas production function and finding numerical solutions, are left for future work.
VI. ACKNOWLEDGMENTS

We are grateful to Nicolas Lechopier and Eric Tannier for their inspiration and the en-
gaging conversations that contributed to this work.

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