Irreversibility of change in a model of boundedly
rational agents exchanging goods and capital
Takuma Tanaka
Graduate School of Data Science, Shiga University, 1-1-1 Banba, Hikone, Shiga
522-8522, Japan
E-mail: tanaka.takuma@gmail.com
January 2020
Abstract. This paper investigates the direction of change in a model of agentsexchanging goods and capital. Agents have heterogeneous utility functions and are
not perfectly rational but can refuse exchanges that decrease their own utility. In thismodel, an exchange is shown to always increase a quantity defined by the amount ofproduction and the time preference of the agents.

Keywords: Irreversibility, agent-based model, utility function

Submitted to: New J. Phys.

1. Introduction
Predicting the behavior of agents is one of the most important tasks in social sciences
and sociophysics. Economics has succeeded in making quantitative predictions byassuming optimal behaviors of perfectly rational agents, which can maximize theexpected outcomes [1, 2]. However, experiments have suggested that humans are not
perfectly rational in many situations [3]. Humans are instead boundedly rational [4].
To describe human behavior more realistically, a formulation of boundedly rational
agents assumes that agents cannot necessarily choose the action that maximizes theexpected outcome but can choose actions that lead to an outcome exceeding a predefined
threshold [4]. Making these assumptions might preclude quantitative predictions, butthey should allow qualitative predictions, such as those of the direction of social andeconomic change. A direction of change can be predicted in various cases. For instance,
if economic change is unidirectional (i.e., irreversible), the direction is predictable. Inthis case, the direction may be predicted by a measurable quantity, such as entropy
in thermodynamics. This is because, if the quantity increases for any possible socialand economic change, any change that decreases the quantity should not to happen.
Whether or not economic change is irreversible, the quantity measuring irreversibility

Irreversibility of an economic model2

exists, and human behavior is irreversible should be useful in social modeling. Theseissues have not been investigated intensively [5, 6].
In this paper, I demonstrate the irreversibility of change using a model ofheterogeneous boundedly rational agents whose utility is a function of the consumptionof goods in each period. The changes taking place in the model are the exchangeof goods and capital among agents, which has been studied intensively in sociophysics[7, 8, 9, 10]. I show that a quantity defined by the amount of production never decreases.
I also discuss the potential extensions of the model.
2. Results
Denote the set of agents by A = {1, . . . , A}. At the beginning of each period, agentsproduce goods, such as foods. We assume that goods cannot be saved, and consequently,goods produced in a period must be consumed during the same period. We also assume
that there is capital, such as land, which cannot be directly consumed but can be used
to produce goods. Although we need no assumption on the functional dependency ofthe production of goods with respect to capital (i.e., a production function) to support
the main result, it is convenient for us to assume that the production of goods is anincreasing function of capital. Agents can exchange goods for capital with others. The
total amount of capital remains constant. The increase and decrease of capital owned
by an agent affect the agent’s production of goods over the subsequent periods. Thus,
the agents can increase goods production during subsequent periods by decreasing thegoods in the present period and vice versa.
For simplicity, we concentrate on a case in which agents exchange goods andcapital only once. Specifically, at the beginning of period 1, agents produce goods
by using capital. Then, agents exchange goods and capital and consume goods. During
subsequent periods, each agent consumes only the goods that are self-produced. Thus,
the amount of goods produced by agent a in period 1, ya , can be different from theamount of goods consumed by it in period 1, ca , whereas the amount of goods producedby agent a in each of the subsequent periods, Ya , is equal to the amount of goods
consumed by it, Ca . We assume that the exchange conserves the total amount of goods:XXya =ca .
(1)a∈Aa∈A

As a result of the exchange of capital used for production, Ya can be greater or less thanP
Pya , and that a∈A ya = a∈A Ya does not necessarily hold.
Let us assume that the utility of an agent is a function of the amount of goods
it consumes. As is typical in the literature of economics and sociophysics [11, 9], we
assume that an agent’s utility is an increasing concave function of the consumptionof the goods. This is a natural assumption, because increasing the amount of goods
improves our lives, but the degree of improvement is diminished if we already have alarge amount of goods. In this case, the utility of agent a is given byua (ca ) + βa ua (Ca ),(2)Irreversibility of an economic model

3

where ua (·) is a strictly increasing concave function, and βa > 0 is the parameter that
determines the relative preference of future consumption over present consumption.
This can be regarded as an additive, separable, intertemporal utility function,
P∞ t−1t=1 γa ua (ca,t ) [11], where(ca if t = 1
ca,t =(3)Ca if t > 1
is agent a’s consumption in period t and γa = βa /(1 + βa ).
If agent a does not engage in the exchange of goods and capital, its utility isua (c̃a ) + βa ua (c̃a ) = (1 + βa )ua (c̃a ),(4)where c̃a = ya is the consumption and production in the periods before the exchangetakes place, because it consumes exactly the same amount of goods as that produced byitself. Here, we assume that, without exchange, the amount of produced goods is kept
constant because the amount of capital owned by the agent is constant. Let us assume
that agents can refuse an exchange decreasing their own utility. Then,(1 + βa )ũa ≤ ua + βa Ua(5)holds, where we abbreviate ua (c̃a ), ua (ca ), and ua (Ca ) to ũa , ua , and Ua , respectively.
In other words, the predefined threshold that a boundedly rational agent in this model
wants to exceed is the utility that the agent would achieve if it did not engage in theexchange.
Now, I show the inequalityXX
βa ya ≤
βa Ya .
(6)a∈Aa∈A

Remarkably, this result holds for agents with any strictly increasing concave utilityfunctions and any production functions.
From Eq. (5), agent a′ with ua′ = ũa′ , ua′ = Ua′ , or, ũa′ = Ua′ satisfies ya′ ≤ Ya′ ,and, consequently, A = {a′ } satisfies Eq. (6). Hence, it is sufficient for us to support
Eq. (6) in excluding agents that satisfy one of these conditions from A and replacing
Eq. (1) withXXc̃a ≥ca(7)
a∈Aa∈Abecause c̃a′ ≤ ca′ holds for each excluded agent, a′ . Thus, for a ∈ A, ũa , ua , and Ua are
distinct values, and c̃a , ca , and Ca are distinct values.
Because ua (·) is a strictly increasing concave function, we note thatũa − Ua> 0,c̃a − Caua − ũa> 0,ca − c̃aUa − ua> 0,
C a − ca

(8)
(9)
(10)

4

Irreversibility of an economic modelũa − Uaua − ũa≥
if ca < Ca
ca − c̃ac̃a − Ca.
ũa − Uaua − ũa≤
if ca > Caca − c̃ac̃a − Ca(11)

We define1,ũa − Ua1d2a =,ua − ũa1,
d3a =Ua − uaz1a = c̃a − Ca ,

d1a =

(12)
(13)
(14)
(15)

z2a = ca − c̃a .

(16)

In addition, we define 2A × (4A + 1)-dimensional matrix M and 2A-dimensional vectors
b and ζ as

M=0 −D 1O
D 3 −D 1 D 3
1 O−D 2 D 3 D 2 D 3!

,(17)

b = [β1 , . . . , βA , 0, . . . , 0],(18)ζ = [z11 , . . . , z1A , z21 , . . . , z2A ].

(19)

Here, 0 and 1 denote the A-dimensional vectors whose elements are all 0 and 1,respectively, O denotes the A × A null matrix, and
di1 0 · · · 0 0 di2 · · · 0 
(20)Di =  .
..  .
.. . .
 ..
. . .
00· · · diA

Equations (7)–(11) lead toM T ζ ≤ 0.

(21)

As we show in the following, there is a 4A + 1-dimensional nonnegative vector, χ,satisfying M χ = b. It follows from Farkas’ lemma that there is no ζ satisfying M T ζ ≤ 0and bT ζ > 0. Hence, we have bT ζ ≤ 0, which is identical to Eq. (6).
The task is now to find
χ = [x0 , x1 , x2 , x3 , x4 ]satisfying M χ = b, x0 ≥ 0, and xi ≥ 0 (1 ≤ i ≤ 4). Because(22)Irreversibility of an economic model

d3aβax3a − d3a x4a −= x1a ,
d1ad1ad3a
1x0 +x3a + d3a x4a = x2ad2ad2a5(23)(24)

hold for 1 ≤ a ≤ A, the inequalities x1a ≥ 0 and x2a ≥ 0 are satisfied byx0 = 1,(25)x3a = 0,((26)

x4a =−β/(d3a d1a ) if d1a > 0 or (d1a < 0 and d2a < 0).
0
if d1a < 0 and d2a > 0(27)Note that d1a > 0 implies d2a > 0 and d3a < 0 and that d1a < 0 and d2a < 0 imply
d3a > 0 because−11+ βa≤ 0,d2ad1a(28)which follows from Eq. (5), and1
11++= 0,d1a d2a d3a(29)which is a direct result of Eqs. (12)–(14). This completes the proof.
Let us state a few consequences of this result. First, most change is irreversiblebecause
XS=βa ya(30)a∈A

increases with an exchange, unless the equality holds in Eq. (6). Although the amountof production of agent a, ya , can increase and decrease, S, calculated from the amountof production of all agents, never decreases. Second, selling capital to agents increases
S. To support this, let us consider another agent, α, whose production of the goods isan increasing function of capital i. Let us assume that boundedly rational agent α sells
capital i to agents in A and buys the goods. Because the goods produced by agent αdecrease, S must increase in order for
S ′ = βα yα + S
to increase.

(31)

Irreversibility of an economic model6

3. Conclusion
In this paper, I have shown that change is irreversible if agents can avoid decreasing
their utility. The model does not assume perfectly rational agents, perfect information,agent budget constraints, a competitive market, or Pareto-efficient allocations [1].
Additionally, we do not need any assumption on the dependency of the productionof goods on capital. The main result was derived without using the assumption thatthe production functions were increasing functions. It is noteworthy that these weak
assumptions are sufficient to derive the irreversibility of change. Although we assumed
that the exchange takes place only once, the present model can be applied to multiple
exchanges in multiple periods if we assume that agents do not know what happens in thesubsequent periods and that they expect no exchange during the subsequent periods.
The present model can be a widely applicable toy model of goods-exchanging agents,
including perfectly rational agents.
In this model, irreversibility originates from the loss-avoiding behavior of agents[12, 6]. Although irreversibility has not been emphasized in economics [5], it plays
a vital role in physics, particularly in thermodynamics [13, 14]. This model suggeststhat there may be a novel correspondence between physical and economic systems.
The present results suggest that S may be a quantity corresponding to entropy. This
correspondence might enable us to formulate economic and social models from theviewpoint of irreversibility.
Extending the present model could provide great insight into social and economicdynamics. For example, a study could investigate whether agents trading goods whoseamounts are represented by two or more scalar values exhibit irreversibility. Although
the present study showed that any changes decreasing S are impossible, whether allchanges increasing S are possible or not remains unanswered. For thermodynamic
entropy S, the answer is yes. Assuming that agents are at the competitive equilibrium[1] would allow us to define an adiabatic process and to answer this question.
Acknowledgments
I thank Takashi Hayakawa for his comments on the draft of this manuscript.
References[1] Hal R Varian. Intermediate microeconomics with calculus: a modern approach. WW Norton &Company, 2014.
[2] Refet S Gürkaynak and Cédric Tille. DSGE models in the conduct of policy: Use as intended.
CEPR Press, 2017.
[3] Daniel Kahneman. Maps of bounded rationality: Psychology for behavioral economics. American
Economic Review, 93(5):1449–1475, 2003.
[4] Herbert Alexander Simon and Chester Irving Barnard. Administrative behavior: a study ofdecision-making processes in administrative organization. Macmillan, 2nd edition, 1957.

Irreversibility of an economic model

7[5] Paul A. Samuelson.
Rigorous observational positivism: Klein’s envelope aggregation;
thermodynamics and economic isomorphisms. In Lawrence Robert Klein, F. Gerard Adams,and Bert G. Hickman, editors, Global Econometrics: Essays in Honor of Lawrence R. Klein,
pages 1–38. The MIT Press, Cambridge, Massachusetts, 1983.
[6] Katalin Martinás. About irreversibility in economics. Open Systems & Information Dynamics,
7(4):349–364, 2000.
[7] Anirban Chakraborti and Marco Patriarca. Variational principle for the pareto power law.
Physical Review Letters, 103(22):228701, 2009.
[8] Marco Patriarca and Anirban Chakraborti. Kinetic exchange models: From molecular physics tosocial science. American Journal of Physics, 81(8):618–623, 2013.
[9] Giuseppe Toscani, Carlo Brugna, and Stefano Demichelis. Kinetic models for the trading of goods.
Journal of Statistical Physics, 151(3-4):549–566, 2013.
[10] Pierre Degond, Jian-Guo Liu, and Christian Ringhofer. Evolution of the distribution of wealthin an economic environment driven by local nash equilibria. Journal of Statistical Physics,
154(3):751–780, 2014.
[11] Fabio C Bagliano, Giuseppe Bertola, et al. Models for Dynamic Macroeconomics. Oxford
University Press on Demand, 2004.
[12] Carl Menger. Grundsätze der Volkswirtschaftslehre (Principles of Economics). W. Braumüller,
1871.
[13] Elliott H Lieb and Jakob Yngvason. A guide to entropy and the second law of thermodynamics.
In Statistical Mechanics, pages 353–363. Springer, 1998.
[14] Hal Tasaki and Glenn Paquette. Thermodynamics–A Modern Point of View. 2019. (unpublished).