Predicting agents’ direction of change in a non-stochastic economy using apotential function
Takuma Tanaka
Graduate School of Data Science, Shiga University, 1-1-1, Banba, Hikone, Shiga 522-8522,
Japantanaka.takuma@gmail.com

Abstract
This study examines an optimal agent producing a consumption good and depreciating capital and trading capital without depreciation. Assuming that the prices of the depreciating
capital are ﬁxed and that the future prices of undepreciated capital are announced, this study
demonstrates that the agent never loses proﬁt on trading undepreciated capital if the agent’s
state converges to the initial state. Corollary to this result, we found a scalar potential thatpredicts the change direction of agents trading the undepreciated capital exclusively among
them. The similarity between the scalar potential and the Helmholtz free energy suggeststhat stochastic economic models could be characterized by a framework similar to information thermodynamics.
Keywords: Non-stochastic model, intertemporal choice, capital, planning
JEL: D15, D25

1. Introduction
The characterization of an agent’s optimal decision-making (Bertsekas, 2005) without using
the agent’s detailed information is of great value. This type of characterization would allow
for a prediction applicable to any agent. The amount of goods traded by the agent is acharacteristic that can be measured without the detailed information. Thus, in this note, wefocus on the total quantity traded by an agent, as a step toward the characterization of optimal decision-making. We investigate a model in which an imaginary government can control
the price and supply of land, water rights, and ﬁshery rights, that is, the capital withoutdepreciation, of the region where the agent is living. This note demonstrates whether thegovernment can gain proﬁt from trading the undepreciated capital at previously announced
prices.
This note is organized as follows. Section 2 describes the model and states the main
result; that is, the government never gains proﬁt. Section 3 sketches the implication ofthe result. We obtain a function that predicts the direction of change of agents trading
the undepreciated capital exclusively among them. Section 4 summarizes the results andsuggests future directions. Finally, Appendix A presents the proof of the main result.

2. Model and the main result
In this model, an agent produces a consumption good and depreciating capital, and theagent trades capital without depreciation. For brevity, we refer to depreciating capital
simply as “capital” and to capital without depreciation as “land.” We assume that a unitof consumption good can be exchanged with ﬁxed amounts of capital. In other words,
the agent may trade the consumption good and capital in a large market, in which theirprices are constant. In contrast, the prices of land are not ﬁxed. An entity referred to as
government announces the time series of the land prices and trades the land with the agent.
The government prohibits the agent from selling land to the large market. This model does
not consider stochasticity.
The capital and land owned by the agent in period t are denoted by kt ∈ RNcapital andat ∈ RNland , respectively. For the land price vector, the government announces its sequencein period t = 0, measured in the unit of the consumption good in period t, q t , and it is
known to the agent. Meanwhile, the price vector of the capital is ﬁxed to r. The agentmaximizes
Xβ t u[f (at , kt ) − q t · (at+1 − at ) − r · (kt+1 − ρ ◦ kt )]
(1)
0≤t≤∞

with a0 and k0 given at t = 0, where ◦ is the element-wise product. The utility function u(·)
and the production function f (·) are concave. The discount factor β satisﬁes 0 < β < 1. ρiis deﬁned by 1 − δi , where 0 < δi < 1 is the depreciation rate of capital i.
Appendix A demonstrates thatXq t · (at+1 − at ) ≤ 0(2)0≤t≤∞if the state of the agent converges to the initial state in the limit of t → ∞ (i.e., limt→∞ at =a0 and limt→∞ kt = k0 ). This means that the government cannot gain proﬁt from tradingland at the previously announced prices, making the agent’s ﬁnal state identical to its initialstate. Here we refer to a as the agent’s state because k is determined by a at the steadystate if f (·) is a strictly concave function.

3. Implication of the main result
At the steady state, the price vector of the land isq=1
β −1 − 1

∇a f.

(3)

Therefore,
H the net gain
H of the government in an inﬁnitely slow
H cyclic change C of the land
prices is C q · da.
H If C q · da > 0, it contradicts Eq. (2). If C q · da < 0, reversing the pricesequence yields −C q · da > 0, which contradicts Eq. (2). Hence,
Iq · da = 0(4)
Cholds for any cyclic integral. This means thatZ aref
Z aref
1∇a f · da,F (a) =q · da = −1
β −1 aa(5)where aref is an arbitrary reference point, does not depend on the path of integration connecting a and aref . That is, F (·) is a scalar potential. F (a) can be viewed as the government’s proﬁt gained by slowly changing the agent’s state from a to aref . Moreover, thegovernment’s proﬁt gained by slowly changing the agent’s state from aref to a is −F (a).
Meanwhile, F (a) − F (a′ ) is the maximal proﬁt that the government can gain by changing
the agent’s state from a to a′ because if the government can gain F̃ > F (a) − F (a′ ) bychanging the agent’s state from a to a′ , it can gain F̃ + F (a′ ) − F (a) > 0 in a process inwhich the agent’s state visits a, a′ , aref , and a in sequence. This contradicts Eq. (2).
Next, we examine the convexity of F (a). Let us denote the vector whose i-th element2∂f (a,k)by fxy (a, k). Deﬁning
is ∂xi by fx (a, k) and the matrix whose (i, j) element is ∂∂xf i(a,k)∂yi−1k(a) by the solution of fk (a, k) = b ◦ r, where bi = 1/(β − ρi ), yields
Z aref
1F (a) = −1
fa [a, k(a)] · da.
(6)
β −1 aIts Hessian with respect to a is given by∇a ⊗ ∇a F (a) = −Because1{faa [a, k(a)] + fak [a, k(a)]∇a k(a)}.
β −1 − 1(7)∇a k = −fkk (a, k)−1 fka (a, k),(8)1[faa (a, k) − fak (a, k)fkk (a, k)−1 fka (a, k)].
β −1 − 1(9)we obtain∇a ⊗ ∇a F (a) = −The concavity of f (a, k) implies the positive deﬁniteness of
−faa (a, k) −fak (a, k)−fka (a, k) −fkk (a, k)(10)

and, consequently, the positive deﬁniteness of −faa (a, k) + fak (a, k)fkk (a, k)−1 fka (a, k).
Combining it with Eq. (9) demonstrates the convexity of F (a).
Let us investigate the interaction among more than one agent by using F (a). We assume that Nagent agents have heterogeneous production functions, utility functions, anddiscount factors. For each agent i, F (i) (a(i) ) can be deﬁned. The government can let theagents trade the land among themselves withoutintervention. This situation is achievedP(i)(i)
by designing the sequence of q t , so that 1≤i≤Nagent (at+1 − at ) = 0 for t ≥ 0. Let us
(i)

(i)

examine what happens if the agents’ states start from a0 and converge to a∞ without(i)(i)
government intervention and then return from a∞ to a0 in trading the land with governP
P
(i)
(i)
ment intervention. If 1≤i≤Nagent F (i) (a0 ) < 1≤i≤Nagent F (i) (a∞ ), the government can gainP(i)(i)(i)(i)1≤i≤Nagent [F (a∞ ) − F (a0 )] > 0 by its intervention to the agents. This contradicts
Eq. (2). Hence,XX(i)F (i) (a0 ) ≥F (i) (a(i)
(11)
∞ ).
1≤i≤Nagent

1≤i≤Nagent

This means that trading the land exclusively among agents never increases the summationof F (i) (a(i) ). This argument holds even if the land is traded by only a fraction of agents for
only a few periods.

4. Conclusion
This note has demonstrated that the government, which is capable of controlling the landprices as previously announced, cannot proﬁt if the capital price is ﬁxed and the agent’s
state converges to the initial state. This result holds for an agent with any utility function,production function, and discount factor. A scalar potential that predicts the direction ofchange in agents’ states, F (·), is derived from the result.
This study’s result is an inﬁnite-period extension of a previous result on an exchange
economy with two periods (Tanaka, 2020). The result can be extended to an agent thattrades the consumption goods and capital with ﬁxed prices and can produce them with one
unit of labor endowment.
Is the present result applicable to an economy with consumption goods and capital withchanging prices? Numerically, counterexamples to Eq. (2) are found for this type of economy.
In other words, this type of economy allows for an agent proﬁting from the cyclic trade ofthe land. Thus, in the real economy, one possibility is that some agents gain proﬁt bycyclic trade. Another possibility is that the real economy oscillates to prohibit agents fromreaching steady states where agents can gain proﬁt. Further work must determine whichof these two possibilities the case is. Moreover, extension to stochastic models would be ofinterest because this extension enables us to apply the result to a broader range of settings.
This approach is diﬀerent from the previous approaches to economic dynamics (Stokeyet al., 1989; Ljungqvist and Sargent, 2012). Moreover, this note does not give the solution

to the optimization problem and the stability of the dynamics, but the condition that thesolutions must satisfy. Integrating both approaches will be useful to investigate the behaviorof an agent if this approach could be extended to stochastic dynamics.
The similarity of the mathematical frameworks of this model and thermodynamics is alsoof interest. Considering the analogy between the negativity of the government’s gain and thenegativity of the work done by a system in an isothermal environment, F (·) is the economic
counterpart of the Helmholtz free energy. Helmholtz free energy is also a convex functionthat determines the direction of change in physical systems (Callen, 1985). Recently, thebehavior of stochastic systems has been described by information thermodynamics (Ito andSagawa, 2013). Thus, stochastic economic models could be characterized by a framework
similar to information thermodynamics.

A. Proof
We demonstrate the theorem for the case of Nland = Ncapital = 1 to simplify the proof,
although extending the derivation to the case of Nland > 1 and Ncapital > 1 is trivial. Accordingly, we replace at , kt , q t , r, ρ with at , kt , qt , r, and ρ, respectively, in the following.
The necessary conditions for the maximization of Eq. (1) are∂f (at , kt ) + u′t qt − β −1 u′t−1 qt−1 = 0,∂at∂u′tf (at , kt ) + ρu′t r − β −1 u′t−1 r = 0∂ktu′t(12)(13)

for t ≥ 1, where we deﬁnedu′t =∂u(c).
∂c c=f (at ,kt )−qt (at+1 −at )−r(kt+1 −ρkt )(14)

We do not assume the nonnegativity of the amount of capital, land, and consumption.
However, Inada conditions can guarantee the nonnegativity.
Using Eqs. (12) and (13) leads the concavity of u(·) and f (·) toxt,s =u′t ft − u′t fs + (u′t qt − β −1 u′t−1 qt−1 )(at − as )+ (ρu′t r − β −1 u′t−1 r)(kt − ks ) ≥ 0,yt,s =ut − us − u′t [ft − qt (at+1 − at ) − r(kt+1 − ρkt )− fs + qs (as+1 − as ) + r(ks+1 − ρks )] ≥ 0,

(15)
(16)

where
ft = f (at , kt ),ut = u[f (at , kt ) − qt (at+1 − at ) − r(kt+1 − ρkt )].

(17)
(18)

In the following, we showlim

T →∞X0≤i≤Tzi = −X0≤t≤∞qt (at+1 − at ), ,(19)where(−1Pt ′(1 − β) ∞βui=0,wi =P∞ t ′ t=0−1 t+ii≥1t=0 β ut+iXX
zi = wiβ t−i xt,i + wi (1 − β)β s+t−2i yt,s .
i+1≤t≤∞

(20)
(21)

i≤s≤∞
i≤t≤∞

Equation (19) proves Eq. (2) because the left-hand-side of Eq. (19) is the summation of thenonnegative values xt,s and yt,s with nonnegative weights.
CombiningX
β t−i xt,ii+1≤t≤∞X=

β t−i {u′t ft + (u′t qt − β −1 u′t−1 qt−1 )at + (ρu′t r − β −1 u′t−1 r)kt }i+1≤t≤∞

X−β t−i u′t fi + u′i qi ai + u′i rki +i+1≤t≤∞X

β t−i u′t (1 − ρ)rki(22)i+1≤t∞

andX

β s+t−2i yt,si≤s≤∞
i≤t≤∞

X
1
β t−i u′t [ft − qt (at+1 − at ) − r(kt+1 − ρkt )]1 − β i≤t≤∞XX−β t−i u′t
β s−i [−fs + qs (as+1 − as ) + r(ks+1 − ρks )],=−i≤t≤∞(23)i≤s≤∞yields
wi−1 zi = −X

β t−i u′t fi +i≤t≤∞− (1 − β)Xi≤t≤∞

X

β t−i u′t (1 − ρ)rkii≤t≤∞X

β t−i u′t

β s−i [−fs + qs (as+1 − as ) + r(ks+1 − ρks )].

(24)

i≤s≤∞

Thus,z0 = −X1
1f0 +(1 − ρ)rk0 −β s [−fs + qs (as+1 − as ) + r(ks+1 − ρks )]1−β1−β0≤s≤∞(25)

andzi = −fi + (1 − ρ)rki − (1 − β)Xi≤s≤∞

β s−i [−fs + qs (as+1 − as ) + r(ks+1 − ρks )](26)for i ≥ 1. Moreover,XXzi = −qs (as+1 − as ) + rk1 − rkT +11≤i≤T1≤s≤T+X

β s [−fs + qs (as+1 − as ) + r(ks+1 − ρks )]1≤s≤∞

X−β s−T [−fs + qs (as+1 − as ) + r(ks+1 − ρks )];(27)

T +1≤s≤∞thus, the summation is given byXX
β
1f0 +(1 − ρ)rk0 + rρk0 − rkT +1 −qs (as+1 − as )1−β1−β0≤s≤T
Xs−T−
β [−fs + qs (as+1 − as ) + r(ks+1 − ρks )].

zi = −0≤i≤T(28)T +1≤s≤∞Taking the limit of T → ∞, we obtainlimT →∞X0≤i≤Tzi = −X0≤s≤∞qs (as+1 − as ) −1βf0 +(1 − ρ)rk0
1−β1−ββ+ rρk0 − rk0 −[−f0 + r(1 − ρ)k0 ]1−βX=−qs (as+1 − as ),(29)0≤s≤∞where we used limt→∞ at = a0 , limt→∞ kt = k0 , and limt→∞ ft = f0 . This completes theproof.

References
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