Abstract
Marx (1894) introduced the concept of "production price" as a result of the full mobility of capital between different sectors of production toward the equalisation of its gain rates. Nevertheless, he doesn't explain this process of equalisation between heterogeneous sectors, and its Law of Decreasing Tendency of the Gain Rate supposes an unjustified equalisation of the plusvalue rates (quotient between the gain and the capital invested in work power).
After, Sweezy (1972) and SalamaValier (1973) intended to explain the process, but their explanations weren't satisfactory for treating with static situations in equilibrium, when the real evolution is produced from situations of nonequilibrium.
PlaLopez (1986) presented a model of System Dynamics of the equalisation of the Gain Rate, which explained the possibility of evolution toward a greater proportion of inversion in production means, and concluded that the Marx's Law of Decreasing Tendency of the Gain Rate was wrong.
Now, we are going to complete the model by introducing the
adaptation
of the Demand to the Incomes, which we name Solvent Demand, and
studying
by computer simulation the results of the model in the evolution of the
Main Gain Rate.
Adjustment of Supply and Demand
We work with classical curves of Supply and Demand (see for example
Lipsey (1967)), with its "elasticity" as exponent. These curves ideally
would determine the Supply and Demand in function of the prices,
Of = k_{O}pr^{Eo}
Dem = k_{D}
pr^{Ed}
Of course, usually the elasticity of the Supply will be positive,
Eo>0,
and the elasticity of the Demand will be negative, Ed<0. But the
condition
of equilibrium is that the elasticity of the Demand would be greater in
absolute value than the elasticity of the Supply, Ed>Eo.
In order to simulate the evolution of the Supply, we suppose that the
Ideal Supply is determined by the real price, and the real Supply Of
tends
lineally toward the Ideal Supply Of with a delay t1,
so that
(1) Of_{i }= k_{O}^{i
}pr_{i}^{Eo}
(2) Of_{i}(t+1) = Of_{i}(t) + (Of^{I}_{i}(t)
 Of_{i}(t))/t_{1}
for every sector "i". Of course, if the delay t1
is greater, the evolution of the Supply to its Ideal value will be
slower,
and if t_{1}=1 then the Supply equals
its
Ideal value in one step.
Also, in order to simulate the evolution of the Demand to its Ideal
value with equalisation of Supply and Demand, we will suppose that the
Ideal Demand is equal to the real Supply, the Ideal Price is determined
by the Ideal Demand according to the relation (1'), and the real Price
tends lineally toward the Ideal Price with a delay t_{2},
that is to say
(3) Dem^{I}_{i} = Of_{i}
(4) pr^{I}_{i} = (Dem^{I}_{i}
/ k_{D}^{i})^{1/Ed}
(5) pr_{i}(t+1) = pr_{i}(t)
+ (pr^{I}_{i}(t)  pr_{i}(t))/t_{2}
(6) Dem_{i }= k_{D}^{i}pr
^{Ed}
If the condition of equilibrium Ed>Eo is fulfilled, then this
System
will evolve toward an situation of equilibrium in which
Dem = Dem ^{I}
=
Of = Of ^{I}^{ }= (k_{D}^{Eo}
k_{O}^{Ed})^{1/(EoEd)}
and
pr = pr^{I }= (k_{D}/k_{O})^{1/(EoEd)}.
Equalisation of the Gain Rates
We will suppose that to produce a unity of the product of the sector
"i" we need a^{ij}
unities
of the sector "j", so that its unitary cost is
(7) k^{i} = Σ_{j} a^{ij
}pr_{j}
for every sector "i". We consider work power as a sector, so that wages
are included in this unitary cost.
The total cost in a sector "i" will be k^{i }Of_{i
},
but its total incomes will be pr_{i }m^{i},
where m^{i }will be the minimum between
its
Supply and its Demand,
(8) m^{i }= min ( Of_{i},
Dem_{i} )
Therefore, the gain rate in this sector will be g^{i}
= (pr_{i }m^{i
}
k^{i }Of_{i})/
k^{i }Of_{i }=
pr_{i }m^{i }/
k^{i }Of_{i }
1 .
Because m^{i }≤ Of_{i}
, g^{i }≤
pr_{i} / k^{i }
1 , and therefore, g^{i }> 0 only if
Σ_{j} a^{ij}
pr_{j}< pr_{i}
. So, in order to get g^{i}>0 for
every i,
we can take a^{ij}so
that a<1, and therefore a
pr<pr. We will work with infinite norm, and so we will take a^{ij} so
that Σ_{j} a^{ij}
< 1 (the coefficients a^{ij}
are always positive). This condition is not necessary, but if
the
condition
å_{j
}a^{ij}
pr_{j}
< pr_{i }is fulfilled we can
select
the unity of each product in order to get it.
The Mean Gain Rate will be g_{m} = Σ_{i }g^{i}
k^{i}Of_{i }/Σ_{j }k^{j}Of_{j} , and
therefore
(9) g_{m} = Σ_{i
}pr_{i
}m^{i}/Σ_{j
}k^{j
}Of_{j
} 1
If you launch the model with the relations (1) to (5) with the
condition
of equilibrium Ed>Eo and random coefficients a^{ij} so
that Σ_{j} a^{ij
}< 1 , the Mean Gain Rate increases until reaching the
equilibrium
between Supply of Demand (in fact, out of this equilibrium can
inefficiently
be m^{i} < Of_{i
}, so that g^{i }< pr_{i}/
k^{i } 1 and g_{m}
< Σ_{i }pr_{i
}Of_{i
}/Σ_{j
}k^{j}Of_{j}
1 which is its value in
this equilibrium). We can see an example in the Figure 1.
Figure 1 Evolution of Mean Gain Rate with a single adjustment of Supply and Demand 
Nevertheless, in this simple equilibrium between Supply and
Demand,
with constant parameters k_{O}^{i }and
k_{D}^{i} , the Gain Rates of
the
different sectors are not equalised. This equalisation requires a
movement
of capital between sectors which will produce a displacement of the
curves
of Supply, through a change of the parameters k_{O}^{i
},
so that the prices of equilibrium between Supply and Demand will tend
to
the Production Prices pp_{i} in each
sector
"i" so that its Gain Rate were equal to the Main Gain Rate,
g^{i }= pp_{i }/ k^{i}  1 = g_{m} , and therefore 
In order to simulate the evolution of the parameters k_{O}^{i
},
we will suppose that they tend lineally with a delay t_{3}
to an ideal value k_{O}^{I}_{i}
which corresponds to the Production Price pp_{i}
with the present parameter k_{D}^{i }of
the curve of Demand, so that
(11) k_{O}^{I}_{i
}=
k_{D}^{i} /pp_{i}^{EoEd}
(12) k_{O}^{i} (t+1)
= k_{O}^{i} (t) + (k_{O}^{I}_{i}(t)
 k_{O}^{i} (t))/t_{3}
Note that, in conditions of equilibrium, Σ_{j} a^{ij} pp_{j} = pp_{i} / (1+g_{m}), and therefore 1/(1+g_{m}) has to be a selfvalue of the matrix a^{ij} , and the prices of production form its corresponding selfvector. Therefore, the possible values of Mean Gain Rate in equilibrium, and also the relative proportion of the prices in equilibrium, depend only on the intersectorial coefficients a^{ij} (PlaLopez 1986).
The relative proportions of the parameters in equilibrium of the
curves
of Supply, k_{O}^{i }, will
also depend
on the parameters of the curves of Demand, k_{D}^{i}.
Of course, according to the relations (11), if the prices in
equilibrium
change in a proportion λ from different initial
conditions, these parameters in equilibrium will change in a proportion
λ^{EdEo}.
Adjustment to the Solvent Demand
Nevertheless, a Demand is only sustainable if it is Solvent, that is
to say, it can be paid for the corresponding Incomes. In order to study
the adaptation to the Solvent Demand, we will consider only three
sectors:
Sector 0: production of means of production.
Sector 1: production of goods for workers.
Sector 2: production of goods for owners of means of production.
In order to simplify the model, we will suppose that there is a full
division between workers and owners of means of production, and between
sector 1 and sector 2.
In the relations (6), the prices of sector 2 don't contribute to the
unitary cost of any product, and therefore a^{i2}=0
for every sector "i". By the way, the prices of sector 1 contribute to
the unitary cost through the wages (see Sraffa 1960). So, the total
incomes
of the workers (its wages) will be equal to the total cost from the
sector
1, Σ_{i }Ofi
a^{i1}
pr_{1} . Therefore, the Solvent Demand
of
the sector 1 will be Dem^{S}_{i}
= Σ_{i
}Of_{i
}a^{i1}
pr_{1}/ pr_{1}
= Σ_{i }Of_{i
}a^{i1}
and according to the relation Dem = k_{D}
pr^{Ed} its parameter of Solvent
Demand will
be
(13) k_{SD}^{1}
= Dem^{S}_{1}/pr_{1}^{Ed}
= Σ_{i }Of_{i
}a^{i1}
/ pr_{1}^{Ed}
and we will suppose that the parameter k_{D}^{1}
tends lineally to this parameter of Solvent Demand with a delay t_{4}
,
(14) k_{D}^{1}(t+1)
= k_{D}^{1}(t) + (k_{SD}^{1}(t)
 k_{D}^{1}(t))/t_{4}
On the other hand, the total incomes of the owners of means of
production
will be its total gain,
Σ_{i }pr_{i
}m^{i
} Σ_{i
}k^{i
}Of_{i}. Therefore, the
Solvent
Demand of the sector 2 will be
Dem^{S}_{2}_{ }= (Σ_{i
}pr_{i
}m^{i} Σ_{i
}k^{i
}Of_{i
})/ pr_{2} , and the corresponding
parameter
of Solvent Demand will be
(15) k_{SD}^{2}
= Dem^{S}_{2} / pr_{2}^{Eo}
= (Σ_{i }pr_{i
}m^{i}
 Σ_{i
}k^{i
}Of_{i})/ pr_{2}^{Eo+1}
We will also suppose that the parameter k_{O}^{2}
tends lineally to this parameter of Solvent Demand with a delay t_{4}
,
(16) k_{D}^{2}(t+1)
= k_{D}^{2}(t) + (k_{SD}^{2}(t)
 k_{D}^{2}(t))/t_{4}
Figure 2 Evolution of Mean Gain Rate with static technology 
The interaction between the adjustment of Supply and Demand,
the equalisation
of the Gain Rates and the adjustment to the Solvent Demand not always
tends
toward equilibrium. But if so, the Mean Gain Rate oscillates toward its
value of equilibrium, which only depend on the intersectorial
coefficients
which express the technological conditions. In Figure 2 you can see an
example.
On the other hand, in a solvent equilibrium

Remember, but, that the unitary cost of a product is k^{i }= a^{i0} pr_{0} + a^{i1} pr_{1}, where a^{i0} expresses the contribution of the means of production, and a^{i1} expresses the contribution of the work power through the wages (equal to a^{i1} pr_{1}). Therefore, these coefficients also express the social relations between workers and owners of means of production, and the unionist struggle can carry to increase the wages, and so it could carry to increase the coefficients ai1. Nevertheless, we can plausibly suppose that the wages increase lesser than the productivity, and so that the coefficients ai1 do not increase, but decrease through technological progress, by means of saving in work power.
The Model uses Ed=.5 and Eo=.3 (but these values can be easily changed), and demands to fix the values of the delays (we have used t_{1}=t_{2}=t_{4}=10 and t_{3}=t_{5}=20). And, in order to simulate the process without more presuppositions, the Model can fix randomly the initial values of intersectorial coefficients, prices, amount of Supply and parameters of the curves of Supply and Demand. Then, the dynamical relations of the Model can change these values, eventually toward some value of equilibrium of the Mean Gain Rate.
Figure 3
Evolution of Mean Gain Rate through technological
progress
Nevertheless, the technological progress, through a random decreasing of the intersectorial coefficients, can break this equilibrium, and begin an evolution toward a new equilibrium. You can see a typical process in Figure 3: through this technological progress, the Mean Gain Rate increases over its previous value of equilibrium.
You can find the source code of our Model in http://www.uv.es/~pla/models/modpp.c
, and a DOS executable program in http://www.uv.es/~pla/models/modpp.exe
. This is an ANSI C program with only text output, but, of course, you
can adapt the source code to your compiler in order to get a graphical
output, like we have done.
Conclusions and open questions
Through our computer simulation, we have tested and ratified our previous conclusion (PlaLopez 1986) about the increasing of the Mean Gain Rate through technological progress. Moreover, its oscillations can explain some economic crisis.
Nevertheless, our Model explores a theoretical situation, without
monopolies
nor ecological bounds of the economic growth. These questions remain
open
to later studies, applications and developments of our Model.
Bibliography
Forrester, J:W. (1968), "Principles of systems", WrightAllen Press, Cambridge.
Lipsey, R.G. (1967), "Introducción a la economía positiva", VicensVives, Barcelona.
Marx, K. (1894), "El Capital", vol.III, Fondo de Cultura Económica, Mexico, 1959.
PlaLopez, R. (1986), "Study by System Dynamics of the Problem of the Equalization of the Gain Rate" in "System Dynamics: on the move", proceedings of the 1986 International Conference of the System Dynamics Society, Sevilla.
Salama, P. and Valier, J. (1973), "Une introduction a l'économie politique", Maspero, Paris.
Sraffa, P. (1960), "Production of Commodities by means of Commodities", Cambridge University Press, Cambridge.
Sweezy, P.M. (1972), "Teoría del desarrollo capitalista",
Fondo
de Cultura Económica, Mexico.