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The article reviews the idea of calculation in kind. It is argued that Kantorovich and subsequent mathematicians essentially validated the idea of in-kind calculation. This has not been evident because Kantorovich nowhere deals with the Austrian school and they for their part have ignored him. The article continues by examining improvements in linear optimisation since Kantorovich and the implications these have for economic planning. Finally it discusses the problem of deriving the plan ray in the context of markets for consumer goods.
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Von Mises, Kantorovich and in-natura calculation
W. Paul Cockshott*
e article reviews the idea of calculation in kind. It is argued that Kantoro-
vich and subsequent mathematicians essentially validated the idea of in-kind
calculation.  is has not been evident because Kantorovich nowhere deals with
the Austrian school and they for their part have ignored him.  e article con-
tinues by examining improvements in linear optimisation since Kantorovich
and the implications these have for economic planning. Finally it discusses the
problem of deriving the plan ray in the context of markets for consumer goods.
JEL classifi cations: C61, B14
Keywords: planning, calculation debate, linear optimisation
1. Introduction
is paper presents a historical review and extended tutorial on the work of Kantoro vich
and his position with respect to the famous economic calculation debate. It focuses on
Kantorovich because he is the most signifi cant Soviet contributor to the question, and be-
cause his ideas are less well known to modern Western economists than those of the Aus-
trian school. A Western readership is more likely to be familiar with neo-Classical or Sraf-
an approaches to economic calculation, so it is perhaps worth saying a little about how
Kantorovich’s approach will be seen to diff er from these. We shall argue that in one sense
Kantorovich’s methods are a generalisation of those of Ricardo, but one aspect of Ricardo’s
work that Kantorovich shares, Ricardo’s analysis of foreign trade, is not one that the mod-
ern neo-Ricardian school lays much emphasis on. In another aspect of his work though,
* University of Glasgow.
Correspondence Address:
W. Paul Cockshott, Dept. Computing Science,  Lilybank Gardens, Univerity of Glasgow,
Glasgow, G QQ.
Received  July , accepted  March 
© INTERVENTION 7 (1), 2010, 167 – 199
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the use of linear production functions, Kantorovich has a certain amount in common with
Sraff a, but with an important diff erence. Kantorovich assumes that there are multiple pos-
sible linear techniques, the optimal intensities of which have to be determined.  e exist-
ence of a multiplicity of techniques, a combination of which will be used, means that the
production frontier for Kantorovich is piecewise linear.  is contrasts with the continu-
ously curved production frontier assumed by the Cobb Douglas function typically used in
neo-Classical work. At a micro level, at the level of detailed production planning, we know
that what happens has to be strictly linear: the output of cars will be constrained by engine
production in a linear rather than an exponential way.  is implies that the stylised curved
production frontier of neo-Classical theory is probably best seen as a conceptual approxi-
mation to a piecewise linear reality. We will start, however, by situating Kantorovich in re-
lation to the Austrian school.
2. What is economic calculation?
In contemporary society the answer seems simple enough: economic calculation involves
adding up costs in terms of money. By comparing money costs with money benefi ts one
may arrive at a rational – wealth maximising – course of action.
In a famous paper (von Mises ) the Austrian economist von Mises argued that it
was only in a market economy in which money and money prices existed, that this sort of
economic rationality was possible.
His claims were striking, and, if they could be sustained, apparently devastating to
the cause of socialism.  e dominant Marxian conception of socialism involved the abo-
lition of private property in the means of production and the abolition of money, but von
Mises argued that
»every step that takes us away from private ownership of the means of production
and the use of money also takes us away from rational economics« (von Mises :
).
e planned economy of Marx and Engels would inevitably fi nd itself »groping in the
dark«, producing »the absurd output of a senseless apparatus« (von Mises : ). Marx-
ists had counterposed rational planning to the alleged ›anarchy‹ of the market, but accord-
ing to von Mises such claims were wholly baseless; rather, the abolition of market relations
would destroy the only adequate basis for economic calculation, namely market prices.
However well-meaning the socialist planners might be, they would simply lack any basis
for taking sensible economic decisions: socialism was nothing other than the »abolition of
rational economy«.
As regards the nature of economic rationality, it is clear that von Mises has in mind the
problem of producing the maximum possible useful eff ect (satisfaction of wants) on the basis
of a given set of economic resources. Alternatively, the problem may be stated in terms of its
dual: how to choose the most effi cient method of production in order to minimize the cost
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
of producing a given useful eff ect. Von Mises repeatedly returns to the latter formulation in
his critique of socialism, with the examples of building a railway or building a house: How
can the socialist planners calculate the least-cost method of achieving these objects?
As regards the means for rational decision-making, von Mises identifi es three possi-
ble candidates:
. Planning in kind (in natura). is he rejects, and the validity of this rejection will be
the main subject of this article.
. Planning with the aid of an »objectively recognizable unit of value« independent of
market prices and money, such as labour time.  is too he rejects.
. Economic calculation based on market prices.
It is clear that monetary calculation lends itself well to problems of the minimising or max-
imising sort. We can use money to fi nd out which of several alternatives is cheaper, or which
sale will yield us the most profi t. But if we look in more detail at what is involved here, we
shall see that a lot of calculation has to be done prior to the use of money. If an architect
is planning a house, she must do a large amount of calculation in physical terms: estimat-
ing how much timber of each diff erent type, how many bricks, how many tiles, window
frames etc. will be required. Only once all the physical calculation has been done, once the
bill of materials and the work schedules have been determined, then a costing can be done
and presented to the client.
e architect would, in a capitalist economy, have prices of materials in mind when she
chose them, but even in a capitalist economy price can not be the only factor. Actual avail-
ability of the supplies, lead time on delivery etc. are just as important. For an architect in a
pre-capitalist economy, the designer of the Great Pyramid for example, these physical con-
straints would have been all that she had to go on.  e architect in an earlier society would
have done her calculations directly in terms of the available labour and natural resources, so
such in-natura calculation has obviously been possible in the past.  e question is whether
it is still possible in modern societies with an extensive division of labour.
3. Planning in kind
e organisational task that faced a pyramid architect was vast.  at it was possible without
money was an indication that monetary calculation was not a sine qua non of calculation. But
as the project being planned becomes more complex, then planning it in material units will
become more complex. Von Mises is in eff ect arguing that optimization in complex systems
necessarily involves arithmetic, in the form of the explicit maximization of a scalar objective
  e railway example is in von Mises ().  e house-building example is in von Mises
().
In talking about planning »in kind«, von Mises was responding to the proposals of Neurath
().
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function (profi t under capitalism being the paradigmatic case), and that maximising the
money return on output, or minimising money cost of inputs is the only possible such sca-
lar objective function. Von Mises argued for the impossibility of planning in kind because,
he said, the human mind is limited in the degree of complexity that it can handle.
So might the employment of means other than a human mind make possible plan-
ning in kind for complex systems?
ere are two »inhuman« systems to consider:
. Bureacracies. A bureacracy is made up of individual humans, but by collaborating on
information processing tasks, they can carry out tasks that are impossible to one indi-
vidual.
. Computer networks. Nobody familiar with the power of Google to consolidate and
analyse information will need persuading that computers can handle volumes and com-
plexities of information that would stupify a single human mind, so a computer net-
work could clearly do economic calculations far beyond an individual human mind.
More generally as Turing pointed out (Turing ) any extensive calculation by human be-
ings depends on artifi cial aides-memoir, papyrus, clay tablets, slates, etc. With the existence
of such aides to memory, algorithmic calculation becomes possible, and at this point the
diff erence between what can be calculated by a human using paper and pencil methods or
a digital computer comes down only to matters of speed (Turing  and ).
ere is no question that the procedure of economic calculation considered by von
Mises was primarily algorithmic. It involves a fi xed process of:
. For each possible technique of production
a) form a physical bill of materials,
b) use a price list to convert this into a list of money expenditures,
c) then add up the list to form a fi nal cost
. Select the cheapest fi nal cost out of all the costs of techniques of production
e question then arises as to whether there exist in-natura algorithms with an analogous
function?
3.1 Kantorovich’s method
In the s and early s when von Mises fi rst advanced his arguments, no such algo-
rithmic techniques were known. But in  (Kantorovich ) the Soviet mathematician
V. Kantorovich came up with a method which later came to be known as linear program-
ming or linear optimisation, for which he was later awarded both Stalin and Nobel prizes.
Describing his discovery he wrote:
»I discovered that a whole range of problems of the most diverse character relating to
the scientifi c organization of production (questions of the optimum distribution of
the work of machines and mechanisms, the minimization of scrap, the best utiliza-
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
tion of raw materials and local materials, fuel, transportation, and so on) lead to the
formulation of a single group of mathematical problems (extremal problems).  ese
problems are not directly comparable to problems considered in mathematical anal-
ysis. It is more correct to say that they are formally similar, and even turn out to be
formally very simple, but the process of solving them with which one is faced [i.e., by
mathematical analysis] is practically completely unusable, since it requires the solu-
tion of tens of thousands or even millions of systems of equations for completion.
I have succeeded in fi nding a comparatively simple general method of solving this
group of problems which is applicable to all the problems I have mentioned, and is
suffi ciently simple and eff ective for their solution to be made completely achievable
under practical conditions.« (Kantorovich : )
What was signifi cant about Kantorovich’s work was that he showed that it was possible, start-
ing out from a description in purely physical terms of the various production techniques
available, to use a determinate mathematical procedure to determine which combination of
techniques will best meet plan targets. He indirectly challenged von Mises, both by prov-
ing that in-natura calculation is possible, and by showing that there can be a non monetary
scalar objective function: the degree to which plan targets are met.
e practical problems with which he was concerned came up whilst working in the
plywood industry. He wanted to determine the most eff ective way of utilising a set of ma-
chines to maximise output. Suppose we are making a fi nal product that requires two com-
ponents, an A and a B. Altogether these must be supplied in equal numbers. We also have
three types of machines whose productivities are shown in the Table .
Table 1: Kantorovich’s fi rst example
Type of machine # of machines Output per machine Total output
As Bs As Bs
Milling machines 3 10 20 30 60
Turre t lath es 3 20 3 0 60 90
Automatic turret lathes 1 30 80 30 80
Max total 120 230
Suppose we set each machine to produce equal numbers of As and Bs.  e three milling
machines can produce  As per hour or  Bs per hour. If the  machine produce As for
 minutes in the hour and Bs for  minutes then they can produce  of each. Applying
similar divisions of time we can produce  As and Bs on the Turret lathes and  As and Bs
on the automatic turret lathe (see Table ).
  ere is no indication that he was aware of von Mises at the time.
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Table 2: Kantorovich’s examples of output assignments
Type of machine Simple solution Best solution
As Bs As Bs
Milling
machines
20 20 26 6
Turret lathes 36 36 60 0
Automatic turret
lathes
21 21 0 80
Max total 77 77 86 86
But Kantorovich goes on to show that this assignment of machines is not the best. If we as-
sign the automatic lathe to producing only Bs, the turret lathe to producing only As and split
the time of the milling machines so that they spend  minutes per hour producing Bs and
the rest producing As, the total output per hour rises from  As and Bs to  As and Bs.
e key concept here is that each machine should be preferentially assigned to pro-
ducing the part for which it is relatively most effi cient. e relative effi ciency of producing
As/Bs of the three machines was milling machine = 1/2, turret lathes = 2/3, and automatic
lathe = 3/8. Clearly the turret lathe is relatively most effi cient at producing As, the auto-
matic lath relatively most effi cient at producing Bs and the milling machine stands in be-
tween.  us the automatic lathe is set to produce only Bs, the turret lathes to make only As
and the time of the milling machines is split so as to ensure that an equal number of each
product is turned out.
e decision process is shown diagrammatically in Figure .  e key to the construc-
tion of the diagram, and to the decision algorithm is to rank the machines in order of their
relative productivities. If one does this, one obtains a convex polygon whose line segments
represent the diff erent machines.  e slopes of the line segments are the relative produc-
tivities of the machines. One starts out on the left with the machine that is relatively best
at producing Bs, then moves through the machines in descending order of relative produc-
tivity. Because relative productivity is monotonically decreasing one is guaranteed that the
boundary will be convex. One then computes the intersection of the  degree line repre-
senting equal output of As and Bs with the boundary of this polygon.  is intersection point
is the optimal way of meeting the plan.  e term linear programming stems from the fact
that the production functions are represented by straight lines in the case of two products,
planes for three products, and for the general higher dimensional case by linear functions.
at is to say, functions in which variables only appear raised to the power one.
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Figure 1: Kantorovich’s example as a diagram
auto turret lathe
turret lathe
milling machine
tops
bottoms
Plan Ray
Note:  e plan ray is the locus of all points where the output of As equals the output of Bs.  e
production possibility frontier is made of straight line segments whose slopes represent the relative
productivities of the various machines for the two products. As a whole these make a polygon.  e
plan objective is best met where the plan ray intersects the boundary of this polygon.
e slope of the boundary where the plan ray intersects was called by Kantorovich the re-
solving ratio. Any machine whose slope is less than this should be assigned to produce Bs
any machine whose slope is greater, should be assigned to produce As.
When there are only two products being considered, the method is easy and lends it-
self to diagramatic representation. But it can handle problems of higher dimensions, in-
volving three or more products. In these cases we can not use graphical solutions, but Kan-
torovich provided an algorithmic by which the resolving ratios for diff erent pairs of outputs
could be arrived at by successive approximations. Kantorovich’s work was unknown out-
side of the USSR until the late s and prior to that Dantzig had independently devel-
oped a similar algorithm for solving linear programming problems, the so called simplex
method (Dantzig/Wolfe ).  is has subsequently been incorporated into freely avail-
able software tools. ese packages allow you to enter the problem as a set of linear equa-
tions or linear inequalities which they then solve.  e constraints of the problem have to
be expressed as a series of equations and the software package can then be treated as ›black
box‹ to solve these equations. For Kantorovich’s example problem we can express the pro-
duction constraints as shown in Table .
For example lp_solve and GLPK.
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Table 3:  e constraints of Kantorovich’s original problem expressed as equations
3
3
1
m
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎣⎦
Number of machines constraint
11
10 20
11
20 30
11
30 80
⎡⎤ ⎡⎤
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
=−
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
ab
mx x
Productivity
xa(i) number of As made on ith machine
xb(i) number of Bs made on ith machine
=
a
xA Total A production equals production on
each machine
=
b
xB Total B production equals production on
each machine
Note: For the vectors above, index  means milling machines,  means turret lathes, and  automatic
turret lathes.
In the West, linear programming was used to optimise the use of production facilities op-
erating within a capitalist market.  is meant that the objective function that was maxim-
ised was not a fi xed mix of outputs, in Kantorovich’s fi rst example equal numbers of parts
A and B, but the money that would be obtained from selling the output: price A × number
of As + price B × number of Bs as expressed in Algorithm . Manuals and textbooks pro-
duced in association with Western linear programming software assumes this sort of ob-
jective.  us one can get formulations which say that the task of linear programming is to
maximise the objective function f (x):
f (x) = c · x ,
where x is a vector of inputs or outputs, c is a unit cost or price vector. Maximisation is sub-
ject to the constraints
Ax b and x 0 ,
where A is a technology matrix and b a vector of available stocks. It is obvious that this for-
mulation of linear programming is not in-natura calculation since it relies on the price vec-
tor c, readily available in a market economy, but which can not be assumed to exist in a
planned economy. But the general formulation of linear programming that Kantorovich
gives for the economy as a whole is an extension of the one he gave for his initial machine
tool example. It again involves fi nding the intersection between the production possibility
frontier given by the linear constraints, and a multi-dimensional plan ray.  e diff erence
between the two approaches is highlighted by Figure .
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
Figure 2: Comparison of the Western and Soviet versions of linear programming
price lines
A
B
P
Q
Plan Ray
constant relative
Note: In the Western formulation the problem is to fi nd P the maximal intersection of the produc-
tion possibility frontier with lines of constant relative price for the outputs (A and B). In the Soviet
formulation the problem is to fi nd Q the intersection of the plan ray with the production possibility
frontier.
In the Soviet formulation of the linear programming problem, there is no initial assump-
tion of the existence of a set of relative prices.  ere is also a diff erence in where the solu-
tion point will occur. In the Western version of the problem the solution occurs at a ver-
tex of the production possibility frontier, whereas in the Soviet formulation it occurs at a
face of the frontier.
Let us fi rst look at how one could apply modern software to solve a Western style lin-
ear programming problem with the same production constraints as those given by Kan-
torovich. If the factory he was dealing with faced prices such that an A sold for  roubles
and a B sold for  roubles, we could express the problem faced by the factory manager
as: Maximise A + 0.6B subject to the production constraints in Table . In Algorithm  we
show how this objective function and constraints can be expressed in the notation required
by lp_solve.
To get Kantorovich’s type of optimisation we replace maximising A + 0.6B with just
maximising A and add the constraint that
AB = 0 .
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Algorithm 1: Western factory facing Kantorovich’s problem would formulate it as follows
/* Objective function */
max:A+0.6 B ;
/* Variable bounds */
m1<=3;m2<=3;m3<=1;
m1-0.1 x1a - 0.05 x1b=0;
m2-0.05 x2a - 0.033333 x2b=0;
m3- 0.033333 x3a - 0.0125 x3b=0;
x1a+x2a+x3a - A=0;
x1b+x2b+x3b -B =0;
int A;
Algorithm  shows how to express Kantorovich’s problem in lp_solve notation. When this
algorithm is run it exactly reproduces the solution originally given in Kantorovich ().
Algorithm 2: Kantorovich’s example as equations input to lp_solve
/* Objective function */
max:A;
/* plan ray constraint */
A-B=0;
/* Variable bounds speci ed as in algorithm 1*/
3.2 Kantorovich and Ricardo
ere is a strong parallel between the arguments that Kantorovich uses and those that Ricar-
do used in his Principles to explain the benefi ts of international trade. He constructed an
argument to the eff ect that if it took Portugal  men to produce one unit of wine and it
took England  men to do the same (Ricardo : ). On the other hand in Portugal it
took  men to produce one unit of cloth but in England . Under these circumstances
he said it was advantageous for England to export cloth to Portugal and import wine.  e
argument was that labour in each country should be used to produce what it is relatively
best at.  is can now be seen as a specifi c case of linear optimisation.
We can set up a plan ray requiring the production of equal quantities of wine and cloth
for similarity with Kantorovich’s example. We will also assume that both England and Por-
tugal have  million workers.
We can then express Ricardo’s example as a linear program:
max: wine;
/* Constraints */
wine -pwine -ewine = 0;
cloth -pcloth -ecloth = 0;
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
wine -cloth = 0;
-90 pcloth +pcl = 0;
-80 pwine +pwl = 0;
pcl +pwl <= 1000000;
-100 ecloth +ecl = 0;
-120 ewine +ewl = 0;
ecl +ewl <= 1000000;
Where the variables are:
wine: total wine production, cloth similarly
pwine: Portuguese wine production, ewine, pcloth, ecloth similarly
ecl: English cloth producing labour, pcl, ewl, pwl similarly
Solving for the equations with lp-solve (Table ) we fi nd a net production of both wine and
cloth of , units. If we now prevent trade by forcing each country separately to equate
its wine and cloth production (e.g. pcloth=pwine), we fi nd that total production of each
falls to , units, demonstrating that Ricardo was right: overall production turns out to
be seven percent higher with trade between the two countries. But whilst we can say that
Ricardo recognised a specifi c instance of linear optimisation, it was not until Kantorovich
that a general mechanism for formulating economic problems in this way was arrived at.
Table 4: Solving Ricardo’s problem with lp_solve
Variables trade allowed trade not allowed
obje cti ve -11176.47 -10 427.8 0
wine 11176.47 10 427.8 0
pwine 11176.47 5 882 .35
ewine 0.00 4545.45
cloth 11176.47 10 427.8 0
pcloth 1176.47 5882.35
ecloth 10000.00 4545.45
pcl 105882.35 529411.76
pwl 894117.64 470588.23
ecl 1000000.00 454545.45
ewl 0.00 54545 4.54
3.3 Generalising Kantorovich’s approach
In his fi rst example Kantorovich deals with a very simple problem, producing two goods in
equal proportions using a small set of machines. He was aware, even in  that the poten-
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tial applications of mathematical planning were much wider. We will look at two issues that
he considered which are important for the more general application of the method.
. Producing outputs in a defi nite ratio rather than in strictly equal quantities.
. Taking into account consumption of raw materials and other inputs.
Suppose that instead of wanting to produce one unit of A for every unit of B, as might be
the case if we were matching car engines to car bodies, we want to produce four units of
A for every unit of B, as would be the case if we were matching wheels to car engines. Can
Kantorovich’s method deal with this as well? Consider Figure  again. In that the plan ray
is shown at an angle of  degrees a slope of one to one. If we drew the plan ray at a slope
of four to one, the intersection with the production frontier would provide the solution.
Since this geometric approach only works for two products, let us consider the algebraic
implications.
You should now be convinced that it is possible to solve Kantorovich’s original prob-
lem by algebraic means. In Algorithm  we specifi ed that AB = 0 or in other words A = B,
if one wanted four units of A for every B we would have to specify A = 4B or, expressing
it in the standard form used in linear optimisation, A – 4B = 0. Suppose A stands for en-
gines, B stands for wheels. If we now say wheels come in packs of , then we can repose the
problem in terms of producing equal numbers of packs of wheels and engines. Introduce
a new variable β = 4B to stand for packs of wheels, and rewrite the equations in terms of β
and we can return to an equation specifying the output mix in the form Aβ = 0, which
we know to be soluble.
In order to use standard linear programming packages to solve a »Soviet type« prob-
lem with a plan ray and n products we introduce n – 1 additional constraints of the form
AkbB = 0, AkcC = 0, ...AknN and maximise on A.  e constants kb, kc, ...kn specify
the ratios in which the goods are to be produced in the plan.
How do we deal with consumption of raw materials or intermediate products?
In our previous example we had variables like x1b which stood for the output of prod-
uct B on machine .  is was always a positive quantity. Suppose that there is a third good
to be considered – electricity, and that each machine consumes electricity at a diff erent rate
depending on what it is turning out. Call electricity C and introduce new variables x1ac,
x1bc etc. referring to how much electricity is consumed by machine  producing outputs A
and B.  en add equations specifying how much electricity is consumed by each machine
doing each task, and the model will specify the total amount of electricity consumed.
We now know how to...
. ...use Kantorovich’s approach to specify that outputs must be produced in a defi nite
ratio.
. ...use it to take into account consumption of raw materials and other inputs.
Actually this was his »problem A«
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
If we can do these two tasks, we can in principle perform in-natura calculations for an en-
tire planned economy. Given a fi nal output bundle of consumer and investment goods to
maximise (the plan ray) and given our current resources, a system of linear equations and
inequalities can be solved to yield the structure of the plan. From simple beginnings, op-
timising the output of plywood on diff erent machines, Kantorovich had come up with a
mathematical approach which could be extended to the problem of optimising the opera-
tion of the economy as a whole.
3.4 A second example
Let us consider a more complicated example, where we have to draw up a plan for a sim-
ple economy. We imagine an economy that produces three outputs: energy, food, and ma-
chines.  e production uses labour, wind and river power, and two types of land: fertile
valley land, and poorer highlands. If we build dams to tap hydro power, some fertile land is
ooded. Wind power on the other hand, can be produced on hilly land without compro-
mising its use for agriculture. We want to draw up a plan that will make the most rational
use of our scarce resources of people, rivers and land.
In order to plan rationally, we must know what the composition of the fi nal output
is to be – Kantorovich’s ray. For simplicity we will assume that fi nal consumption is to be
made up of food and energy, and that we want to consume these in the ratio three units of
food per unit of energy. We also need to provide equations relating to the productivities
of our various technologies and the total resources available to us. Valleys are more fertile.
When we grow food in a valley, each valley requires , workers and , machines
and , units of energy to produce , units of food. If we grow food on high land,
then each area of high land produces only , units of food using , workers, 
machines and , units of energy. Electricity can be produced in two ways. A dam pro-
duces , units of energy, using one valley and  workers and  machines. A wind-
mill produces  units of electricity, using four workers and six machines, but the land on
which it is sited can still be used for farming. We will assume that machine production uses
 units of electricity and ten workers per machine produced. Finally we are constrained
by the total workforce, which we shall assume to be , people.
Tables  and  show how to express the constraints on the economy and the plan in
equational form. If we feed these into lp_solve we obtain the plan shown in Table .  e
equation solver shows that the plan targets can best be met by building no dams, generating
all electricity using  windmills, and devoting the river valleys to agriculture.
It also shows how labour should be best allocated between activities: , people
should be employed in agriculture in the valleys,  people should work as farmers in the
highlands, , people should work on energy production, and , people should work
building machines.
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Table 5: Variables in the example economy
etotal energy output
echousehold energy consumption
ffood
vvalleys
wwindmills
mmachines
ddams
uundammed valleys
hhighland
fhfood produced on high land
fvfood produced in valleys
Table 6: Resource constraints and productivities in our example economy
nal output mix f = 3ec
number of valleys v = 4
dams use valleys vu = d
valley food output fv = 50,000u
valley farm labour lv = 10,000u
valley energy use ev = 20,000u
valley farm machines mv = 1,000u
highland food output fh = 20,000h
highland farm labour lh = 10,000h
highland energy use eh = 10,000h
highland farm machines mh = 800h
energy production e = 500w + 60,000d
energy workers le = 100d + 4w
machines in energy prod me = 80d + 6w
workers making machines lm = 10m
energy used to make machines em = 20m
energy consumption em + ev + eh + ec e
machine use me + mh + mv m
total food prod f = fh + fv
workforce lm + le + lv + lh 104,000
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
Table 7: Economic plan for the example economy using lp_solve
d (dams) 0
e270,500
f200,218
h0.0108889
m6172.71
u4
v4
w (windmills) 541
ec66739.3
eh108.889
em123,454
ev80,000
fh217.778
fv200,000
le2,164
lh108.889
lm61727.1
lv40,000
me2,164
mh8.71111
mv4,000
e results that we have obtained were by no means obvious at the outset. It was not in-
itially clear that it would be better to use all the river valleys for agriculture rather than
building dams on some of them. In fact, whether dams or windmills are preferred turns
out to depend on the whole system, not just on their individual rates of producing electric-
ity. We can illustrate this by considering what happens if we cut the labour supply in half
to , people?
If we feed this constraint into the system of equations we fi nd the optimal use of re-
sources has changed.  e plan now involves one dam and  windmills. Cut the working
population slightly further, down to , people and the optimal plan involves fl ooding
two valleys with dams and building only  windmills. Why?
As the population is reduced, there are no longer enough people available to both farm
the valleys and produce agricultural machinery. Under these circumstances the higher fer-
tility of lowland valleys is of no importance, it is better to use one or more of them to gen-
erate electricity. By applying Kantorovich’s approach it becomes possible for a socialist plan
to do two things that von Mises had believed impossible:
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. It allows the plan to take into account natural resource constraints – in this case the
shortage of land in river valleys which can be put to alternative uses.
. It allows rational choices to be made between diff erent technologies – in this case be-
tween windmills and hydro power and between lowland and highland agriculture.
Contrary to what von Mises claimed, the whole calculation can be done in physical units
without any recourse to money or to prices.
4. Valuation
e core of von Mises’s argument relates to the use of prices to arrive at a rational use of in-
termediate or capital goods. Von Mises argues that, in practice, only money prices will do
for this, but concedes that, in principle, other systems of valuation, such as labour values
would also be applicable. Kantorovich too, was very concerned with the problem of rela-
tive valuation (Kantorovich ), and developed what he called objectively determined val-
uations (ODV).
He considered a situation where planners have to deal with several diff erent types of
factories (A..E) each able to produce products one and two, and where the intended ratio
of output of product one and two are fi xed in the plan. Each class of factory A..E has dif-
ferent relative productivities for the two products.
He next looked at the apparent profi tability of producing products one and two un-
der diff erent relative valuations. Under some schemes of relative price, all factories would
nd product one to be unprofi table relative to product two, under other the reverse would
occur. Intermediate price schemes would allow both products to be produced, with some
classes of factories specializing on one and others on two. He gives the example of children’s
clothing as something which, under the administratively determined valuations then used
in the USSR, were unprofi table to produce, and unless factories were specifi cally instruct-
ed to ignore local profi tability, too few children’s clothes would be made.
He asks if there exists a relative valuation structure which would allow factories to con-
centrate on the most valuable output, and at the same time meet the specifi ed plan targets
and arrives at certain conclusions:
.  at among the very large number of possible plans there is always an optimal one
which maximises output of plan goals with current resources.
.  at in the optimal plan there exists a set of objectively determined valuations (ODV)
of goods which will ensure that each factory
a) produces the output which will contribute most to maximising the plan goals;
b) each factory also fi nds that the output which contributes to maximising plan targets
is also the output which is most profi table.
. With arbitrary valuations which diff er from ODV, these conditions can not be met,
and profi t maximising factories will not specialise in a way that meets plan goals op-
timally.
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
It is important to understand that his ODVs are valuations that apply only for a plan
which optimally meets a specifi c plan target. Kantorovichs procedure for arriving at an opti-
mal plan involved successive adjustments to the ODVs and factory specialisation until both
the appropriate mix of goods is reached, and at the same time each factory is producing its
most profi table good. He actually gave several diff erent mathematical procedures for arriv-
ing at such a plan and system of ODVs.  e ODVs basically specify the derivatives of the
production possibility frontier in the region of its intersection with the plan ray.
Table 8: Example optimal plan with technology matrix, plan ray and net output
outputs inputs gross outputs
labour corn machines coal
corn 1
3
1
10
1
20 1416.76
machines 23
20 93.43
coal 1
10
1
100 858.49
bread 1
5
1
10 1275.08
plan ray
coal bread
12
net output
coal bread
1275.08 637.54
labour force
1000
Let us use another example to explore the idea of ODVs.
Although Kantorovich asserts that labour is ultimately the only source of value, his
ODVs are short term valuations and diff er from the classical labour theory of value, which
gave valuations in terms of the long term labour reproduction costs of goods – including
the reproduction costs of capital goods. Kantorovich, in contrast, is concerned with valu-
ations which should apply with the current stock of means of production and labour re-
sources. For example, he considers the situation of giving a valuation to electric power rel-
ative to labour. Instead of valuing it in terms of the labour required to produce electricity,
he fi rst assumes that the total electrical power available is fi xed – i.e., power-stations oper-
ating at full capacity, and then works out how many person hours of labour is saved by us-
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ing an additional kilowatt hour of electricity.  is defi nition of the value of electricity in
terms of labour is clearly diff erent from the way labour value was defi ned by the classical
economists. In their formulation the labour value of a kilowatt hour, for example, was the
mean labour expended to produce a kilowatt. One would expect the classical labour value
to be lower than Kantorovich’s labour ODV, since otherwise the use of electricity would
not be worth while.
Nemchinov (: ) criticised Kantorovich for raising what the former saw as
just:
»indices expressing defi ciency, limitation, and scarcity of resources […] applicable to
the economic calculations involved in discovering how best to use resources to en-
sure maximum fullfi llment of a production programme.«,
to concepts of a universal character. For this he was »gravely at fault«.
Kantorovich’s insistence on considering short term, very material constraints – so
many megawatts of power, such and such a number of cutting machines, etc., gives his
work an intensely practical and pragmatic character, quite diff erent from that of most the-
oretical economists.
Why is Kantorovich so concerned with valuations and profi tability? ere seem to be
two reasons. We should fi rst note that by profi t maximising Kantorovich actually meant
maximising the value of output.  is must be understood in the context of Soviet prac-
tice where mines and factories were given incentives to over-fulfi ll plan targets. If the out-
put was a single good – say coal, the target could be specifi ed in tons. But if the factory
produced several goods, then the target had to be set in terms of x rubles worth of a mix of
goods. With the »wrong« price structure, plants would attempt to maximise the production
of the goods which were of the highest value, ignoring those of lower value, with the result
that the aggregate supply of all goods was often not in the proportions that the planners
intended.  is practice of setting plan targets in money terms refl ected the limited ability
of GOS PLAN to specify detailed targets in kind as described by Nove (). We have to
understand that he was engaging in a wider debate during the s about the appropriate
pricing structure for a socialist economy.  us Nove () identifi es three alternative re-
forms being proposed for Soviet prices: the suggestion of Strumlin that prices should cor-
respond to labour values; that of Novozilov who argued for Marxian »production prices«
(Novozilov : ); and that of Kantorovich (: ) who proposed »objectively
determined valuations«. Nove argues that Novozilov and Kantorovich were both trying to
develop rational costing models.
e second reason relates to his particular algorithm for solving linear programming
problems which used an iterative adjustment to initial ODVs (resolving multipliers) until
an optimal plan is achieved.
ese two aspects seem intimately linked in his presentation, but the presuppositions
about the incentives to factories are not brought to the fore which owes something to the
cryptic language in which economic debates in the USSR were conducted. Swann ()
relates that some of the Soviet optimal planning school, he cites Volonsky, argued that the
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
ODVs were all that had to be exchanged between distinct units of production each follow-
ing their own optimisation goal. In this his arguments for come close to those of the Aus-
trian school with respect to prices. In retrospect though we can see that the use resolving
multipliers, however much it infl uenced Kantorovich’s own thought, is incidental to the is-
sue. With computer algorithms, the process of solving a linear program becomes a »black
box«.  e user need not concern herself with details such as the method of calculation –
whether it uses Kantorovich’s approach Dantzig’s or Karmarkar’s, except insofar as this af-
fects the size of problem that can be handled, as we discuss in Section . With computer
packages, ODVs would no longer be needed for computing a plan, but would they still be
needed for specifying targets to factories?
is depends on the information processing capacity of the planning system. If it were
capable of specifying fully disaggregated plans, then it could in principle just place orders
with factories for specifi c quantities of each good. In these circumstances, the factories could
not cheat by producing more of high value items and less of low value ones. Indeed, the very
information that would be required to compute Kantorovich’s ODVs, would have been suf-
cient for GOS PLAN to specify disaggregated orders in kind for the products that would
have had valuations attached.  us were it possible to compute ODVs then they would have
been redundant for the purpose that Kantorovich originally proposed them!
ere remains another level at which valuations would have been useful – when prod-
uct designs were being drawn up at a local level. If a refrigerator designer was deciding on
what components to use in a planned new model, she would need some way of telling which
components would, from a social standpoint, have been the most economical, which im-
plies a system of valuations. However it is not clear that the full apparatus of ODVs would
be either necessary or appropriate here. ODVs correspond to a system of marginal cost,
rather than average cost pricing.  ey refl ect current marginal costs with the immediately
current constraints on production.  e use of such marginal costing was criticized by other
Soviet economists (Grossman , Menshikov ).
It is not clear, in retrospect, that ODVs would have been more appropriate than a sys-
tem of average cost valuation if one was projecting ahead a year or so. If one draws up ex-
ample plans in which all goods are reproducible from scratch, then the ODVs will just be
the same as labour values. If there are some other »free« inputs left over from the past, then
the ODVs begin to deviate from labour values. So in the short term ODVs could be useful,
but it is not clear that they are so useful in the preparation of long term plans. Indeed, giv-
en the stochastic properties of prices in a real capitalist economy (Farjoun/Machover ),
it is questionable whether, with the exception of certain constrained products like oil, the
diff erence between average and marginal costs is signifi cant in the West.
5. Complexity
Linear programming, originally pioneered by Kantorovich, provides an answer in principle
to von Mises’ claim that rational economic calculation is impossible without money. But
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this is an answer only in principle. Linear programming would only be a practical solution
to the problem if it were possible, in practice, to solve the equations required in a socialist
plan.  is in turn requires the existence of a practical algorithm for solving them, and suf-
cient computational resources to implement the algorithm. Kantorovich, in an appen-
dix to Kantorovich (), gave a practical algorithm, to be executed by paper and pencil
mathematics.  e algorithm was suffi ciently tractable for these techniques to be used to
solve practical problems of a modest scale. When tackling larger problems he advised the
use of approximative techniques like aggregating similar production processes and treating
them as a single composite process. Whilst Kantorovich’s algorithm uses his ODVs, which
he earlier called resolving multipliers, subsequent algorithms for linear programming do
not, so the ODVs should not be considered as fundamental to the fi eld.
Since the pioneering work on linear programming in the s, computing has been
transformed from something done by human »computors« to something done by electron-
ic ones.  e speed at which calculations can be done has increased many billion-fold. It is
now possible to use software packages to solve huge systems of linear equations. But are
computers powerful enough to be used to plan an entire economy?
In a large economy like the former USSR there were probably several million distinct
types of industrial products, ranging from the various sorts of screws, washers and types of
electronic components to large fi nal products like ships and airliners. Although there was
great enthusiasm for Kantorovich’s methods in the USSR during the s, the scale of the
economy was too great for his techniques to be used for detailed planning with the then
available computer technology. Instead they were used either in optimising particular pro-
duction plants, or, in drawing up aggregated sectoral plans for the economy as a whole.
How has the situation changed today, given that the power of computers has contin-
ued to grow at an exponential rate since the fall of the USSR? In other words to what com-
plexity class (Sipser , Part III) does linear programming belong?
»For a long time it was not known whether or not linear programs belonged to a
non-polynomial class called ›hard‹ (such as the one the traveling salesman problem
belongs to) or to an ›easy‹ polynomial class (like the one that the shortest path prob-
lem belongs to). In , Victor Klee and George Minty created an example that
showed that the classical simplex algorithm would require an exponential number
of steps to solve a worst-case linear program (Klee/Minty ). In , the Russian
mathematician L.G. Khachian developed a polynomial-time algorithm for solving
linear programs (Khachian ). It is an interior method using ellipsoids inscribed
in the feasible region. He proved that the computing time is guaranteed to be less
that a polynomial expression in the dimensions of the problem and the number of
digits of input data. Although polynomial, the bound he established turned out to
be too high for his algorithm to be used to solve practical problems. Karmarkar’s al-
gorithm (Karmarkar ) was an important improvement on the theoretical result
of Khachian that a linear program can be solved in polynomial time. Moreover his
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
algorithm turned out to be one which could be used to solve practical linear pro-
grams.« (Dantzig )
Modern linear programming packages tend to combine Dantzig’s simplex method with the
more recent interior point methods.  is allows the most modern implementations to solve
programming problems involving up to one billion variables (Gondzio/Grothey a and
b). For such huge problems large parallel supercomputers with over a thousand proc-
essor chips are used. But even with much more modest  CPU computers, linear program-
ming problems in the million variable class were being solved in half an hour using inte-
rior point methods.
ese advances in linear programming algorithms and in computer technology mean
that linear programming could now be applied to detailed planning at the whole economy
level, rather than just at an aggregate level.
6. Deriving the plan ray
Kantorovich assumed that the plan had a given target to optimise in the form of a particu-
lar mix of goods: the plan ray.  is refl ected the social reality for those engaged in manag-
ing Soviet industry, in that they were given a mix of products to produce by GOS PLAN.
e planning authorities themselves however, needed to specify what this ultimate output
mix would be. In the early phases of Soviet planning, when Kantorovich wrote his original
paper, the goals set by the planners were primarily directed at achieving rapid industriali-
zation and building up a defence base against the threat of invasion.  e planning process
was successful in achieving these goals. But in an already industrialised country, in times
of peace, the meeting of current social needs becomes the fi rst priority and so the plan vec-
tor has to be pointed in that direction. A criticism commonly levelled at the Soviet-type
economies – and not only by their Western detractors – is that they were unresponsive to
consumer demand. It is therefore important to our general argument to demonstrate that
a planned economy can be responsive to the changing pattern of consumer preferences –
that the shortages, queues and surpluses of unwanted goods of which we hear so much are
not an inherent feature of socialist planning.  e economists Dickinson and Lange, writ-
ing just prior to Kantorovich, outlined a practical mechanism by which this could be done
(Lange , Dickinson ).
ey proposed that the state wholesale sector should operate on a break-even basis with
exible prices. Wholesale managers would set market clearing prices for the products on sale
as consumer goods.  ese wholesale prices would then act as a guide to the plan authori-
ties, telling them whether to increase or decrease production of particular lines of product.
If prices were high, then that line of product would have its output increased, otherwise its
planned output would be reduced.
See Bienstock (: Chap ).  e Harmony Algorithm for constructing plans, given in Cot-
trell/Cockshott (), is an instance of the class of algorithm discussed by Bienstock.
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e basic idea is clear, the same principle that adjusts production of consumer goods
in a capitalist economy was to be employed. But this then raises the problem of how one de-
termines that a price is high or low. High or low relative to what? What would be the basis
of valuation used? Although Marx and Engels had laid great stress on planning as an alloca-
tion of labour time, this conception had been more or less abandoned by English speaking
socialist economists by the late s. Neither Lange nor Dickinson relied on the classical
theory of value in their arguments. Writing in , Appel et al. () had laid great stress
on the relevance of the labour theory of value for socialist economics, but their ideas were
largely ignored. More recent writers have again laid emphasis on Marx’s theory of value as
a guide to socialist planning (Dieterich , Peters  and Peters ). It has been ar-
gued that labour values are eff ectively calculable and that in combination with Dickinson’s
proposals for socialist markets they provide a pragmatic way of obtaining a plan ray that
conforms to consumer demand (Cottrell/Cockshott ).
Whilst the task of determining the plan ray itself can be solved by Dickinson’s meth-
od, determining its intercept with the production frontier remains problematic unless all
the technical coeffi cients of production are available to the planning system. In this context
the Austrian school has tended to emphasise the importance of tacit or private knowledge.
ey have argued that a key role of the price system is its ability to make public, data that
was previously held privately by fi rms.  e gist of the argument is that although fi rms may
wish to keep private the technical details of production, they are forced by the market to
make public such portions of the data as relate to their interaction with other fi rms through
the prices that they bid for inputs. But this view of the price system as the principle chan-
nel of inter fi rm information is demonstrably wrong. Prices are only a small part of the in-
formation that is exchanged between fi rms. Details about quantities, specifi cation of com-
ponents, delivery times etc., all have to be exchanged between supplier and consumer. In
quantity, this other information far outweighs the information content of the price that is
nally agreed upon. If you were to gather together the information of this sort that a fi rm
communicates to its suppliers and to its customers, you would be able to reconstruct a pret-
ty accurate linear model of the production processes it was undertaking. In a modern econ-
omy this information is already largely computerised. Orders are entered into purchasing
systems that record the purchases in a database and typically transmit the details electroni-
cally to suppliers. It requires no great feat of imagination to envisage a planned economy in
which a set of standardised order control packages are generally used.  ese packages, in-
stead of sending the orders directly to supplier, could route them via central servers which
record the information in databases used by the planning computers. In the process, checks
could be made to see if the anticipated use of critical inputs was likely to exceed availabil-
ity. A generalised solution to the linear programming problem on a national scale can then
be invoked to adjust up or down the intensities of outputs of diff erent processes in order to
converge on the Kantorovich ray.
  e metric used for measuring information in this context could either be that Shannon ()
or Li/Vitanyi ().
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
7. Conclusion
e Soviet mathematical school founded by Kantorovich and the Austrian school exempli-
ed by von Mises took radically diff erent positions on the feasibility of socialist economic
calculation. To a large extent they ignored one another.  e Austrian school largely concen-
trated on criticising Western trained socialist economists like Lange and the Soviet school
appears to have ignored von Mises completely. Even when the key participants met, the is-
sue was not raised. Menshikov writes:
»It is interesting that in the account of his trip to Sweden for receiving the Nobel
Prize, Kantorovich mentions an informal reception with the participation of several
American economists – Nobel Prize laureates – including Hayek, Leontief, and Sam-
uelson. But, apparently, neither at this reception, nor during other meetings, this is-
sue was never raised. In January , when I worked in USA as the Director of the
United Nation Projections and Perspective Studies Branch, I was asked to present
L.V. Kantorovich as a new Nobel Prize laureate at the annual meeting of the Amer-
ican Economic Association in Atlantic City. Of course, I put the emphasis on the
economic discovery of the laureate. In the discussion, none of the audience, which
included T. Koopmans and L. Klein, a future Nobel Prize laureate, mentioned the
question of actual Kantorovich’s answer to a part of Hayek’s argumentation.« (Men-
shikov : )
With the political demise of the USSR, the Austrian school have tended to assume that von
Mises arguments have been vindicated, but theoretical economic arguments are not fi nally
resolved by politics. No, one has to bring economic arguments head to head in their own
terms. Kantorovich, an absent participant in the Western debate on socialist calculation, is
still worth paying attention to.
A. Appendix: Kantorovich’s Algorithm
We refer in the main text of the article to Kantorovich’s method of resolving multipliers. In
Kantorovich () he gives what is essentially a paper and pencil algorithm for his prob-
lems.  e algorithm described there requires a certain residual of human intelligence to im-
plement. In what follows I give a representation of his algorithm in a form suffi ciently un-
ambiguous as to allow computer implementation.
What follows is a program written in Vector Pascal (Cockshott ) a dialect of Pascal
(Jensen/Wirth ) extended with elements of Iversons notation (Iverson ). Pascal is a
strongly typed language which helps guard against programming errors. Iverson developed
his notation whilst he was a PhD student of Leontief and was looking for a notation suit-
able for unambiguous expression of algorithms, initially algorithms needed for computer-
ised prepartion of Input/Output tables.  e program has been processed by a literate pro-
gramming tool similar to that described in Knuth (a) and typeset using TEX (Knuth
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b).  e text in roman font that follow are comments describing the algorithm.  e pro-
gram code is generally in san-serif font.  e text in roman font that follows are comments.
e program code is generally in san-serif font, and the whole, is in the literate program-
ming output format generated by the compiler.
program excavate;
e objective of the program is to solve Kantorovich’s soil excavation problem by his meth-
od of resolving multipliers. It starts out from the data provided in Table . In the table the
norms for the excavator types are shown in italics. In Soviet parlance, a norm appears to have
meant the expected output per hour of the A-machine applied to a particular type of work.
us Excavator A can dig out  m3/hr of soil type I,  m3/hr of soil of type II etc.
e objective is given in the last column: , m3 of each type of soil.
Table 9: Simplifi ed version of Kantorovich (1939: Table 5)
Kinds of soil Machiner y for the work
Excavator A Excavator B Excavator C
I 105 190 107 0 64 0 20,000
II 56 92 66 222 38 0 20,000
III 56 0 83 60 53 282 20,000
Total Hours 282 282 282
Following the columns of the norms Kantorovich gives the optimal allocation of machine
times to activities to minimise overall time taken to do the digging.  e program will repro-
duce this result by applying his method of resolving multipliers or objectively determined
valuations. We rst introduce our domain of discourse: the types of soil, the types of ma-
chine and the units of measurement we are using.
type
soil = (I, II, III);
excavator = (A, B, C);
units =(hr, meter);
Now we introduce the dimensions in which volume, time and norms are specifi ed. For in-
stance norms are real numbers denoting cubic meter per hour.  e word pow in what fol-
lows means raised to the power, and * is the multiplication operator.
type
volume = real of meter pow 3;
duration = real of hr;
norm = real of meter pow 3 * hr pow –1;
const
hour: duration = 1.0;
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
cubicmeter: volume = 1.0;
epsilon = 0.001;
e production norms for the machines working on each kind of soil and targets for soil to
be moved are copied from Kantorovichs Table  and stored in an appropriate matrix called
norms, and a scalar called targets. In a more general algorithm this could be a vector, but
since all targets are the same I use a scalar.
const
norms : array [soil, excavator] of norm =
((105, 107, 64),
(56, 66,38),
(56, 83, 53));
identity: array [soil, soil] of real =
((1, 0, 0),
(0, 1, 0),
(0, 0, 1));
target: volume = 20000 ;
We now introduce the variables of the problem: a matrix x which will encode the time each
machine spends on each type of soil; L, a vector of objectively determined valuations of dif-
ferent soils.  e standardised output of each machine for each soil type is obtained by ap-
plying resolving multipliers to the soil types.
var
x: array [soil ,excavator ] of duration;
Let dx duration;
L: array [soil] of real;
standardisedoutput: array [excavator, soil] of norm;
outputs: array [soil] of norm;
totals: array [soil] of volume;
Let ok boolean;
Let greatestsoil, leastsoil, deltam volume;
Let best norm;
Let e, Scoop excavator;
Let s, least, m, j soil;
Let λ, f real;
Let count integer;
equated: array [soil] of real;
procedure ComputeTotalsEtc; (see Section A)
function marginalgain (s: soil; d: excavator): volume; (see Section A)
function mainSoilProducedBy (e: excavator): soil; (see Section A)
function ndScoop (var s: soil): excavator; (see Section A)
begin
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L 1;
ok false;
count 0;
f 0.3;
Iterate the following steps until we have a satisfactory answer.
while not ok do
begin
x 0 × hour;
Use the L to get a standardised performance for each machine.
standardisedoutput (norms × LT )T ;
For each machine fi nd the soil for which it has the best performance
for e A to C do
begin
nd the best performance of the machine on any soil
outputs standardisedoutpute;
best \max outputs;
set each machine to work on the soil it is best at
for s I to III do
if standardisedoutpute,s = best then xs,e hour;
end;
totals (norms × x);
greatestsoil \max totals;
leastsoil \min totals;
if leastsoil 0.0 × cubicmeter then
begin
check if any soil has a zero output and raise its value if it has
for s I to III do
if totalss < greatestsoil then Ls Ls(1.02 + (ord(s)/10))
end
else ok true;
end;
count 0;
At this point our estimate of the resolving multipliers is accurate enough to ensure that some
of each soil is now being moved, but we have not yet met the requirement that the same
amount of each soil must be moved. We now try to get a more precise estimate of the re-
solving multipliers and in the process we adjust the amounts of each soil being moved. It is
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important to note at this point that any further adjustments must come by de-specialising
some of the excavators so that they move more than one soil type.  e resolving multipliers
have until now been used to weight the outputs of diff erent soil types in order to assign each
digger to the soil it is best suited to. If a machine is no longer specialised, that is if it moves
more than one soil, then the weights must be such that it is no longer best at one particular
soil type.  e multipliers must be set so that the marginal weighted output of the excavator
on any of the soils on which it is employed are the same.  us if a machine k is employed
on two soils i, j then standardisedoutput[k, j] = standardisedoutput[k, i].
In turn this implies that for any machine that is employed to move two soils the ratio
of the resolving multipliers must be the inverse of the ratio of the norms.
e algorithm will work soil a time bringing ever more soil outputs into equality.
We defi ne the set of soils whose outputs has been brought into equality as the equated set.
For those soils in the equated set, the resolving multipliers of the soils will have been
corrected so that for any machine moving more than one soil they stand in inverse ratio to
that digger’s norms.
computeTotalsEtc;
repeat
Find which machine not in the equated set is nd best at producing this soil under current
resolving multipliers. Call this machine Scoop.
Scoop ndScoop (least);
m mainSoilProducedBy (Scoop);
Adjust the resolving multiplier ratio between Scoops soil and the least produced soil to ra-
tio of Scoops norms.
least least,scoop
m
m,scoop
norms
L;
norms
It is now necessary to reduce the output of scoop on scoops soil and increase it on the least
produced soil. It is necessary to compute how much to reduce scoops soil by.  e resolv-
ing multipliers give us substitution ratios between diff erent soil outputs. Suppose that we
want to reduce output of soil m by one unit and increase the output of soil j, the increase
in j we get is
m
j
j
L
L
Δ=
.
If we want to reduce the output of i and we have two other soils j, k which we want to in-
crease equally then we have
Δk = Δj
where Δk means change in x and
ΔmLm = ΔkLk + ΔjLj = Δk (Lj + Lk) .
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us
jk
mk
m
LL
L
ΔΔ
+
=−
.
Let Cm, Cj, Ck be the current outputs of each soil; given that Cj = Ck we have to chose the
Δs so that
Cm + Δm = Cj + Δj = Ck + Δk .
It follows that
jk
mk kk
m
LL
CC
L
ΔΔ
+
−=+
.
and
(1 )
jk jk
mk k k k
mm
LL LL
CC LL
ΔΔΔ
++
−= + = +
so
1
mj
j
jk
m
CC
LL
L
Δ
=+
+
.
We next compute the reduction to be made in soil m from the formula
jk
mk
m
LL
L
ΔΔ
+
=−
substituting we get
()
1
mj jk
m
jk m
m
CC LL
LL L
L
Δ−+
=− +
+
.
Translating this to the variables used in the program we have:
j least;
L.equated
L;
deltam (-1)× ×(totals -totals )
1+ ;
m
mj
λ
λ
λ
Note that in the line above we are generalising the term Lj + Lk to an arbitrary number of
multipliers ( or  in this program) by computing the inner product between the equated
vector and the multipliers.  is works because the equated vector has a 1 for all soils in the
equated set. We now compute the change in duration that Scoop spends on its best soil (dx)
by scaling deltam by Scoops norm for soil m.
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
m,Scoop
m,Scoop m,Scoop
deltam
dx ;
norms
xx+dx;
reallocate this time to Scoops best soil in the equated set which we will now call j
best normsI,Scoop × 0;
j I;
for s I to III do
if normss,Scoop × equateds > best then
begin
best normss,Scoop;
j s;
end ;
xj,Scoop xj,Scoopdx;
computeTotalsEtc;
count count + 1;
until (( equated ) = 3) (count > 10);
writeln(‘answer arrived at after’‚ count, ‘trys‘);
writeln(‘allocation’, /I×x target totals
hour );
end;
A.1 ComputeTotalsEtc
procedure ComputeTotalsEtc;
Work out how much is being produced, which soil is being produced least and which soils
outputs are equals to this,
var
d: array [soil] of real;
begin
totals ←∑ (norms × x);
leastsoil \min totals;
Find the soil that is least produced.
for s I to III do
if totalss = leastsoil then least s;
Find the ones on the plan ray
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1.0 if
0.0 otherwise
εε
totals-leastsoil
d;
cubicmeter
(d< ) (d>- )
equated ;
end;
A.2 marginalgain
function marginalgain (s: soil; d: excavator): volume;
is computes the marginal gain, under the weighting imposed by the current resolving
multipliers, of a small shift of the digger ds time to the specifi ed soil type s. We compute
the eff ect of multiplying all current time allocations to 1 – ε whilst increasing the alloca-
tion of time to soil s by ε.  e assumption here is that for now each machine has only one
hour to allocate.
const
epsilon = 0.001;
var
Let currentoutput volume;
begin
currentoutput Σ xd × normsι0,d;
marginalgain ((ε × hour) × normss,d) - ε × currentoutput;
end;
A.3 mainSoilProducedBy
function mainSoilProducedBy (e: excavator): soil;
is determines which soil excavator e produces the most of.
var
Let v volume;
Let j, s soil;
begin
v 0 × cubicmeter;
s i;
for j I to III do
begin
if v < normsj,e × xj,e then
begin
v normsj,e × xj,e;
s j;
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Paul Cockshott: Von Mises, Kantorovich and in-natura calculation 
end;
end;
mainSoilProducedBy s;
end;
A.4 fi ndScoop
function ndScoop (var s: soil): excavator;
Find which machine not currently fully committed is best at producing the soil s.  e pa-
rameter s is updated by the call.  e soil s must be drawn from one of those in the equated
set. We call this machine Scoop.  e algorithm searches to fi nd which machine will have
the greatest marginal output of the soil in the equated set per unit of other soil it gives up
by switching to produce s.
var
Let gain volume;
Let j, m soil;
Let d, Scoop excavator;
begin
Scoop A;
gain (–maxint) × cubicmeter;
for d A to C do
for j I to III do
if equatedj > 0 then
begin
m mainsoilproducedby (d);
if marginalgain (j, d) > gain then
begin
if equatedm < 1 then
begin
gain marginalgain (j, d);
Scoop d;
s j;
end;
end;
end;
ndScoop Scoop;
end;
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... Esta apreciación se basa en la confusión típica de la teoría subjetiva del valor, pues cuando hay competencia por el lado de la oferta, los precios tienden a ajustarse a los costes medios de producción (Guerrero 2000y 2007, Cockshott & Cottrell 1997 donde la demanda (la utilidad del consumidor) solo interviene para fijar la cantidad que se compra a dicho precio. Y a diferencia de lo que sucedía en tiempos de la URSS, hoy esos costes medios sí pueden determinarse directamente de forma exhaustiva en una economía socializada gracias a la capacidad informática actual (Cockshott & Cottrell 1993Cockshott 2010), algo que ya reconocen también autores austriacos (Brewster, 2004) o social-liberales (Agafonow, 2008). ...
... Precisamente una economía planificada con base informatizada como la que hoy ya es posible, estaría capacitada para responder de manera automática y con mayor eficiencia a cualquier cambio que se registre en cualquier punto del aparato productivo global o en la demanda final de los consumidores, pues la información se trasmite en tiempo real a lo largo de toda la cadena de interdependencias productivas sin que el proceso de ajuste quede distorsionado por la incertidumbre, las expectativas o la rentabilidad de las empresas. La planificación es, sencillamente, otra forma de satisfacer las necesidades sociales (de manera democrática, no plutocrática) y de coordinación empresarial mediante técnicas matemáticas de optimización (Cockshott, 2010;Cockshott & Cottrell, 1993. La estructura institucional para llevarlo a cabo puede ser tan sofisticada como se desee, involucrando a agencias de inno-vación para proponer nuevos productos y técnicas, comités de consumidores y usuarios, espacios para la experimentación empresarial y el emprendimiento (incubadoras y aceleradoras de empresas, con posibilidad de dirigir un proyecto propio reglado con un sistema de incentivos), etc. ...
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In the present research we make a comparative study in detail of the evolution of several commercial indicators linked to the activity of the car dealers located in territories economically depressed during the crises of 1993 and 2008, as well as an analysis of the evolution in the same periods of its financial results and of various socioeconomic variables, at national and local levels, that could affect its activity. It is concluded that the observation and systematic analysis of the deterioration of these data in the pre-crisis phases should help to predict the coming of an economic crisis that will affect to other economic sectors and more favoured geographical areas.
... Esta apreciación se basa en la confusión típica de la teoría subjetiva del valor, pues cuando hay competencia por el lado de la oferta, los precios tienden a ajustarse a los costes medios de producción (Guerrero 2000y 2007, Cockshott & Cottrell 1997 donde la demanda (la utilidad del consumidor) solo interviene para fijar la cantidad que se compra a dicho precio. Y a diferencia de lo que sucedía en tiempos de la URSS, hoy esos costes medios sí pueden determinarse directamente de forma exhaustiva en una economía socializada gracias a la capacidad informática actual (Cockshott & Cottrell 1993y 2008Cockshott 2010), algo que ya reconocen también autores austriacos (Brewster, 2004) o social-liberales (Agafonow, 2008). ...
... Precisamente una economía planificada con base informatizada como la que hoy ya es posible, estaría capacitada para responder de manera automática y con mayor eficiencia a cualquier cambio que se registre en cualquier punto del aparato productivo global o en la demanda final de los consumidores, pues la información se trasmite en tiempo real a lo largo de toda la cadena de interdependencias productivas sin que el proceso de ajuste quede distorsionado por la incertidumbre, las expectativas o la rentabilidad de las empresas. La planificación es, sencillamente, otra forma de satisfacer las necesidades sociales (de manera democrática, no plutocrática) y de coordinación empresarial mediante técnicas matemáticas de optimización (Cockshott, 2010;Cockshott & Cottrell, 1993y 2008. La estructura institucional para llevarlo a cabo puede ser tan sofisticada como se desee, involucrando a agencias de inno-vación para proponer nuevos productos y técnicas, comités de consumidores y usuarios, espacios para la experimentación empresarial y el emprendimiento (incubadoras y aceleradoras de empresas, con posibilidad de dirigir un proyecto propio reglado con un sistema de incentivos), etc. ...
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La crítica a la posibilidad del cálculo económico en el socialismo ha señalado dos tipos de problemas presuntamente insolubles: uno de computación, ligado a la escala de una economía compleja (esta sería la objeción de la escuela neoclásica) y otro de disponibilidad de información (según apunta la escuela austriaca). El artículo hace balance de estas dos críticas a partir de los avances técnicos en las últimas décadas en informática, telecomunicaciones e inteligencia artificial, y defiende que ya es técnicamente posible balancear y calcular exhaustivamente costes en una economía con millones de productos distintos, sin que la existencia de procesos de mercado sea condición necesaria para generar la información que requiere una asignación eficiente de los recursos. La conclusión es que el desarrollo técnico actual abre posibilidades reales, por primera vez en la historia, para una genuina planificación socialista de la económica inspirada en las ideas de Marx.
Chapter
Covid-19 has shown that governments with monetary sovereignty can turn the tap off quickly, if they must, and just as easily turn the tap back on. This has been coupled with a new appreciation for the ability of a sovereign economy to operate effectively despite large levels of net government (and net foreign) debt as a proportion of GDP, reconfirming the experience of those governments during WWII, when debt was used as an instrument to curb consumption and to redirect productive resources and research activity into investment in new capacity and new technology to support the war effort (viz the Agenda 30 strategic policy goals).
Article
The paper presents both the key arguments and the historical context of the socialist economic calculation debate. I argue that Oskar Lange presented the most developed strategy to deal with bourgeois economics, decisively helping to create the scientific consensus that rational economic calculation under socialism is possible. Lange’s arguments based on standard economic theory reveal that the most ardent defenders of capitalism cannot reject socialism on technical terms and that, as a consequence, the Austrian School was left with no choice but to diverge from mainstream economics in its search to develop a framework that could support its political position. This shows that Mises’ challenge from 1920 was solved and has been replaced by a political posture developed by Hayek and leading Austrians economists, who have been struggling since the 1980s to revise the standard interpretation of the socialist economic calculation debate. I argue that this revision should not be uncritically accepted and conclude that socialism cannot be scientifically rejected; it can only be politically rejected, by those whose economic interests it opposes.
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A probabilistic approach to economy using labor value of products as a basic measure. many variables are defined as random variables and their distribution is shown to have some important implications. Profits, accumulation of capital and wealth and wages are discussed in a historic and present day framework.
Conference Paper
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Multistage stochastic programming is a popular technique to deal with uncertainty in optimization models. However, the need to adequately capture the underlying distributions leads to large problems that are usually beyond the scope of general purpose solvers. Dedicated methods exist but pose restrictions on the type of model they can be applied to. Parallelism makes these problems potentially tractable, but is generally not exploited in today’s general purpose solvers. We apply a structure-exploiting parallel primal-dual interior-point solver for linear, quadratic and nonlinear programming problems. The solver efficiently exploits the structure of these models. Its design relies on object-oriented programming principles, treating each substructure of the problem as an object carrying its own dedicated linear algebra routines.We demonstrate its effectiveness on a wide range of financial planning problems, resulting in linear, quadratic or non-linear formulations. Also coarse grain parallelism is exploited in a generic way that is efficient on any parallel architecture from ethernet linked PCs to massively parallel computers. On a 1280-processormachine with a peak performance of 6.2 TFlops we can solve a quadratic financial planning problem exceeding 109 decision variables. Keywords: asset and liability management, interior point, massive parallelism, structure exploitation. 1 Introduction Decision making under uncertainty is an important consideration in financial planning. A promising approach to the problem is the multistage stochastic
Book
The same rule which regulates the relative value of commodities in one country does not regulate the relative value of the commodities exchanged between two or more countries. Under a system of perfectly free commerce, each country naturally devotes its capital and labor to such employments as are most beneficial to each. This pursuit of individual advantage is admirably connected with the universal good of the whole. By stimulating industry, by rewarding ingenuity, and by using most efficaciously the peculiar powers bestowed by nature, it distributes labor most effectively and most economically: while, by increasing the general mass of productions, it diffuses general benefit, and binds together, by one common tie of interest and intercourse, the universal society of nations throughout the civilised world. It is this principle which determines that wine shall be made in France and Portugal, that corn sell be grown in America and Poland, and that hardware and other goods shall be manufactured in England…
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An Introduction to Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject. Economic and financial applications are explained in detail before students learn how mathematics can be used, enabling students to learn how to put mathematics into practice. Starting with a revision of basic mathematical principles the second half of the book introduces calculus, emphasising economic applications throughout. Appendices on matrix algebra and difference/differential equations are included for the benefit of more advanced students. Other features, including worked examples and exercises, help to underpin the readers' knowledge and learning. Akihito Asano has drawn upon his own extensive teaching experience to create an unintimidating yet rigorous textbook.