#./complex.tex#
One of Hayek's most fundamental arguments is that the efficient
functioning of an economy involves making use of a great deal of
distributed information, and that the task of centralising
this information is practically impossible.
In what follows we attempt to put this argument to
a quantitative test.
We compare the communications costs implicit in a market
system and a planned system, and examine how the respective costs
grow as a function of the scale of the economy.
Communications cost is a measure of work done
to centralise or disseminate economic information: we
will use the conceptual apparatus of algorithmic information
theory (Chaitin, 1982) to measure this cost.
Our strategy is first to consider the dynamic problem
of how fast, and with what communications overhead, an economy can
converge on equilibrium. We will demonstrate that this can be done
faster and at less communications cost by the planned system.
We consider initially the dynamics of convergence
on a fixed target, since the control system with the faster
impulse response will also be faster at tracking a moving target.
Consider an economy #math14##tex2html_wrap_inline434# with #tex2html_wrap_inline436#
producers each producing distinct products under constant returns to
scale using technology matrix #tex2html_wrap_inline438#, with a well defined vector of
final consumption expenditure #tex2html_wrap_inline440# that is independent of
the prices of the #tex2html_wrap_inline442# products, an exogenously given wage rate #tex2html_wrap_inline444# and
a compatible rate of profit #tex2html_wrap_inline446#. Then there exists a possible Sraffian
equilibrium #tex2html_wrap_inline448# where <#370#>U<#370#> is the commodity flow
matrix and <#371#>p<#371#> a price vector.
We will assume, as is the case in commercial arithmetic,
that all quantities are expressed to some finite precision
rather than being real numbers. How much information is required to
specify this equilibrium point?
Assuming that we have some efficient binary encoding method and
that #tex2html_wrap_inline450# is a measure in bits of the information content of the data
structure #tex2html_wrap_inline452# using this method, then the equilibrium can be specified
by #tex2html_wrap_inline454#, or, since the equilibrium is in a sense given in the starting
conditions, it can be specified by #tex2html_wrap_inline456# where #tex2html_wrap_inline458# is a
program to solve an arbitrary system of Sraffian equations. In
general we have #math15##tex2html_wrap_inline460#. In the following we
will assume that #tex2html_wrap_inline462# is specified by #tex2html_wrap_inline464#.
Let #tex2html_wrap_inline466# be the conditional or relative information (Chaitin, 1982) of
#tex2html_wrap_inline468# given #tex2html_wrap_inline470#. The conditional information associated with any arbitrary
configuration of the economy, #math16##tex2html_wrap_inline472#, may then be
expressed relative to the equilibrium state, #tex2html_wrap_inline474#, as
#tex2html_wrap_inline476#. If #tex2html_wrap_inline478# is in the neighbourhood of #tex2html_wrap_inline480# we should expect that
#math17##tex2html_wrap_inline482#. For instance, suppose that we can derive #tex2html_wrap_inline484# from <#375#>A<#375#> and an intensity vector #tex2html_wrap_inline486# which
specifies the rate at which each industry operates then
#math18#
#displaymath488#
where #tex2html_wrap_inline490# is a program to compute #tex2html_wrap_inline492# from some <#380#>
A<#380#> and some #tex2html_wrap_inline494#. Since #tex2html_wrap_inline496# is a matrix and #tex2html_wrap_inline498# a vector, each of scale #tex2html_wrap_inline500#, we can assume that #math19##tex2html_wrap_inline502#.
As the economy nears equilibrium the conditional
information required to specify it will shrink, since #tex2html_wrap_inline504# starts to
approximate to #tex2html_wrap_inline506#.6
Intuitively we only have to supply the
difference vector between the two, and this will require less and less
information to encode, the smaller the distance between #tex2html_wrap_inline508#
and #tex2html_wrap_inline510#. A similar argument applies to the two price vectors
#tex2html_wrap_inline512# and #tex2html_wrap_inline514#. If we assume that the system
follows a dynamic
law that causes it to converge on equilibrium then we should have the
relation #math20##tex2html_wrap_inline516#.
We now construct a model of the amount of information that
has to be transmitted between the producers of a market
economy in order to move it towards equilibrium. We make the
simplifying assumptions that all production process take one timestep to
operate, and that the whole process evolves synchronously. We
assume the process starts just after production has finished, with the
economy in some random non-equilibrium state. We further assume
that each firm starts out with a given selling price for its product.
Each firm #tex2html_wrap_inline518# carries out the following procedure.
<#429#>7
- It writes to all its suppliers asking them their current prices.
- It replies to all price requests that it gets, quoting its current
price #tex2html_wrap_inline520#.
- It opens and reads all price quotes from its suppliers.
- It estimates its current per-unit cost of production.
- It calculates the anticipated profitability of production.
- If this is above #tex2html_wrap_inline522# it increases its target production rate #tex2html_wrap_inline524# by some fraction. If profitability is below #tex2html_wrap_inline526# a proportionate
reduction is made.
- It now calculates how much of each input #tex2html_wrap_inline528# is required to
sustain that production.
- It sends off to each of its suppliers #tex2html_wrap_inline530#, an order for amount
#tex2html_wrap_inline532# of their product.
- It opens all orders that it has received and
- totals them up.
- If the total is greater than the available product it scales
down each order proportionately to ensure that what it can supply is
fairly distributed among its customers.
- It dispatches the (partially) filled orders to its customers.
- If it has no remaining stocks it increases its selling price
by some increasing function of the level of excess orders, while if it has
stocks left over it reduces its price by some increasing function of the
remaining stock.
- It receives all deliveries of inputs and determines at what scale
it can actually proceed with production.
- It commences production for the next period.
<#429#> 8
Experience with computer models of this type of system
indicates that if the readiness of producers to change prices is too
great, the system could be grossly unstable. We will assume that the
price changes are sufficiently small to ensure that only
damped oscillations occur. The condition for movement towards
equilibrium is then that over a sufficiently large ensemble of points
#tex2html_wrap_inline534# in phase space, the mean effect of an iteration of the above
procedure is to decrease the mean error for each economic variable
by some factor #tex2html_wrap_inline536#.
Under such circumstances, while the convergence time
in vector space will clearly follow a
logarithmic law---to converge by a factor of #tex2html_wrap_inline538# in
in vector space will take time of order #math21##tex2html_wrap_inline540#---9
in information space the convergence time will be linear.
Thus if at time #tex2html_wrap_inline542# the distance from equilibrium is #tex2html_wrap_inline544#,
convergence to within a distance #tex2html_wrap_inline546# will take a take a time
of order
#math22#
#displaymath548#
where #tex2html_wrap_inline550# is a constant related to the number of economic
variables that alter by a mean factor of #tex2html_wrap_inline552# each step.
The convergence time
in information space, for small #tex2html_wrap_inline554#, will thus approximate to a linear
function of #tex2html_wrap_inline556# which we can write as #tex2html_wrap_inline558#.
We are now in a position to express the communications costs of
reducing the conditional entropy of the economy to some level #tex2html_wrap_inline560#.
Communication takes place at steps 1, 2, 8 and 9c of the procedure.
How many messages does each supplier have to send, and how
much information must they contain?
Letters through the mail contain much redundant pro
forma information: we will assume that this is eliminated and the
messages reduced to their bare essentials. The whole of the pro
forma will be treated as a single symbol in a limited alphabet of
message types. A request for a quote would thus be the pair
#tex2html_wrap_inline562# where #tex2html_wrap_inline564# is a symbol indicating that the message is a quotation
request, and #tex2html_wrap_inline566# the home address of the requestor. A quote would
be the pair #tex2html_wrap_inline568# with #tex2html_wrap_inline570# indicating the message is a quote and #tex2html_wrap_inline572#
being the price. An order would similarly be represented by
#math23##tex2html_wrap_inline574#, and with each delivery would go a dispatch note
#tex2html_wrap_inline576# indicating the actual amount delivered,
where #math24##tex2html_wrap_inline578#.
If we assume that each of #tex2html_wrap_inline580# firms has on average #tex2html_wrap_inline582# suppliers,
the number of messages of each type per iteration of the procedure
will be #tex2html_wrap_inline584#. Since we have an alphabet of message types #math25##tex2html_wrap_inline586# with cardinality 4, these symbols can be encoded in 2 bits
each. We will further assume that #math26##tex2html_wrap_inline588# can
each be encoded in binary numbers of #tex2html_wrap_inline590# bits. We thus obtain an
expression for the communications cost of an iteration of
#tex2html_wrap_inline592#. Taking into account the number of iterations, the cost of
approaching the equilibrium will be
#math27##tex2html_wrap_inline594#.
Let us now contrast this with what would be required in a planned
economy. Here the procedure involves two distinct procedures, that
followed by the (state-owned) firm and that followed by the planning bureau.
The firms do the following:
<#432#>
- In the first time period:
- They send to the planners a message listing their
address, their technical input coefficients and their current output
stocks.
- They receive instructions from the planners about how
much of each of their output is to be sent to each of their users.
- They send the goods with appropriate dispatch notes to
their users.
- They receive goods inward, read the dispatch notes and
calculate their new production level.
- They commence production.
- They then repeatedly perform the same sequence replacing
step 1a with:
- They send to the planners a message giving their
current output stocks.
<#432#> 10
The planning bureau performs the complementary procedure:
<#433#>
- In the first period:
- They read the details of stocks and technical
coefficients from all of their producers.
- They compute the equilibrium point #tex2html_wrap_inline596# from technical
coeffients and the final demand.
- They compute a turnpike path (Dorfman, Samuelson and
Solow, 1958) from the current output structure to the equilibrium
output structure.
- They send out for firms to make deliveries consistent
with moving along that path.
- In the second and subsequent periods:
- They read messages giving the extent to which output
targets have been met.
- They compute a turnpike path from the current output
structure to the equilibrium output structure.
- They send out for firms to make deliveries consistent
with moving along that path.
<#433#>
We assume that with computer technology the steps b and c can be
undertaken in a time that is small relative to the production period
(Cockshott 1990, Cockshott and Cottrell 1993).
Comparing the repsective information flows, it is clear that the
number of orders and dispatch notes sent per iteration
is invariant between the two modes of
organisation of production. The only difference is that in the planned
case the orders come from the center whereas in the market they
come from the customers. These messages will again account for a
communications load of #tex2html_wrap_inline598#. The difference is that in the
planned system there is no exchange of price information. Instead, on
the first iteration there is a transmission of information about stocks
and technical coefficients. Since any coefficient takes two numbers
to specify, the communications
load per firm will be: #tex2html_wrap_inline600#. For #tex2html_wrap_inline602# firms this approximates to
the #tex2html_wrap_inline604# that was required to communicate the price data.
The difference comes on subsequent iterations, where, assuming no
technical change, there is no need to update the planners' record of
the technology matrix. On #tex2html_wrap_inline606# subsequent iterations, the planning
system has therefore to exchange only about half as much
information as the market system. Furthermore, since the planned
economy moves on a turnpike path to equilibrium, its convergence time
will be less than that of the market economy. The consequent
communications cost is #math28##tex2html_wrap_inline608# where
#math29##tex2html_wrap_inline610#.
The consequence is that, contrary to Hayek's claims, the
amount of information that would have to be
transmitted in a planned system is substantially lower than for a
market system. The centralised
gathering of information is less onerous than the commercial
correspondence required by the market. In addition, the convergence
time of the market system is slower. The implication of faster
convergence for adaptation to changing rather than stable conditions
of production and consumption are obvious.
In addition, it should be noted that in our model for the market, we have ignored
any information that has to be sent around the system in order to make
payments. In practice, with the sending of invoices, cheques, receipts,
clearing of cheques etc., the information flow in the market system is
likely to be twice as high as our estimates.
The higher communications overheads of market economies are
reflected in the numbers of office workers they employ,
which in turn leaves its mark on the architecture of cities---11
witness the skylines of Moscow and New York.
#./complex.tex#