information flows under market and plan

#./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

  1. It writes to all its suppliers asking them their current prices.
  2. It replies to all price requests that it gets, quoting its current price #tex2html_wrap_inline520#.
  3. It opens and reads all price quotes from its suppliers.
  4. It estimates its current per-unit cost of production.
  5. It calculates the anticipated profitability of production.
  6. 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.
  7. It now calculates how much of each input #tex2html_wrap_inline528# is required to sustain that production.
  8. It sends off to each of its suppliers #tex2html_wrap_inline530#, an order for amount #tex2html_wrap_inline532# of their product.
  9. It opens all orders that it has received and
    1. totals them up.
    2. 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.
    3. It dispatches the (partially) filled orders to its customers.
    4. 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.
  10. It receives all deliveries of inputs and determines at what scale it can actually proceed with production.
  11. 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#>

  1. In the first time period:
    1. They send to the planners a message listing their address, their technical input coefficients and their current output stocks.
    2. They receive instructions from the planners about how much of each of their output is to be sent to each of their users.
    3. They send the goods with appropriate dispatch notes to their users.
    4. They receive goods inward, read the dispatch notes and calculate their new production level.
    5. They commence production.
  2. They then repeatedly perform the same sequence replacing step 1a with:
    1. They send to the planners a message giving their current output stocks.
<#432#> 10

The planning bureau performs the complementary procedure:

<#433#>

  1. In the first period:
    1. They read the details of stocks and technical coefficients from all of their producers.
    2. They compute the equilibrium point #tex2html_wrap_inline596# from technical coeffients and the final demand.
    3. They compute a turnpike path (Dorfman, Samuelson and Solow, 1958) from the current output structure to the equilibrium output structure.
    4. They send out for firms to make deliveries consistent with moving along that path.
  2. In the second and subsequent periods:
    1. They read messages giving the extent to which output targets have been met.
    2. They compute a turnpike path from the current output structure to the equilibrium output structure.
    3. 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#