Consider an economy E = [A, c, r, w]characterised by: n producers each producing distinct products under constant return to scale using technology matrix A, a well defined vector of final consumption expenditure c_{1..n} that is independent of the prices of the n products, an exogenously given wage rate w and rate of profit r. Then there exists a possible Sraffian equilibrium e = [U, p] where U is the commodity flow matrix, p a price vector. How much information is required to specify this equilibrium point?

Assuming that we have some efficient binary encoding method and that I(s) is a measure in bits of the information content of the data structure s using this method, then the equilibrium can be specified by I(e), or since the equilibrium is in a sense given in the starting conditions it can be specified by I(E) + I(p_{s}) where p_{s} is a program to solve an arbitrary system of Sraffian equations. In general we must have:

I(e) ≤ I(E) + I(p_{s})

so in what follows we will assume that I(e) is specified by I(E) + I(p_{s}).

Let I(x|y) be the conditional or relative information[1] of x given y. Then for some arbitrary configuration of the economy k = [U_{k}, p_{k}] we can specify it relative to e as I(k|e). If k is in the neighbourhood of e we should expect that I(k|e) ≤ I(k). For instance, suppose that we can derive U_{k} from A and an intensity vector u_{k} which specifies the rate at which each industry operates then

I(k|e) ≤ I(u_{k}) + I(p_{k}) + I(p_{u})

where p_{u} is a program to compute U_{k} from some A and some u_{k}. Since U_{k} is a matrix and u_{k} a vector each of scale n we can assume that I(U_{k}) > I(u_{k}).

In general as the economy nears equilibrium the conditional information^{1} required to specify it will shrink since u_{k} starts to approximate to u_{e}. 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 u_{k} and u_{e}. A similar argument applies to the two price vectors p_{k}, p_{e}.If we assume that the system has a dynamics that cause it to converge on equilibrium then we should have the relation I(k_{t+1}|e) < I(k_{t}|e).

We will now construct a model of the amount of information that would have to be transmitted between the producers of a market economy in order to move it towards equilibrium. We make the simplying assumption 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 current selling price for their product. Each firm i carries out the following procedure

1This information measure of the distance from equilibrium differs from a simple euclidean one. The latter is based on a sum of squares, the information theoretic one on a sum of logarithms, so it is more sensitive to a multiplicity of small errors than to one large error.

Because of the equivalence between information and entropy it also measures the conditional entropy of the system.

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 p_{i}.

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 r it increases its target production rate u_{i} by some fraction. If profitability is below r a proportionate reduction is made.

7. It now calculates how much of each input j is required to sustain that production: U_{ij}.

8. It sends off to each of its suppliers j, an order for amount U_{ij} of their product.

9. It opens all orders that it has recieved and

(a) Totals them up.

(b) 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.

(c) It dispatches the [partially]filled orders to its customers.

(d) If it has no remaining stocks it increase its selling price by some increasing function of the level of excess orders, if it has stocks left over it reduces its price by some increasing function of the remaining stock.

10. It recieves all deliveries of inputs and determines at what scale it can actually proceed with production.

11. It commences production for the next period.

It is clear 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 at most only damped oscillations occur. The condition for movement towards equilibrium then is, that over a sufficiently large ensemble of points k in phase space, the mean effect of an iteration of the above procedure is I(k_{t+1}|e) = g.I(k_{t}|e) where 1 < g ≤ 0. Under such circumstances the convergence time will clearly follow a logarithmic, law such that to converge by a factor of D in information space will take time of order log_{\frac{1}{g}} D.

We are now in a position to express the communications costs of reducing the conditional entropy of the economy by a factor D. 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?

Since letters through the mail contain much redundant pro-forma information, we will assume that this is eliminated and that they are 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 [R, H] where R was a symbol indicating that it was a quotation request, and H the home address of the requestor. A quote would be the pair [Q, P] with Q indicating that it was a quote and P being the price. An order would similarly be [O, U_{ij}] and with each delivery would go a dispatch note [N, U_{ij}] indicating the actual ammount delivered.

If we assume that each of n firms has on average m suppliers, the number of messages of each type per iteration of the procedure will be nm. Since we have an alphabet of message types (R, Q, O, N) with cardinality 4, these symbols can be encoded in 2 bits each. We will further assume that (H, P, U_{ij}, U_{ij}) can each be encoded in binary numbers of b bits. We thus obtain an expression for the communications cost of an iteration of nm(b + 2). Taking into account the number of iterations, the cost of approaching the equilibrium by a factor of D will be nm(b + 2) log_{\frac{1}{g}} D.

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)firm and that followed by the planning bureau. The firms do the following:

1. In the first time period:

(a) They send to the planners a message listing their address, their technical input coefficients and their current output stocks.

(b) They recieve instructions from the planners about how much of each of their output is to be sent to each of their users.

(c) They send the goods with appropriate dispatch notes to their users.

(d) They recieve goods inward, read the dispatch notes and calculate their new production level.

(e) They commence production.

2. They then repeatedly perform the same sequence replacing step 1a with:

(a) They send to the planners a message giving their current output stocks.

The planning bureau performs the complementary procedure:

1. In the first period they:

(a) They read the details of stocks and technical coefficients from all of their producers.

(b) They compute the equilibrium point e from technical coeffients and and the final demand.

(c) They compute a turnpike path from the current output structure to the equilibrium output structure.

(d) They send out for firms to make deliveries consistent with moving along that path.

2. In the second and subsequent periods they:

(a) They read messages giving the extent to which output targets have been met.

(b) They compute a turnpike path from the current output structure to the equilibrium output structure.

(c) They send out for firms to make deliveries consistent with moving along that path.

References

[1] Chaitin, G. J., Algorithmic Information Theory, Encyclopedia of Statistical Sciences, vol 1, Wiley, New York, 1982, pp.38-41.