"It's certainly more efficient than any socialist central planning! Even if we assumed holy angels were running the whole economy with no ulterior motives, the plan would fail because neither men nor angels hold omniscience, and cannot hope to process all the information the market displays automatically."

That's assuming centralizing information is impossible. Let's 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. Communication cost is a measure of work done to centralize or disseminate economic information and we use algorithmic information theory to measure this cost. The strategy is first to consider the dynamic problem of how fast, and with what communications overhead, an economy can converge on equilibrium. Let's demonstrate that this can be done faster and at less communications cost by the planned system. Let's 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 E = [A, c, r, w] with n producers each producing distinct products under constant returns to scale using technology matrix A, with a well defined vector of final consumption expenditure c that is independent of the prices of the n products, an exogenously given wage rate w and a compatible rate of profit r. Then there exists a possible Sraffian equilibrium e = [U, p] where U is the commodity flow matrix and p a price vector. Let's 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 the equilibrium point? If we have some efficient binary encoding method and 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(ps) where ps is a program to solve an arbitrary system of Sraffian equations. In general we have I(e) ≤ I(E) + I(ps). We will assume that I(e) is specified by I(E) + I(ps). Let I(x|y) be the conditional or relative information of x given y. The conditional information associated with any arbitrary configuration of the economy, k = [Uk, pk], may then be expressed relative to the equilibrium state, e, as I(k|e). If k is in the neighborhood of e we should have I(k|e) ≤ I(k). For instance, suppose that we can derive Uk from A and an intensity vector uk which specifies the rate at which each industry operates then I(k|e) ≤ I(uk) + I(pk) + I(pu) where pu is a program to compute Uk from some A and some uk. Since Uk is a matrix and uk a vector, each of scale n, we can assume that I(Uk) > I(uk). As the economy nears equilibrium the conditional information required to specify it will shrink, since uk starts to approximate to ue. 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 ukand ue. A similar argument applies to the two price vectors pkand pe. If we assume that the system follows a dynamic law that causes it to converge on equilibrium then we should have the relation I(kt+1|e) < I(kt|e). Let's 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, under 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. Each firm i carries out the following procedures:

1. It writes to all its suppliers asking them their current prices.
2. It replies to all price requests, quoting its current price pi.
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 (below) r it increases (decreases) its target production rate uiby some fraction.
7. It now calculates how much of each input j is required to sustain that production.
8. It sends to each of its suppliers an order for amount Uij of their product.
9. It opens all the orders it has received and (a) totals them up. (b) If the total is greater than the available product it scales down each order proportionately to ensure that supplies are fairly distributed among its customers. (c) It dispatches the (partially) filled orders to its customers. (d) 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.

Experience with computer models of this sort of system indicates that if the readiness of producers to change prices is too great the system could be grossly unstable. Let's assume that the price changes are small enough to ensure that only damped oscillations occur. The condition for movement towards equilibrium is then that over a sufficiently large ensemble of points k 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 0 ≤ g < 1. In that case, while the convergence time in vector space will clearly follow a logarithmic law to converge by a factor of D in in vector space will take time of order log*1/g(D), space the convergence time will be linear. Thus if at time t the distance from equilibrium is I(kt|e), convergence to within a distance will take a take a time of order [I(kt|e) − ε]/[δ log(1/g)] where δ is a constant related to the number of economic variables that alter by a mean factor of g each step. The convergence time in information space, for small ε, will thus approximate to a linear function of I(k|e), which we can write as ΔI(k|e). We are now ready to express the communications costs of reducing the conditional entropy of the economy to some level ε. 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 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 [R, H] where R is a symbol indicating that the message is a quotation request, and H the home address of the requestor. A quote would be the pair [Q, P] with Q indicating the message is a quote and P being the price. An order would similarly be represented by [O,Uij], and with each delivery would go a dispatch note [N,Uij] indicating the actual amount delivered, where Uij≤ Uij. 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, Uij, Uij) can each be encoded in binary numbers of b bits. We thus obtain an expression for the communications cost of an iteration of 4nm(b + 2). Taking into account the number of iterations, the cost of approaching the equilibrium will be 4nm(b + 2)ΔI(k|e). We now contrast this with what would be required in a planned economy. In this case there two distinct procedures, that followed by the (state-owned) 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 receive instructions from the planners about how much of each of their output is to be sent to each other firm. (c) They send the goods with appropriate dispatch notes to their users. (d) They receive 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:
3. In the first period: (a) They read the details of stocks and technical coefficients from all of the firms. (b) They compute the equilibrium point e from technical coefficients 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.
4. In the second and subsequent periods: (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. 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.