A common talking point from Austrian economists is that prices encode people's subjective preferences. This notion has not been examined from an information theoretical point of view as far as I can tell. So let's look at it.
The price of a commodity is a definite value in some currency. Prices are discrete values that correspond to a definite amount of information. Prices cannot be real-valued, they must be discrete, because humans and computers have finite information processing capabilities. The world market runs on actual computers sending actual bits around, on people sending letters and invoices with discrete characters written on them.
If for example we have a market where things are priced in cents, and the range of prices are from zero to say $999,999.99, then the information content of any price is at most log2(10⁸) ≃ 26.6 bits. Actual prices will contain less information than this since prices are not uniformly distributed in any given range. Let's say we do statistics on prices in the market and we come up with a number of 20 bits. If we have a good predictor for the value of a commodity, then the information content of each price goes down further still. In the Marxist camp the go-to predictor of prices are labour values, which account for some 93% of variations seen in prices last time I checked the literature. This means that the residual amount of information is 20 + log2(1 - 0.93) ≃ 16.2 bits. If we find a better predictor then the information content drops even more. In practice there will be some lower bound. A 99.9% accurate predictor still leaves 10.0 bits unaccounted for in this example.
Prices change over time, and such changes correspond also to definite amounts of information being signalled. They are less than the information content in the price itself, because prices don't fluctuate uniformly randomly. If the value of a commodity changes then its price tends to change as well, sooner or later. If our predictor is good then the information contained in the price change is small. If the price changes frequently then the mass of information signalled will be large. This does not necessarily mean that the information is actually useful, merely that some amount of information is being sent around. For all we know the price may be functionally equivalent to a Ouija board.
If the price changes frequently but the range within which it fluctuates is very small, then the mass of information contained will be smaller than if the price changes over a larger range. It may even be smaller than much more infrequent price changes of larger magnitude.
The Austrians are partly correct in that the price system is an approximate way of signalling supply and demand in the market. It is a feedback mechanism. But the amount of information being signalled at any one time is quite small, and it is only a scalar value. At every point in time the subjective preferences of every actor in the market must be projected down and quantized to a definite price. Prices don't "encode" things in the sense that we can decode them later. It is a lossy process.
In planning (calculation in kind) the loss of information from a projection down to a scalar price does not happen. We also have available information of supply and demand explicitly, rather than implicitly. Supply is determined by the means of production currently available. Demand is measured and predicted. If necessary we can put limits on demand by a renumeration system, to prevent people from making demands of the system that would make the plan infeasible.
Since the demand vector comes from all consumers, each consumer participates in steering production. On the supply side we similarly want as many people as possible to at least be able to test their ideas for new productive methods. Everyone then participates in shaping the feasible region of the mathematical program from which the plan is derived. These are two ways in which democratic planning addresses the local knowledge problem. Democracy is important, not for moral reasons, but for being able to capture as much and as fine-grained demand information as possible, and to capture good ideas that are currently ignored by the market system. This allows democratic planning to surpass the market in terms of meeting demands and enabling human flourishing.