Today our goal is to derive an exact formulation of an uncertainty principle in markets. To that purpose we have established in earlier posts a commutation relation between demand and price of a good in a market. I state it again:
Prices and demands interact according to
for a fixed real .
What I didn’t tell you so far is how measurement of market observables is supposed to work. Let me just close this gap. Measurement of an observable, e.g. the price of good , in a market in state (e.g. in this case selling a small quantity of good ) will result in a jump of the market into a new state being an eigenvector of the observable. The outcome of the measurement will be a real number (e.g. the price), the eigenvalue of the observable corresponding to with probability
For an observable on one can show that its mean value at state is given as
The dispersion of an observable on is given as
Now we are in the shape to state the uncertainty principle in markets. In essence it claims that prices and demands of a good cannot be measured with arbitrary precision. Moreover, an explicit lower bound on the maximal simultaneous precision is given. Its proof is essentially a straight forward application of Cauchy-Schwarz inequality.
Proposition. For a market in state the dispersions of and satisfy
In the asymmetric case , the right-hand side is strictly larger than zero.
Proof. Since dispersion and mean do not depend on the norm of a state we can, without loss of generality, assume that and obtain
Now Cauchy-Schwarz inequality implies
Since with we obtain
and since the first term is positive
Now (1) and the fact that positive observables have a square root yields the final inequality
Since can only be zero if is zero the proposition is proved.
Admittedly, that was a bit dry, but it does the job and that is sometimes all that is necessary in mathematics. Now the pace increases and we are heading with giant leaps towards the time evolution equations for markets …