Supply and demand from information transfer

At this point we will take our information transfer process and apply it the the economic problem of supply and demand. In that case, we will identify the information process source as the demand $Q^d$, the information transfer process destination as the supply $Q^s$, and the process signal detector as the price $p$. The price detector  relates the demand signal $\delta Q^d$ emitted from the demand $Q^d$ to a supply signal $\delta Q^s$ that is detected at the supply $Q^s$ and delivers a price $P$.

We translate Condition 1 in [1] for the applicability of our information theoretical description into the language of supply and demand:

Condition 1: The considered economic process can be sufficiently described by only two independent process variables (supply and demand: $Q^d, Q^s$) and is able to transfer information.

We are now going to look for functions $\langle Q^s \rangle = F(Q^d)$ or  $\langle Q^d \rangle = F(Q^s)$ where the angle brackets denote an expected value. But first we assume ideal information transfer $I_{Q^s} = I_{Q^d}$ such that:

$$(4) \space P= \frac{1}{\kappa} \frac{Q^d}{Q^s}$$
$$(5) \space \frac{dQ^d}{dQ^s}= \frac{1}{\kappa} \frac{Q^d}{Q^s}$$

Note that Eq. (4) represents movement of the supply and demand curves where  $Q^d$ is a "floating" information source (in the language of Ref [1]), as opposed to movement along the supply and demand curves where $Q^d =Q^d_0$ is a "constant information source".

If we do take $Q^d =Q^d_0$ to be a constant information source and integrate the differential equation Eq. (5)

$$(6) \space \frac{\kappa }{Q_0^d}\int _{Q_{\text{ref}}^d}^{Q^d}d\left(Q^d\right)'=\int_{Q_{\text{ref}}^s}^{\left\langle Q^s\right\rangle } \frac{1}{Q^s} \, d\left(Q^s\right)$$

We find

$$
(7) \space \Delta Q^d=Q^d-Q_{\text{ref}}^d=\frac{Q_0^d}{\kappa }\log \left(\frac{\left\langle Q^s\right\rangle }{Q_{\text{ref}}^s}\right)
$$

Equation (7) represents movement along the demand curve, and the equilibrium price $P$ moves according to Eq. (4) based on the expected value of the supply and our constant demand source:

$$
\text{(8a) }P=
\frac{1}{\kappa }\frac{Q_0^d}{\left\langle Q^s\right\rangle }
$$
$$
\text{(8b) }
\Delta Q^d=\frac{Q_0^d}{\kappa }\log \left(\frac{\left\langle Q^s\right\rangle }{Q_{\text{ref}}^s}\right)
$$

Equations (8a,b) define a demand curve. A family of demand curves can be generated by taking different values for $Q_0^d$ assuming a constant information transfer index $\kappa$.

Analogously, we can define a supply curve by using a constant information destination $Q_0^s$ and follow the above procedure to find:

$$ \text{(9a) }P= \frac{1}{\kappa }\frac{\left\langle Q^d\right\rangle }{Q_0^s}$$ 

$$ \text{(9b) }\Delta Q^s = \kappa Q_0^s \log \left(\frac{\left\langle Q^d\right\rangle}{Q_{\text{ref}}^d}\right)$$

So that equations (9a,b) define a supply curve. Again, a family of supply curves can be generated by taking different values for $Q_0^s$.

Note that equations (8) and (9) linearize (Taylor series around $Q^x=Q_\text{ref}^x$)

$$
Q^d =Q_{\text{ref}}^d +\frac{Q_0^d}{\kappa }-Q_{\text{ref}}^s P
$$
$$
Q^s = Q_{\text{ref}}^s-\kappa Q_0^s+\frac{Q_0^s{}^2\kappa ^2}{Q_{\text{ref}}^d}P
$$

plus terms of order $(Q^x)^2$ such that

$$
Q^d=\alpha -\beta P
$$
$$
Q^s=\gamma +\delta P
$$

where  $\alpha =Q_{\text{ref}}^d+\left.Q_0^d\right/\kappa$, $ \beta  = Q_{\text{ref}}^s$ ,$ \gamma  = Q_{\text{ref}}^s-\kappa  Q_0^s $ and $ \delta =\kappa ^2 Q_0^s{}^2/Q_{\text{ref}}^d$. This recovers a simple linear model of supply and demand (where you can add a time dependence to the price e.g. $ \frac{dP}{dt} \propto Q^s - Q^d $).

We can explicitly show the supply and demand curves using equations (8a,b) and (9a,b) and plotting price $P$ vs change in quantity $\Delta Q^x=\text{$\Delta $Q}^s $ or $\text{$\Delta $Q}^d$. Here we take $\kappa = 1$ and $Q_{\text{ref}}^x=1$ and show a few curves of $Q_0^x = 1 \pm  0.1$. For example, for $x = s$ and +0.1, we are shifting the supply curve to the right. In the figure we show a shift in the supply curve (red) to the right and to the left (top two graphs) and a shift in the demand curve (blue) to the right and to the left (bottom two graphs). The new equilibrium price is the intersection of the new colored (supply or demand) curve and the unchanged (demand or supply, respectively) curve.

References

[1] Information transfer model of natural processes: from the ideal gas law to the distance dependent redshift P. Fielitz, G. Borchardt http://arxiv.org/abs/0905.0610v2

[2] http://en.wikipedia.org/wiki/Gronwall's_inequality

[3] http://en.wikipedia.org/wiki/Noisy_channel_coding_theorem#Mathematical_statement

[4] http://en.wikipedia.org/wiki/Entropic_force

[5] http://en.wikipedia.org/wiki/Sticky_(economics)