# Can Linear Supply-Demand Equilibria Be Understood as a Feedback-Control Process?

I'm not an economist, but rather an applied mathematician working in control theory. I've recently been watching Berkeley's intro to econ course in a personal project to better understand economics and finance. The first few lectures are on linear supply-demand curves and how they create price equilibria.

To wit, suppose $\sigma(p)$ and $\delta(p)$ and the quantity supplied and demanded at price $p$ resp. Then the assumption of linearity implies

$$\sigma(p) = S_\sigma p+\sigma_0,\ \ \delta(p) = -S_\delta p+\delta_0$$

(this is my notation). The numbers $S_{\sigma,\delta}$, $\delta_0,\sigma_0$ are presumed positive and correspond to the slope (what I believe is called "elasticity" in econ jargon) and the intercepts, which I've interpreted as the amount supplied and demanded at $p = 0$ (e.g. how much people are willing to supply or would demand if the good were free). Clearly we assume $p \geq 0$.

The equilibrium is met when supply and demand are equal--$\delta = \sigma$--and give the equilibrium price $$p_e = \frac{\delta_0-\sigma_0}{S_\delta+S_\sigma}.$$

This much is trivial to a real economist.

Now to the interesting part:

Through the lecturer explaining how an out of equilibrium price eventually comes to equilibirium, I recognized the basic idea of feedback at work: the change in price is forced by the signed difference between supply and demand. I propose this is modeled by the equation

$$\dot{p} = k(\delta-\sigma),$$

or exactly a feedback control law ($P$-controller) with feedback gain $k$. Furthermore, since we assume the supply and demand curves are linear, the entire equation is now a linear differential equation

$$\dot{p} = k(\delta_0-\sigma_0)-k(S_\delta+S_\sigma)p.$$

The equilibrium occurs when $\dot{p} = 0$ and is identical to that obtained using the graphical method.

But wait--there's more! Since we now have a differential equation, we can get a time-domain solution: $$p(t) = e^{-t/\tau}p_i+p_e(1-e^{-t/\tau})$$ Here $p_i$ is the initial (out-of-equilibrium) price, $\tau = \frac{1}{k(S_\delta+S_\sigma)}$, and $p_e$ is the equilibrium price.

This implies that not only can we predict what equilibrium prices will be, but we can also, subject to understanding $k$, predict the amount of time it will take to settle into the equilibrium. In fact, the form of this equation means that equilibrium is asmyptotic ($p_e = p(t= \infty)$) and percentages--63%, 86%, 95%, etc. for $t/\tau = 1,2,3,$ etc.--of equilibrium will be obtained. $t/\tau = 6$ is a good approximation of actual equilibrium (99.8%), so we gain the ability to state how long it will take for a price initially at $p_i$ to settle to $p_e$ (call this $t_e$, the "equilibirum time")

$$t_e \approx \frac{6}{k(S_\delta+S_\sigma)}.$$

Thoughts on $k$

So what is $k$ really? The short answer is I don't know. Mathematically it is a feedback gain, but that's not really so helpful.

It has the units of price/(quantity$\cdot$time), so I have pulled it into two factors $k = f_t/S_X$. I suppose $f_t$ to be the frequency at which transactions are made of the good. It makes sense that a less often traded good will reach equilibrium more slowly than a more often traded good. $S_X$ has the units of $S_{\delta,\sigma}$ but I have no good interpretation. I've considered that we might have $S_X \approx S_\delta + S_\sigma$ to produce $\tau \approx 1/f_t$ but haven't got much farther than this.

The Bottom Line

I think I've found an interesting interpretation of supply-demand dynamics in terms of control theoretic principles, namely proportional feedback. Unfortunately I am so unknowledgeable in econ that I'm unable to know whether or not this is a fruitful line of inquiry which I should pursue, or whether I have just reconstructed a longstanding theory.

Can anyone help provide some context?

• I'll try to write a slightly more detailed answer later, but what you describe sounds a lot like the process of tâtonnement in general equilibrium theory. It may be worth looking into that. – Theoretical Economist Feb 24 '17 at 22:44
• Is there any standard resource for understanding the time-dependent response of a market? Most of my amateur research has only uncovered equilibrium analyses which are by definition a stationary, possibly asymptotic solution. – JMJ Feb 24 '17 at 22:49
• I know there is a literature on equilibrium adjustment processes (and some of this work will posit a law of motion for the price similar to what you propose) but unfortunately that material is not usually a standard part of the economics curriculum, even at the graduate level, which makes standard resources hard to come by. – Theoretical Economist Feb 24 '17 at 23:44
• @TheoreticalEconomist I would like very much to review some of this literature; can you point me to a journal or set of papers which might contain more information on equilibrium adjustment and laws of motion for the price? Thanks for your time man! – JMJ Feb 24 '17 at 23:47
• If I could do that off the top of my head, I would've written an answer instead of leaving some comments. :) Alas, the fact that this topic isn't currently taught in university, and it not being a particularly active research area means that what I've mentioned now is essentially all I know on this subject. My guess is you'll be looking for papers at least a decade (if not more) old. Journals to look at are Econometrica, the American Economic Review, the Quarterly Journal of Economics, the Journal of Political Economy, or the Review of Economic Studies. <continued> – Theoretical Economist Feb 25 '17 at 1:53

I think there might be an issue with this as currently constructed:

$\delta(p)= \infty$, $\forall P \in \mathbb{R_+}$ and $P_e = \infty =\dot{p},$ $\forall P \in \mathbb{R_+}$ This is because $\delta_0 = \infty$. Further, $\sigma(p) = S_\sigma p+\sigma_0= S_\sigma p$ since $\sigma_0 = 0$.

In words - there should be infinite demand for any normal good whenever P is 0 and there should be zero supply of any good whenever P is 0.

That said, your intuition here is solid but I do think you're working on reinventing the tatonement wheel. Tatonement, in a nutshell, adjusts prices up when there is excess demand and prices down when there is excess supply.

If you are able to do this after watching a few lectures online, then I say keep going. You could make meaningful contributions once you have a better understanding of the field.

edit: If you'd like to message me privately, I would be more than happy to pass along my graduate school notes/materials to help guide your studies. At minimum, I recommend you followup your current study with some more advanced micro theory courses.

• Thanks for your comment. Tatonement was mentioned in the comments as well, and I've done a little bit of research into it. It does seem very similar to this, so I wonder if there could be an interesting connection between Tatonement and feedback (at a more rigorous level). I'd love to get a hold of your resources but unfortunately do not know if private messaging is possible on SE. – JMJ Mar 12 '17 at 20:46

As Theoretical Economist says in his comments, this process is called Walrasian tatonnement. Here and here are two citations in this literature. Start by reading them and their references, and then use SSCI to go backwards and forwards in the literature.

My (somewhat vague) recollection is that, somewhere in this literature, is a negative result basically saying that tatonnement can't be shown to converge to an equilibrium except under very restrictive assumptions on demand (like no income effects).

There is a related literature on the Cobbweb Theorem. Here is a recent paper which contains a brief lit review. You can search in either SSCI or EconLit for Cobbweb Theorem to get into this literature.

The answer to your question, however, is that your insight does not appear to be original. The question of how price-theoretic markets get to equilibrium is old, and the solution you are proposing (that prices adjust up in markets with excess demand and down in markets with excess supply) has been extensively investigated.