# Refinements of Walrasian equilibrium

In Walrasian equilibrium there is a market clearing price, that is $$D(p) = S(p)$$ or the supply is worthless, so you can have $$D(p) < S(p) \mbox{ if } p = 0.$$ This can be summarized as $$p \cdot D(p) = p \cdot S(p)$$ If the demand function is not always downward sloping (e.g. Giffen goods, markets with asymmetric information) or the supply function is not always upward sloping (e.g. economies of scope) multiple Walrasian equilibria may exist. If this is the case, are these all considered equally 'valid' equilibria or are there some established refinements to the equilibrium concept?

An illustrative example: Are $A$, $B$, and $C$ all equilibria?

(Personally I dislike $B$. You may well have no problems with the bugger.)

• I'd argue that they are all equilibria. Why not? However, they may or may not be stable. – clueless Apr 11 '16 at 18:32
• @clueless And how would you define stability for Walrasian equilibria? Dynamic equations for price or supply adjustment? – Giskard Apr 11 '16 at 18:34
• Well, we may argue that for $q < q_A$ $D > S$ and thus the price increases until $A$ is rechead. For $q_A < q < q_B$ $D < S$ and thus the price decreases until $B$ is reached. However, for $q_B < q < q_C$ $D > S$ and thus the price increases...So we may conclude that $B$ is stable and $A$ instable (one sided stable)? I don't know what concept applies here. In general we should define some differential/difference equation which motivates the dynamic price adjustment. – clueless Apr 12 '16 at 8:50
• @clueless Which brings us back to my original question: "are there some established refinements to the equilibrium concept?" – Giskard Apr 12 '16 at 8:57

## 1 Answer

Forgive me in advance if this is already familiar. I'll be talking about stability of equilibrium the way I (briefly) learned it.

Let $\xi^j(p)$ denote the excess demand for good $j$ given price $p$.

$$\xi^j(p) = \sum^m_{h=1}\left( x^j_h(p) - e^j_h \right) = x^j(p) - r^j$$

For $m$ households/consumers, $e$ being an endowment to a household, and $r^j$ being the total resource available of good $j$. We say the rate of price adjustment over time $\left(\frac{\partial p^j}{\partial t}\right)$ is proportional to excess demand.

For either of your market clearing prices $p^*$, we then know that $x^j(p^*) = r^j \ \forall j \implies \xi^j(p) = 0$ Otherwise,

$\xi^j(p)> 0 \implies \text{excess demand} \implies \frac{\partial p^j}{\partial t} > 0$

$\xi^j(p)< 0 \implies \text{excess supply} \implies \frac{\partial p^j}{\partial t} < 0$

Out of the three equilibrium you have up there, we'd probably want to find one that was locally stable and say that's where the market is more likely to realistically go.

Steady state $p^*$ is locally stable if $\exists \epsilon > 0 \ \text{s.t.} \ \forall \ p(0) \in B_\epsilon (p^*)$, an open neighborhood with radius $\epsilon$ around $p^*$, then we have $p(t, p(0)) \rightarrow p^*$

I don't want to fuddle this too much with math (particularly when you linearize the system and have to deal with eigenvalues/eigenvectors), so let's just take a look at your supply and demand curves above. We can peruse the math on our own time.

Case A: Let's say you are at point A, your nice equilibrium. Suddenly there is a downwards quantity shock and you are now to the left of $q_A$. Since there is excess demand, the change in price over time should be positive, so the price will rise, and you will arrive back at point A.

What if you are shocked to the right of $q_A$ in the short run? Now there is excess supply and price over time should fall. If you were still above $p^*_A$ at this point, you will arrive back at point A. Otherwise price will fall all the way until you reach point B.

Case B: We know what happens with a shock to the left of point B now. What about to the right? If point C looks like a global minimum of the supply curve from what it looks like, then any shock that gets you before point C gets you excess demand, which means the price will rise and you will arrive back at point B.

Case C: To the right of point C, there is excess supply, and prices will fall back down until you reach C again.

None of these equilibria seem globally stable. It seems C is not locally stable on the left, so I wouldn't say the market would hang out there for very long. Points A and B are locally stable, but it seems B has greater room to get shocked and still move back towards B.

So for this scenario, I personally like B as your long run equilibrium.

If you had extra information about how shocks in the market happen (how often, variance, etc.), it may allow you to describe what percentage of the time you stick around at each equilibria, and you could describe an aggregate equilibrium like that as well.

• Sorry for the slow response. I am not sure if I agree with your reasoning that we always move 'locally' on the supply curve when price changes. Even if we do so it seems to me that an individual supplier would gain at point B by increasing production because at that point someone has lower marginal cost than the current market price (the supply curve is downward sloping there). Would you not agree that this disqualifies it as an equilibrium? – Giskard Aug 18 '16 at 15:51
• I never thought of the possibility of somewhere between B and C along the supply curve being profit maximizing compared to just B. The existence of excess demand supposedly means the price would rise over time, but this is an odd scenario... – Kitsune Cavalry Aug 18 '16 at 16:54
• @denesp It's been around a year since discussing this question, but my mind keeps going back to it. If the supply curve is downward sloping, that more production requires less money due to decreasing costs of some sort, then I guess I do have to agree that the notion of stability I've offered is contradictory. Price being above marginal cost for all firms should imply production increases in a normal economy. So this is all to say I am still not sure of what to think with your scenario. – Kitsune Cavalry Aug 27 '17 at 23:41
• It is unfortunate that microeconomics focuses a lot on equilibrium results and not much on dynamics. If I knew a little more about tâtonnement I might be willing to give this another crack, but otherwise, I was thinking of putting a bounty on the question. I'd probably only want to do that if you were still interested in an answer to this question as well. :P – Kitsune Cavalry Aug 27 '17 at 23:44
• Once you provide an equation for price changes (tâtonnement) you get a dynamic system. I don't think you can have any reasonable equation that would make equilibrium $B$ stable. I have a related question that I think shows this is a known issue in markets with incomplete information. – Giskard Aug 28 '17 at 7:31