## Friday, April 17, 2015

### Macro prices are sticky, not micro prices

Two not very sticky prices ...

David Glasner, great as always:
While I am not hostile to the idea of price stickiness — one of the most popular posts I have written being an attempt to provide a rationale for the stylized (though controversial) fact that wages are stickier than other input, and most output, prices — it does seem to me that there is something ad hoc and superficial about the idea of price stickiness ...
The usual argument is that if prices are free to adjust in response to market forces, they will adjust to balance supply and demand, and an equilibrium will be restored by the automatic adjustment of prices. ...
Now it’s pretty easy to show that in a single market with an upward-sloping supply curve and a downward-sloping demand curve, that a price-adjustment rule that raises price when there’s an excess demand and reduces price when there’s an excess supply will lead to an equilibrium market price. But that simple price-adjustment rule is hard to generalize when many markets — not just one — are in disequilibrium, because reducing disequilibrium in one market may actually exacerbate disequilibrium, or create a disequilibrium that wasn’t there before, in another market. Thus, even if there is an equilibrium price vector out there, which, if it were announced to all economic agents, would sustain a general equilibrium in all markets, there is no guarantee that following the standard price-adjustment rule of raising price in markets with an excess demand and reducing price in markets with an excess supply will ultimately lead to the equilibrium price vector. ...
This doesn’t mean that an economy out of equilibrium has no stabilizing tendencies; it does mean that those stabilizing tendencies are not very well understood, and we have almost no formal theory with which to describe how such an adjustment process leading from disequilibrium to equilibrium actually works. We just assume that such a process exists.

Calvo pricing is an ad hoc attempt to model an entropic force with a microeconomic effect (see here and here). As I commented below his post, assuming ignorance of this process is actually the first step ... if equilibrium is the most likely state, then it can be achieved by random processes:
Another way out of requiring sticky micro prices is that if there are millions of prices, it is simply unlikely that the millions of (non-sticky) adjustments will happen in a way that brings aggregate demand into equilibrium with aggregate supply.
Imagine that each price is a stochastic process, moving up or down +/- 1 unit per time interval according to the forces in that specific market. If you have two markets and assume ignorance of the specific market forces, there are 2^n with n = 2 or 4 total possibilities
{+1, +1}, {+1 -1}, {-1, +1}, {-1 -1}
The most likely possibility is no net total movement (the “price level” stays the same) — present in 2 of those choices: {+1 -1}, {-1, +1}. However with two markets, the error is ~1/sqrt(n) = 0.7 or 70%.
Now if you have 1000 prices, you have 2^1000 possibilities. The most common possibility is still no net movement, but in this case the error (assuming all possibilities are equal) is ~1/sqrt(n) = 0.03 or 3%. In a real market with millions of prices, this is ~ 0.1% or smaller.
In this model, there are no sticky individual prices — every price moves up or down in every time step. However, the aggregate price p = Σ p_i moves a fraction of a percent.
Now the process is not necessarily stochastic — humans are making decisions in their markets, but those decisions are likely so complicated (and dependent e.g. on their expectations of others expectations) that they could appear stochastic at the macro level.
This also gives us a mechanism to find the equilibrium price vector — if the price is the most likely (maximum entropy) price though “dither” — individuals feeling around for local entropy gradients (i.e. “unlikely conditions” … you see a price that is out of the ordinary on the low side, you buy).
This process only works if the equilibrium price vector is the maximum entropy (most likely) price vector consistent with macro observations like nominal output or employment.
http://informationtransfereconomics.blogspot.com/2015/03/entropy-and-walrasian-auctioneer.html

1. Despite the frequent small changes, it seems to me that there is a clear price that the potato chips and spaghetti keep returning to. Maybe the prices aren't sticky in the fact that they change very rapidly, but that the firms only re-optimize every ~100 weeks. Even though the prices aren't technically "sticky", I'm pretty this could lead to some nominal rigidity. In terms of microfoundations, couldn't the data be replicated by simply adding a stochastic element to calvo pricing so that $$p_t = \theta \bar p_t + (1-\theta)p_{t-1} + \epsilon_t$$ where epsilon is a "sale" shock that occurs each period? (I'm not sure having the stochastic element would even be relevant to business cycle fluctuations anyway).

1. P.S. sorry about the equation, I couldn't find your settings for mathjax by using Chrome's debug menu

2. Hi John,

I think this line gets at what I am trying to say:

Even though the prices aren't technically "sticky", I'm pretty [sure] this could lead to some nominal rigidity.

That's what I am saying -- it would lead to nominal rigidity if they re-optimized every 100 days ... in fact, it would lead to nominal rigidity if they re-optimized every day. Nominal rigidity is a macro phenomenon that is independent of the micro behavior (as long as no firm dominates prices or there is a 'representative firm').

I have some pictures of what the macro/micro four scenarios (sticky/sticky, sticky/flexible, flexible/sticky and flexible/flexible) here:

http://informationtransfereconomics.blogspot.com/2015/04/micro-stickiness-versus-macro-stickiness.html

And I agree, I'm sure you could add a Calvo mechanism that produces the exact micro fluctuations we see in the graphs above and lead to macro prices being sticky. My main point is that this isn't necessary ... sticky prices come for free (without any micro mechanism) if you aggregate using a maximum entropy framework.

PS You have the mathjax right I just don't always add the js to each page depending on what platform I'm using to write -- and I had some issues trying to put it in the template.

3. With regards to mathjax, putting the javascript in the post template does nothing, but if you paste the javascript under the first head section of the HTML for the whole site it should work. Just make sure to avoid numbering your equations or mathjax will number your entire blog which I'm sure you can imagine is a pain.