The lack of uniqueness of Arrow-Debreu equilibria

A visualization of a macroeconomic equilibrium.

In the "ensemble of many markets" picture where the diagram above indicates a "statistical equilibrium" of a set of growth states, we have a definition of a general economic equilibrium that is describable in terms of a partition function.

First, this is one way you can make sense of economic equilibrium that Steve Keen says is impossible using an overly strict mathematical construct to represent neoclassical economics. Equilibrium is not a case where all relative prices are always the same (a ridiculous definition that we can see is not true of even a normally operating economy by inspection because e.g. sometimes things go on sale), but rather a case where the distribution of prices remains relatively unchanged.

We can observe examples of the distribution in profit rates, the prices of goods, as well as stocks, for example. Disequilibrium can be seen as strong deviations from the distribution (illustrated in the stock example) and macroeconomic forces as entropic forces (here, in the paper, or here as causal entropic forces) maintaining it.

Now let's go to a bit I wrote about the lack of uniqueness of the Arrow-Debreu-McKenzie (ADM) equilibrium before:

Let's look at what the ADM equilibrium says with regards to a partition function in thermodynamics. It effectively says there exists some set of occupation numbers [i.e. growth states] so that the energy of the system is the total energy, or more generally, there exists a microstate consistent with an observed macrostate. The SMD theorem then tells us that there are only limited properties of that microstate that survive to the macrostate. ... The other consequence of the SMD theorem should also be intuitive. If your macro system appears to be described by n << N degrees of freedom, then it seems highly likely that among the total number of microstates, large subsets of the microstates are going to be described by a given macro state -- i.e. the equilibrium (the microstate satisfying macro constraints) is not going to be unique.

Basically, since every re-labeling of the boxes in the diagram above is another macroeconomy with the same growth state distribution, every re-labeling represents an equivalent macrostate. Since each growth state (i.e. IT index k state) also indicates a price growth state [1], the price clearing vector is highly non-unique. This is a good thing, too. It means that macroeconomics is somewhat easier than applied microeconomics ‒ it's a different theory (like particle kinematics versus the ideal gas law). Additionally, consistent with the SMD theorem, the detailed properties of the microstates (the individual boxes) are not absolutely necessary to describe the macrostate. (It also means that agent based modeling is fine, but unnecessary.)

Instead of a single price vector pointing motionless in a single direction, we should visualize a rapidly changing price vector with elements drawn from a stable distribution (possibly even a Stable Distribution per here).

...

Footnotes

[1] If $n_{i}$ is the nominal output of the $i^{th}$ market with common "factor of production" ($m$ money, or it could be labor $\ell$), we have:

$$
\begin{align}
\log n_{i} \sim & k \log m\\
\log p_{i} \sim & (k - 1) \log m
\end{align}
$$

Basically, the distribution of $k$ states describes both the nominal output of the $i^{th}$ market as well as the price $p_{i}$ of the good in that market.

PS Snow day! Seattle tends to be useless with even a few inches of snow, so I took the day off.

Randomly generated economies (a work in progress)

I have been in the process of constructing randomly generated unemployment time series based on the dynamic equilibrium model (partially inspired by writing my qualitative analysis post). I'm still in the process of working out the parameters (or rather, the distributions the parameters are drawn from). However, the work in progress looks neat on its own.

Here's what I mean by a randomly generated time series: effectively a series of Gaussian shocks with random centers, amplitudes, and widths. 

Here is the actual unemployment rate time series (red) alongside a randomly generated path (blue):

In constructing the model (still in progress), I came across some interesting mistakes. For example, if you (accidentally) use a Poisson distribution to yield integer recession transitions:

If you run it for 500 paths you get this:

And here is what happens if you use quarters:

It would be fascinating if there was some resonance with the annual cycle ... Again, this is a work in progress. I will keep you updated.

Qualitative economics done right, part 1

This post is Part 1 of a series. Part 2 (on Keen), part 2a, and part 3 (on Godley) are also available. Added in December of 2020: Part N (on Farmer).

I entered into a discussion with UnlearningEcon on Twitter about qualitative analysis in economics. Different people mean different things by "qualitative" ranging from vague handwaving to detailed order of magnitude estimates of magnitudes and directions of effects. However since our discussion concerned the models of Steve Keen and Wynne Godley, which are both mathematical models, I think it is safe to take qualitative analysis to mean:

• Identifying scales (size) separating large effects from small effects
• Identifying the scale (size) of error and noise in the data
• The order of magnitudes of effects
• The direction of effects
• The relevant variable dependencies of effects

Qualitative analysis plays a major role in theory development. All theories must go through four tests:

1. Qualitative description of the data
2. Qualitative predictions
3. Quantitative description of the data
4. Quantitative predictions 

Now you could switch 2 and 3 in the ordering. And you could jump right to 3 and 4 (essentially using 3 and 4 to "test out" of 1 and 2). Sometimes you never hear about a theory because it fails 1 or 2. Sometimes you only hear about it when it reaches 3. However, under no circumstances should you ever start with 2. A theory that does not describe data even qualitatively should not be validated by qualitative predictions, and theories that do not describe the data qualitatively should not be used to make qualitative predictions. 

I think a lot of economic theory out there from RBC and DSGE to SFC and MMT, from the mainstream to the heterodox is suffering from something I'd like to call "data disease" using a metaphor with Baumol's cost disease. With the increasing availability of economic data, the productivity of theory (how much data a theory of a given complexity can explain) has noticeably decreased. Cobb and Douglas had very little data, but the theoretical model quantitatively captured a lot of the variation of the data:

That's two parameters for >20 data points over > 20 years. Today we have things like this (reference) that have 40-50 parameters looking out over three years:

Cobb and Douglas wanted you to know their result was quantitative. I think we're supposed to look at the the latter model as a qualitative model. I know; the latter model works out a lot more variables. I'd actually like to commend them on comparing the results to data at all! That was the first thing to go as a result of "data disease". And because the productivity of mainstream theory became so low, a lot of non-mainstream theorists decided they also didn't need to compare theory to data. I mean, why hold yourself to a higher standard than mainstream economists hold themselves?

I'd like to push back against this tide of pretending knowledge where none actually exists. Saying a model is a qualitative model ought to mean something. At least it should mean something besides "please don't compare this model to data". So what should qualitative description of the data mean for economics? 

The "mainstream separation"

There are a couple of possible starting points that I think can be illustrated using employment data. The first is what I'll call the "mainstream separation". As frequently happens in physical theories, in this case there is a scale that separates large effects (e.g. long run economic growth) and small effects (fluctuations, e.g. the business cycle). In economics, this actually separates two sub-fields (growth economics and macroeconomics). Qualitative analysis in this picture sets up one or more scales for "the long run" versus "the short run", as well as a scale for fluctuations and the size of expected error ‒ I drew a diagram using the unemployment level to illustrate this:

In this picture it can actually make sense to talk about fluctuations separately from economic growth (i.e. the two sub-fields don't have to talk to each other much). There could be one theory for the long run (e.g. Solow model) and another for the short run (e.g. DSGE). Additionally, the errors are separable as well. The errors in the former are set by the scale of the latter theory (and the latter's errors are irrelevant to the former). One theory's signal is another theory's noise.

However, if you've made this separation, your short run model is only valid (only has scope) in the neighborhood of the long run equilibrium. The rational expectations in your DSGE model may then only be valid if the fluctuations are small (discussed more extensively here).

This view can easily encompass most ideas from Minsky's credit cycles to Real Business Cycle models (RBC) using random AR processes. That is to say that this picture doesn't really eliminate a lot of theories since the next picture can be represented as this picture at least locally.

Multi-scale holism

The other possibility is that your long run and short run scales are too close to each other to be separated, or there exist many different scales. This approach sees a single complex system (nonlinear dynamics, or linear dynamics with multiple scales) and the relevant diagram looks like this:

In this view there is no independent "business cycle" and "the long run" and "the short run" cease to have well-defined meanings. The fluctuations are part and parcel of the long run process.

There is a particular interpretation of Minsky's credit cycles that sees credit cycles as having scales on the order of a generation (e.g. culminating in the Great Depression and Great Recession). You could even posit a self-similar fractal system where there are multiple scales from generational down to the typical time between recessions of about 8 years in the US.

Aside: dynamic equilibrium

One thing I'd like to point out is that the dynamic equilibrium approach I've been looking at recently is mostly an attempt to understand which of the two approaches above is the most economical way to understand the data and in the case of the former what the long run equilibrium is. For employment, the system really looks like a single dynamic equilibrium subjected to stochastic shocks that is deeply connected to matching theory.

What does the data say?

First, since we really only have about 60 years of decent data, we only have information about regular cycles that have a period of less than 20 years. You can immediately discount e.g. Kondratiev waves as a theory that either will be considered lucky (much like Democritus's atoms or Newton's photons) or pareidolia. This is a fundamental result of mathematics and information theory; enough information to determine long regular cycles simply does not exist.

The way out of this limit is to say we are not looking at regular cycles, but rather stochastic "cycles" (e.g. autoregressive processes). In that case, the limit of complexity is set by the number of data points. With at best monthly data over 60 years, that is 720 data points limiting complexity to about 36 parameters for the entire series (20 data points per parameter is a good heuristic for a qualitative analysis ‒ we'd expect an error on the order of 20% ~ 1/√20).

I pause to note that the dynamic equilibrium description of the unemployment data uses 22 parameters (one for the dynamic equilibrium plus 3 for each of seven shocks).

You may ask what the details of the data and the number of parameters in the theory have to do with a qualitative analysis. Let me illustrate with another diagram:

Theories above the heuristic complexity limit (i.e. cycles greater than 20 years and theories with more than about 40 parameters) lose explanatory power through overfitting (too many parameters given the data) or unwarranted extrapolation (which is actually just another kind of overfitting that uses too many Fourier components). Theories below the heuristic complexity limit have some explanatory power. However: theories pushing up against the complexity limit have sufficient complexity (a sufficient number of degrees of freedom) that they should be able to fit the data fairly well. Therefore we should expect a complex theory to be a quantitative theory. Qualitative theories that push the complexity limit are seriously underperforming.

This is why I tend to chuckle when 40 parameter DSGE models looking at 10 years of data are purported to be considered qualitative ‒ just a way to organize thinking. It would probably be best to re-organize your thinking around a completely different paradigm.

The data also tells us that the scale of macroeconomic fluctuations is on the order of a few percent for most large economies (the US, EU, Japan, etc) ‒ for example the unemployment rate fluctuates between 5 and 15% nearly all of the time. This means that with natural coefficients an expansion around a given state x (equilibrium) will be of the form

f(x) ~ c₀ + c₁ δx/x + c₂ (δx/x)²

and it will be good quantitatively to about the 10% level keeping just the (log-)linear terms. This sets a pretty high bar: most of the data can be quantitatively described by a fluctuations around a couple of log-linear macro equilibria (simple AR/ARMA processes do pretty well at forecasting too).

The performance of a simple line on a log graph along with the short length of decent time series data come together to tell us that the two pictures above (the mainstream separation and the multi-scale holism) cannot be distinguished at the qualitative level. They also cannot be distinguished at the quantitative level unless your precision reaches the 1% level (e.g. the second order terms above). This was the key point of Roger Farmer's argument against non-linear models that I discuss here:

But there is currently not enough data to tell a low dimensional chaotic system [ed.: nonlinear system] apart from a linear model hit by random shocks. Until we have better data, Occam’s razor argues for the linear stochastic model.

Summary

So where does this leave us? What should a qualitative economic model that fails to be a quantitative model look like? A qualitative model ...

• ... should have only a few parameters.
• ... should have a linear or log-linear macro equilibrium (or have one as a limit).
• ... if it has cycles, those cycles should have periods of at most 10-20 years.
• ... if it is just a theory of macro fluctuations, it should be qualitatively consistent with the shape of the fluctuations.
• ... if it is a nonlinear theory, it should be consistent with the full time series data at the 10% level (i.e. have the aforementioned log-linear limit).

In part 2 and part 3, I will show how both Steve Keen's and Wynne Godley's models (the ones UnlearningEcon suggested I accept as qualitative models, but not quantitative ones) fail to meet this standard. Failing to meet this standard isn't necessarily conclusive. Most of the arguments above provide heuristics. A simple example of a similar heuristic you are probably familiar with is a p-value. We tend to look for p-values near 1% or 5% because that was set up as a heuristic. Now the heuristic became a problem in journals when it became codified (leading to p-hacking), so we shouldn't just blindly use heuristics to cut off discussion of some subjects (with p > 5%) and focus on others (with p < 1%). The idea behind the p-value heuristic is to do science ‒ i.e. ask questions. Does the experiment really test what it says it tests? Is there a sample bias? The thing is that we forget a p-value that is exceedingly low is also a sign of potential problems.

In the same vein, failure of qualitative models to meet these heuristics just means we should ask a series of questions:

Why aren't these models delivering quantitative results? 

Why are these models under-performing simplistic log-linear stochastic models? 

What is the model complexity helping us achieve? 

What is the model's scope? 

What does the model tell us we should get wrong versus what it actually gets wrong?

However there are also two questions we definitely should not ask: What are the policy implications? and What does the model predict?

Worthwhile Canadian prediction comes true

A tweet from Lars Christensen reminded me of a prediction I made over two years ago. Christensen said:

[Vancouver home sales falling] is not the result of a "bubble bursting" - this is Bank of Canada failing to deliver on its 2% inflation target.

In my prediction, I said that Canada would start to undershoot its inflation target due to factors beyond the control of the central bank. Here's me:

I'm going to put forward a prediction, using the information transfer model, that Canada will either undershoot its inflation targets or will have produced significantly more currency than the current log-linear trend. ... The interesting thing about this prediction is that it should start to become apparent by the end of next year (Dec 2015).

Well, the currency (monetary base) is right at its log-linear trend. So where is CPI? Well below that 2% trend ...

This undershooting effect is exactly as expected for an ensemble average of markets as presented in my paper:

This is a pretty big success. The US was already undershooting an implicit inflation target back in 2014, so saying Canada will start to undershoot in the future is a genuine test of the theory. Additionally, the reason for the undershooting has nothing to do with monetary policy ‒ there are simply more ways a complex economy can be organized as many low growth markets than as a few high growth markets. This means the former becomes the more likely state as an economy grows. A more thorough and technical explanation can be found here.

Update + 20 minutes

I would also like to say that this success is also a major failure of most other approaches from theories where central banks set expectations, to quantity theories (well, unless they're modified quantity theories like the information equilibrium model), to Taylor rules. It's because this undershooting was predictable with zero weight placed on any of the actions of the central bank in the model.

Heterogeneous labor supply shocks

FRED posted a tweet linking to a graph of the unemployment rate by education level. I thought I'd try the dynamic equilibrium model out on it:

One of the interesting things is that while the dot-com bust and the housing bust/financial crisis hit all levels at roughly the same time (vertical lines in 2001 and 2008), the 2014 boom hits the lowest education levels first, followed by the higher levels. That boom might have already passed for those with less than a high school diploma (and might be a leading indicator of a future recession).

Another interesting thing is that the logarithmic rate of decline is roughly the same in each case (from highest education to lowest it is 0.087, 0.070, 0.091, and 0.081 (multiply these by 100 to get a rough estimate of percent decline per year). That is to say that while the shocks may appear at different times, the process of the unemployed finding work is roughly the same for all education levels.

[Update 10 October 2018: In thinking about this a bit more, while these are roughly the same, the matching rate for "some college" (0.070/y) is significantly less than the others. Discussed more here.]

...

Update 3 February 2017

I just wanted to note that the U6 unemployment rate tells basically the same story (rate of decline is 0.087 just like the others):

This means that given the U3, I could derive the U6 rate (as well as the rates for different education levels).

Unemployment forecast update

With the new unemployment data out today, I thought I'd update the forecasts I've made. The original dynamic equilibrium model (which didn't use the logarithm) continues to fall in uncertainty (I put the two graphs together so you can toggle between them ‒ the new one is second/on the right):

I also added a deviation forecast (like the one above) to the improved dynamic equilibrium model. Since I didn't show this before, I thought I'd show several different versions choosing a different baseline year from 2016 to 2016.8 in steps of 0.1 (i.e. fit the data up to the baseline, then look at deviations afterwards). Thankfully, the result is relatively stable (it just becomes more uncertain if the baseline is later). The red curve is the baseline fit (with 90% confidence intervals as the red band). The gray line is the deviation fit (the horizontal line indicates the "onset" of a new shock); it shows 50-70-90 percent confidence bands. The deviation fit always uses data from 2015 to the present, it is just the baseline fit that causes these to be difference.

Nearly all select a mid-2017 center for the recession shock (consistent with the naive linear model at the top of the post). Of course, the unemployment rate has paused and then continued downward before. But as I noted in the post linked at the top of this post, these metastable states usually don't persist for very long.

...

Update: I'm going to use 2016.3 as the "official" version instead of updating multiple graphs. The three graphs 2016.1, 2016.2 and 2016.3 are stable, they all show the greatest separation from the counterfactual (red), and the last one has the least amount of data it is being projected from. Basically, if it's going to be wrong, then the 2016.3 will become wrong the fastest. Also, the graph of the data looks like it breaks between the straight line piece (blue) and the beginning of the curve (yellow) pretty well.

 

Monetary base and interest rate forecast updates

I am updating some forecasts that I last updated here, here, and here (see those posts for descriptions, but for the most part they are pretty self-explanatory). One thing to note is that while the monetary base has been pretty slow to respond to the interest rate hikes of Dec 2015 and Dec 2016, it nonetheless is pretty clear that its path has changed direction as predicted by the IE model.

A desired wealth to income ratio as a dynamic equilibrium

Part of stock-flow consistent modeling that doesn't actually have much to do with stock-flow consistency (see here) is positing that agents have a desired ratio of wealth $W$ to income $I$. Some unknown source with a Mark Thoma- or Scott Sumner-scale following recently linked to that post and that has suddenly made it the second most viewed post on my blog ever. It's not clear whether it was in a positive or negative context (regardless: thanks for the link!), but in any case it reminded me of the wealth to income ratio at a time when I've been looking at several different applications of the information equilibrium/dynamic equilibrium approach.

A desired constant wealth to income ratio $W/I$ would be

$$
\frac{W}{I} = \frac{W}{dW/dt} = t_{0}
$$

Note that the constant would have units of time: [dollars]/[dollars/time] = [time]. You can think of it as a "time horizon". For example a $1/t_{0}$ = 0.05/year (5% growth) represents a time horizon of 20 years. We can solve this differential equation:

$$
\begin{align}
W(t) & = W_{0} e^{t/t_{0}}\\
I(t) & = \frac{W_{0}}{t_{0}} e^{t/t_{0}}
\end{align}
$$

Note that

$$
\begin{align}
\frac{d}{dt} \log W(t) & = \frac{1}{t_{0}}\\
\frac{d}{dt} \log I(t)  & = \frac{1}{t_{0}}
\end{align}
$$

and

$$
t_{0} dI = dW
$$

So we have an information equilibrium relationship with $k = 1$

$$
\frac{dW}{dI} = \frac{W}{I}
$$

This means that

$$
\frac{d}{dt} \log \frac{W}{I} = \frac{k - 1}{t_{0}} = 0
$$

making it a special case of dynamic equilibrium/information equilibrium. So what happens if we try to fit the data (this, this, and this) to a dynamic equilibrium plus a series of shocks? We don't need to do the entropy minimization process since we already know the slope should be zero. The result involves a lot of shocks:

What is interesting is if we overlay the positive and negative shocks to the stock market (SP500 shocks, blue) and the positive and negative shocks to housing prices (Case-Shiller index shocks, orange), a general picture emerges (positive shocks come up from the bottom, negative down from the top):

If there is a desired wealth to income ratio, it tends to be afloat on a sea of markets. The decline from the 1960s to the 70s is associated with the general stagnation of the stock market. The boom and bust of the late 90s and early 2000s is the dot-com bubble. And finally, the boom and bust of the later half of the '00s is the boom and bust of the housing bubble (and subsequent financial crash). The 2013 rise in W/I might be associated with a rise in Case-Shiller index relative to the dynamic equilibrium around the same time. There is also a hint of a stock market rise around the same time. It could be a bit of both. I didn't think either of these were significant enough in the individual time series to warrant inclusion, but the additional evidence from the wealth to income ratio makes me re-think that.

However, the data appears to show that the causal mechanism is that fluctuations in asset markets (housing, stocks) cause the wealth to income ratio to change. There does not appear to be a "restorative force" keeping it at a specific level (a desired W/I ratio).

Additionally, this is written entirely in terms of information equilibrium rather than as a stock-flow consistent model. The only time I used any accounting is when I calculated net wealth to be assets minus liabilities. However, even this isn't necessary as the same story can be told using assets instead of wealth (A/I):

I am sure I will get comments from the SFC club that say the data series I am using is wrong (it includes non-profit organizations, and income is just disposable income) or that income includes revaluations of assets (note that I actually wrote it out that way first, but it reduces to the form above). Feel free to point me to the correct data sources -- but at least see how well the "correct" W/I ratio corresponds to the W/I ratio I show above first.

Aren't we looking for pluralism, anyway? Isn't it interesting you can analyze the macro effects and the wealth to income ratio with a completely different approach?

...

Update

I wanted to note that the information equilibrium condition is less restrictive than the wealth-to-income ratio equation (first equation at the top of this post). The former effectively allows any function $I(t)$:

$$
W(t) = c I(t)
$$

where

$$
I(t) = \frac{W_{0}}{t_{0}} e^{t/t_{0}}
$$

is a special case.

What about the S&P 500?

The S&P 500 is a price index, right? How well does the dynamic information equilibrium approach (previous link) describe the S&P 500? Remarkably well, actually:

There are only four major "shocks". Three are negative: 1971.9,  2001.7, and 2008.5. The former is a very broad slow shock that may well be related to the high inflation of the 1970s (that's pure speculation on my part at this point). The latter two are the so-called "dot-com bust" and the global financial crisis.

There is one positive shock associated with the dot-com boom: 1997.6.

The second graph smooths the S&P 500 data slightly (a local average derivative) since the index is noisy and well-resolved (daily measurements).

But really, the stock market is highly complex [1].

...

Update 1 February 2017

I decided to do a bit more with this. If we look a the difference between the series (difference of the logs, i.e. a geometric random process) we can see the dynamic equilibrium takes out nearly all of the drift:

As expected, it passes all the unit root tests (although they have low power). What if we use this series to estimate a process? Mathematica chooses a second order ARMA process ‒ ARMA(2,1) to be precise ‒ if you use TimeSeriesModelFit with default options on the last 7 years of data [2]. Taking that forecast into the future we basically figure out that the S&P 500 should return to the dynamic equilibrium trend:

So now we have a test. Since I only grabbed Mathematica's financial data up to 1 January 2017, we actually have a few data points (black, above) to show against the forecast already. Here's a zoomed-in version:

...

Update 1 February 2017, the second

I added a couple more potential "shocks" to the S&P 500 model (gray ball and sticks) and looked at the unemployment shocks (green ball and sticks), the Case-Shiller shocks (orange ball and sticks), as well as the NBER recessions (blue):

Around 1970, 1974, 2001 and 2008 we have pretty good alignment of an NBER recession, an unemployment shock, and an S&P 500 shock. The 1980s and 90s show unemployment shocks associated with NBER recessions but not S&P 500 shocks.

The sizes also don't match up completely. The 1974, 2001 and 2008 S&P 500 shocks are "large", but all of the unemployment shocks are of comparable size. Basically, the 1980s and 90s recessions happen without a large signal in the S&P 500. Therefore it's hard to say there is causality happening in either direction, but rather just a loose association.

However the 1980s and 90s recessions (as well as the one in 2008) are associated with shocks to housing prices (1974 probably would be as well given the longer Case-Shiller time series).

There's no real conclusion to be drawn from so few events. We can just generally say that most recessions are associated with falling housing prices, rising unemployment, and a falling S&P 500. This is not very illuminating. 

I should also show the derivative picture for the additional shocks:

...

Footnotes:

[1] See here and here.

[2] The results are pretty robust to fiddling around with the length of data used to estimate as well as restricting to ARMA processes.

Housing prices and dynamic equilibrium

I apologize if the dynamic equilibrium [1] posts are getting monotonous, but as the blog's primary purpose is as a "working paper" (one that is now apparently a few hundred pages long) I must continue!

The latest Case-Shiller price index data was released earlier today showing a continued rise in housing prices. In looking at the data, I noticed it has the telltale signs of a a dynamic equilibrium in the presence of shocks. However as the previous derivation looked at ratios of quantities in information equilibrium, I thought I needed to expand the theory a bit.

If we have housing demand $H_{d}$ in information equilibrium with housing supply $H_{s}$ with abstract price $P$ (i.e. $P : H_{d} \rightleftarrows H_{s}$), we can say:

$$
P \equiv \frac{dH_{d}}{dH_{s}}  = \; k \; \frac{H_{d}}{H_{s}}
$$

We can solve the differential equation to obtain

$$
\begin{align}
H_{d} & = \; H_{d}^{(0)} \left( \frac{H_{s}}{H_{s}^{(0)}} \right)^{k}\\
P & = \; k \frac{H_{d}^{(0)}}{H_{s}^{(0)}} \left( \frac{H_{s}}{H_{s}^{(0)}} \right)^{k-1}
\end{align}
$$

Now if housing supply grows at some rate $r$ such that $H_{s} \sim e^{rt}$, then

$$
\frac{d}{dt} \log P \approx \; (k-1) r
$$

Note that this is basically identical to the result for the ratios of quantities in information equilibrium in [1]. This should be apparent because the RHS of the first equation above is such a ratio and the LHS is the abstract price. Now let's use our procedure in [1] and say that the Case-Shiller index is our abstract price. The results are pretty decent:

The vertical lines again represent the centroids of the shocks. The negative shocks are at 1982.5, 1993.1, and 2007.7 (each associated with recessions). The positive shocks are at 1978.5 and 2005.6 (likely the California housing bubble and the global housing bubble, respectively).

...

PS I did want to note that we get increased prices with increased supply per the equations above. That is because we are assuming equilibrium (general equilibrium). If the housing supply increased quickly relative to housing demand, then we would get the standard economics 101 result. I discussed this more extensively here.