My comments here are tentative, as I am in the process of working through the mathematics of the paper. My objective is to spend some time with models like the one in this paper in the second volume of

(c) Brian Romanchuk 2020

*Recessions*.## Model Microstructure

The underlying story of the model is similar to that of other DSGE models, which is not a strength from my perspective. A few issues stand out from my perspective.

- This is an overlapping generations model. Since the time frame of the model is unknown -- how long is "one generation"? -- it cannot be fit to data. However, any alternative that is mathematically similar would be far more complex, and harder to understand. Since my objective is to discuss the model qualitatively, I am interested in an easier-to-explain model, and one that it will be easier for a reader to look at independently.
- There are the usual concerns about equilibrium. I will ignore the metaphysical aspects of "equilibrium," and just translate that term to "model solution." The term model solution is a concrete mathematical concept, and can be translated to the real world (does the model solution look like observed data?).
- There are niggles like having a continuum of firms. A continuum of firms cannot be thought of as taking the limit of a finite number of firms, and so the microfoundations cannot be interpreted in the real world. As far as I can tell, the continuum of firms is just a backstory to explain how all entities end up being price takers.

Economists have spent an incredible amount of time and effort discussing equilibrium. All that literature should be burned to the ground, and replaced with concrete mathematics. From that perspective, the issue that matters is: does the mathematical system have a solution, and is that solution unique?

If we want to relate a mathematical model to the real world, we need there to be a solution.

- If there is no solution, the model tells us nothing. There are an infinite number of mathematical systems that have no solution. We need to be able to relate model solutions to observed data, and see whether there is any relationship we can discern.
- If there are multiple solutions, there are two problems. The first is practical applied mathematics: unless the model is very simple and can be solved by hand, numerical algorithms will generally not converge to any solution. The solutions need to be "isolated" in some sense, so that the algorithm will converge to at least one. The second issue is that we need to relate observed data to the model solution. If there are multiple solutions, which one do we compare to real world data?

In the case of the Farmer/Platanov model, most of the equations of the system are similar to standard DSGE models. However, those equations do not provide enough constraints to pin down the solution. An added constraint needs to be imposed to ensure uniqueness.

From the perspective of standard applied mathematics, such a situation is completely unremarkable. However, economists insist on discussing "equilibrium," and adding the constraint is framed as "picking an equilibrium." Since mathematical systems are ultimately a set of mathematical statements about sets (such as the set of time series, which are elements in the set of sequences), we should not project human-like behaviour like "picking" onto them.

## A Digression About Equations...

If I were to submit to an academic journal, I could typeset equations easily in LaTex. However, I am writing handbooks where about half of my sales are ebooks. The retail outlets that sell fixed format ebooks -- which are essentially PDFs -- are very limited. So I want to stick with reflowable text -- in which chapters of a book are essentially web pages. It is possible to do simple equations like:

*x = 2y + 3,*

and even include some Greek letters, and superscripts/subscripts. However, that would not be enough to cover the Farmer/Platonov model.

My preferred solution is to create "boxes" with the equations and explanatory text. These can be typeset, and then converted into an image that is embedded in the book. However, creating them is fiddly, and one of the last things to be done in a manuscript. For now, I will just verbally hand-wave about equations, and the reader is free to consult the article to see the glorious equations.

## The Belief Function

Farmer and Platonov describe the belief function as follows:

The belief function distinguishes our model from the New Keynesian approach and it replaces the New Keynesian Phillips curve. In the absence of this new element, the other six equations would not uniquely determine the seven endogenous variables [variables omitted -BR].The belief function is an equation that determines how much households are willing to pay for claims on the economy’s capital stock.[Emphasis mine - BR] It represents the aggregate state of confidence or ‘animal spirits’ and, in combination with the other six equations of the model, the belief function selects an equilibrium.

The meaning of emphasised sentence is not obvious since I skipped over writing out the full model specification. The model is simplified versus the real world, and so one should be careful about drawing implications.

The stock of capital is fixed (which eliminates the theoretical issues of how capital is measured), and the entire stock of capital is owned by the "old" generation, which is sold to the "young." These sales are used to finance consumption by the old generation, which does not receive wage income.

This simplification is critical for the model. Since the entire stock of capital has to be exchanged in one vaguely defined time period, the effect of valuation changes is much simpler. In the real world, only a small fraction of ownership shares are exchanged in a time period, allowing for much more complex asset price behaviour.

## Static (Steady State) Analysis

The easiest part of the results to describe is the steady state analysis, which can be viewed as a "static" model such as seen in Economics 101 textbooks. The model is called the IS-LM-NAC model, which as the usual IS and LM curves, as well as the NAC curve. The NAC curve equates the return of capital to the nominal interest rate.

There are three equations with 3 unknowns: real GDP, the nominal interest rate, and the price level. These are determined by the equations, along with two variables: the money supply, and the "animal spirits" variable (denoted by the Greek capital theta: Θ).

I am highly averse to the IS-LM model and its variants, but I am likely in the minority, so this would probably be of interest.

## Dynamic Version

The equations of the dynamic model are indeterminate. They state that there is "a one-dimensional continuum of dynamic paths" that are consistent with a given steady state. (There were no details provided as to how this was determined or the mathematical specification of such a continuum; I would need to chase after references.)

An added constraint needed to be added to ensure uniqueness: that the realised future goods price is equal to the expected price. This implies that prices are set one period in advance.

Given that added constraint, it is possible to see the effects of various shocks to model variables.

## Advantages Versus New Keynesian Models

The claimed advantages for the model versus standard approaches include two points of general interest, as well some more technical ones.

- Persistence of unemployment is better explained.
- Output is dependent upon the beliefs about the future value of the stock market.

These are of importance in discussing the business cycle (which explains why I am looking at this class of models).

## Drawbacks

The immediate drawback for my purposes is that I am interested in the analysis of recessions. Recessions typically last less than a year -- not a "generation." There is no way that we can truly fit this particular model to data and see what it says about recessions. I would need to look at other articles by Roger Farmer if I wanted to attempt such an exercise. However, the simplicity of the model does make it easier to explain (particularly the static version), which I need for my book.

Stepping away from my needs, the next issue I see with this model is that fiscal policy does not exist. All production is being split between two generations, there is no government consumption. This limitation is related to the dynamic assumption of prices being set one period in advance: price level shocks due to fiscal policy are impossible. That is, the government can increase its demand for goods in the present by any amount, and there is no immediate effect on prices. From the perspective of Functional Finance, this means that the model cannot be used to discuss fiscal policy, even if government consumption is shoehorned into the model.

The final concern I have is in the interpretation of the model. The model has a mechanism to ensure that the market valuation of capital is tied to the real economy, which is conflated with the stock market. In the real world, there is no hard link between the stock market and the real economy. The ownership of the entire stock market is not turned over periodically, which means that the market value need not correspond to the value of underlying assets.

Although I am a believer in the need for an "animal spirits" variable inside macro models, my argument is that the stock market is an imperfect proxy for this variable. In particular, even if the authorities prop up the stock market, there is no reason to believe that real activity must follow.

## Next Steps...

This is likely to be my last theoretical post of the year. I will return to doing a final editing pass on my manuscript, and then move to the next steps of the publishing process. I will be taking a look at other article by Roger Farmer, and see whether there are other candidates to do a deeper dive into. Once that is done, I hope to lay out a sequence of more detailed articles that would be part of the second volume of my book on recessions.

*(To recapitulate my objective, I covered heterodox economic theories about recessions in volume one. I wanted to have a chapter or two on mainstream approaches. The issue with neoclassical macro -- as was recognised by neoclassicals themselves -- as the core theory did not have a deeply convincing theory of recessions. Extra theory needed to be bolted onto the core to generate recognisable recession dynamics. In my view, Roger Farmer's work is one of the more interesting candidates, and I want to focus on that, rather than attempt to survey the various strands of theory that have popped up since the Financial Crisis.)*

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