## Sunday, September 17, 2017

### DSGE Wars (Again)

There's been a small revival in the arguments around Dynamic Stochastic General Equilibrium (DSGE) models. My thinking on these models has evolved somewhat; I think they are not particularly interesting, but the arguments over there use are somewhat misplaced.

(I am getting ready to head out to the Modern Monetary Theory conference in Kansas City, so my comments here are brief.)

## Recursive Macroeconomic Theory

The approach by Ljungqvist and Sargent is much cleaner than every other exposition of DSGE mathematics that I have seen. They make clear the root mathematics behind DSGE macro -- optimal control theory. This is obviously attractive to someone with a doctorate in control systems theory.

## Optimal Control Theory versus Economist Mathematics

There is a stark difference between the presentation in Recursive Macroeconomic Theory and most "introductory" DSGE texts. The optimisation problems to be solved is clearly stated in Recursive Macroeconomic Theory, and the solution techniques generally follow accepted mathematical norms. Meanwhile, standard introductions right down objective functions, and blindly differentiate everything in sight, asserting that this leads to the optimal solution.

It is obvious that we cannot rely on taking derivatives of objective functions in the presence of constraints. For example, let us assume that we want to find the maximum of the function f which is defined by:
f(x) = -(x-10)^2.
If there are no constraints on the solution variable x, then we can differentiate f and show that x=10 as the maximum. However, this does not apply if x is constrained to lie on on the interval [0,1], in which case the maximum is x=1.

Therefore, we need to take the various constraints on the optimisation problem seriously, and that is not done in most of the treatments of DSGE macro I have run across. The proofs rely on the "Everybody Knows That Theorem," which is not a feature of real academic mathematics. Conversely, Ljungqvist and Sargent (generally) do not skip over the details.

In the next section, I will discuss the "standard" DSGE approaches, and then get back to Ljungqvist and Sargent's text later.

## Should We Care About DSGE Macro?

In practice, the DSGE macro methodology consists of:
• Decide upon the desired conclusions.
• Write down a complicated mathematical backstory that has lots of symbols and fancy equations.
• Assume that the backstory leads to the desired conclusions.
Although this sounds extremely harsh, it is the only way to describe aspects of DSGE macro such as the assumption that the level of interest rates is a key determinant of economic behaviour. In practice, this assumption is built into all mainstream models, and the empirical methodologies have no way of rejecting the assumption. It is not entirely an accident that the consensus has been shocked by the slow pace of recovery after modern recessions -- after all, it was believed that the level of interest rates was "unsustainably low." Indeed, the natural rate of interest had to be revised lower in order for the data to fit the theory.

In other words, the whole panoply of mathematics used is a gigantic red herring.

From the perspective of wanting to understand how the economy functions, there is only one real question: are the desired theoretical outcomes of DSGE macro practitioners useful? The fact that DSGE macro is roundly ignored by everyone whose job depends upon being right about the economy is probably the best answer to that question. (In theory, central bankers are supposed to care about being right about the economy, but in practice, even the raw incompetence displayed heading into 2007 did not cut into retired senior central bankers' subsequent speaking fees.)

From the perspective of academic economics, it is an obvious problem that this methodology has to be used in the "top journals." This is only a surprise if you assume that the academic system shows a tendency to progress towards the truth.

## Recursive Macroeconomic Theory?

The first chapter of Recursive Macroeconomic Theory is a "literary" description (their word) of how the authors see what the whole point of DSGE macro is. It does not appear to correspond to how pretty much everyone else describes the debate. It is hard to see the methodology as being obviously wrong, although that could be my bias as a result of my background in control theory.

I prefer to return to their arguments at a later time. However, I would argue that the weakness is the embedded assumption that the only way to approach mathematical problems like this is to cast them as optimisation problem. That was the psychology of control systems engineering in the 1960s, and I still see that attitude in some economists and people in finance. Basically, you say that you are finding an "optimal solution," and that is supposed to end all debate on the question whether the solution is any good. The reality is that any solution is the "optimal solution" for some objective function, and so we need to look for more sensible decision criteria. Control engineering dropped its infatuation with optimal control by the early 1980s, and so seeing economists favouring the approach is an intellectual time warp for me.

The debates about methodology are interesting, but unlikely to be resolved. As for the mathematical treatment, even Recursive Macroeconomic Theory gets squirrely when confronting the area that I have the most complaints about DSGE mathematics: equilibrium. In Section 7.3, recursive competitive equilibrium is defined. It starts with a lot of mathematics, but even then, some economist hand-waving sneaks into the definition. They use terms that do not appear to correspond to standard mathematical concepts, and hope the reader knows what they mean. Mathematics largely consists of statements about sets and the property of sets; it is unclear what set properties they are describing at key sections of their definition.

I need to return to the text later, but I just wanted to highlight that even the cleanest bit of mainstream economist mathematics relies on faith being extended by the reader.

## Concluding Remarks

It is unclear whether DSGE macro is worth studying. The mathematics is not clearly explained, and it is unclear whether it bears any relation to the time series being studied.

(c) Brian Romanchuk 2017

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