Thursday, September 13, 2018

What do equations mean?


Arjun Jayadev and J.W. Mason have an article out on INET on what MMT purportedly is along with the basic fact that MMT policy prescriptions do not differ too much from that advised by the average macroeconomist in surveys. My post on this had been sitting around in my drafts since the day I read their article, but was boring (to me) and seemed likely to set off the MMT hive-mind. However, Jo Michell put up a thread about it and there was a subsequent discussion between he and  J. W. Mason that brought up an interesting general question: What do equations mean?

Michell says that it looks like you can get the same results Jayadev and Mason give from a standard DSGE treatment — the resulting equations in both cases are formally similar. I agree with that entirely. After bristling from being told the results are formally equivalent to a DSGE model (the three equation New Keynesian DSGE model), Mason says:
“A can be derived from B (with various ancillary assumptions)” is not the same as “B is just a form of A”, any more than fact that a text can be translated into Latin means it was written in Latin all along.
In a sense, I agree with that as well! I'll get into it later — first I want to talk about what Mason said next (I added some brackets to add context back in because it's lost when quoting a single tweet out of a thread):
People like Cochrane, who [r]eally do believe the sacred texts [i.e. microfoundations], are appropriately scathing on this [i.e. that it's not just the final form that matters]
with a link to a blog post by John Cochrane. Cochrane claims that Old Keynesian (OK) and New Keynesian (NK) models may have the same policy prescriptions, but have entirely different mechanisms leading to them. In the OK model, government spending increases income and each household's marginal propensity to consume means that increased income yields more output than the original government outlay (a "multiplier"). In the NK model, the additional output arises because government spending increases inflation and households spend now because their income in later periods is going to be worth less because of that inflation. It's essentially due to the equilibrating effect of the "Permanent Income Hypothesis" (PIE). The best blog post on this kind of legerdemain was at Mean Squared Errors, calling it "Houdini's Straitjacket":
Consider the macroeconomist.  She constructs a rigorously micro-founded model, grounded purely in representative agents solving intertemporal dynamic optimization problems in a context of strict rational expectations.  Then, in a dazzling display of mathematical sophistication, theoretical acuity, and showmanship (some things never change), she derives results and policy implications that are exactly what the IS-LM model has been telling us all along.  Crowd — such as it is — goes wild.
The NK DSGE model comes up with the same policy prescriptions as the OK model by essentially escaping the straitjacket of microfoundations.

Of course, John Cochrane is being disingenuous — the requirement for microfoundations was created explicitly to try and prevent fiscal policy from having any effect in macro models. The NK DSGE model comes along and says it can get the same results even if it plays his game. Cochrane complaining that the model doesn't get the same result for the same reasons is effectively admitting that the microfoundations weren't some hard-nosed theoretical approach but rather just a means to stop fiscal policy from having an effect.

When Mason says that it's the "various ancillary assumptions" or Cochrane says it's the mechanisms that make these theories different, they're right. That the OK model, the NK DSGE model, the IS-AS-MP model (per Michell), the Information Equilibrium (IE) NK DSGE model, and MMT say fiscal expansion can raise GDP doesn't mean they are the same theory.

It does mean they are the same effective theory, though.

And that means that the "various ancillary assumptions" don't matter except for the scope conditions (the limits of validity of the model) they imply, the empirical tests they suggest, as well as giving us a hint as to what might happen when a theory fails empirically. It only matters how you arrive at a result if you failed to fit the data or try to make further progress because you succeeded in fitting the data. How you arrive at a result tells you whether contrary empirical observations are out of scope, what the possible failure mechanisms are, what assumptions you can relax to get more general results, or address other questions. For example, DSGE approaches tend to make assumptions like the PIE. Failures of DSGE models will potentially show up as deviations from the PIE. The IE NK DSGE model I link to above contains assumptions about information equilibrium relationships and maximum entropy; the relationships will fail to hold if agents decide to correlate in the state space (opportunity set) — e.g. panic in a financial crisis.

This is all to say that the "various ancillary assumptions" don't matter unless you make empirical tests of your theory. And in this particular case, I want to show that those empirical tests have to be tests of the underlying assumptions, not the resulting policy conclusions. All of those models contain an equation that looks something like Jayadev and Mason's equation (2) relating interest rates, fiscal balance, and output — motivating the policy prescriptions. I'm going to show that equation (2) arises with really no assumptions about output or interest rates, or even what the variables $Y$, $i$, or $b$ mean at all. They could stand for yakisoba, iridium, and bourbon.

This is where we get into my original draft post. Let's say I have an arbitrary function $Y = Y(i, b)$ where $i$ and $b$ are independent. It will turn out that the conclusion and policy prescriptions will depend entirely on choosing these variables (and not others) to correspond to the interest rate and fiscal balance. Varying $Y$ with respect to $i$ and $b$ gives me (to leading order in small deviations, i.e. the total differential):

$$
\delta Y = \frac{\delta Y}{\delta i} \delta i + \frac{\delta Y}{\delta b} \delta b
$$

These infinitesimals can be rewritten in terms of deviation from some arbitrary point $(i_{0}, b_{0} ,Y_{0})$, so let's rewrite the previous equation:

$$
Y- Y_{0} = \frac{\delta Y}{\delta i} (i - i_{0}) + \frac{\delta Y}{\delta b} (b - b_{0})
$$

Let's re-arrange:

$$
Y = Y_{0} - \frac{\delta Y}{\delta i}i_{0} - \frac{\delta Y}{\delta b}b_{0} +  \frac{\delta Y}{\delta i} i + \frac{\delta Y}{\delta b} b
$$

The first three terms are what they define to be $A$ (i.e. the value of $Y$ when $b$ and $i$ are zero). Let's add in $Y/Y \equiv 1$ (i.e. an identity):

$$
Y = A + \frac{1}{Y}\frac{\delta Y}{\delta i} i Y + \frac{1}{Y}\frac{\delta Y}{\delta b} b Y
$$

As stated in their article, "$\eta$ is the percentage increase in output resulting from a point reduction in the interest rate", which means:

$$
\eta \equiv - \frac{100}{100} \frac{1}{Y}\frac{\delta Y}{\delta i} =  - \frac{1}{Y}\frac{\delta Y}{\delta i}
$$

Likewise, the multiplier $\gamma$ is (based on the sign convention for $b$ with deficits being negative):

$$
\gamma \equiv - \frac{1}{Y}\frac{\delta Y}{\delta b}
$$

Therefore:

$$
Y = A - \eta i Y - \gamma b Y
$$

based entirely on the definition of the total differential and some relabeling. This is to say that this equation is entirely content-less [1] in terms of the real world aside from the assertion that the variables correspond to real world observables. It's usefulness (and ability to lead to MMT policy prescriptions) would come from not just estimating the values of $\gamma$ and $\eta$ empirically, but finding that they are constant and positive. Of course this is going to describe a relationship between $Y$, $i$, and $b$ for changing $\gamma$ and $\eta$ because it essentially reiterates what we mean by functions that change (i.e. calculus). That's why all those models I listed above — such as the NK DSGE — come to a roughly isomorphic result. If you're trying to show fiscal expansion increases GDP you're going to arrive at something like this equation to leading order.

It's the assumption that output depends on fiscal balance and interest rates that leads us here, and so it's only useful if we find empirically that the coefficients are constant — otherwise we can always find some $\eta$ and some $\gamma$ that works. It's the same way $PY = MV$, the monetarist equation of exchange, would be useful if $V$ is constant. Otherwise, it is just a definition of $V$. This fiscal equation is actually a bit worse because it doesn't unambiguously identify $\eta$ and $\gamma$ (different values work for different values of $A$).

Defining terms is often a useful start of a scientific endeavor, but what we have here is a mathematical codification of the assumption that fiscal policy affects GDP of almost exactly the same form as the old school monetarist assumption that "printing money" affects GDP. The problem is that this is exactly the problem in question: How does a macroeconomy respond to various interventions? MMT prescribes deficit spending in the face of recession "because the output responds positively to fiscal expansion". It's question begging in the same way that people seem to conduct research into recessions by assuming what a recession is.

In MMT, there appears to be a lot of representing assumptions about how an economy works as math, and then using that math to justify policy prescriptions that is effectively equivalent to justifying policy prescriptions by assuming they are correct. I've discussed this before (e.g. here, here, or here) — whether the "various ancillary assumptions" are that government expenditures are private income, that there is no money multiplier, or that the desired wealth to income ratio is reasonably constant, these assumptions are translated into math and then that math is said to justify the assumptions.

The point of Jayadev and Mason's article is that these assumptions are completely in line with mainstream assumptions and models in macroeconomics — and they are. That's the problem. Macro models like DSGE models also make a bunch of assumptions about how an economy works — some of the same assumptions — but then don't end up describing the data very well. But then the equations are formally similar, which implies the MMT model won't describe the data very well either. There's more parametric freedom in the MMT approach, so maybe it will do better. What we really need to see is some empirical validation, not arguments that "it actually has things in common with mainstream macro". Mainstream macro isn't very good empirically, so that's not very encouraging.

I have no problems with the policy prescriptions of MMT proponents (austerity is bad and deficits don't matter unless inflation gets crazy), but I do have a problem with the idea that these policy prescriptions arise from something called "Modern Monetary Theory" purported to be a well-defined theory instead of a collection of assumptions. It would go a long, long way towards being useful if it empirically validated some of those equations. Without that empirical validation, all the equations really mean is that you were able to find a way out of your own Houdini's straitjacket constructed from your own assumptions in order to arrive at something you probably already believed.

...

Footnotes:

[1] There's nothing wrong with content-less theory on the surface. Quantum field theory, one of the most successful frameworks in human history for explaining observations, is essentially content-less aside from analyticity and unitarity [pdf]. It's the particle content (electrons, photons, etc) you put in it that gives it its empirical power. A similar situation arises with Kirchoff's laws: content-less accounting until you add real world circuit elements.

9 comments:

  1. Jason,

    I cannot claim anything on behalf of MMTers, so I may be wrong here, but I think that MMT people, when faced with such equations (like Y = A - eta*i*Y - gamma*b*Y) would claim either that 1) eta and gamma are not constant, so this would not be a very useful equation, or 2) that equation is incomplete, in the sense that there are many relevant variables in that relationship that are not taken into account by orthodox economists. So if you write like Y = A - eta*i*Y - gamma*b*Y + beta*x*Y, where x is the lacking variable (or a vector with all lacking variables?), then maybe eta and gamma would be constant, but probably x is much more relevant to the discussion than i, or than b, or both.

    For example, I think MMT claims that if government spends $ 1 billion in a booming sector, you probably will not accelerate growth and may cause inflation. If government spends the same $ 1 billion in an underutilised sector, you probably will accelerate growth without causing inflation. So it is not just the "b" or the "i" that counts, you have a lot of important information in many other variables not usually considered by economists.

    What I am saying is that, in my opinion, it is NOT a matter of whether MMT considers that gamma is big (and beta is low) while orthodox economists believes that gamma is small (and beta is big), as Jayadev and J W Manson claims. It is much more than that. They don't seem to understand the issues at stake, I think.

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    Replies
    1. I agree that other variables could easily enter — GDP depends on many, many factors. And that is kind of the point I am making: if you just assume Y only depends on i and b, then what you get out is exactly the equation they were saying describes MMT. If you assume other factors are important, you can add any number of extra terms. The question becomes: which version describes the data?

      Delete
    2. "The question becomes: which version describes the data?"

      Answer: the one that is best supported by empirical evidence.

      And that is another issue altogether, because in a very complex world it is not easy to understand whether empirical observations are indeed supporting one kind of hypothesis (equations) and not others.

      However, in my mind, it is clear that most economists do not confess that their hypothesis are wrong, in the sense that they are not supported by empirical evidence. Maybe it is pride, maybe ignorance, maybe they believe that bad theories are better than no theories at all (which I disagree), maybe it is political ideology, I don't know.

      Arjun Jayadev and J.W. Mason, for example, seem to believe that orthodox theories are good and that MMTers just have a little difference in opinion regarding some paraneters, which is cleary not the case. Are they ignorant? Or is it political motivation? I don't know...

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  2. Oops, I used the dollar sign and the formatting went crazy. That is what I meant:

    "For example, I think MMT claims that if government spends USD 1 billion in a booming sector, you probably will not accelerate growth and may cause inflation. If government spends the same USD 1 billion in an underutilised sector, you probably will accelerate growth without causing inflation. So it is not just the "b" or the "i" that counts, you have a lot of important information in many other variables not usually considered by economists."

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  3. Really enjoyed this one. Thanks Jason.

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  4. I’ve been reading your blog only sporadically this year, but this post is thought-provoking for several reasons. The essence of what you are saying seems to be summed up in these two extremely good quotes:

    Jason: “The problem is that this is exactly the problem in question: How does a macroeconomy respond to various interventions”?

    Jason: “I want to show that those empirical tests have to be tests of the underlying assumptions, not the resulting policy conclusions”

    I agree with both quotes – I suspect that I agree with them more than you do, particularly the second. However, they both suggest challenges for everyone - including you and me.

    Regarding the first quote, the challenge for you is that your technique does not meet your own challenge, as policy interventions are EXCLUDED from your own models. As a result, you have little to say about the effects of any specific policy intervention or policy option. That is why I suspect that, as it stands, your approach is a dead-end, even though your unconditional forecasts are at least as good as those of professional economists. We have discussed this before on several occasions.

    You should write a post about what quality criteria ANY economist (or physicist) would have to meet to fulfil the requirements of your own challenge. The recent post about the forecasts you would have made in 2008/9 would be a good starting point. The core exam questions regarding 2008/9 are:

    “what real-time advice would you have given President Obama, based on your models, from the point he was elected in November 2008 onwards, about how he should respond to the crisis”, and
    “how could you amend your models to INCLUDE the potential impacts of various options for policy intervention to assist the President as he considered his options”, and
    “how would you verify your models against the outturn of whatever policy option the President chose to implement”?

    Regarding your second quote, underlying assumptions are central to almost all human disputes in three respects.

    First, many assumptions are unstated, so the real disputes are not articulated. I know that this is true for many disputes in business and government. I suspect that it is also at the heart of many personal disputes. It’s certainly true in economics.

    Second, many assumptions are stated as though they are facts, when they are just assumptions. The people most likely to do this are middle-aged white men (as they feel entitled) who have high intellectual intelligence (so they value their own modes of thought) but low emotional intelligence (so they don’t value other people’s modes of thought).

    Third, many assumptions remain as assumptions precisely because, as you argue in your post, no-one tests them empirically. That is often because, when assumptions are unstated or viewed as facts, people imagine that there is no need to test them. Assumptions are only assumptions because they have not been tested empirically. If they pass empirical tests they cease to be assumptions. Instead, they become facts or theories. That’s what science and other forms of detective-work are for!

    (Cont’d)

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    Replies
    1. Hi Jamie,

      It has been awhile ...

      You said:

      Regarding the first quote, the challenge for you is that your technique does not meet your own challenge, as policy interventions are EXCLUDED from your own models. As a result, you have little to say about the effects of any specific policy intervention or policy option.

      This is not true, and in fact there appears to be e.g. a reduction in unemployment with the implementation of the Affordable Care Act, and there is a plausible case the ARRA (the "Obama stimulus") lessened the severity of the Great Recession in the US. The presidential election appears to have bumped interest rates up (though not attributable to a specific policy, rather the general fiscal profligacy of Republican controlled administrations), and Brexit appears to have sent interest rates down in the UK. WWII and the Korean War boosted GDP in the US.

      With particular assumptions, the framework reproduces the IS-LM model as well as the NK DSGE model and the AD-AS model, all of which allow monetary and fiscal policy to increase GDP.

      I think you might be referring to the fact that it turns out *empirically* that versions of the models constructed using the framework that neglect most policy interventions are the most accurate. (In fact, this finding goes against my own personal bias in favor of interventions.)

      That is to say I find empirically that a lot of policy interventions seem to have little noticeable effect on aggregate macro measures. Cutting the monetary base in India had little aggregate effect on GDP, just like increasing the base in the US during QE had no observable effect on inflation. But because policies do not improve GDP does not mean that policies do not improve quality of life (e.g. unemployment insurance may not affect the rate of improvement of the unemployment rate — constant over roughly the post-war period in the US — but it does help people who are unemployed). My own advice to politicians would be to generally ignore GDP and try and fix problems.

      But there is no aspect of the framework that precludes policy interventions from having effects. Finding they have no effect is not the same as saying I have nothing to say about the effects.

      I also find that it is completely different variables that seem to affect observables than are typically talked about in discussion of macro. Inflation appears to be primarily a demographic/labor force effect, not monetary or fiscal policy. Recessions appear to be social phenomena that build over years rather than being caused by specific events.

      Mainstream macro, MMT, post-Keynesianism, market monetarism: all of these are in my view somewhat misguided because they continue to look at economies using a framework established in the 1800s with Wicksell, talking about interest rates, money, and inflation. It doesn't seem to be a big empirical improvement over the mercantilism that came before it.

      At least the information equilibrium approach gets observables right. If you can show me some empirically accurate models based on e.g. MMT, I might be interested in checking them out. But they don't exist. And it's not because I haven't looked — I've been looking long and hard for models to compare my models to. Any models.

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  5. There is no question in my mind that these points are at the heart of most of the problems in economics. I blame Milton Friedman. That gives us all the challenge of figuring out what assumptions people are making, where inconsistent assumptions cause disputes, and how we might test at least some of the assumptions empirically. I suspect that this process would be traumatic for most economists. For example, the behaviour of supply & demand curves is an assumption which is not borne out by the facts. Even economists know this. They have an assumption that prices change with supply & demand, and another assumption that prices are sticky and don’t change with supply & demand. I guess that these two assumptions cover all the bases, but they don’t represent science, not even remotely. Non-economists often argue that supply & demand curves don’t work in specific areas of the real world, but many economists talk as though the people who say such things are ignorant. Here’s Bill Gates being “ignorant” about the software industry.

    https://twitter.com/BillGates/status/1029414776735059968

    Finally, let’s focus on the specifics of the mainstream versus MMT disputes (which I would say are really the mainstream versus Post-Keynesian disputes). These disputes are typical of those where assumptions are unstated or subsumed as points of detail. On the surface, the disputes often appear petty, but they are obviously not, or they would blow over. The challenge for the rest of us is to figure out the small number of core underlying assumptions which provoke these disputes, even though neither side will articulate them, and then to assess their relative merits.

    I will articulate one set of assumptions which I am pretty sure about. If you re-read JW Mason and Jo Michell, you’ll see oblique references to these assumptions. The assumptions are as follows.

    Mainstream economists assume that democratically elected governments cannot be trusted to run the macroeconomy, so control should be moved, as far as is practical, to unelected bankers and academic economists in Central Banks.

    Post-Keynesian economists assume that democratically elected governments should run the macro-economy, even if this occasionally results in sub-optimal policy, and that the role of unelected bankers and academic economists should be neutered to a greater or lesser extent.

    Here’s Jo Michell: “the only substantive difference between MMT and the orthodox analysis of fiscal policy is a change in the assignment of instruments”

    The only substantive difference … the ONLY substantive difference! … the ONLY difference!! … ONLY!!! … you cannot be serious!!!! This aspect of these disputes is there, in plain sight, but no-one can see it as both sides assume they are right and that their underlying assumptions do not need to be either stated or justified.

    By the way, disputes on who is responsible for what aspects of policy are also central to the problems of the Euro (as first articulated in the 1990s by Wynne Godley) and Brexit. I imagine that they are also partly responsible for Trump as the unresponsiveness of Central Bank-dominated policy to the needs of the man-in-the-street means that many people probably feel that it makes next to no difference who is the impotent figure-head in the White House.

    ONLY!!!!!

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