## Thursday, June 18, 2015

### Fiscal austerity logic fail

I'm still sufficiently angry about the total analytical garbage that is this post by Scott Sumner that I want to write it as pedantic logic:
Proposition: Fiscal austerity is contractionary at the zero bound regardless of whether you have an independent central bank.
Let's get some symbols in here.

A(X) = ∀x ∈ X, x has engaged in fiscal austerity
I(X) = ∀x ∈ X, x has an independent central bank
Z(X) = ∀x ∈ X, x is at the zero lower bound
C(X) = ∀x ∈ X, x has experienced economic contraction

So Sumner's Keynesian view is:

Proposition
1. A ∧I  ∧ Z → C
2. A ∧¬I ∧ Z → C

He wants to set out to disprove this pair of statements (i.e. show the proposition is false). One thing he does is throw out all of the countries without independent central bank:

X = {x | I(x)}

That basically assumes that ¬I = F₀ so we have

A ∧¬I ∧ Z → C
A ∧ F₀ ∧ Z → C
F₀→ C
T₀

So now statement 2 is a tautology. Remember, Sumner wanted to prove statement the proposition was false! It gets better. Statement 1 becomes:

A ∧ I ∧ Z → C
A ∧ T₀ ∧ Z → C
A ∧ Z → C

But note that in the data we have some countries not at the zero lower bound, i.e.

{x | ¬Z(x)} ⊆ X

Therefore, Z = F₀ so statement 1 becomes:

A ∧ Z → C
A ∧ F₀ → C
F₀ → C
T₀

So statement 1 is now a tautology on the set X and Sumner's Keynesian view is:

Proposition
1. T₀ ∀x ∈ X
2. T₀ ∀x ∈ X

∴ T₀ ∧ T₀ = T₀

QED

Both statements Sumner set out to show were false are now tautologically true on the data set he selected. Either Sumner is a Keynesian now, he's not convinced by his own logic, or he wasn't using logic.

I'm going with the last one.

Update:

There's also the monetarist part:
Proposition: Fiscal austerity is contractionary if you lack an independent central bank. Fiscal austerity would not be expected to have much effect if you have an independent central bank, due to monetary offset.

In symbols:

1. A ∧ ¬I  → C
2. A ∧ I  → ¬ C

Now the set X = {x | I(x)} so:

1. A ∧ F₀  → C
2. A ∧ T₀  → ¬ C

Simplifying

1. T₀
2. A  → ¬ C

We've reduced Sumner's statement to whether or not 2 is true. Back in English:
All countries kept in the data set X engaged in fiscal austerity do no experience economic contraction.
This means that for all countries exhibiting austerity, there should be no contractionary economic effect. That is (H/T to Tom Brown below for some corrections in bold, I said it in words but got sloppy with the symbols):

∀x ∈ X, A(X) ∧ ¬ C(X)

Which means that to show monetarism is false, all you need to show is there is one point in the data set that engaged in austerity and experienced contraction, i.e.

∃ x ∈ X, A(x)  C(X)

EU ∈ X

∴ ∃ x ∈ X | A(x) C(X)

∴ F₀

for statement 2.

∴ T₀ ∧ F₀ (statements 1 and 2)

F₀ (statements 1 and 2)

QED

So Sumner not only proved Keynesianism is tautologically true on the set of points he shows (X), but that market monetarism is tautologically false. I guess it is falsifiable!

Sumner's framing of the problem actually backs him into this corner. His assumption that all the data points have independent central banks means that any country engaging in austerity can't experience contraction. It only takes one case! You can weasel your way out through the "would not be expected" qualifier. However, that basically makes statement 2 useless (austerity with an independent central bank may or may not be contractionary) and then you're left with statement 1: fiscal austerity is contractionary if you don't have an independent central bank. Which is exactly the Keynesian view! You'd get no argument from Krugman about that!

Maybe Sumner can throw the EU out of the data set.

1. Jason, thanks for posting. I'm not good at deciphering the symbols so bear with me. Let me take the first line:

A(X) = ∀x ∈ X, x has engaged in fiscal austerity

So A is then a selector on the set X. It produces a subset of X of countries that have engaged in austerity. True? Etc for the other lines similar to this (I, Z and C).

A ∧I ∧ Z → C

I read this as "The intersection of subsets A, I and Z is a subset of subset C." Is that correct?

A ∧¬I ∧ Z → C

The intersection of subsets A not-I and Z is also a subset of C. We could combine the two and say the intersection of A and Z is a subset of C, but you keep them separate for your following development.

X = {x | I(x)}

I read this as "redefine the set X to be only those elements that are in subset I." I'm curious, how do you read the vertical bar "|" in English? As "given" like in conditional probabilities (e.g. P(A | B))?

OK, moving on:

Z = F₀

What is F₀? I'm not sure if I should read that as "false" or an empty set. Z I'm assuming is a subset, so since there's an equal sign I'm inclined to think that F₀ must be a subset as well. Likewise for T₀ except that it's either "true" or ... what? Everything?

"EU" is clearly this (being consistent with your previous notation):

EU(X) = ∀x ∈ X, x is a member of the European Union.

True? Or is "EU" simply a single element of X?

OK, let me back up slightly because this whole bit confuses me:

"This means that for all countries exhibiting austerity, there should be no contractionary economic effect. That is:

∀x ∈ X, A(X)"

I'm not getting how your English words map to those symbols. You write about "contractionary" in English, but the symbol C doesn't show up. How should that line of symbols be read? I read it as "For all x an element of X, subset A." I must be wrong. What does the comma (",") operator do there?

Which means that to show monetarism is false, all you need to show is there is one point in the data set that engaged in austerity and experienced contraction, i.e.

∃ x ∈ X, ¬A(x)"

OK, now for this bit:

"Which means that to show monetarism is false, all you need to show is there is one point in the data set that engaged in austerity and experienced contraction, i.e.

∃ x ∈ X, ¬A(x)"

Again, how do I read those symbols. I'm seeing "There exists x an element of X, not A." Again, I'm not sure how to read the comma.

And finally:

"∴ ∃ x ∈ X | ¬A(x)"

I read that as "Therefore there exists x an element of X" but what do I do with the vertical bar "|?" "Given" doesn't seem to work for that.

Oddly enough, I *think* I see what you're saying overall. You're saying that we only need to demonstrate that one element of the set of X has experienced both austerity and a contraction and we've disproved monetarism. That element (or subset?) is "EU." But I'm getting tripped up on the notation here which I want to learn, because I love the idea of translating these sentences into symbols.

In fact before I read this tonight I was going to post a question on Sumner's post to Sadowski asking him if we can consider the entire EU to be a single country (in a weighted average, weighted by population or size of their economies). So I think I was getting at your point here.

1. You could interpret the logical statements in terms of set theory, but it's better to use the language of logic where ∧ is "and" and the ∨ is "or". The vertical line | is read "such that" so:

X = {x | I(x)}

means "X is the set of objects x such that I is true for x" ... i.e. X is the set of countries in the data set that have independent central banks.

The predicate A(x)

A(X) = ∀x ∈ X, x has engaged in fiscal austerity

Says: "A is true of X if for all x in X, x has engaged in fiscal austerity".

And:

∴ ∃ x ∈ X | ¬A(x)

should be read: "therefore there exists x in the set X such that A is not true of x".

But in reading that I realized there is an error that I will fix. It should be a combination of A and C.

2. Also, for these two lines:

∃ x ∈ X, ¬A(x)

and

∴ ∃ x ∈ X | ¬A(x)

Why is it "A(x)?" In other words, why did the "X" become an "x" (lower case)?

3. The x becomes lowercase when it is about an element of the set X. A(x) means A is true of x, A(X) means A is true for all x in X.

I also made the corrections I mentioned above.

EU refers to the Eurozone so the element EU is just one element of X, i.e. X = {EU, US, ... (other countries on Sumner's graph) }.

Also, the F0 and T0 are universal false (falsehood, always false) and universal true (tautology, always true). In set theory they are the empty set and the "universe".

4. The Z = F0, which should actually be Z(X), is the statement that for every element in X, that country is at the zero lower bound. This isn't true (there are countries that aren't at the ZLB in X), so Z(X) is false, or:

Z(X) = F0

5. And yes, the overall picture is that the EU engaged in austerity and experienced contraction, therefore there exists a country that has an independent central bank but no monetary offset.

Sumner adds the bit about "expected to" but that just turns the statement into the wishy-washy "austerity may or may not result in contraction in a country with an independent central bank".

If we allow the wishy-washy statement, then monetarism becomes a tautology on the dataset as well -- both Keynesianism and monetarism are true!

In the next post, I show that Sumner's characterizations of monetarism and Keynesianism do in fact say the same thing!

6. Also the right arrow means "implies" or for p → q:

"p implies q"

or

"if p then q"

Also important to the logic is that

False → True
False → False

are true statements. "If black is white, then all ants can talk." But black isn't white, so it doesn't matter what the rest of the statement is.

7. Thanks Jason: much appreciated. I knew about → being "implies" but I was trying to force a reading along the lines of set theory. You decoder ring is a big help.

8. Also, in the last plot Mark did include "Euro Area" so I guess that might be the "EU" point you were referring to? I asked him what his last plot would look like if he also removed the countries which were not at the zero bound.

2. BTW, Sumner's proposition,

1. A ∧I ∧ Z → C
2. A ∧¬I ∧ Z → C

simplifies to

A ∧ Z → C

To disprove that, all he needs to do is to find a counterexample. Throwing out countries without an independent central bank ties one hand behind his back.

There is also a question about the meaning of A, since "austerity" was not a technical term in economics as of a few years ago, and, as far as I know, does not have a consensus definition yet. It obviously means more than running a government surplus, but what? (It may also mean attempting to run a government surplus, even if the result is a deficit.)

3. Here is Sumner:

"let’s review the two competing theories:

"1. Keynesian: Fiscal austerity is contractionary at the zero bound regardless of whether you have an independent central bank.

"2. Market monetarist: Fiscal austerity is contractionary if you lack an independent central bank."

OK. Let A = fiscal austerity, B = independent central bank, C = contraction, Z = zero lower bound. That gives us this.

1. Keynesian theory: A . Z -> C
2. Market monetarist theory: A . -B -> C

To test these theories against each other we are interested in instances where 1 is false but 2 is true, and vice versa.

1 true, 2 false: (A . Z -> C) . -(A . B -> C) , or
(-A v -Z v C) . ( A . B . -C) , or
A . -C . B . -Z

Likewise, 1 false, 2 true: A . -C . -B . Z

IOW, we look for countries that have engaged in fiscal austerity without experiencing contraction. If that country had an independent central bank but was not at the zero lower bound, then Keynesian theory trumps money marketist theory. But if that country did not have an independent central bank but was at the zero lower bound, then money marketist theory trumps Keynesian theory.

Logic is sometimes useful. :)

1. Oops! I miscopied 2.

It should be this.

1 true, 2 false: (A . Z -> C) . -(A . -B -> C) , or
(-A v -Z v C) . ( A . -B . -C) , or
A . -C . -B . -Z

Likewise, 1 false, 2 true: A . -C . B . Z

And therefore this:

IOW, we look for countries that have engaged in fiscal austerity without experiencing contraction. If that country did not have an independent central bank and was not at the zero lower bound, then Keynesian theory trumps money marketist theory. But if that country did have an independent central bank and was at the zero lower bound, then money marketist theory trumps Keynesian theory.

2. I completely agree, but the thing is that the Eurozone already shows this. But the MM answer is that the Eurozone wants low growth. This is the "no true scotsman" logical fallacy:

https://en.wikipedia.org/wiki/No_true_Scotsman

Basically, the ECB isn't a true scotsman (a proper central bank) so the austerity does have an impact on growth.

The reason is that they use NGDP as an indicator of what monetary policy is -- if NGDP is bad, it's because the central bank wants bad NGDP; if NGDP is good, it's because the central bank wants good NGDP.

It's not falsifiable since whatever the data is, it is an indicator of what it's supposed to be testing!

3. As for the Eurozone, first, it is not a country, and it has no fiscal authority, so A is neither true nor not true for it. IIUC, each Eurozone country has both an independent central bank and a fiscal authority, but the ECB controls monetary policy for the whole Eurozone. That is like the gold standard, so I guess that the Eurozone countries should not be considered to have a central bank.

4. "His assumption that all the data points have independent central banks means that any country engaging in austerity can't experience contraction". This is wildly dishonest. By you.