Some of you have commented that the economist's chain of deductions in Message 6 are superficially convincing. I agree. I'm going to go through them here with a fine-tooth comb and show the problems.

But first some general thoughts on deduction. Deductive reasoning is a powerful tool for doing science. Einstein's general theory of relative is the pinnacle of this kind of thought. Einstein started with a single assumption called the "equivalence principle". According to the equivalence principle, a gravitational acceleration is indistinguishable from any form of uniform acceleration. This means that if you are in gravitational free fall, the observable laws of physics must be the same to you as they would be if you were at rest (uniform acceleration = 0). In other words, there is no (local) experiment that can detect if you are in gravitational free fall.

From the equivalence principle, Einstein deduced that gravity was not a force at all, but the warping of space-time. Particles do not "pull" on one another through gravity. Instead, matter and energy warp space-time, and then particles travel on the shortest path through this warped space. Einstein deduced the equations that govern this warping, which led to many predictions that have now been confirmed. What an achievement!

But let's look closely at it. All of Einstein's reasoning hinges on his first assumption --- the equivalence principle. If this goes, everything else goes. Fortunately, the equivalence principle is testable, and has been subject to immense scrutiny. No test has ever contradicted it. So general relativity is on a sound footing.

But while it is a powerful tool for science, deductive logic is also a powerful tool for persuasion and misdirection. Why? Because steps of deductive logic require assumptions. And when we are engaged in following the deductive logic, we often fail to see the assumptions. Critically thinking about deductive reasoning requires practice and training. It requires maintain two separate trains of thought. First, we need to follow the reasoner's logic, which implicitly requires agreeing with the deductive assumptions. But while doing this, we must also question the assumptions themselves.

This is not an easy task. It is why John Maynard Keynes was skeptical of using math in economics. Math tends to hide one's deductive assumptions, making them even more difficult to question then when the argument is stated in words. We get entranced by following the algebra and fail to see the dubious assumptions that lie underneath. As Sabine Hossenfelder observes "You can't lie with math. But it greatly aids obfuscation".

So let's look at how deductive reasoning can be used as a tool for persuasion by misdirecting us from subjective assumptions. Many theologians advanced "proofs" that God exists. Here is a famous one from Anselm of Canterbury:

1. God is the greatest being that can be imagined.

2. God exists as an idea in the mind.

3. A being that exists both in the mind and in reality is greater than a being that exists only in the mind.

4. Thus, if God exists only in the mind, we can imagine something that is greater than God.

5. But we cannot imagine something that is greater than God. Therefore, God exists.

Wow ... God exists! We have proved it! Except we have done no such thing. Why? Step 1 is an assumption. It is an empirical statement about God that requires observing God to actually test it (and agreeing on the correct interpretation of "greatest"). Step 3 is also an assumption. Why is something that exists in reality and the mind "greater" than something that exists only in the mind? This assumption is dubious because there are no empirical grounds for testing it.

So let's reverse these assumptions and "prove" that God does not exist:

1. God is the greatest being that can be imagined.

2. God exists as an idea in the mind.

3. A being that exists only in the mind is greater than a being that exists both in the mind and in reality.

4. Thus, if God exists only in the mind, we cannot imagine something greater.

5. Therefore, God exists only in the mind.

Of course, this says nothing about the actual existence of God. The existence of something is an empirical matter. This example simply illustrates how deductive logic hinges on the underlying assumptions.

So let's look at the economist's chain of reasoning.

1. We have to make a normative choice of what constitutes output. Does it include housework? Rent?

This is true. It is the core of my argument in the "Aggregation Problem". When we aggregate, we make subjective decisions that are informed by our goals. Each different decision opens up a "possibility space" for measuring output. So keep your eye on this. From the very beginning, the economist has admitted that measuring output is "normative", not objective. It depends on our predefined goals.

2. *Given the choice of definition of output* we have any number of objective measures respecting aggregate output.

This is a lesson in misdirection. In step 1, the economist admitted that our metrics for output are normative because they depend on our goals and aggregation decisions. In this step, the economist slyly adopts the following redefinition:

normative decisions ---> objective measure

After making normative decisions, our measure of growth suddenly becomes "objective". No it doesn't. Here is what actually happens:

normative decisions ---> normative measure whose underlying calculation is objective

When we make aggregation decisions, we collapse the possibility space. The resulting calculation crucially depends on our decisions, so it is normative. But having made the decisions, the CALCULATION of our metric can (usually) be done objectively. It is important to make this distinction. We only get to call our metric an "objective" measure of output if we agree that the decisions in step 1 are "objective".

3. Note that— regardless of the definition of output-- none of these measures are indices of output. Rather, they are indices of certain characteristics of output.

This is correct. Aggregate measures capture one characteristic at a time. This is true of all measurement. It adopts a dimension of analysis, which allows the quantification of a specific characteristic. The point of science is to point unambiguously at the appropriate characteristic to measure. Newton's laws unambiguously declare that inertia is to be measured using mass. When aggregating economic activity, there is no equivalent science that points to the "correct" dimension of analysis.

4. None of these objective indices are entirely defensible as measures of output. If we take mass as our measure of output, then services do not count toward our index of output. If we take volume, then producing smaller phones reduces our index of output. And so on.

This is my entire point in "The Aggregation Problem". When it comes to economic growth, there is no consensus on the appropriate dimension of analysis. Nor are there any scientific laws that tell us the "correct" dimension. So when we aggregate output we must make subjective decisions on the dimension. This opens a possibility space for our measurement.

5. Expanding further on the second example in item 4, it is a matter of intuition that if production of every type of good and every type of service is unchanged from one period to the next that a "good” measure of output would also be unchanged. Call this the "item-5 problem"

This is not a matter of "intuition". It is a strict requirement for a unit to be valid. A unit must measure equivalent things as being equivalent. If it does not, then it FAILS as a unit of analysis.

Take this equivalence:

A + B = A + B

If we measure using mass, we must find that:

mass A + mass B = mass A + mass B

If we find the following, we have a problem:

mass A + mass B = 2*mass A + mass B

For objective measurement to be possible, the unit must be UNIFORM. It must measure equivalents as equivalents. If it does not, it fails as a unit of analysis. Keep your eye on this. It means prices fail as a unit of analysis.

6. So, given any nontrivial definition of output, there is no “objective” measure of output per se. We cannot construct some instrument which will reveal the objective “truth” about growth in output. We can seek merely to perfect instruments to measure objectively the growth in particular characteristics of output.

This is absolutely true. But it means that everything depends on our choices of the characteristic of output that we want to measure. And we need to be able to measure this characteristic objectively.

7. GDP is— by construction— the realized exchange value of all an economy’s currently-produced goods and services defined as output. In the US, the exchange value is denominated in USD. Some of the transactions are non-market and require imputation of the dollar value they *would* have fetched in the market. In this sense among others, GDP is objectively but imperfectly measured.

Here we are talking about nominal GDP. It is the aggregate market value of all economic activity deemed to be within the boundaries of measurement.

No, we cannot objectively "impute" the price of non-market activity. Doing so requires many assumptions. See Assa's article below. So no, nominal GDP cannot be objectively measured in this sense.

https://ideas.repec.org/p/new/wpaper/1813.html8. Consistent with item 5, we don’t care for GDP as a proxy for output itself. In part— but hardly exclusively— this is because even if production of every single type of good or service is unchanged, the dollars for which that production is exchanged may increase. An objective increase, perhaps, but also not intuitive as a measure of output.

Pay attention here. In step 7, the economist talks about aggregating using market price. This means we are making the normative decision to aggregate economic activity using prices. This gives nominal GDP.

In step 8, the economist admits that nominal GDP is not a good measure of output because prices change over time. In the economist's words, we should not "care for [nominal] gdp as a proxy for output".

But we must go much further than this. The problem resides in our UNIT. Prices fail to measure equivalents as equivalents. This is absolutely crucial. It means that prices fail as an objective unit of analysis. This is why we reject nominal GDP as a measure of output.

So let's unpack the reasoning so far. The economist advances prices as the unit of aggregation, but then (implicitly) rejects them because they fail as a unit. Let's be absolutely clear about this. The deductive chain should STOP here. Prices fail as unit of analysis ... end of story.

But the economist does not stop here. To follow the rest of the deduction, one must "forget" that prices fail as a unit. Then you get hoodwinked when prices are reintroduced as a criteria for a "good" measure. Remember that this is an absurdity. It means prices fail to measure output objectively, but we will still appeal to them to judge our success at measuring output.

9. We can fix this "item-5 problem" with GDP as a proxy for output by howsoever fixing the prices of all goods and services in the measure— irrespective of when they were produced.

The terminology here is misleading. We cannot "fix" this problem. The problem resides in our unit --- prices --- which are unstable. The only way to fix this problem is if price change was uniform. Since we cannot change history, we cannot "fix" this problem. From this step onward, we are in the subjective territory of deciding how to "deal" with instability in our unit.

The most objective thing to do here is go no further. Our unit is unstable. This means it fails the only requirement of a unit --- to be UNIFORM. We go past this point at our own peril.

10. The fix of item 9 creates its own problem— that growth in the measure depends of the choice of prices employed. Call this base-price problem the “item-10 problem”

No. The "fix" does not "create problems". The problems reside in the unit, which is unstable. Choosing different base years simply reveals this problem. It does not create it.

11. If we select prices from (as well as possible) a consistent survey of the economy at a certain time, then the “item-10 problem” becomes the more specific base-period problem. Call this the “item-11 problem”

This is just a restatement of step 10.

12. Now, let us back up a bit. Let’s also make a normative judgement that the relative price of goods and services reflect “real” relative value of each item’s marginal product. So we can pick absolutely any good or service as a basis for the calculation. By declaring that item as the basis, we find that GDP divided by the price of that good is “real” in any period— in that it is consistent with the normative choice. This “real” GDP is the (normative) value of total output relative to the (normative) value of one additional unit of production of the base unit. Unfortunately, this creates a problem in the sense that the growth of “real” GDP varies according to the choice of base item. Again this is due to relative prices changing over time. Call this the “item-12 problem”

This is a doozy. We are assuming that relative prices = marginal product. This is certainly normative. It is also untestable and circular.

Marginal productivity sets out to show that productivity explains income. To do this, you need an objective way to measure productivity. But how do we do this? How do we compare the output of a lawyer to the output of a farmer? There are not objective criteria for doing this.

In practice, neoclassical economists test marginal productivity by measuring productivity in terms of INCOME. If this seems circular ... it's because it is. Income is the thing that is to be explained. We then explain it ... in terms of itself!

So the assumption that relative prices = marginal product is right up there with the assumption that something that exists both in the mind and reality is "greater" than something that exists only in the mind. It is immune to evidence. That's a nice feature to have in dogma.

We could just as easily assume that prices NEVER reflect marginal productivity. This kills the rest of the reasoning in the economist's chain of deduction.

But let's look closer. The economist writes:

"So we can pick absolutely any good or service as a basis for the calculation. By declaring that item as the basis, we find that GDP divided by the price of that good is “real” in any period— in that it is consistent with the normative choice."

This reasoning requires that relative prices = marginal productivity. The "productivity" is what is "real". It is what is revealed by prices. Without this assumption all we have are a ratio of two prices.

This what Nitzan and Bichler call a "differential" measure. It compares one set of prices to another. It is important if we are interested in prices. But it tells us absolutely nothing about output. The "real" part only comes when we assume that prices reflect marginal productivity.

13. Fortunately, no matter how fast relative prices are growing, then over a small enough time period, growth in “real” GDP over the period is insensitive to the choice of prices so long as the relative prices reasonably apply sometime in the period.

This is a favourite argument of neoclassical economists. If we have a problem when we look at the "big picture", we zoom into the very "small picture" and the problem miraculously goes away. Except it doesn't.

Here's an example from geometry. Suppose we are doing geometry on the Earth's surface. But we find that Euclidean geometry doesn't work. In Euclidean geometry, parallel lines never intersect. But because the Earth is round, parallel lines on the Earth's surface can intersect (think of longitudinal lines).

The correct interpretation is that Euclidean geometry does not apply on the surface of the Earth. But now an economist comes and says "let's look at curvature in the limit of a short space". Over short distances, the Earth is roughly flat, and Euclidean geometry roughly applies. The economist then takes this truism and goes in the wrong direction. He concludes that the Earth is flat!

Our economist here is doing the same thing. Over long periods of time, prices are incredibly unstable as unit. This means that there is incredible uncertainty in the growth of real GDP, again over long periods of time. But over very short periods of time, prices change very little. So over this imagined short period of time, there is no problem with our unit. It is stable.

The problem is that this says nothing about the long-term change in price. It is precisely the change in the long-term that matters. The economist here is essentially arguing that the Earth is flat. To solve the long-term instability of prices, we look only at the short-term. Our unit miraculously becomes stable! Except that it doesn't. This is a lesson in fallacious reasoning. We are essentially fooling ourselves into thinking that prices are a stable unit.

And yet another problem. The growth over this short period does not reveal the growth of "real" GDP, in the sense of productive value. We are still just looking at price ratios. But now we are doing so over very short periods. The "real" part only comes if we assume marginal productivity is true.

14. So we may— for example— pick not prices at the start of period, nor those at the end of period, but pick average price of each item throughout the period. We can compute the rate of growth in “real” GDP over the period holding constant those prices. And we can do the same for the next period using the average next-period prices of each item. Etc. We can simply chain together these small-period growth rates to produce a long-period index of “real” GDP which much better estimates changes in the relative value of production over time. (That is, we have adjusted our measure to work around the item-12 problem.)

Here is the pièce de résistance, the moment when the round Earth becomes flat. To calculate real GDP, we look at successive short-term windows of prices. Within each window, prices are (relatively) stable. We then "chain" together these windows and viola --- we get a single measure of "real GDP".

We get this because chaining short-term windows together is a recipe for hiding information. It hides the instability in prices.

Let's unpack the problem. First, it is OK to look at the small picture ... if you want to draw conclusions within the small picture.

Say we are doing geometry on the surface of the Earth, but only over very short distances. At this small scale, it is fine to approximate the surface as flat, giving approximate Euclidean geometry.

But we are NOT allowed to take our small-scale approximation and extrapolate it back to "infer" the large scale trend. In our geometry example, this means zooming in and approximating the Earth as flat over the small scale. We then take this approximation and extrapolate it back to the large scale. And we find that the Earth is approximately flat! The problem is that the small scale picture is blind to the large scale trend.

It is logically indefensible to use the small-scale picture to say anything about the big picture WHEN WE KNOW THE BIG PICTURE ALREADY. So back to prices. Chain indexing is a clever algorithm for hiding the instability in prices. The chart below shows this.

*US Annual Price Change*
- prices_annual.png (665.12 KiB) Viewed 302 times

Panel A shows price change in all CPI commodities. Panel B normalizes this so that we see price change relative to the basket average. This shows us the long term dispersion in prices. This is our changing meter stick --- our unstable unit. It is clearly visible over the long term.

Now we use the chaining trick. We look only at short-term windows of time --- in this case one year. We observe how much each commodity changes in price over this period. In each new window, we re-index each commodity's price to 1 and observe how much it changes in a year. Panel C shows this chained change in price. By looking only at the one year spans of time, we blind ourselves to the long-term changes in price. Over this short time, prices are relatively stable.

Now we take our short-term windows and chain them together to "reconstruct" price change over the long term. We measure price change in terms of the relative standard deviation of prices within our basket (standard deviation / basket mean ). We do this for both the actual price change and for our chained index of price change.

See the difference in Panel D? In the actual data, price dispersion increases over time. But in our chained series it does not. Price dispersion stays constant over time. The instability in our unit has disappeared! Our problem is solved!

No. The problem remains in the long term trend. We have just blinded ourselves to this trend by piecing together short term trends. We have found that the Earth is flat!

The shorter the window that we look at, the more stable prices will appear. And the more inaccurate our reconstruction of long-term price dispersion will be. Here is what happens when we look at monthly windows:

*US Monthly Price Change*
- prices_monthly.png (682.88 KiB) Viewed 302 times

To summarize, chaining is a technical recipe for hiding information about long term instability in prices. So as I claim in "The Aggregation Problem", chaining just hides the problem.

16. It is an improvement because it better reflects growth in output as valued by observed prices. That is, it better reflects the normative choice of value.

Hold on. We've already rejected "observed prices" as not being useful for measuring output (step 8). "Observed prices" lead to nominal GDP, which the economist already rejected as a measure of output.

It is logically indefensible to now claim that our index of output is "better" because it gets closer to nominal prices. Once we start "adjusting" for inflation, we can no longer use nominal prices as a criteria to judge our metric. The fact that we are adjusting prices means we no longer trust them to tell us what we want.

It's like saying we will estimate fossil fuel consumption using carbon emissions. But we reject this because of problems with the measure of carbon emissions. We then devise another technique and then declare that it is successful because it gets close to our measure of carbon emissions ... a measure we have already rejected!

17. The “uncertainty” in real GDP you describe arising from the base-year problem is not uncertainty in the sense of not knowing what the length of our ruler is. Not is it uncertainty in the sense of not knowing what output is. It is uncertainty regarding whomever’s choice of values (as reflected by the choice of base year.)

The last sentence is exactly correct. It is uncertainty caused by the normative decisions made by the observer.

18. Chaining does nothing to hide this. In a very true sense it merely reflects a different normative choice. If anything, then, it *adds* to your “uncertainty” rather than hides it.

The economist is correct that chaining "merely reflects a different normative choice". But the economist misses the point. The uncertainty in real GDP is only visible when we look at the possibility space for our normative decisions. This means calculating real GDP using ALL the possible normative decisions. This is where the uncertainty resides.

The crucial point is that the government reports ONLY one value for real GDP --- the chain-weighted value. This is the whole problem. By reporting only one value, the government hides the uncertainty. It is irrelevant which particular method the government uses for the official metric. We have already admitted that this is a normative decision. The important thing is that OTHER METHODS ARE POSSIBLE. If these are not published, we are hiding the uncertainty in real GDP.

19. Alternatively, one could— normatively— select any specific base year prices as the “true” measure of value just as one could select relative prices. And there would be zero “uncertainty” in the measure because such a choice would not be subject to the base-period problem. A Laspeyres index of output value based on the price of that base period would do a great job of reflecting that choice.

This paragraph is a mess of confusion. The uncertainty in GDP is caused by the existence of many possible normative choices for the way we can calculate it. Now the economist wants us to subjectively choose which normative method to use. And then we report only this method. Does this get rid of the uncertainty? No! It hides it.

It's like measuring the temperature with an inaccurate thermometer. The resulting measures will be uncertain. Then we pretend that we can remove this uncertainty by reporting only one measure of our choosing. Every other branch of science knows that this is bad science. Economists seem to be oblivious to this ... but that's because they have been indoctrinated in a secular cult.

20. So yes, none of the base-year indices reflect the normative choices implicit in real market-valuation of GDP. This is not “uncertainty”, but a collection of inferior estimates.

This is another mess of logic. The normative choice for calculating GDP is to measure output using nominal prices. The problem is that we immediately reject this choice the moment we start adjusting prices for inflation.

The economist is implying here that fixing prices in a single base year does not "reflect the normative choices implicit in real market-valuation of GDP". OF COURSE IT DOES NOT! We rejected our normative choice in step 8! Choosing nominal prices gives us nominal GDP. But because prices change over time, so we rejected this as a measure of output.

So here is what the economist is saying:

1. The normative choice is to use nominal prices to value output.

2. We reject this because prices change over time.

3. But we will continue to judge our metrics as "inferior" or "superior" based on how closely they get to nominal prices ... our normative choice that we already rejected.

21: In short, if one wants to estimate real market value of output, then chaining is important. If one wants to estimate growth in output as opposed to growth in some observable characteristic of output, then one is up a creek. That is what I meant by “the truth of things”— that the chained measure better reflects growth in real market value of output. The “truth” of output growth is… nonsense.

This is another lesson in misdirection. The "real" part here is supposed to represent marginal productivity. But this is unobservable. The economist simply assumes marginal productivity is revealed by relative prices.

This logically means that ALL PRICES AT ALL TIMES reveal marginal productivity. Oh wait ... they can't. This would mean "real" GDP equals nominal GDP. But we reject this as a measure for growth because it does not measure equivalents as equivalents.

The economist openly rejects nominal GDP. But this should mean that we also reject the assumption that relative prices reveal marginal productivity. The moment we start adjusting prices, we are assuming that the ADJUSTED price (not the nominal price) reveals "real" marginal productivity.

So at this point, any method we use to deflate prices should reveal "real" market value. But again the economist uses a bait and switch. The economist rejects nominal GDP, but then uses nominal prices as a criteria for measuring "real" production.

This is logically indefensible. It is a duality hidden in the deduction. Nominal prices are both the "wrong" and "right" measure of productive value. It's nice to have dogma that covers all the possible bases.

22: Yes, economists are sloppy about this in the sense that GDP and output are often used interchangeably. I’m certainly guilty of this. But that is where the normative choice is hidden— not in the chaining.

Hmm. Why are economists guilty of this? It's because they are plagued by cognitive dissonance. In the back of their minds they know that a host of subjective choices are required to measure real GDP. But they want to give their discipline the appearance of doing objective science. They want deterministic theories of why output grows.

Neoclassical growth theory is built on an unambiguous notion of output. It posits that output can be objectively measured ... because it assumes a single commodity. Now legions of economists used the resulting theory to "explain" the growth of real GDP. But to do this, you need to wilfully ignore the subjective element of GDP. Otherwise your regressions mean nothing! They simply reflect a particular set of normative choices that underlying the chosen method for calculating real GDP.

Can you imagine what macroeconomics would look like if the uncertainty and ambiguity in real GDP was published in official reports? Economists might realize they had been fooling themselves all along.