The fallacy of composition and graph algebras.

We run across economists who claim two things, one, the sum is not always equal to the whole, and two we can measure the sum in macro economics. Seems like a contradiction? The math the economy uses is not the same math the macroeconomists use.

 If a business reports output-input = surplus, it is not the case that surplus + input = output. The later computation is invalid because the economy cannot do that, in general. We cannot take a car apart and return the parts into inventory, no general invertability. The second problem is the change of units accountants tend to use. Rather than count carburators in inventory, they report that number in units of carburator prices, which may have changed considerably. The other problem is round off error, units are always rounded to the nearest integer in inventory management. Probably all three fallacies are related.

If the fallacy of composition relates more to macroeconomists using the wrong rules of computation, then what should macroeconomists count? Units of goods moved over a specified cycle of exchanges, the count of inventory exchanges from the slowest transaction rate to the fastest. This composition works, it makes no assumptions about asymmetry of trade, it deals with integer units, and insures mass flow is consistent.

A lot of economists get this, mainly sector economists. They count the flow of automobiles to the consumer, and compare car sales totals from one business cycle to the next. Calculated risk does this all the time. I see it done with volumes flows for the stock market.

If we count the flows of money, we can add it up and compare it to flows of money from a similar business cycle.  If we compute on a business cycle that is one generation long, that is the longest inventory cycle is a generation then we have to wait one generation before we get an accurate measure of NGDP.  I see historical economists compute real gdp based on hundred year cycles.

Lets call NGDP(i) the measure of integer flows of money up to some cycle i.

The ensembale of inventory cycles is an integer set. So it is perfectly reasonable to estimate nominal gdp relative to a specified maximum cycle and cycles less than. So we have GDP relative to the yearly cycle and the quarterly cycle. If I have the data, I can get real time NGDP on a daily cycle, and the number is accurate, to the daily cycle, as long as I measure short enough to avoid inerference with the next integer inventory cycle up stream, and as long as I measure unit flows. But I can always capture the sum components down stream, because they have smaller inventory cycle.

What is real GDP then?  It is total GDP measured in money using a complete algebra, it is remotely related to NGDP.  Take a NGDP measure accurate to a specified inventory cycle, NGDP(i).  Estimate the discount rate from the selected inventory cycle, i ,  out to a maximum known cycle, and apply that estimate to convert to  RGDP.

 So, when Scott Sumner wants to increase NGDP he really means he wants to increase the rank of money distribution,  he wants to extend NGDP out to the next cyle.