Units of Measure

We should have learned about units of measure by the sixth grade, but frequently those who move on to use higher levels of math forget their importance. The practitioners (and followers) of "mathematical economics" seem to ignore units of measure by intent. The principles of systems thinking attempt to bring their importance back to mind. Programs like Insight Maker, which I have used here, even check for appropriate use of units of measure, but, regrettably, some modelers build models using only unitless variables, thereby masking problems of mismatched units in the model.

Let me demonstrate the importance of units of measure with a simple example:

An abstract and unitless formula: 95 + 17 + 50 + 71 + 7 = 240

however,

Less abstract, using units: 95 lbs + 17 ozs + 50 gallons + 71 tons + 7grams ≠ 240 lbs-ozs-gallons-tons-grams

You need no more discussion here, and you will see the importance of this concept in the next tab (above).

Avoiding Errors of Aggregation

(The following comments apply to the bottom row of the diagram above. I will describe the upper rows in my discussion about levels of abstraction.)

In economics, and the broader discipline systems thinking, the practitioner must avoid committing errors of aggregation. Simply put, you cannot add dissimilar units and arrive at meaningful conclusions. In economics you must avoid errors of aggregation in three categories of units: items, units of measure, and prices.

Items

Valid Aggregations

One can logically aggregate like items (or fungible items). Valid e.g. 40 pigs or 120 spoons

 

Invalid Aggregations

Attempting to add (or aggregate) different items violates both the rules of logic and of physics. You cannot, for example, add bacon and peaches and Chevrolets. Thus, in order to create a more generalized model, we need a way to overcome the dilemma of too many variables created in our prior model, for we cannot add wheat, shoes, bread, and iPods. (See the first row in the table below.) Invalid e.g. 40 wheat-shoe-bread-Pods

Units of Measure

Valid Aggregations

One can logically aggregate quantities of the same unit of measure. Valid e.g. 69 tons or 96 miles

Invalid Aggregations

In stating economic facts or relating economic theory it also makes no sense to attempt to add (or aggregate) different units of measure—as we discussed in the prior tab. Even within fungible items we cannot add different units of measure. We cannot add pints, gallons, liters, pounds and grams—in any order—and achieve a meaningful result. (See the second row in the table below.) Invalid e.g. 47 pint-gallon-liter-pound-grams

Prices

Valid Aggregations

One can logically aggregate prices of like (or fungible) items or the prices of like (or fungible) items using the same unit of measure. Valid e.g. $5 per pound of ham

Invalid Aggregations

Finally, we must avoid the aggregation error which occurs commonly in economic statistics – the error of summing the prices of different economic goods. A money price consists of the ratio of dollars given up for a quantity of goods received; therefore, we cannot add $1,000/150 pounds of bacon and $2,000/400 bushels of peaches and realize a meaningful sum of $3,000. We can say that $3000 has been exchanged, but the unit of measure here is not a price but a statement of the number of dollars used in the transactions. Invalid e.g. $450 per lb-oz-bushels of bacon-soda-peaches.

	Valid Aggregations					Invalid Aggregations
Items	4 Peaches	5 Shirts	3 Houses	10 Dollars		12 Peach-Shirt-Houses
Units of Measure	40 bushels	2 dozen	1,200 sq. ft.	100 Dollars (Items=Units)		1,242 bushels-dozen-sq. ft.
Prices	$10/ 4 Peaches	$30/ 5 Shirts	$450,000/ 3 Houses	$50,000/ $50,000		$450,040/ (12 Peach- Shirt-Houses)

Conclusion

In order to make meaningful statements about larger and larger segments of an economy, we need to stop aggregating items, units, and prices, and move to higher levels of abstraction. I will describe levels of abstraction in the next tab (above).

Levels of Abstraction

In the last tab, I said that we cannot add wheat, shoes, bread, and iPods. Similarly—referring to the diagram above—we cannot add bacon, peaches, corn flakes, etc. So, how can we make meaningful statements or build meaningful models that represent larger economic systems?

The answer lies in higher levels of abstraction. I have given just a few examples of levels of abstraction in the diagram above. "Food" represents a higher level of abstraction than bacon, peaches, and cornflakes. And you can see from the diagram other examples of slightly higher levels of abstraction applied to other economic goods. At higher levels of abstraction we must first use a term that encompasses terms at a lower level and second apply a different unit of measure from those used at lower levels in order to make meaningful statements. Whether that unit be tons or gallons or some other measure, you can see that a certain amount of precision gets lost. We must, therefore, take care in interpreting the results of any model that we create at a higher level of abstraction.

In this diagram, I have moved up first to two very general categories – consumer goods and capital goods – and finally, to an all-encompassing category for which I have created an artificial name and unit of measure. By using this high level of abstraction, I can make statements that apply to general principles and theories of economics without violating either the laws of physics or logic.

Why not money?

At this point some might ask, why not use money is an abstraction for, and thereby a way to quantify, the general economy?

We cannot use money for two reasons. First, money by definition lies at the lowest level of abstraction along with the other items on this diagram. Second, the dollar amounts summed in most economic statistics represent shorthand for the aggregation of dollar prices. Doing this creates the same logical error discussed in the previous section, i.e. you cannot sum $10/bacon, $100/peaches, and $25/cornflakes and create a meaningful answer – $135/bacon-peach-cornflakes makes no sense.

Conclusion

In order to overcome the problems of aggregation in the creation of a generalized model, I have replaced all other units with hypothetical units I have called "economic units" (for stocks) and "economic units/year"(for flows). By using these units the model remains logically and theoretically consistent.

The aggregations that naturally occur with simulations of the models on succeeding pages all occur within the same items and the same units of measure, and prices of the same items (or units). The models remain valid at all levels of abstraction.

Emphasis

Models that use dollars or dollar prices (or any other monetary denomination) have created an error of aggregation. An aggregate of any good (or claim on any good) cannot represent the goods for which market actors trade that good. And, the exchange ratios of that good with the goods for which it has been traded (i.e. prices) cannot represent an aggregate of those other goods.

Does this make measures such as GDP and CPI invalid? Yes.