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In a hypothetical situation, every person produces some product but instead of consuming it sells it to another person.

This situation would increase the GDP whereas self-consumption doesn't.

Say, person x cleans person y's car and vice versa, and both pays each other some amount Z. Or maybe person x cleans y's car, y cleans a's car, a cleans b's...and so on. Or let's say x sells eggs to y, y sells to x.

Either way, the persons involved got some product or service and is left with no extra money.

This situations increased the GDP, whereas self-consumption doesn't. Is it a flaw? If so, what's it called?

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Yes this can be considered flaw of GDP since GDP’s intended purpose is to measure the value of output produced by a nation. In fact in his report to U.S. Congress Kuznets (who is the person who developed the modern concept of GDP) mentions in a preface that national product (GDP) in his view should encompass all production at its hypothetical market value (not just a production actually sold at market):

If all commodities produce and all personal services rendered during the year are added at their market value, and from the resulting total we subtract the value of that part of the nation's stock of goods which was expended (both as raw materials and as capital equipment) in producing this total, then the remainder constitutes the net product of the national economy during the year. It is referred to as national income produced, and may be defined briefly as that part of the economy's end-product which is attributable to the efforts of the individuals who comprise a nation. [emphasis is mine]

Kuznets even explicitly mentions house production should be part of this income, but he concluded it’s omission is ‘unavoidable’ due to the fact that it is problematic to measure it:

Services of housewives and other members of the family. —The volume of services rendered by housewives and other members of the household toward the satisfaction of wants must be imposing indeed, when totaled for the 30 million families comprising the population of this country; and the item is thus large enough to affect materially any estimate of national income. ... It was considered best to omit this large group of services from national income, especially since no reliable basis is available for estimating their value. This omission, unavoidable though it is, lowers the value of national income measurements as indexes of the nation's productivity in conditions of recent years when the contraction of the market economy was accompanied by an expansion of activity within the family [emphasis is mine].

So already the original creator of GDP arguably considered this omission as a ‘flaw’ but an ‘unavoidable’ flaw.

In addition historically GDP was from its inception used also for welfare comparisons and economists generally agree that home production and self-consumption increases welfare (see England; 1998).

Consequently, it is valid from economic perspective to consider omission of non-market activity (such as home production) as a failure.

This being said from purely accounting perspective this would not be flaw because GDP is defined as “the total monetary or market value of all the finished goods and services produced within a country's borders in a specific time period” (see here). So from pure accounting/statistical perspective it’s by definition not intended to measure non-market production, but as mentioned above this accounting/statistical definition was a result of expediency and not because it’s intended purpose was to omit non-market activity.

I am not aware of there being any special name within economics for this flaw. Generally, the above would be just called limitation of GDP, but limitation here is not some economic jargon.

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