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My understanding is that consumption = consumption of final goods = whatever consumers buy from companies (with an adjustment for export-import-process + maybe housing is slightly different (not important here). So if you were to buy a fully US-made table it wouldn't really matter that first one company plants and harvests the tree, another then buys the whole tree and transforms it into useable wood (cuts into useable sizes, sorts out bad parts, possibly treats wood etc), and then the last step somebody buys that wood and assembles a table. The only thing that matters is when this table is actually sold to an private customer (for 100\$ or whatever)(assuming its directly sold by the joiner and leaving out dedicated retailer).
So far seems somewhat understandable that only the use of the activity to real persons/consumers would ultimately be relevant and seen as economic productivity (use being measured by $spent). However I don't understand why some expenditures made that are called "investments" still increase GDP (and aren't lumped in with other precursors as non-final goods)?(1)


(1)(Maybe expenditures that are classified as investments, like a new production machine, can be seen as not directly tied to a specific output-product (unlike e.g. a concrete amount of wood bought per output-table to be assembled into a table)? But i would not see why this is directly relevant when measuring the quality/productivity of the whole economy as GDP probably should (just because somebody invests into something doesn't mean that's reasonable, e.g. investing into gold-plated machines etc or just machines not fit for their intended use, while you could somewhat reasonably argue that everythng bought by customers should be defined as a useful product and GDP-increasing because apparently the customer found it useful and satisfying customers is ultimately the purpose of the economy?
Also where would that leave sandpaper and similar things? (Machines detoriating very little with each table produced, wood deterioirating/disappearing completely (into the table), and sandpaper in between depending a bit on luck with the wood and not clear whether its more non-final good or investment))

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Well first of all the story why we only record final consumption is not completely straight - it’s not because we only care about activity of “real person” or consumer. But because counting it all the way would mean we double count the value of the good.

For example imagine the following situation. A wood is produced and sold for 15\$ to furniture company and then furniture is sold to consumers for 30\$. Well it’s not that we don’t care about the 15\$ because it was bought by company, but because the 15\$ value is already “hidden” in the value of the final good. Counting both would give you value of 45\$ but thats double counting the 15\$ is already "hidden" in the final price (of course final price could be sometimes lower if the second business by mismanagement manages to destroy value - i.e. producing something with lower value then value of the inputs - then you would want to record only the final price still to account for the fact that the process destroyed part of the old value).

Even when business purchases something which is not used for further consumption it counts as final good, purchasing catering for some firm celebration would be reflected in consumption since the catering is final good.

However, while we can’t really count let’s say wood that will later (same year) be used for furniture production we can count spending on machinery because there would not be any double counting since it’s not like you just modify the machine and sell it later as some final good.

This is because GDP basically tries to measure output at current prices. You can do that either by income approach since by definition output is equal to income or by spending approach because all output is going to be used or “spent” in one way or other. Investment also represents spending the produced output so it should definitely be there.

The same question you are asking about gold plated machines can be asked about any final good. If a consumer wants gold plated PC instead of normal PC is that any less or more rational than business owner wanting gold plated machine? The answer is if you want to assume people are rational the business owner will only invest in golden machine if that’s somehow relevant to business process otherwise it would be outcompeted away by other business. If you want to relax the rationality assumption then why should we assume consumers are rational and business not? If we count irrational purchases by consumers we should not discriminate irrational investment.

Regarding the auxiliary question about depreciation. Depreciation actually is not reflected on GDP because it is Gross Domestic Output. However, there is also NDP - net domestic product that controls for depreciation.

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