This post follows Fallacy: “If imports rise, then GDP falls”.
Consider this erroneous argument: We have a can of beans. Let

S = Net Weight (of beans only);

T = Total Weight (of beans + can); and

U = Can Weight (can only).
So, S = T − U.
Now suppose we increase U (Can Weight) by 100 g.
Then “ceteris paribus”, S = T − U falls by 100 g.
Simply by increasing the U (Can Weight), we magically decreased S (Net Weight of beans only) by 100 g.
The error above is the same as that in the fallacy, “If M rises, then ceteris paribus, Y = C + I + G + X − M falls”.
Above, in “S = T − U“, the only reason we subtract U (Can Weight) is because T (Total Weight of beans + can) must already include all of U (which we don’t want to count in S = Net Weight of beans). So, if we increase U (Can Weight) by 100 g, then T (Total Weight) also increases by 100 g (it’s absurd to claim “ceteris paribus”, i.e. hold T the same as before).
Similarly, in “Y = C + I + G + X − M “, the only reason we subtract M (imports) is because C + I + G + X must already include all of M (which we don’t want to count in Y = Domestic valueadded). So, if we increase M by $100, then C + I + G + X also increases by $100 (it’s absurd to claim “ceteris paribus”, i.e. hold C + I + G + X the same as before).
In both cases, we have a pure accounting identity where the direct effect of any change in U (or M) must be an equal (and hence offsetting) change in T (or C + I + G + X).
Objections.
A. Why is it that “C + I + G + X must already include all of M”?
B. What about indirect effects?
Responses to Objections.
A. Actually, the more correct statement should be this:
C + I + G + X + Intermediate Consumption
must already include all of M.
To see why, let’s review some basic national accounting principles:
I. Input: Every product (good or service) must be produced either at home (Domestic Production) or abroad (imports). So,
Input = Domestic Production + M.
II. Output: Every product must be used for final consumption (C + G), capital formation (I), exports (X), or intermediate consumption (IC). So,
Output = C + G + I + X + IC.
Note: The terms GDP and output are often used as interchangeable synonyms. While this is fine in loose and informal usage, this is actually incorrect—the System of National Accounts defines these two terms differently (output as above and GDP as below).
III. Input = Output. So,
Domestic Production + M = C + G + I + X + IC.
IV. Define GDP (Y) = Total domestic valueadded = Domestic Production − IC.
Do the algebra (III into IV) to get our usual Y = C + I + G + X − M.
Again, the only reason for the “− M” on the right is that C + G, I, X, and IC already fully include M, which we don’t want to include in GDP (domestic valueadded).
Unfortunately, teachers often skip over the above principles and jump straight to “Y = C + I + G + X − M“, a simplistic formula that can easily be assigned as homework or exam questions, but can also easily lead to error.
Objection B. What about indirect effects?
Through pure accounting principles and identities, we’ve shown above that any increase in M must have zero direct effect on Y.
But of course, there can be indirect effects:
Example 1. Imports of $1M of alcohol lead to a Great Moral Downfall and fall in productivity and GDP.
Example 2. Imports of $1M of books lead to a Great Intellectual Awakening and rise in productivity and GDP.
Example 3. “Unsustainable” imports of $1M of Ferraris lead to a loss of confidence, capital outflows, mass panic, and subsequent fall in GDP.
Example 4. Imports of $1M of raw materials allow for production of $3M of semiconductors for exports and hence a rise of $2M in domestic valueadded (and GDP).
Example 5. (Common claim by laypersons and Trump advisor Peter Navarro). Imports of $1M of kimonos from Japan lead to an exact and precise reduction of $1M in US GDP.
Such indirect effects can occur. But they must be explained and justified through stories (or “models”) beyond pure accounting principles and identities.
In particular, they cannot be explained or justified through an incorrect, handwaving, and mechanistic application of “ceteris paribus” to Y = C + I + G + X − M. This “works” only through a lack of understanding of what this accounting identity means or where it comes from.
Example 5 is particularly dubious and requires detailed explanation—how does importing $1M of kimonos from Japan lead to a fall elsewhere in US domestic valueadded (in dresses, Big Macs, subway rides, etc.) of exactly and precisely $1M?
Another argument. If you’re still not persuaded, here’s a separate and more “formulaic” argument that might:
Decompose each of C, I, G, and X into their domestic valueadded and imported valueadded:
C = Cᵈ + Cⁱ, I = Iᵈ + Iⁱ, G = Gᵈ + Gⁱ, and X = Xᵈ + Xⁱ.
We have M = Cⁱ + Iⁱ + Gⁱ + Xⁱ. Hence,
Y  = C + I + G + X − M 
= Cᵈ + Cⁱ + Iᵈ + Iⁱ + Gᵈ + Gⁱ + Xᵈ + Xⁱ − (Cⁱ + Iⁱ + Gⁱ + Xⁱ)  
= Cᵈ + Iᵈ + Gᵈ + Xᵈ. 
So, GDP (Y) does not (directly) depend on M.
(Again, as already explained above, there can be indirect effects. But any such effects must be explained by stories beyond pure accounting identities.)
Originally posted 20220116. Slight edits made on 20220117 and 20220118.
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