In the Wild: Identities Deceive

I have argued before that accounting identities can be deceiving. I specifically argued early on here that the GDP identity $$ Y=C+I+G+X-M $$ does not by itself imply that increased imports $(M)$ reduce Gross Domestic Product $(Y)$; the presentation merely invites the reader to form a model in which it is true. But this, from Noah Smith, is just painful:

@imalatkinson @jodiecongirl @Jacob_Atkinson1 Imports neither add to GDP nor subtract from it. Exports add to it. Econ PhD 2012. :D

— Noah Smith (@Noahpinion) January 30, 2017

To be clear, Smith’s argument is that that this is true “mechanically”— distinct from any model. But this just requires us to ask what Smith means by “mechanically.”

Imports-in-GDP is a correction to avoid double-counting when measuring GDP based on final sales. We start with domestic sales of consumption, fixed investment, and government goods and services $(C+I^*+G)$. To this, we add sales of all goods and services to foreign economies— that is, exports $(X)$ are considered final sales in terms of their disposal with respect to the domestic economy. To get domestic production from final sales requires two adjustments. First, we add in net unsold production— that is, changes in inventories $(\Delta inv)$; second, we subtract foreign production sold domestically— imports. Thus, $$ Y=C+I^*+G+X+\Delta inv-M $$ Changes in inventories are grouped with fixed investment into gross investment $(I=I^*+\Delta inv)$ so we get $$ Y=C+I+G+X-M $$ But this does $not$ tell us one way or another whether imports add or subtract from GDP. It merely tells us that if $M$ does change, that something else must also change. If imports increase, then $Y$ must fall or $C+I+G+X$ must rise. We need a model to tell us anything more.

For example, it could be that in the long run, imports today increase GDP by increasing pressure on domestic industries to become more productive. One might argue that \$10 of additional $M$ means an additional \$1 of $Y$ and \$11 of $C$. But this is clearly not what Smith has in mind.

For imports to “neither add to GDP nor subtract from it” the change in $Y$ must be zero from any change in $M$. The identity thus tells us Smith believes that any change in $M$ is balanced by an corresponding movement in $C+I+G+X$. A \$10 increase in $M$ must “mechanically” raise $C+I+G+X$ by \$10.

Let us suppose for a moment that this makes sense. What in $C+I+G+X$ can be so definitively affected? For imported goods, the only reasonable answer is $\Delta inv$. When I import a consumer good, I might hope to sell that good and therefore have it counted later in final domestic sales of consumption. I might even have an order for something specific and so be extremely confident that eventually that the extra $M$ will become $C$. However, the immediate effect is that I have increased my inventories. Thus, every dollar of goods imports adds directly to gross investment and so the net effect on GDP is zero.

Given that, well, the net effect of goods exports on GDP is also zero by similar logic. Between production and exportation, goods pass through inventory. I might increase production (increasing $Y$) to compensate for the loss of inventory, but every goods export dollar is immediately a dollar taken out of inventories. In this sense, exports do not add to GDP as Smith argues.

At the very least, we cannot argue that exports definitively add to GDP in any immediate sense. Contra Smith, There is no mechanism by which an increase in exports requires an increase in production. And that is what makes him so painful to read.