Do circular invoices count in GDP? I'm trying to understand GDP and what it means.
Say that there are two US software companies, Foo Inc and Bar Inc. Foo Inc produces software and charges Bar \$1 million for its usage, while Bar Inc produces its own software product and charges Foo Inc the same amount of \$1 million.
While money did change hands, the profits of both companies remained the same, as they both incurred the same amount in revenue and expenses.
Would the national GDP increase by \$2 million because of these transactions?
A firm's contribution to GDP isn't revenues. A firm's contribution to GDP is roughly the sum of the wage bill and the money earned by capital (debt (interest) and equity (economic and not accounting profits)). That is, profits + interest payments + wages (look elsewhere for a more precise definition). Your example doesn't really have enough to go on to estimate the GDP impact.
The contribution could be zero. This can and did happen in the dot com bubble when companies would agree to buy advertisements on each other's sites of equal amounts to boost sales figures. This increases revenues and expenses by the same amount with no changes in wages, profits, or interest expense (maybe minimal changes in practice).
The contribution could also be arbitrarily large. Imagine a firm that makes a buys a piece of software for \$2 million that allows them to make \$10 billion in sales (no other expenses and sales and no profit without this expense). That would make a GDP contribution $9.998 billion. Now imagine splitting that company in half so that two different firms each make \$5 billion in additional sales after buying \$1m in software.
In summary, the impact on GDP depends and there isn't enough detail here to be more specific.