GDP Will Reflect "AI" (Response to Dwarkesh Patel's Concern)

Should we care about Gross Domestic Product (GDP)? Does GDP reflect what matters? Is GDP a good measure? Recently, an influential tech podcaster and interviewer, Dwarkesh Patel, brought up the topic in the following context:

"As measured by GDP, AI will be super undervalued. How would the datacenter of geniuses show up in GDP? GDP would show raw inputs (aka chip & energy), and raw outputs (aka cost of tokens). But wouldn't clearly reflect the value of the crazy new [stuff] that's being cooked up in those tokens. Similar problem to how the Internet's value is undercounted today (since many products are free, and thus contribute nothing to measured GDP).”

Dwarkesh Patel is both right and wrong. To understand why, we need go over what is the GDP measure, why we measure it, and does it capture what matters to us.