Core PCE and the other widely cited measure of inflation, the consumer price index, use two different methods of tracking healthcare expenses,which we've noted before. CPI tracks only what Americans are spending out of pocket on their healthcare, whereas PCE takes into account the insurance costs picked up by the government via Medicaid or Medicare, and the costs private insurers pay.
picked up means acquired in this case, meaning that they have received these costs. This is similar to sponsored, but is not the best word to describe the case. Aother interpretation of this would be: PCE takes into account the insurance costs paid for by the government via medicaid or medicare.
When we pick up the tab at a bar or restaurant, we're paying the bill for everyone in the group. It means "to assume responsibility for paying for something, to pay for goods or services consumed by others".