A few brief excerpts from some analysis by Goldman Sachs economists:

The Treasury projects that the debt limit will need to be raised by September 29, and we expect Congress to take at least that long to raise it.
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What happens if Congress does not raise it in time?

Scheduled federal payments would be delayed and financial markets would be disrupted. The primary consequence of a failure to raise the debt limit before October 2 would be a likely failure to make some payments on that day or soon thereafter. This could include, for example, the main monthly payment to Social Security beneficiaries, which occurs on the 3rd of every month. All told, the decline in federal payments that would be necessary to avoid breaching the debt limit would be roughly equal to the budget deficit, resulting in a temporary fiscal contraction of 3-4% of GDP for the period the debt limit is binding. A sharp decline like this would likely be disruptive to financial markets and could have consequences for the real economy, which is why Congress has managed to avoid such an outcome in the past.
CR note: I'll have much more on the "debt limit" as the deadline approaches. I've written about this nonsense before, from 2011:
Unfortunately "debt ceiling" sounds virtuous, but it isn't - it is actually a question of "paying the bills".
And here from 2014:
There are certain politicians who think it is OK to not pay the bills as long as the U.S. makes interest and principal payments on the debt.  That is crazy talk.  There is a name for people who don't pay their bills: deadbeats.  If politicians don't pay their personal bills, they are deadbeats.  But if they stop the government from paying the bills, we are all deadbeats.  And there will be serious economic consequences for not paying the bills on time.  The consequences will build over time, but in a few months, not "paying the bills" will ripple through the entire economy.
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The only smart vote would be to eliminate the debt ceiling vote in the future, and let the annual budget process automatically set the debt limit. But then there wouldn't be as much camera time ...