There is a lot of discussion about the debt ceiling, most of it somewhat exaggerated and panicky. In a recent WAPO op-ed it was called Financial Armageddon. From a run on the dollar to the complete collapse of the economy, one can find almost anything in the news. And sure enough there are reasons to be more concerned this time than in previous disputes between a Republican House and a Democratic White House, which is always the pattern when it comes to the debt ceiling, an institutional feature that very few countries have, btw (old post here, and piece in Dollars & Sense).*
First, let me say a few things about the consequences. If the debt limit is breached, and a default occurs, the first and most direct consequence is that a series of government functions that require government spending cannot go on, and these activities will stop. That is essentially like a shutdown. Government shutdowns have occurred several times, but not as a result of a default, and this will have further implications. Last time we were close to breaching the debt ceiling, the credit rating agencies (Standards & Poor's, to be specific; old post on that here), that determine whether public and private agents are creditworthy, downgraded the US debt for the first time. It was unnecessary at that time, because there was no doubt that the government could pay its bills in its own currency. But that is likely to happen again, more so if there is a default. This would reflect not only the budgetary inability to spend, like in a shutdown, but also the fact that the Treasury will, most likely, stop paying interest on its debt.
Normally, countries that default pay a significant price. Argentina, for example, has defaulted, and that has led to a run on the currency, as agents seek to sell government bonds and try to buy foreign denominated bonds, mostly in dollars. That is inflationary, as the currency depreciates and the price of imported goods go up, and contractionary, since the inflation reduces the ability of consumers to spend. Depreciation, inflation and recession, are the likely outcomes of a sovereign default. Note that Argentina normally defaults on its foreign obligations in dollars, not the domestic ones in pesos (there would be no reason for that, even though Macri did it once), as the United States might do soon.
In the case of the United States, that holds the global reserve currency the consequences would be considerably milder. Everybody knows that the Treasury, besides the political bickering, can always pay its bills in dollars. This crisis does not represent a fundamental inability of the government to pay its bills, but simply the decision to not pay them for calculated political gains by Republicans. The likely effect might be a mild recession associated to the inability of the government to spend, that would add to the already contractionary monetary policy. I should note that a negotiation between McCarthy and Biden to reduce spending, which will be contractionary for sure, is in my view worse than breaching the debt ceiling, since Biden can always find some solution and continue to spend. A recession would be politically disastrous for Biden.
Besides a possible recession, some depreciation of the dollar might or might not occur. Sure enough it is possible that some agents would go to Euro denominated assets, or some other alternative currency, but since the interest rate is relatively high in the US, and it was raised last week again as a result of the preoccupation with inflation, that might attract economic agents into holding dollar denominated assets. The results are ambiguous. Certainly there is no danger of the dollar losing its international position, as Larry Summers correctly pointed out. And of course, even if there is some mild depreciation, its inflationary impact in the US is negligible as compared to developing countries like Argentina. In a developing country, if the currency depreciates 30 percent with respect to the dollar, the price of oil (priced in dollars in international markets) in domestic currency goes up tantamount, but that is not the case in the US.
I am skeptical that a default, if it happens, would be a prolonged problem, since it most likely will backfire politically for Republicans, as shutdowns normally do. So Biden should ignore this, and go on paying, as I suggested a while ago.
* Proof of that is that both Paul Krugman and Laurence Tribe have written in favor of alternative ways of dealing with the issue. I, for what's worth, always thought the one trillion dollar coin exceedingly idiotic, and would prefer the 14th Amendment solution. I think if it went to the SCOTUS, even this group of corrupt, pro-business, conservative justices would muster a narrow majority (Roberts and Kavannaugh perhaps) for the unconstitutionality of the debt ceiling.