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Airliners like Boeing 747 and even Airbus 380 are at times considered as personal jets for extremely wealthy people. Looking from the side, Concorde was ideal for this role: while not very big, it probably could provide luxury transportation for ten VIPs or thereabout, adding to that something that looks more valuable and more exceptional than just a "flying palace" - speed like no other. New supersonic aircraft like Aerion AS2 are often seen in this role. However, it looks like rich individuals only bought various bones of Concorde after retirement.

The fate of F-BVFD looks especially strange. As I understand, it has been retired very early, in airworthy condition, just for being a surplus aircraft. After standing for many years in the open air, it deteriorated and then has been scrapped. I cannot imagine it to cost a lot during these years of standing. Why did nobody try it for personal use? Also, some pre-production aircraft have been retired after logging not so many hours and were airworthy.

Surely, Concorde uses lots of fuel, but probably the A380 burns no much less if per flight (not per passenger). And it only could fly supersonic over sea, but was surely capable of the fastest possible subsonic flight over the land.

Private business jets have a few requirements:

1. Operational costs must be within budget. The concorde is pretty expensive to run as demonstrated by the high ticketprices you had to pay for a trip on one of them. Fuel is only part of that costs; maintenance is a pretty big part of that as well.




2. Be able to land and take off close to where the CEO has his meeting. This means making use of small airfields with their short runways and noise restrictions. Concorde needed a lot of runway and was not a quiet plane.




3. Account for airfields sometimes not having fuel available or the fuel being too expensive. This may mean doing the roundtrip on a single tank or doing another short hop to an airfield with fuel.

Business jets would also mostly operate on a domestic scale. Doing mostly short hops where the concorde would not even reach supersonic cruise altitude before it's over the destination. While plenty of business jets can easily do 0.75 mach which is already 3/4 the speed of sound.

While it was never explicitly sold as such (although BAE would have happily delivered it as such) it was operated in such a manner on many occasions. As detailed by this former pilot of the Concorde, the plane regularly flew as a charter with small passenger loads of VIPs often. So even though no one ever owned it specifically for private business it served that purpose quite well.

There was also not a huge reason to own it privately for business as that was to be its main market. Glitz and glam aside, the Concorde was a serious way for American and European based entities to have a face-to-face meeting on either turf and still be home for dinner. Offloading the maintenance and ownership to an airline was a nice plus.

Some people also speculate that the ultimately canceled Iranian order by the then Shah was perhaps for his personal use:

I can’t blame Iran for canceling their orders, as like most airlines
at the time, Iran Air was more a symbol of national identity, and the
Concorde’s were most certainly toys for the Shah rather than aircraft
for his people. Still, it’s a pity nonetherless.

Although they never bought the whole aircraft, Pepsi did buy the rights to paint one of the Concordes.

It's also worth noting that Concordes sales occurred mostly in the mid 1960s (approximately 24 minutes in) a time in which business aircraft were different than what we currently think of. Commercial flying was still a fancier affair than it is now and lots of business individuals flew commercial. Smaller private planes lived more in the propeller-driven space than the jet space. Keep in mind the Lear 23 had also just come out and the whole space of business/private jets was quite literally in the process of being defined. It would not be until 1972 that we would even start to see rock stars toted around in private jets. The late 1960s and early 1970s were pretty much considered the haydays of general aviation as post-war pilots were in a position to buy planes. The focus in the private market was heavily on selling single-engine piston planes.

At the time the jet was in production private jets were still in their infancy. The largest purpose built one was the Grumman Gulfstream II. I have a feeling the Concorde was a bit out of the reach of all firms at the time.

In addition by the time the SST had gone into full production, it had largely fallen out of disfavor for noise and environmental reasons. Concorde was also, dimensionally a big airplane with a length over 200 ft, making it awkward and unruly to maneuver at many smaller airports. It also had a gross takeoff weight of over 400,000 lbs, again limiting what runways it can fly from. Supersonic aircraft also don’t typically feature good short field capabilities which limits the airports you could fly a Concorde into and out of. SUD/BAC quoted a balanced field length at STP for the Concorde of 9400 ft. That really limits where where you can fly from.

So unpopular was the SST by 1980 that SUD/BAC had only produced a total of 14 production Concordes, all earmarked for either British Airways or Air France. Most likely all of these factors combined together deterred even the most rabid buyer.

Another factor to consider: Aircraft require a lot of support. It's not enough to just sell a few aircraft. There's support that must be factored in - spare parts, training of maintenance crews, etc.