The Big Lebowski was backed by PolyGram Filmed Entertainment and Working Title Films, with a not-too-big-but-not-too-small budget of $15 million.
The Coen Brothers already had a reputation for being indie auteurs: they wanted to make high-quality, cerebral films that would appeal to sophisticated audiences. The fewer explosions and car chases, the better. (There's that wood-chipper scene in Fargo, but anyway …)
PolyGram Filmed Entertainment was about as indie as they came. It started as a Dutch music company, partnering with indie producer Peter Guber in 1980 to create London-based PolyGram Filmed Entertainment (PFE). The goal was to create a European film studio that could distribute films internationally on the same scale as the big Hollywood studios.
Working Title Films was a small production company working out of the PFE London studios. Their jam was quirky, intelligent comedies. They churned out a lot of hits for PFE in the '90s, including Four Weddings and a Funeral (1994), Dead Man Walking (1995), and Notting Hill (1999). By the time they bought The Big Lebowski, both PFE and Working Title Films were on the upswing, positioned to become serious threats to their meat-and-potatoes American competitors.
PFE had already backed the Coen Brothers' Fargo, which was a huge critical and commercial success, with two Oscars and a ton of other awards. At that point, they were ready for anything the Coen Brothers could throw at them.
From the beginning, both Working Title and PFE knew that the movie would be targeted at a fairly niche audience, the kind that didn't care if any of the characters had superpowers or could blow up buildings. They also knew that this audience wouldn't include children; most of the Coen Brothers' films have enough violence in them to qualify them as "dealing with adult themes." And what kid would really care whether or not a rug ties a room together?
With the knowledge that the movie wouldn't exactly be a summer blockbuster, PFE and Working Title set out to expand the audience as best they could. Believe it or not, the amount of swearing in the screenplay was actually reduced before the movie was filmed. The production company also wanted to preserve the retro feel of the screenplay without introducing too many drugs into the picture. This is why The Dude sticks to alcohol and weed as his main vices (although there are massive quantities of each); only once, in conversation with Maude Lebowski, does he mention LSD use in the past.
The final product ended up being pretty close to the Coen Brothers' vision, down to the littlest details, like Walter's "chin strap" beard, The Dude's dancing landlord, and the way John Turturro's character licks his bowling ball before sending it down the lane. These guys know exactly what they want.