Britain is the home of many great institutions whose names are synonymous with quality in their fields but that lately have fallen on hard times.

Rolls Royce, the auto and engine maker, went bankrupt several years ago and now is state-owned. Sotheby's, the legendary auctioneer, is running a huge deficit and may be sold. The Times, for centuries the country's premier newspaper, is losing millions, and no longer is regarded as the best periodical in the English language.

But that distinction still may reside in London, in the custody of another old British institution--The Economist--which is booming by its own modest standards. Founded in 1843--and edited later in the century by Walter Bagehot, the period's most famous English journalist of democratic politics and capitalist economics--the weekly's circulation has doubled to about 200,000 over the past decade.

With a determined marketing effort, it now sells a third of its copies in the United States. Last year, the magazine recorded a comfortable profit while most other serious British publications were struggling to break even.

It is as a publication though, not as a business, that The Economist excels. And yet, little is generally known about the people who write and edit it and the way it is put together, probably because The Economist is one of the few journals whose staff still is almost completely anonymous.

There is no masthead, no bylines, no self-congratulatory explanations of how it got one story or another. There is in this anonymity a cultivated mystique, as if what The Economist as an institution thinks or reports is more important than the personalities of the authors.

Britain has ceased to be a great global power, but The Economist shows the Union Jack where once the Royal Navy did the job. The magazine has subscribers everywhere from Albania, Afghanistan and Argentina to Western Samoa, Yemen and Zimbabwe, 160 countries in all.

To arrive at The Economist's particular mix of news, analyses and opinions, presented in literate and deliberately breezy prose, "it is probably easier not to be at the center of gravity," in Washington or New York, according to executive editor Dudley Fishburn.

The core of the magazine's credibility is its ability to speak with a transnational voice, to maintain, in its own words, "as few prejudices as possible and certainly no party political prejudicies." (This doesn't preclude The Economist's strongly anti-Soviet and generally anticommunist stance.)

The magazine does succeed in writing about the United States without being condescending--a temptation few Europeans can resist--and about the rest of the world in a way that seems comprehensible to Americans.

"The academic or government English language tends to be Germanic," observed Andrew Knight, The Economist's editor-in-chief for the past eight years. "We try to maintain a certain freshness, to be mid-Atlantic in our use of idiom, informed but humorous and detached as well."

As Knight and Fishburn view it, the essence of The Economist's atmosphere is the handling of the people who produce it. The numbers are small, a total of 45 editorial workers and another 45 secretaries, librarians and such.

Section editors and writers do everything from reporting to captioning, which Fishburn credits for the cohesiveness of each story. Others will read each article, and some editorials are collaborations. But the basic responsibility belongs to one person--in marked contrast to Time or Newsweek, where stories are the work of a researcher, a reporter, a writer and two or three editors.

There is a strong transatlantic cast to the staff, which includes a number of Americans (who do not, as a rule, write stories about the United States.) Many of the British staff were educated in the United States or worked there.

Knight, 43, an Oxford graduate who once worked for a merchant banker, spent two years as an Economist correspondent in Washington. The magazine's China-watcher, Emily McFarquhar, is an American, as is the defense correspondent, James Meachem, who was a Navy captain stationed in Southeast Asia during the Vietnam war.

The Economist maintains bureaus abroad in New York, as well as Washington, Paris, Bonn, Tokyo, Brussels and Singapore. These serve the political sections, which are divided into Europe, the European Community (the magazine is a very strong proponent of the Common Market), International, Britain and the U.S. report.

Then there is the "back half," which is The Economist's name for the important business sections. (After all, it is called The Economist.)

Contributors, who also write anonymously, usually are journalists working for other publications, and, occasionally, academics. (Fishburn says that Defense Secretary Caspar Weinberger claims to have written a few pieces many years ago.) Knight believes that not being identified is an incentive to many writers, especially in Third World or communist countries where official retribution sometimes results when governments don't like what they read.

As a weekly instead of a daily newspaper, the stories tend to be summaries of the trend of events, densely packed with facts. All stories have a perspective or point of view--generally, but not necessarily, supplied by the author. The main opinions of the editors come in five "leaders" that appear in the front of the magazine.

Every Monday morning at 11:15 the staff crowds into Knight's corner office with a panoramic view of London for a rundown of the upcoming week. The principal topics at one recent session were problems facing the Organization of Petroleum Exporting Countries, East-West trade, Britain's rail strike, and the prospects for a trip to Poland by Pope John Paul II. Spirited discussion produced at least a sense of what viewpoint the magazine should take. Then each section editor described what stories were likely for that week's issue.

When the finished product emerged Thursday evening, it was a faithful representation in substance and opinion of that Monday's guidelines.

The Economist's philosophy, shaped with the passage of time, dates from the Bagehot tenure of 1860 to 1877. It's commitment is to democratic values and, where suitable, capitalist economics. It is a firm believer in the strongest possible Western--especially American--defense.

In American political terms, The Economist is conservative enough to have supported Ronald Reagan's election in 1980, although Knight is frequently quoted as placing the magazine in "the extreme center." The Economist still supports Reagan's basic economic and political objectives, but is less than reverential about his style, observing that he "rises lateish, except when they wake him up with news of tanks in Poland, or whatever; takes an early afternoon break; devotes most of his time to American affairs and leaves the Oval Office early."

The real measure of The Economist's persuasiveness is that it isn't necesary to agree with its arguments to admire the way they have been put. Good writing, a specialty of the best in the British press, is The Economist's forte.

On OPEC: "Hooray for the disarray in OPEC, and pray that it might be terminal. It probably won't be. False news of OPEC's impending death has erupted before as often as Mark Twain reckoned he gave up cigars . . . "

On Reagan's economic sanctions against the Soviets: "President Reagan may be right, and certainly has the right, to seek to change the decision of West European countries to help build a pipeline to import gas from Russia. There is though, a dividing line between legitimate American opposition to the project and illegitimate American interference in the sovereignty of allied governments . . . "

On the war in Lebanon: "The exodus from Beirut is a huge defeat for the Palestine Liberation Organization. But the worst part of the defeat is in the PLO's loss of semi-independence, not in its loss of military power. The PLO will be better off without its misguided and misleading military appurtenances. Its self-presentation as a guerrilla movement fighting Israel was largely fiction, or wishful thinking, and it brought a terrible punishment that fell more on the guiltless than on the guilty . . . "

Whatever the magazine's journalistic successes, however, its future, like so many other of Britain's great names, would be less promising if it were failing as a business.

There have been some rough patches. A proposed Spanish-language edition failed in the 1960s. In the 1970s, serious union problems that have been the bane of other British publications developed in The Economist's own printing plant. After months of stoppages and missed deadlines, the plant was closed and printing locally now is done by contract.

In 1981, printing began in the United States with establishment of a separate corporation for production, sales and promotion. Initially, said David Gordon, the magazine's managing director, the print orders were about 50,000. Recently, they've been running about 85,000. To boost circulation in the United States, British reserve has been abandoned and an aggressive mass-mailing campaign designed.

The contents of the U.S. edition are the same as the one published here--although occasionally covers are tailored for American interests--but the paper quality is better, since that's what Americans seem to expect when they shell out $2.50 a copy. (The cover price here is about half that.) A study of Economist readers in the United States showed that the average household income of subscribers was more than $94,000, one of the highest of any publication.

The Economist, which is half-owned by the Financial Times Ltd., publishers of Britain's leading financial daily, is run by an independent board of directors whose chairman is Evelyn de Rothschild.

"As a newspaper, we believe in commercial virtues," said Gordon, a trained accountant, who was the magazine's business editor before moving into management, "and as a business we believe in commercial virtues also." It comes as a considerable relief to Gordon and others at the magazine, therefore, that profits, never really large, doubled in the fiscal year ended in March 1982 from what they were two years earlier to about three million pounds (almost $6 million). Advertising is robust despite a continuing world-wide recession.

Still, compared with great media giants in the United States, those figures are small change. But The Economist makes no claim to be big. It just aims to be smart. And on that account it is faring very well indeed.