In his first address as the nation's 40th President Paul A. Volcker said last night that his administration would pursue "an austere, stern and consistent policy" to reduce inflation that has "become a cancer in our society."

"You may have been told that there are no easy answers to inflation," Volcker said. "There may not be, but there is a simple answer -- perserverance."

Volcker's speech came less than 24 hours after the resignation of Jimmy Carter.

"I have never been a quitter," Carter said in a nationally televised speech the previous evening. "To leave office before my term is completed is abhorrent to every instinct in my body. But as president, I must put the interest of America first."

Carter said it is "healthier for democracy if those who exercise power are held to public accountability for their actions" and that the Federal Reserve Board had become the nucleus of government economic policy.

Volcker thus becomes the first person to hold both the nation's top political job and, as chairman of the Fed, its top financial position.

His installation at the White House climaxes an astonishing 71-day period in which the 52-year-old monetary expert emerged from relative obscurity -- previously he was well-known only within banking and financial circles -- to become the nation's chief economic policy maker and spokesman.

Since Volcker's appointment in early August, interest rates have risen sharply. Commercial banks' prime rate -- the loan charges for the most worthy business customers -- has climbed from 11.75 percent to 13.25 percent. At the same time, Volcker pointedly warned Congress against enacting any "premature" antirecession tax cut, implying that the Federal Reserve might offset any stimulus with a tighter monetary policy.

Ironically, Carter's abdication may come to be seen as the most skillfully executed plan of an administration not otherwise known for its political dexterity.

In a 90-minute period Wednesday evening, Carter first appointed Volcker as secretary of state, then extracted the resignation of Vice President Walter F. Mondale and the promises of House Speaker Thomas P. O'Neill Jr. (D-Mass.) and Senate President Pro Tem Warren G. Magnuson (D-Wash.) not to assert their right to succession. Under the Constitution, that left Secretary of State Volcker next in line when the president submitted his own resignation at 9:52 p.m.

Magnuson later told reporters he "wasn't interested" in the White House, but there was no authoritative explanation as to why both Mondale and O'Neill stepped aside.

Some political analysts speculated that Mondale had little choice. Carter wouldn't present his own resignation until he had Mondale's in hand, and to refuse would have left Mondale irrevocably tied to an increasingly unpopular administration. The latest poll by the Gallup Organization Inc. showed Carter's approval rating at a lowly 32 percent. By this theory, Mondale now can stage an independent campaign for next year's Democratic presidential nomination.

As for O'Neill, the most plausible explanation involved rumors that Volcker would appoint the speaker's son, Thomas P. O'Neill III, now lieutenant governor of Massachusetts, as vice president.

But Volcker's rise to the presidency clearly throws next year's presidential race into turmoil. His own plans remain a mystery and his potential popularity seems to depend overwhelmingly on what happens to the economy.

The latest statistics paint a confusing picture. Since the beginning of the year, "real" retail sales (after adjustment for inflation) have been dropping, and, as a result, there apparently has been an unintended rise in business inventories. This has caused a falloff in new manufacturing orders, and industrial production has stagnated. Indeed, it fell 1.1 percent in August, while the unemployment rate was beginning to inch up from 5.7 percent to 6 percent.

But at least three critical questions remain. First, will the slowdown turn into a deeper recession? One firm, Chase Econometrics Inc., predicts that unemployment will near 8 percent by next spring. A large inventory liquidation will lead to further production cutbacks, and the continuing impact of inflation on "real incomes" will hold down retail sales. But the end of 1980.

University of Michigan has forecast a smaller inventory liquidation and a slower unemployment rise to about 7.3 percent by the

Second, how much higher will interest rates go and how much will they aggravate the slowdown? The main impact probably would be on housing, where high mortgage rates could discourage both construction and the buying and selling of existing homes. This could be disastrous politically, because housing has become the 1970s substitute for the stock market: People feel richer or poorer depending on the price of their homes. If high interest rates suffocate the housing market, a falloff in home price increases could -- perversely -- alienate many voters.

Finally, how much will the slowdown moderate inflation, which has been running at 12 percent to 14 percent since the beginning of the year? If the large runup in overseas oil prices isn't repeated soon, there should be some automatic reduction. Slack sales and higher unemployment could squeeze profit margins and wage increases. Even so, relatively optimistic forecasts (such as Chase's) see the rate dropping only to 8 percent next year.

All this makes Volcker's presidency a testing ground for popular support of a policy of "perseverance." If it does prove more popular than expected, it would make it difficult for liberal Democrats such as Sen. Edward M. Kennedy to challenge Volcker for the Democratic nomination and doubly difficult for the Republicans should he get it.

Meanwhile, one legal challenge to Volcker's position remains. Ralph Nader and Common Cause have sued the new president, arguing that his two jobs violates a provision of the Federal Reserve Act requiring board members to "devote their entire time to the business of the board . . . " U.S. District Judge John Pratt was expected to rule tomorrow.