In one section of the budget the Reagan administration sent to Congress at the beginning of this month, David A. Stockman, the director of the Office of Management and Budget, laid out his analysis of the "structural" deficit problem that confronts the government.

It is a fascinating discussion--part apology and rationalization for the too-rosy scenario that Stockman and his boss painted in 1981, part soundly conservative interpretation of recent American history. According to Stockman, the villain of the budget deficit is the "vast expansion" from 1963 through 1981 in entitlement program spending for the elderly and needy.

There has been "a nearly five-fold increase in constant-dollar costs" of what Stockman termed "the social contract," mainly retirement and medical benefits for the elderly, and "a parallel expansion of other social programs" that are based on the needs of the recipients. Together, they consumed 10 percent of the Gross National Product in 1981--two-thirds more than they had a decade earlier, Stockman said.

His discussion focused on the budgetary consequences of that rapid buildup-- not its causes. To find some clues to the causes, you might turn to an article by another bright young man round Washington, Peter Edelman, in a new publication by the Center for National Policy called "Budget and Policy Choices 1983."

Edelman, a former assistant to Robert and Edward Kennedy, now teaching law at Georgetown University, compactly traces the forces behind the expansion of the welfare state beyond the New Deal notion that welfare was a temporary palliative for those unable to find work.

"In the postwar period," he wrote, "enormous and unforeseen social and economic changes began to occur. A baby boom of unprecedented proportion took place, creating an inevitable problem down the road for absorption of this larger group into the labor market. And shortly, women began to enter the labor market in increasing numbers.

"Meanwhile, the rural-to-urban migration . . . (was) accelerated" by the war-induced industrial boom, the mechanization of agriculture and the flight of southern blacks from discrimination. Immigration from the Caribbean and Latin America increased, and soon technology began reducing the number of low-skill jobs.

The result was that "unemployment in the major urban areas climbed steadily. Black teen-age youth unemployment--always the highest relative to other groups-- was less than 15 percent in 1950. It is now over 50 percent. . . . Many jobs moved to the suburbs and became relatively inaccessible to residents of the inner-city. . . .

"We now see," Edelman wrote, "that an iron vise was tightening--schools and other institutions that didn't convey sufficient work skills and jobs that weren't there. The more aggressive, resourceful and talented were able to escape its grip. For the rest, help came to consist primarily of cash assistance rather than aid in reaching self- sufficiency."

Edelman makes a point that Stockman manages to minimize or dismiss. While programs for the elderly have increased dramatically in cost, they also have largely achieved their goal, bringing the beneficiaries "to the point where they, in percentage terms, are no more subject to poverty than the population as a whole."

Poverty is now concentrated in female- headed households, and it is there that the Kennedy Democrat and the Reagan Republican agree a new strategy is needed--not just for fiscal but for humanitarian reasons.

The strategy that is explicit in the Edelman article and implicit in the Stockman budget is a closer link between the public schools--particularly the high schools-- and the job market, to help train and employ those minority youths who form the hard core of the hard-core unemployed. Edelman speaks admiringly of the Private Industry Councils (PICs), established in the 1982 bill his old boss, Kennedy, cosponsored with Sen. Dan Quayle (R-Ind.). Training and placement in private-sector jobs is the kind of jobs bill the Reagan administration is willing to expand.

Edelman and Stockman both understand the important point that the big federal aid to education program of the past decade (Title I) has improved basic skills at the elementary-school level. Both understand that the stage is set for a concentration on mathematical, scientific and computer literacy programs in public high schools in the '80s.

They still have basic disagreements on the costs of what Edelman calls "the new entitlement programs," in education, job- training and placement. He is skeptical that subminimum wages or employer subsidies of the kind the Reagan administration is studying will be sufficient to crack the problem. But they are not so far apart in what they are saying that they, and the people they influence, cannot find common ground on which to work.