Buried deep in the new budget is an item that comes with a small price tag and a large message. It's a proposal to cut off aid to any unmarried teen-age mother who doesn't live with her parents.

The money that could be saved by this budget snippet is small, an estimated $19 million out of a $959 billion spending spree. But it's one of the more artistic efforts to reshape social policy with fiscal scissors.

According to the administration, the current system, which adds unwed mothers to the welfare roles, is just "an incentive for minor parents to leave home and use AFDC (Aid to Families With Dependent Children) to establish financial independence." Their plan, on the other hand, would create an economic tie that binds.

Underlying this ideology is at least one murky myth that should be dismissed. It's the subtle and pervasive belief that welfare is an incentive for teen- age motherhood. The notion that girls have babies in order to live on their own in welfare splendor has been proved wrong by every study. Teen-agers who get pregnant haven't planned their parenthood at all. They certainly haven't planned it as a small-business venture.

But once they are on AFDC, the payments can make a difference in where they live. A study of welfare mothers under 24 by Harvard's David Ellwood and Mary Jo Bane showed that the higher the AFDC benefits, the more likely a mother is to leave her parent's household.

In real life, the actual number of teens who do that is very small. Only one out of six mothers under 20 lives alone. Almost 90 percent of the 15- to 17-year-olds live with their families. But we know that the unmarried teen-agers who stay with their parents do better than the others. They are more likely to finish school, more likely to be employed, less likely to have a second child, and their babies are healthier. In short, they have a support system.

Given these facts, it makes humane and common sense to encourage young mothers through public policy to stay at home. But there is a critical difference between encouraging families and coercing them.

The government's attempt to merge a pro-family and pro-savings policy has turned up a package of contradictions. The policy-makers in favor of three-generational togetherness this year are the same ones who worked against it last year. They cut payments to households if the parents' (or should I say grandparents'?) income was too high. They gave teen-age mothers an incentive to leave and now they are forcing them back.

I suppose the only way to squeeze a $19 million budget cut out of the poor is to press them at both ends. If young mothers live at home, cut their AFDC. If they leave home, cut it out altogether.

It's no secret that the Reagan people want to shift the financial burden from public life to private, from the state to the family. They legislate as if they could make all families behave the way the strongest ones do.

This time, it's a kind of national curfew: all children and grandchildren at home or else. The only exemptions would go to a girl whose parents were dead, or whose health and safety would be endangered, or who had already lived on her own. These don't begin to cover the myriad life situations of families in stress.

The government estimates that about 9,000 families will be affected by this proposal. Just a handful. But they are likely to include the least stable and the most troubled.

The University of Pennsylvania's Frank Furstenberg, who has studied teen-age mothers for 17 years, says: "There are a small proportion of adolescents who really cannot get along with their parents. The notion that somehow you should regulate that relationship and insist that the adolescent has to stay with that family to get economic support strikes me as illogical and unwise. Proposals that try to treat all circumstances alike, that bureaucratically try to manage people's lives, end up creating havoc."

Instead of restoring incentives for family success stories, the planners want to punish the failures. In this budget, any family that doesn't fit the Reagan profile can end up on the cutting- room floor.