To approach the OP from a different angle (still hoping for the spirit of General Questions), interpreting the question as “Is the prospect of getting more money the reason that people on welfare tend to have yet more babies?” —
There is a sociological theory associated in the mainstream with Daniel Patrick Moynihan, one not without controversy but for which substantive supporting arguments have been, generally referred to as “culture of poverty”, which explains it more like this:
• There is a social stratum in which people grow up without employed parents in neighborhoods where that is the norm. Many of these are single-parent families (usually single moms, not that that’s relevant in all the ways that some people, incuding Moynihan, have tended to make it).
• When you grow up without role models among peers and parents and parents of your peers and so on who have self-sustaining jobs and careers and whatnot, you are less likely to aspire to same, to assume you’ll have them yourself, and so on. You’re also less likely to acquire the formal and informal skills from your home and neighborhood environment that make you readily employable (watching mom balance the budget and seeing how the math works and discussing investments and capital and savings, going to Tosca with your Dad and discussing the history of religion and politics and of opera’s relationship to modern movies, and simply picking up on the nuances of grammar and intonation and body language that are used by the socioeconomic class of the generally-employed and often-employers) or give you good prospects for getting into college. Or thinking that college is in your future, etc.
• People from puberty through adolescence and early adulthood start seeking ways of demonstrating that they are in fact adults (grown-ups). There are many such observable attainments, including moving out on your own (generally requires an income or admission to a school that provides dorm rooms), getting a job and therefore disposable income, getting a car so you’re independently mobile (generally requires either an income or parents with an income), getting married (doesn’t require economic independence but it tends to be expected); or becoming a parent.
• If you grow up in a social stratum where having a baby without having a job and being able to support yourself and your baby is not outside the norm, there is little stigma attached to doing so; and that tends to go hand in hand with the lack of expectation that you will be self-supporting (it’s the same social stratum). So becoming a parent while (still) being on welfare or obtaining your own welfare (where previously you were supported via your parents’ welfare) is one of the few available and locally-accepted avenues by which you change your status from child to adult.
• And if you’re unemployed and out of school, what ELSE would you be doing with your time? It can look like a more responsible and grown-up way of being in the world than hanging out with your unemployed friends (outdoors on street corners and in parks and malls since most of you have no homes separate from those of your parents yet due to no income etc).
There’s plenty of room to poke holes in bits and pieces of this, but as an overall way of looking at things, I think it rings true. It has often been twisted into a debate over whether or not the existence of unmarried moms is a huge social problem, but I suggest focusing more on the larger picture of what the theory is saying.