Are you looking for general references on Diff-in-Diff (DiD) and Regression Discontinuity (RD)?
If yes, a good starting reference at the undergraduate level is probably
- J. Angrist and J.S. Pischke, Mastering' Metrics: The Path from Cause to Effect, Princeton University Press, 2014. Look at chapters 4 and 5 for RD and DiD.
If who want to dig deeper:
- J. Angrist and J.S. Pischke, Mostly Harmless Econometrics, Princeton University Press, 2009, chapters 5 and 6.
For recent references on RD
- C. Carpenter and C. Dobkin, “The Effect of Alcohol Consumption on Mortality: Regression Discontinuity Evidence from the MLDA, American Economic Journal: Applied Economics 1 (2009), 164-182.
- A. Abdulkadiroglu, et al., “The Elite Illusion: Achievement Effects at Boston and New York Exam Schools,” Econometrica, 2014.
- D. Card and A. Krueger, “Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania,” American Economic Review 90 (1994), 1397- 420.
- D. Card, “Using Regional Variation to Measure the Effect of the Federal Wage,” Industrial and Labor Relations Review (1992) 46, 22-37.
- C. Carpenter and C. Dobkin, “The Minimum Legal Drinking Age and Public Health,” The Journal of Economic Perspectives 25 (2011), 133-156.
Considering your example, I guess that it could be a contribution to show that a small increase in benefit per capita may increase fertility. Is the benefit permanent? If not, I doubt that people will have a strong incentive to get another child. Moreover, it takes time to get a new child! So, there is a time dimension issue. They cannot claim the benefit if they don't have the child! If your design allow for this, you can exploit the time dimension in case some families respond to the incentive.