Thomas A. Garrett (Federal Reserve Bank of St. Louis) & Cletus C. Coughlin (Federal Reserve Bank of St. Louis) have published Inter–temporal Differences in the Income Elasticity of Demand for Lottery Tickets, 62 Nat'l Tax J. 77 (2009).  Here is the abstract:

We estimate annual income elasticities of demand for lottery tickets using county–level panel data for three states and find that the income elasticity of demand (and, thus, the tax burden) for lottery tickets has changed over time. This is due to changes in a state’s lottery game portfolio and the growth in consumer income more so than competition from alternative gambling opportunities. Trends in the income elasticity for instant and online lottery games appear to be different. Our results raise doubts about the long–term growth potential of lottery revenue and have policy implications for state governments and those concerned about regressivity.