Typically, answering this question is difficult as the Medicaid program varies across states and even within states. What Amanda Kowalski and co-authors do in a paper she presented at the 2016 ASSA is collect data on the variation in Medicaid eligibility across states, across demographic groups, and across time from the inception of Medicaid in 1965 to the present.
The authors fine that a 10 percent increase in Medicaid eligibility in a state increases take-up by 1.52%. This factor likely varies by state, over time and by demographic groups, but on average, we see that about 85% of patients don’t take-up Medicaid when offered. Additionally, the authors find that:
Using all legislative variation in our calculator, we find that insurance explains about 20 percent of the overall growth in total health care spending from 1964 to the present.
The authors also find that Medicaid expansions to adults increase health spending much more than for children. It is unclear whether this effect is because children are generally less sick than adults on average (particularly the mean adult rather than the median adult) or because demand for health care for children is more inelastic compared to health care for adults.