by the researcher. Second, treatment effects are likely to differ for observably and unobservably different groups. LATEs can be allowed to differ by observable charac-
teristics by using a linear probability model with interactions. Third, the LATE esti- mates do not measure treatment effects, such as crowd- out, among the entire currently
eligible group. Finally, LATE estimates are not informative about the effects of non- marginal changes in Medicaid eligibility. We now consider an approach that addresses
all of these issues, starting fi rst with a simple economic model of insurance coverage.
III. Economic Model
For simplicity, we consider a family with one child, and when the child is eligible for Medicaid we describe the family as “eligible.”
10
We emphasize that our work differs from previous theoretical work on this issue since our goal is to guide our
empirical work rather than to obtain theoretical results per se. We begin with an ineligible family with fi xed income I
i
whose decision focuses on whether to purchase private insurance at a cost C
pr ,i
. We assume that the family’s utility is given by
4 U
i
D
pr,i
= I
i
− C
pr,i
D
pr,i
+ B
pr,i
D
pr,i
, where D
pr ,i
= 1 if they purchase private insurance and D
pr ,i
= 0 otherwise; hence the direct gross utility produced from having private insurance is B
pr ,i
. Utility maximiza- tion implies that the family will purchase private insurance if the utility from having
this insurance is greater than the utility from not having it, or 5
B
pr,i
− C
pr,i
0. Now consider an eligible family and assume that participating in Medicaid implies
stigma and fi xed costs that are equivalent to a dollar cost of C
pub ,i
. We fi rst assume that this family’s utility is given by
6 U
i
D
pr,i
, D
pub,i
= I
i
− C
pr,i
D
pr,i
− C
pub,i
D
pub,i
+ B
pr,i
D
pr,i
+ B
pub,i
D
pub,i
, where I
i
, D
pr ,i
, C
pr ,i
and B
pr ,i
are defi ned above, B
pub ,i
is the direct gross utility produced from having public insurance, D
pub ,i
= 1 if the family participates in Medicaid and D
pub ,i
= 0 otherwise.
11
Utility maximization implies that the family participates in Medic- aid if
7 B
pub,i
− C
pub,i
0, while the decision rule for purchasing private insurance is still given by Equation 5.
However, one may reasonably argue that our specifi cation of preferences in Equa- tion 5 is too simple since there is no role for crowd- out; eligible and ineligible families
have the same rule for purchasing private insurance although the crowd- out literature has emphasized the substitution possibilities between public and private insurance.
10. We believe that the one child assumption greatly simplifi es the analysis without obscuring the basic message. In our empirical work we allow the observations of children from the same family to be correlated.
11. We assume I
i
C
pr ,i
for both eligible and ineligible families.
To begin an analysis that allows for crowd- out, we specify family preferences when eligible for Medicaid as
8 U
i
D
pr,i
, D
pub,i
= I
i
− C
pr,i
D
pr,i
− C
pub,i
D
pub,i
+ B
pr,i
D
pr,i
+ B
pub,i
D
pub,i
+ INT
i
{D
pr,i
∗ D
pub,i
} , where INT
i
represents the interaction effect on utility of having both types of insur- ance. If there is crowd- out, then INT
i
will be negative. However, we allow for the possibility that public and private insurance are complements such as is the case with
private insurance and Medicare for some families, and thus INT
i
can be positive. We draw two conclusions when we analyze this model in Appendix 1. First, this simple
adjustment to preferences does indeed lead to different decision rules concerning private insurance for eligible and ineligible families. Second, even if we use linear
index functions for B
pr ,i
– C
pr ,i
, B
pub ,i
– C
pub ,i
, and INT
i
, the econometric model be- comes very complicated because there are now three equations that determine which
of the four insurance states the family occupies no insurance, private insurance only, public insurance only, or both. In our empirical work we compromise by requiring
the decision rule for private insurance to have the same functional form for eligible and ineligible families but with different parameter values. The resulting econometric
model described in the next section is still much more general than those currently used in the literature.
12
IV. Econometric Methodology
