In order to analyze the impact that access to birth control has on adolescent females’ education decisions, the modified Gary-Bobo and Trannoy (2015) model’s goal is to find the optimal subsidy for birth control based on the total expense on birth control, degree of avoidance to pregnancy, decision to finish high school or not, type of birth control, the amount of effort put into pregnancy avoidance and either school or work, the probability of nature deciding either pregnancy or not, and the various payoffs in the form of wages. The moral hazard in this case is the risk that adolescents may participate in a higher rate of risky sexual behavior and the adverse selection is that they know how much sex they will have and the vigilance with which they will use the birth control while policymakers do not. This modified Gary-Bobo and Trannoy (2015) model will find the amount of subsidy that should be granted by the government in order to have the greatest utility from birth control, education, and potential wages.
This model follows a timeline of decisions made by the student where every decision is made with them taking into consideration all of their previous choices, as shown below: Variables:
λi (qi, Rij, ri) ei Pi, pi π εij u(ω-Rij), u(w-ri) i k
Explanation of timeline:
The timeline of this model can be broken up into two sections; pre and post-pregnancy. In the first part the student gets assigned their Ex ante type: i. The variable i can be either 1 or 2 depending on if the student is prone to avoid pregnancy (1) or neglect the responsibility of it (2). The next step is where the student self selects the amount of sex they will participate in and chooses what birth control method they will use. The birth control is represented by q (for quantity or quality) and can be either 1 or 2, 1 being none and/or short term options and 2 being highly effective or longer lasting birth control methods. Since students are aware of their Ex ante type these decisions are made with that in mind. They know whether they are likely to avoid the risk of pregnancy and so they choose whether they should have sex and whether they need a short or long term birth control method or none at all based on that. This also takes into account their “Resource constraint” which in this case in the student’s insurance coverage. Next is the effort put into avoiding pregnancy which is the vigilance with which the student takes their chosen mode of birth control. From there nature decides whether the student has become pregnant or not based on probability which is calculated from their previous decisions.
Section two starts with Ex post types k being drawn. This is whether the student stays in school or drops out and joins the workforce. Once the decision is made to stay in school or drop out the student must decide how much effort she will put in to school (if she didn’t drop out) or work (if she did drop out). The final step to this timeline is the realization of payoffs. These payoffs are wages. For instance, if a student became pregnant and dropped out they would not get the payoffs of graduation or higher education or the higher wages that come with those but they can potentially have a longer work history and more experience which could mean higher wages for that reason. If a student were to get pregnant but not drop out they would have the wage benefit of a diploma but would have less income than one who graduated without pregnancy because of the added cost of the pregnancy and/or raising a child.
To find the First-Best Optimality of an insurance subsidy for birth control, the interim expected utility of a non-pregnant student who chose birth control based on their avoidance type is the first step. To find that we take the chance that the student stays in school and multiply it by the utility gains from staying in school minus the cost of effort. This is added to the chance of dropping out times the cost of effort subtracted from the utility gains of not continuing school. This is shown as,
Ui = π(vi − βikεik ) + (1 − π)(Vi − βiiεii)
Where vi is to signify the utility gains from staying in school and Vi is to show the utility gains from choosing to discontinue schooling. They are calculated by taking a function measuring wage based on utility and effort depending of the amount of education chosen based on the avoidance behavior(value of i) and then adding in the value of the subsidy they are given, if any. These are written as,
vi = u[ω(qi, εik ) + Yi],
Vi = u[ω(qi, εii) + Xi],
The utility of the pregnant student whose type indicated they were prone to neglect pregnancy responsibility is represented as ui where ui is utility multiplied by wage minus itemized birth control price: u(w − ri). The ex ante expected utility net expected effort costs is found by adding together the products of non pregnant probability, effort, and utility and pregnancy probability, effort and utility and then subtracting cost times effort. This is defined as follows,
pi(ei)Ui + (1 − pi(ei))ui − ciei
If we assume that welfare is higher when all types i choose education q = i, i = 1, 2, the first-best utilitarian optimum can be obtained as a solution of the following problem,
Maximize i λi[pi(ei)Ui + (1 − pi(ei))ui − ciei]
with respect to {(ei,ri, Rik , εik )}i,k=1,2, subject to the resource constraint RC, and each effort level is chosen in the set {0, 1}. In the above formulation of the first-best problem, it is understood that type i is assigned to education qi = i, for all i. This assumption is justified below. “We call the solution of this problem a standard utilitarian optimum because the expected utility of each type is weighted by its frequency in the population” (Gary-Bobo and Trannoy, 2015).
