We now introduce the NHSC program into the two-location model. Most of the participants in the NHSC program are in the Loan Repayment Program (LRP), so we now focus on this program. We model individuals’ decisions to apply for the program, NHSC selection of applicants into the program, location choices in period 1, the period in which participants are obligated to serve in a HPSA, and then location choices in period 2, the post-obligation period.

Let location 1 represent a HPSA and location 2 represent a non-HPSA. An individual without outstanding student debt and who does not qualify for LRP has value functions and probability of choosing location 1 as shown in equation (1) above. But an individual who has outstanding student debt has the following value function for location 1, where *L* is loan repayment for which the individual qualifies in the presence of the program:

(8)

Assume that if an individual has outstanding debt, the individual always prefers to be in the program and receive LRP over not participating and working in location 1. That is to say, . This assumption simply requires that *L* exceed the wage differential between qualifying jobs and non-qualifying jobs. But if a HPSA is a full location HPSA, this assumption is (almost) surely met because: (1) in a competitive market all employers will be paying the same wage; and (2) because in a full location HPSA all primary care positions qualify for LRP.

Based on this assumption, the condition for an individual with outstanding student loans to want to work in a HPSA and participate in NHSC is given by:

(9)

The probability of applying for NHSC, conditional on having student debt, is thus given by:

(10)

This probability is an increasing function of the wage differential , the LRP amount *L*, and the individual’s preference differential .

Now consider acceptance into the program, which is conditional on application. Let there exist an index function , where X represents observable factors that influence acceptance into the program (e.g., HPSA score and academic record) and the random error measures the net influence of unobservable factors associated with application. (These are factors unobservable to researchers but are observable by NHSC administrators who are evaluating applications.) Conceptually, NHSC rank-orders applicants to the program, establishes some minimum value of A for acceptance into the program (A_{min}) and accepts those candidates for whom A > A_{min}.

Our concern is the potential relationship between net preferences for the HPSA and acceptance into the program. A simple model is to allow to be related to the (standardized) net preference for the program through the following linear equation:

(11)

where is the standardized net preference for the HPSA and is a standard normal random error that captures the influence of unobservables other than location preferences that affect acceptance into the program and is, by definition, uncorrelated with preferences. The parameter governs the correlation between and ; this correlation, denoted , increases with the parameter .^{31}

The probability of acceptance into the program is given by:

(12)

When , acceptance into the program depends on preferences, and the probabilities of application and acceptance are therefore not independent of one another.

Consider a cohort of individuals making their choices in time period 1. There are four distinct groups in this cohort. The first group consists of those who apply for NHSC and are accepted. The second group consists of those who apply but are rejected. The first two groups, of course, have outstanding student loans. The third group consists of those who also have outstanding student loans but do not apply. The last group consists of those who have no outstanding loans and therefore are not eligible to apply.

We wish to know how the retention in the HPSA of NHSC participants upon completion of their program obligations will compare with the retention of these other groups. The retention rate of a given group is simply a weighted average of the retention probabilities of the individuals in the group, and the weights are based on the distribution of in the group. In general, the distribution of at the end of period t will be non-symmetric, and its mean and variance will both depend on how individuals in the population sorted themselves into the different groups at the start of the period. To make this clear, remember that and were assumed to be normally (and thus symmetrically) distributed in the population, which implies that is normally distributed. But individuals with higher values of are attracted to location 1, and individuals with higher values are attracted to apply for NHSC if they have outstanding loans. Different groups will have different mean values of after the period t location choices occur, and the distribution of will vary depending upon factors such as wages in the two locations, the LRP amount, and the NHSC’s minimum standard for acceptance into the NHSC program. Although it is difficult to derive exact analytical answers to the question of how HPSA retention will vary by group, we can make some generally (if not universally) valid statements based on an analysis of how the mean value of varies from group to group at the end of period t.

We can now conceptually compare the retention of NHSC participants with the retention of (1) individuals who have student loans but do not apply, (2) NHSC rejects who work in the HPSA, and (3) individuals who do not have student loans but work in the HPSA.

By the assumption that , anyone with an outstanding student loan who is working in a HPSA must be an NHSC applicant. Thus, anyone who has an outstanding student loan but did not apply must be working in the non-HPSA. This group has no retention in the HPSA because none of them were located in the HPSA to begin with.

Now compare NHSC participants with rejected applicants. Rejected applicants will still choose to locate in the HPSA if

(13)

But because of the above assumption that, for individuals with outstanding student loans, the value of participation is always higher than the value of non-participation (i.e., - ), it follows that, for given values of the other variables in equation (13), the preference differential that renders an NHSC reject willing to locate in the HPSA must be larger than the differential that will make an NHSC participant locate in the HPSA. It is therefore unambiguous that NHSC rejects who nevertheless locate in the HPSA will have higher average preferences for the location than the participants in the program. Other things the same, they will have higher retention. Their higher retention is, of course, due to the fact that NHSC rejects with low preferences select themselves out of the location by choosing the non-HPSA location after program rejection.

Now compare the participants with the individuals who never applied to the program but who work in the HPSA. By assumption, this group is comprised completely of individuals without outstanding student loans. Absent any systematic relationship between wage offers, location preferences, and outstanding student debt, it must also be true that non-participants who locate in a HPSA will *tend to have* higher average preferences for the location than NHSC participants and consequently higher retention. We say *tend to have* because the retention differential between NHSC participants and non-participants without student loans depends on the strength of the relationship between NHSC applicants’ net preferences for the HPSA and acceptance into the program (i.e., the correlation coefficient ). The retention differential will narrow the larger is this correlation.

^{31} We may show that under the assumption that the variables on the right-hand side of equation (11) have standard deviations of 1, . This correlation converges to 1 as increases in value.

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