The retention patterns we observed in Figures III.1 and III.2 appear to indicate that participants have lower HPSA retention than nonparticipants. There may be a suite of factors in addition to NHSC participation that influence the providers’ decisions to first locate in a HPSA and then move out of a HPSA. In this chapter we provide a theoretical model to provide insights into the impact of these additional factors on the observed retention patterns.
Specifically, in this model we aim to: (i) isolate the key factors influencing providers’ location decisions; (ii) explain why some providers locate in areas that others avoid; and (iii) explain when geographic mobility is high or low. We start with a general location choice model and then expand the model to incorporate the incentives offered by the NHSC programs. A technical version of this model is available in Appendix B.

An Economic Model of Location Choice

In general, in any given time period an individual calculates the value (or utility) of each possible location and chooses the location offering the highest value. The value of each location depends on three main factors:

The value that the individual i places on the nonpecuniary factors associated with living in location j (climate, environment, local amenities, spousal employment opportunities, etc.), which is assumed to be timeinvariant (denoted by the symbol ).

The expected present value of money wages if the individual chooses location j in period t. This expected present value is the sum of:

the wage available in location j in period t, (); and

the discounted value of expected future utility if the individual chooses location j in period t (ρΕ(V^{t+1})(where ρ is a oneperiod discount factor. Expected future utility depends on the value of all future wages in all possible locations.^{9}

Finally, a completely random location shock that is unrelated to the individual’s preference for location j in any given period t (denoted by the symbol ). This random shock accounts for unobservable factors that might induce an individual to choose a location she might dislike in period t, or leave a location she likes in period t.
Mathematically, the utility of location j at time t can be written as
(1)
In this model, an individual will choose location j if its utility () exceeds the utilities associated with all other possible locations. Clearly, an individual who has strong nonpecuniary preferences for a particular location is more likely to choose it over other locations. That is to say, the probability of choosing to locate in location j initially, or remaining in location j if the individual is already there, increases with . But dislike for a particular location can be overcome if wages in that location are high enough. That is to say, even if an individual does not like a particular location as given by a negative value of , she may still choose to locate there if the pecuniary advantage of locating in the area, as measured by the value of the current wage plus the expected present value of future wages, is high enough. Given the values of the pecuniary and nonpecuniary factors associated with different locations, an individual’s propensity to move from one location to another is governed by the size of the locationspecific random shocks. If wages were stable and random shocks did not exist, an individual would select his or her best (i.e., utilitymaximizing) location in the first period and remain there forever.
Consider now aggregate (population average) probabilities of choosing a particular location and the aggregate probabilities of remaining in that location. These average probabilities are simply weighted averages of individual probabilities of selecting a location or remaining in it. The weights on which the aggregate averages are based are the fractions of the population with different values of . For example, if there were 5 different values of in the population and each value occurred with equal frequency, each value would receive a 1/5 weight in the calculation of aggregate probabilities. In general, the aggregate probabilities depend on the frequency distribution (probability density) of preferences (the ) in the population as well as the frequency distribution of the random shocks (the ). The parameters of these distributions (means and standard deviations) affect the aggregate probabilities and their sensitivities to changes in wages. We may show that, all else constant:

a smaller standard deviation of the random shock (denoted σ_{ε} reduces the probability of an individual move from location j and increases the expected number of periods an individual stays in the initial location j;

the smaller is σ_{ε}, the smaller is the frequency of moves in a cohort of individuals;

a smaller average preference for location j (denoted ) results in a smaller fraction of individuals choosing a location or remaining in it;

higher current or future pay in location j increases the fraction of the population choosing to locate there and remain in it;

a larger standard deviation of in the population (denoted σ_{θ}) decreases the impact of pay changes.
Stated alternatively, the last proposition says that the more heterogeneous people are in their preferences for different locations, the less influence wage changes will have on their location choices. Conversely, if all individuals placed the same nonpecuniary value on each location, there exists a single set of wages across locations that would make individuals indifferent among locations. In other words, supposing locationspecific random shocks are zero, wages would be the most important determinant of location choices. If wages were insufficiently high in locations with low nonpecuniaries, no one would choose those locations. Heterogeneous preferences ensure that most, if not all, locations will attract or retain some people, even when the average value preferences for those locations (i.e., their μ_{j}) are low or when wages are low.
^{9} Refer to Appendix B for a discussion of how Ε(V^{t+1}) is constructed.



Location Decisions in the Presence of the NHSC LRP

Consider now NHSC’s loan repayment program (LRP). A unique feature of this program is that an individual who applies for the LRP must have an NHSCapproved job in a HPSA and also have outstanding student debt in order to qualify for the program. Applicants are screened by the NHSC and not all applicants are accepted into the program. Importantly, in deciding whether to approve an applicant for the program, NHSC makes a determination regarding the applicant’s fit for the program and for the position the individual accepted. NHSC strives to choose the ‘most qualified’ applicants, but during the acceptance process it gives weight to an applicant’s fit for a particular position. An implication is that NHSC may select an applicant judged to be a ‘good fit’ over other applicants with better academic records. In addition to individual qualifications, the main driver of acceptance into the program is the severity of the shortage of health care providers in a particular area as measured by the HPSA score. Individuals applying for approval of a position in an area with a high HPSA score may have a better chance of approval than individuals applying in an area with a low HPSA score. Prior to 2009, approved individuals received funding only if the HPSA had a score of 14 or above. After the expansion in 2009, all approved individuals were funded regardless of HPSA score. Because of program funding constraints, not all applicants receive approval, even when applying for positions in high HPSA score areas.^{10}
If an individual is accepted into the program and qualifies for the loan repayment amount L_{j}, the utility of location j is given by
(2)
The individual prefers to participate in the program if there exists at least one HPSA location j for which the utility associated with that location is higher than the utilities associated with any other location. If the location that maximizes utility is not a HPSA, the individual of course does not apply. The attractiveness of a given HPSA depends on the loan amount L_{j}, so that the probability of choosing location j increases with the amount L_{j}. Some providers who do not participate in NHSC may still locate in a HPSA. This group will include: (1) individuals without student debt (and therefore ineligible to apply for NHSC program); (2) individuals who have student debt but did not apply (perhaps due to a low expected probability of approval, or lack of knowledge of the program); and (3) individuals who did apply and were not accepted.
^{10} In fiscal years 2012 and 2013 the admission rate into NHSC programs was around one third of applicants in those years.

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