This appendix provides a more detailed discussion of the economic model of location decisions that was sketched out in Chapter IV.

General Model of Location Decisions

An economic model of geographic location choices can be adapted from dynamic programming models of occupational choice (Keane and Wolpin, 1994) and military retention decisions (Asch and Warner, 2001; Asch et al., 2008). The adaptation simply requires substituting “geographic location” for “occupation” or “sector of the economy”. In the dynamic programming approach to the location decision in a given time period, an individual will calculate the value (utility) of each possible location and choose the location offering the highest value (utility). The value of each location has three components. The first component is the value that the individual places on the nonpecuniary factors associated with living in the location (climate, environment, local amenities, etc.). This nonpecuniary value is assumed to be timeinvariant and is denoted by the symbol θ. The second component accounts for the pecuniary value of the location and has two parts: (1) the individual’s current period wage in the location (w^{t}) and (2) the discounted value of expected future utility including wages if the individual chooses the location in period.^{27} The third component consists of a completely random periodspecific location shock that is unrelated to the individual’s preference for the location. Denoted , this shock accounts for the net impact of unobservable factors that might induce an individual to choose a location he or she dislikes or leave a location he or she likes. Any number of factors might have periodspecific (i.e., temporary) effects, including birth of a child, an illness, and death of a parent who was living elsewhere.
More formally, the value (utility) of location j in period t is given by
(1)
where (1) is the nonpecuniary preference for location j, (2) represents the wage available at location j in period t, p is the personal discount rate on future utility, (3) is the expected value of utility as of period t of the optimum location choice in period t + 1 and (4) represents the random shock associated with the choice of location j in period t. Furthermore, is written as:
(2)
is based on the individual choosing the period t+1 location that provides the maximum value in that period. Intuitively, this calculation depends on the size of the random shock to the value of each location. The larger the random shocks are, the more likely it is that the individual will find another location in period t+1 that has a value larger than the current one and the more weight will be placed on the values of other locations. It has been shown (BenAkiva and Lerman, 1985)) that when the random shock follows an extreme value distributed with standard deviation of , the expected value of the choice of location j in period t + 1, equals
(3)
Equation (3) indicates that expected future utility varies positively with the standard deviation of random shocks, and it further shows that expected future utility depends on future utilities at all locations and not just the current one.^{28}
The individual will choose location j, or will remain in location j, if for all . Given the extreme value distribution for the random shocks in each period, the probability of choosing location j in period t is a logistic function of expected values (BenAkiva and Lerman, 1985):
(4)
If the individual is already in location j, equation (4) expresses the probability of remaining in the location. If located elsewhere, equation (4) expresses the probability of moving to the location.
^{27} The calculation of this expected value is discussed below. Because the individual has the option of relocating at each period in the future, the expected present value of future wages does not equal the discounted value of future wages in the current location.
^{28} In equation (3) the parameter , known as Euler’s constant, is approximately equal to 0.5776.


Individual Decisions with Two Locations

To simplify the general location choice model for the purpose of analyzing the NHSC, consider the case of only two locations. With only two locations, an individual chooses location 1 if . This implies
(5)
A further assumption significantly simplifies the condition for choice of location, namely that is independent of location in period t. This assumption is reasonable if job skills are perfectly transferable and the experience gained in one location is just as valuable in other locations as in the current location. To a first approximation, this assumption is plausible in the case of health care providers.^{29} With this assumption, equation (4) can be rearranged as:
(6)
The individual thus chooses location (1) if the location 1 preference differential plus the location 1 wage differential exceeds the (location 2) random shock differential, . Equation (6) implies that individuals may choose location 1 in the face of a negative wage differential if their preference differential is high enough; conversely, individuals with a negative net preference for location 1 may choose it if they receive a wage premium for working there.
In the case of two locations, and with our simplifying assumptions, the probability of choosing location 1 reduces to
(7)
Equation (7) makes it clear that probability of choosing location 1 increases with the net preference for location 1, , and the wage differential between location 1 and location 2, . Notice that, for any given wage differential, there is a preference differential that makes individuals indifferent between location 1 and location 2 (in an expected value sense). According to equation (7), an individual whose preference differential is equal to will have a 50% chance of choosing either location.
^{29} If the future wage path is dependent on the period t location choice, the choice differential in equation (6) would need to include a term for the difference in expected future earnings due to period t choice. For simplicity, we assume that the on both sides of equation (4) is the same.


Cohort Rates with Two Locations

Suppose that a cohort of individuals enters the labor market after schooling and we wish to know what fractions of this cohort will make different choices over time. Again assume there are two locations and two time periods. The individuals in this cohort can choose location 1 in both periods, location 2 in both periods, location 1 in the first period and location 2 in the second period, and location 2 in the first period and location 2 in the second period. Define six expected fractions as follows:
 _{1}= fraction choosing location 1 in period 1 =
 _{2} = fraction choosing location 2 in period 1 =
 _{3 }= fraction choosing location 1 in period 1 and in period 2 =
 _{4} = fraction choosing location 2 in period 1 and in period 2 =
 _{5} = fraction choosing location 1 in period 1 and location 2 in period 2 =
 _{6} = fraction choosing location 2 in period 1 and location 1 in period 2 =
Conceptually, these expected fractions are constructed as weighted averages of the individual probabilities of choosing a given location in each time period, as given by equation (4). In the case of two locations, the weights are based on the distribution of in the cohort. To obtain actual expected fractions, it is necessary to make an assumption about the distribution of in the population. In simulations presented below, we assume that location preferences are independently normally distributed with means and , respectively, and common standard deviation .^{30} With these assumptions, the distribution of is normal with mean  and standard deviation equal to
The retention rate in a location is the fraction of individuals who chose to locate there previously who choose to remain there. Based on the above discussion, the period 2 retention rate in location 1 is given by and the period 2 retention rate in location 2 is given by . It is important to recognize that retention rates are conditional on prior choices and will depend on the composition of the cohort making the prior choices.
^{30} With these assumptions, the distribution of is normal with mean  and standard deviation

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