# Provider Retention in High Need Areas. 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:

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

2. 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(Vt+1)(where ρ is a one-period discount factor. Expected future utility depends on the value of all future wages in all possible locations.9

1. 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 non-pecuniary 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 non-pecuniary factors associated with different locations, an individual’s propensity to move from one location to another is governed by the size of the location-specific random shocks. If wages were stable and random shocks did not exist, an individual would select his or her best (i.e., utility-maximizing) 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 σε 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 σε, 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 σθ) 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 non-pecuniary value on each location, there exists a single set of wages across locations that would make individuals indifferent among locations. In other words, supposing location-specific random shocks are zero, wages would be the most important determinant of location choices. If wages were insufficiently high in locations with low non-pecuniaries, 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 μj) are low or when wages are low.

9 Refer to Appendix B for a discussion of how Ε(Vt+1) is constructed.

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