Provider Retention in High Need Areas. 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 non-pecuniary factors associated with living in the location (climate, environment, local amenities, etc.).  This non-pecuniary value is assumed to be time-invariant 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 (wt) 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 period-specific 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 period-specific (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


where (1)  is the non-pecuniary 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:


 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 (Ben-Akiva 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


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 (Ben-Akiva and Lerman, 1985):


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.


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