19
Figure 3: the mathematics for the income gradient (IG)
For some state h,
;
1
∑
=
×
×
=
m
k
k
k
hk
i
p
d
IG
with m
(
)
(
)
.
:
1
1
1
∑
∑
=
−
=
×
<
<
×
∋
m
k
k
k
m
k
k
k
i
p
s
i
p
where: h
≈ the home state.
k
≈ the k
th
state, where k
= 1 is the nearest state, k = 2 the next, etc.
d
hk
≈ the distance from the home state to the k
th
state.
p
k
≈ the population of the k
th
state.
i
k
≈ the per capita income of the k
th
state.
s
≈ the threshold market size.
m
≈ the threshold state, inclusive.
centers, using the equation of point to point distances along the surface of a sphere.
Figure 4
shows this math.
16
Geographic centers were chosen because they don’t move, they reasonably
calculate income centers of the states, and these data are much easier to find than either
population or income centers. Michigan’s ‘center’ is my one exception. Here the Upper
Peninsula pulls Michigan’s geographic center many miles northwest of the state’s income center.
About 85% of Michigan’s population is in the lower third of the state and 60% is in the eastern
half of that third. Thus, Michigan’s income center is somewhere near and perhaps a bit east of
Lansing, so I use Lansing’s coordinates.
For control variables, which are potential alternative explanations for wage behavior, I
use education levels, age of workers, proportion of women-in-the-workforce, and two tax burden
variables. A higher education level – indicating higher average human capital – generally means
higher wages. I use the proportion of the population in each state that has earned a college
degree as a proxy for worker education levels.
17
A higher ratio of women to men in
manufacturing means lower wages because women historically have been paid less than men.
16
Arctan2 in the equation is a common computer command (sometimes atan2) for calculating the arctangent
and is reliable for radian inputs between -
π/2 to π/2. Latitude and longitude expressed in sexigesimal form
(degrees, minutes, and radians) must be converted to decimal form then into radians.
17
Recent Statistical Data Abstracts represent this data as ‘earned a college degree’. In older Statistical Data
Abstracts the data is represented as ‘4 or more years of college’.
August
2005
Convention