#?
#P[80]&#A*FAmerican^ Studies^ in^ China^ #FKVol.1#FS,^ 1994/_@#a$#P[100]
#J[-100]
#T3THE DISTRIBUTION AND MOVEMENT$
OF U.S. POPULATION AND ITS IMPACT$
ON REGIONAL ECONOMIC DEVELOPMENT#t
#T4MAO Yushi#t
The economic development of the nine regions of the U.S.'s fifty
states has been diversified. The northeastern part where the first
immigrants arrived in North America is still the most developed part
economically and culturally in the U.S. Large cities were needed at
the beginning stage of industrialization, and these cities could only be
supported by water ways as there were few railroads. So, the
configuration of economic development was dependent upon the river and
lake network. As to the late 19th century the accumulation of
capital made it possible to construct railroads in much wider areas. At
the same time, the growth of population created the demand of more
land and natural resources. Hence the great movement
of population and capital to the West that advanced the development of
the western part of America. Early this century, the boom of car industry
changed the whole picture, as railroad can only bring prosperity along
the railroad network, highways spread prosperity all over the
area. During World War II, U.S. armed forces fought in the
Pacific region, and then there were the Korean war and Vietnam war. All
these events helped the economic development of the West coast. The
southern part of the U.S., where oil and gas resources are rich, got
impetus to develop after the energy crisis in 1973. Because of
international competition, U.S. capital and talented people have
been forced to move away from the traditional manufacturing sectors
such as car, steel, textile, towards high technology sectors, like
office equipment, electronics, communication, aviation, defense, etc.
In recent decades the economic efficiency of U.S. economy has been
advanced by better information, less distortion in resources
allocation, thus the percentage of output of service and financial
sectors has kept rising. One result of such change is the
restructuring of regional economy in the U.S. The general tendency is
that the regions with rich natural resources and nice environment
condition and weak labor force and low population density
obtain higher growth rate. The outcome is once again a big trend of
population move westward as shown in Figure 1.#+[1]_ This trend is
still continuing.$
#M1The Distribution and Movement of U.S. Population#m
#M2American Studies in China#m
$$$$$$$$$$$$$$$$$
#FH#P[90]
#i0[Figure 1.__]The speed of movement of population center towards
west in the U.S. since 1820$#FS#P[100]
The diversity in economic growth in the U.S. is important both in
macro- and micro-sense, so there are at least six institutes
involved in research on regional development. The book #FKRegional
Diversity: Growth in the U.S.#FS_ by Jackson et al published in 1981 is
the most renowned one, because of the famous Joint Center for Urban
Studies in MIT and Harvard University and the strong team of the
research group.$
#T4I. A Static Model of Population Distribution$
and Economic Development#t
Jackson and others predicted the regional economic growth in the
80's according to population change and sectoral restructuring
which happened in the past. Relating regional growth with population
movement is a wise method since population is the source of production
as well as the source of consumption which is especially important in
the U.S., an economy often suffering with insufficient aggregate
demand. In addition, the census data are usually more reliable than
others. The point to be made is that different assumption of
population movement leads to different conclusion which has to be
checked by the data of fact.$
Jackson et al emphasized the difference between the
prediction they made and what others had made. The previous prediction
had expected a convergence in regional economic growth while Jackson
expected that difference would persist. Their reasons were that such
difference had existed and grown for decades,#+[2]_ and the U.S. economy
has entered a zero sum period, i.e. the development of some regions
might be at the expense of slowdown of others.#+[3]_ But the future
path of development might not be a simple extrapolation of the past,
particularly their prediction lacks a sound logical foundation.$
A rational assumption could serve as the logical basis of the
convergence postulate and can be reasoned as follows.$
The U.S. is a society with high liquidity due to the mobility of the
people, the widespread communication network, the open posture of
state and the law to protect the free flow of information and
personnel.#+[4]_ Among three production factors, labor, capital and
natural resources, only the last one is fixed to the earth. Under the
U.S. institution most of the natural resources are capitalized, but
they still differ from the man-made capital in regard to the rule of
depreciation - they will never be scrapped, nor be
moved. At a given distribution of natural resources, labor and capital
will gather around them by the promotion of a price system and market
demand to turn out products in the most efficient way under the known
technology. As long as there is a demand for these products, labor and
capital will be continually put in. But by the force of law of
diminishing returns, as more production factors flow in, their
marginal output will decrease. Finally an equilibrium state will be
reached where the marginal output of labor converges to prevailing
wages, and marginal output of capital converges to prevailing interest
rate. Before the equilibrium state has been reached, the marginal
output of labor and capital at different regions are different, and
these differences create a driving force to promote their flow. But
once the equilibrium state is reached, differences in marginal output
among different regions disappear. The static equilibrium thus
established corresponding to a definite distribution of labor and
capital. During this process we can see a convergence of economic
development among#O
$$$$$$$$$$$$$$$$$$
#FH#P[90]
#i0[Figure 2.__]The rule of distribution of labor and capital when the
resources location is a known function of its geographical coordinate
(x,y)$#FS#P[100]
®ions. This should be the logical foundation of the
prediction for convergence.$
The following highly simplified model for labor and capital
distribution can be constructed based on the foregoing reasoning. Let (x,y) denote the geographical coordinate of a place in the U.S. where
a certain kind of resources is located. The amount of the resources R
is a function of its location as shown in figure 2.$
Because of the existence of R, labor L and capital K will flow in, and
output g in terms of dollar per unit of time will be generated. Thus g
is a function of R,L,K, or$
#J[50]#D
g = g(L,K,R)*B(1)@#C
#d
When the equilibrium state is established, the marginal outputs of
labor and capital all over the U.S. will converge to ¦Λ#-[L]_ and
¦Λ#-[K]_ respectively, i.e.$#J[50]#D
*FͺΜg/ͺΜL@=¦Λ*T/L@*B(2)@#C
*FͺΜg/ͺΜK@=¦Λ*T/K@*B(3)@#C
#d
¦Λ#-[L]_ and ¦Λ#-[K]_ is the equilibrium wage rate and equilibrium
interest rate. Under the equilibrium state, L and K both are functions
of geographical coordinate, or L = L(x,y) and K = K(x,y). In fact, R,
L, K occur on a point rather than on an area, while point is an
infinitesimal area, therefore, R, L, K, should be density functions.
Let the given total amount of labor and capital in the U.S. be
#A*S-/L@#a_ and #A*S-/K@,#a _then:$
#J[50]#D
*S-/L@=«ͺ«ͺL(x,y)_ dxdy*B(4)@#C
*S-/K@=«ͺ«ͺK(x,y)_ dxdy*B(5)@#C
#d&
The integration is taken over the boundary of U.S. And the total
output of the U.S. is$
#J[50]#D
G=«ͺ«ͺg(x,y)_ dxdy*B(6)@#C
#d&
Since the output density g depends upon R, L, K which all are
functions of (x,y), so g is a function of (x,y) too as shown in (6).$
As mentioned earlier, the natural resources R is a given function,
while the distribution of L and K, or L(x,y) and K(x,y) are unknown,
which can be determined only after the equilibrium state is
established. The solution of L(x,y) and K(x,y) which must satisfy
equations (2) and (3), depends upon the constants ¦Λ#-[L]_ and ¦Λ#-[K].
Different solutions of L(x,y) and K(x,y) will give different values on
the right side of equations (4) and (5). Only one of them can satisfy
the given value of left side of these equations, and this gives the
solution that we want.$
What is particularly interesting is that the solution as required by
(2) and (3) is exactly the same solution of the following mathematical
programming:$#J[50]#D
max«ͺ«ͺg(L,K,R)_ dxdy*B(7)@#C
s.t.«ͺ«ͺL(x,y)_ dxdy=*S-/L@*B(8)@#C
____«ͺ«ͺK(x,y)_ dxdy=*S-/K@*B(9)@#C
____________R=R(x,y)*B(10)@#C
#d&
Here, g(x,y), R(x,y) are given functions, L(x,y) and K(x,y) are
functions to be solved. When we adjust the distribution of L and K
following the rule: to move L and K creating less marginal output to
the place creating more marginal output, the total output (7) will
increase. Continuing this process and by the force of law of diminishing returns, finally all the marginal outputs of L and K will
converge to ¦Λ#-[L]_ and ¦Λ#-[K]_ respectively. Then all the possibility to
improve the total output has been utilized, or no further improvement
is possible, and the maximum of (7) is obtained.#+[5]_ The conclusion
reveals that free flow of labor and capital to incorporate natural
resources aiming at higher output can achieve the nation's maximum
generation of GNP.$
This is a simplified model, since it ignores the heterogeneity of
labor and capital and the cost of their movement. When people move,
the travel expenses, resettlement of housing and job seeking incur
cost. Unless the expected sum of income increase in a certain period
is big enough to compensate all these costs people would not move.
There are also risk cost and information cost accompanied with capital
flow. The simplified model implies that cut the cost of labor and
capital flow will advance the nation's output.$
#T4II. A Dynamic Model for Population Distribution$
and Regional Economic Development#t
The static model predicts that the wage level and profit level on
different regions will approach uniform. This is the postulate assumed
by the Bureau of Economic Analysis and others. But in fact, such
situation has never occurred. The maximum difference of per capita
income among states was more than _#*[-32]‘η#*[32]1,500 (in 1972 constant dollars) in
1960's and has persisted to very recently.$
Firstly let us investigate if the diversity of per capita income of
fifty states is a random event by comparing the actual frequency
distribution of per capita income with the normal distribution.$
$$$$$$$$$$$$$
#FH#P[90]
#i0[Figure 3.__]A comparison of frequency distribution of per capita
of fifty states with normal distribution$#FS#P[100]
The comparison uses the 1980 income data. Figure 3 shows the result of
comparison where the histogram represents the actual distribution
frequency and the curved line the normal distribution. The comparison
shows that the actual distribution differs significantly from that of
normal distribution. A chi square test also supports such an
impression. In 1980 the mean of per capita income of fifty states was
_#*[-32]‘η#*[32]8,370 with a variance _#*[-32]‘η#*[32]1,129, and the
calculated chi square is 7.63.
If the per capita income is normally distributed, the probability of
chi square exceeding 7.82 is 0.05 and that of exceeding 6.25 is 0.10.
Now chi square falls between 6.25 and 7.82, implying that the
probability of per capita income consistent with normal distribution
is beyond 5% but below 10%. Similar result can be obtained using 1984
data. As we know that the physical and intellectual ability among
individuals is normally distributed, but the distribution of per
capita income skews towards left.#+[6]_ Now we find that even in the
U.S.
economy with high mobility of population the average per capita income
among states deviates significantly from a normal distribution, but
surprisingly, it seems to be skewed towards right.$
Per capita income in the U.S. has increased by 70% within the recent twenty
years, but the relative richness has not changed much. Connecticut,
California, Massachusetts are among the richest; Utah, Arkansas,
Mississippi are among the poorest. Their average per capita income
differs well beyond 50%, sometimes to 100% or even more. Some argue that this is because of difference in the cost of living, or the price level
. But the maximum difference of consumer price index among
big cities (except Miami) has been about 25% in 1980's, and
the price difference among cities is believed to be larger than that
among states. Moreover, cities with higher price index are not
necessary located in high income states. That means the price
difference cannot fully explain this phenomenon. If the fifty states
were isolated from each other as the case the poor countries are isolated
from the rich ones now in the world, the persisting poor and rich
would be natural. But these persisting relative poor and rich occurs
in the U.S. where the population movement has been active.
46.4% of the whole population moved during the period 1975 to 1980. In
the ten years of 70's, Middle Atlantic Region had the record of -
1.1% population growth, the lowest among nine regions. But the natural
growth rate was 3.8%, implying that there were 4.9% of the population
moving away from the region. Mountain Region had a population increase
of 37.2%, the highest record among nine regions, where 12.1% were
natural growth and 25.1% were moved from other regions. If we take
state as a unit, New York state had the lowest growth rate which was -
3.7%, and Nevada the highest which was 63.8%. The big difference in
population growth rate cannot attribute only to the difference in
natural growth rate, but migration was the major cause. Within this
ten years 2.64 million and 2.22 million people moved away
from Middle Atlantic and East North Central respectively which were
the regions losing most of their population. South Atlantic and
Pacific were the regions which gain most population. There were 3.44
million and 2.39 million people moving to these regions respectively.
Figure 1 also shows that after the compensation due to people moving in
opposite directions the net effect of population movement was that the
center of population moved 100 kilometers westward, amounting to every
citizen in the U.S. moving 100 kilometers to the west in that decade.$
Taking population movement as the major factor influencing regional
economic growth is a tenable view in regional economic studies. What
should be the governing power of keeping the general ranking of relative
richness of the fifty states against the background of active population
movement? The answer could be found from Figure 4.4 on page 118 of
Jackson's book. The figure displays the trends of per capita
annual income of the nine regions in the United States from 1960 to
1978. One may be surprised by the fact that these curves are basically
in parallel. If we extend these curves using additional data from 1978
to 1991, such trends are even more obvious as shown in Figure 4.$#!
#FH#P[90]
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
#i0[Figure 4.__]The trends of per capita annual income (in 1972
dollars) of nine regions in the United States$#FS#P[100]#!
The unusual rule shown in Figure 4 that the annual increases in income
for the nine regions are almost the same, could not be caused by
accident. The true reason can be delved as follows.$
Per capita annual income is the regional aggregate annual income Y
divided by the population P that year. For the i-th region we have:$
#J[50]#D
PCI*T/i@=Y*T/i@£―P*T/i@____(i = 1,2,....,9)*B(11)@#C
#d&
As both Y#-[i]_ and P#-[i]_ are functions of time, we can differentiate them
with respect to time so as to find out how PCI changes with Y and P.$
#J[50]#D
*Fd(PCI*T/i@)/dt@=*F1/P*T/i@@*FdY*T/i@/dt@-*FY*T/i@/P*T2/i@@*FdP*T/i@/dt@
*B(12)@#C
&or$
*Fd(PCI*T/i@)/dt@=*F*FdY*T/i@/P*T/i@@/dt@-*FY*T/i@/P*T/i@@
*F*FdP*T/i@/P*T/i@@/dt@*B(13)@#C
#d
&Equation (13) states that the change in per capita income consists of
two parts: that due to the change in aggregate income
population is kept constant (the first term in the right side of the
equation); and that due to change in population aggregated
income is kept constant (the second term in the right side of the equation).
Equation (13) is true for any region at any time. According to Figure 4,
the increase of per capita income for all region at all time are
nearly the same, i.e.$#J[50]#D
*Fd(PCI*T/i@)/dt@=*F*FdY*T/i@/P*T/i@@/dt@-
*FY*T/i@/P*T/i@@*F*FdP*T/i@/P*T/i@@/dt@=constant*B(14)@#C
#d&
Equation (14) manifests that the increase of per capita income due to
the rising aggregate income always surpasses the decrease of per
capita income due to growth of population, and the surpassing amounts
for all regions are equal.$
The variables in (14) involve Y#-[i]_ and P#-[i]. Suppose in a certain year
people move from i-th region to j-th region. If the aggregate income
of both regions do not change, the per capita income of i-th region
will increase, while that of j-th region will decrease. But equation
(14) tells us that in such a case the aggregate income of both regions
will undergo a corresponding change that can offset the effect on
population movement on per capita incomes, thus the resulting change
in per capita income will remain constant.$
Equation (14) reveals a rule of population movement, which states that
the movement of population tends to eliminate the difference on
increase of per capita income, but not to eliminate the difference on
per capita annual income itself as expected by the foregoing static
model. The fact shown in Figure 4 demonstrates that this is not the
way of population movement which assumes that people move from low
income region to high income region resulting in a convergent per capita
income. But they move from slow growth region to high growth region
resulting in a convergent growth rate of per capita annual income, in
spite of the fact that the absolute income level may be higher in the
region where they depart from.$
A fundamental problem in economics is what could be the information
that guides people's decision. The difference in economic development
between regions as expressed by some people richer than others is
observable. but this observable difference did not serve as a guiding
information for people's decision on moving. It needs explanation why
this happens, and moreover, why the unobservable difference in growth
rate becomes the source of decision making, and how are people informed
about the future growth rate.$
The immediate decision such as shopping in supermarket is guided by
price information. The decision which produces long-term effect such
as saving and investment is guided by expectation but not by direct
price. Decision on moving has a long-term effect, therefore, it is
guided by expected change in income but not by wages or the price of
labor. People compare the expected income growth between regions, then
decide to move to region where a higher growth is expected. This is the reason why the static model of population movement is not
supported by the statistics. Equation (14) manifests that in
macroeconomic sense, people move according to the difference in the
expected income growth. They move continuously from the regions where
a slow growth in income is expected to the regions of high expected
growth rate. Owing to the diminishing marginal output of labor, the
growth in income in regions gaining population will dwindle, and
regions losing population will recover their growth rate in income.
So, the high growth rate in some regions will taper down and the slow
growth rate in other regions will be augmented due to migration. The
continuous adjustment will result in a uniform growth in income among
regions. The way how people estimate expected income growth may
consist the following: Job vacancy advertisement, change in price of
real estate, the number of construction sites.$
Equation (14) has a meaning more profound than just
explaining the convergence of growth rate. The constant on the right side
is the growth rate per capita annual income of i-th region, and is
also the growth rate of per capita annual income of the U.S. nation,
as the equation requires that all regions have the same growth
forecast. Let ¦Α denotes the uniform growth rate, and divide the
population change dP into that due to natural growth dP#-[ni]_ and that due
to immigration dP#-[mi]. Then (14) can be written as:$
#J[50]#D
*F*FdP*T/mi@/P*T/i@@/dt@=*F*FdY*T/i@/Y*T/i@@/dt@-*F*FdP*T/ni@/P*T/i@@/dt@
-¦Α*FP*T/i@/Y*T/i@@*B(15)@#C
#d&
where ¦Α is the nation's growth rate of per capita income, whose
long-term trend is shown on Figure 4 ignoring business cyclic fluctuation.
Equation (15) gives the relationship between the regional net
immigration rate and other factors. The higher the rate of increase in
regional aggregate income, the lower the natural population growth
rate and the regional per capita income and there will be a higher
immigration rate for this region when other things being equal. The
natural growth rate of population is relatively stable and can be
estimated by previous data. Because ¦Α, P#-[i], Y#-[i]_ are known variables,
equation (15) gives the value of dP#-[mi]/P#-[i]_ when dY#-[i]/Y#-[i]_
is given, or vice
versa. Thus the degree of freedom can be reduced in forecasting.$
The algebraic sum of immigration of nine regions is the foreign
immigration into U.S.A. dP#-[f], which is a policy variable and can be
seen as known. So we have:$
#J[50]#D
dP*T/mi@=dP*T/f@*B(16)@#C
#d
Equation (16) further reduces degree of freedom in
forecasting.$
Finally, it can be shown mathematically that the pattern of uniform
growth in per capita annual income for all nine regions can ensure
maximum annual growth for the whole nation under certain condition.
Here, we only consider the influence of population movement on
economic growth but not other factors.$
Suppose a problem of mathematical programming is put
forward: The goal is to maximize the nation's economic growth ‘χY#-[T]_ which
is the sum of growth of nine regions, and the constraint is the total
population P#-[T]. The question is to find the optimal pattern of population distribution, i.e.$
#J[50]#D
max¦€Y*T/T@=¦²¦€Y*T/i@(P*T/i@)*B(17)@#C
s.t.__¦²P*T/i@=P*T/T@*B(18)@#C
#d&
where ¦€Y#-[i]_ is the aggregate income growth of i-th region, and it is a
function of population ¦€Y#-[i]_ in this region.$
Make a Lagrange function:$
#J[50]#D
L=¦²¦€Y*T/i@(P*T/i@)+¦Λ(P*T/T@-¦²P*T/i@)*B(19)@#C
#d&
here ¦Λ is the Lagrange multiplier. When the population distribution is
optimal, we have$
#J[50]#D
*FͺΜ¦€Y*T/i@/ͺΜP*T/i@@=¦Λ____(i = 1,2,...,9)*B(20)@#C
#d&
(20) requires that the nation's economic growth can reach maximum only
when the marginal contribution of population to economic growth of
nine regions are uniform and equal to ¦Λ. Now, if the immigrants enjoy
an equal pay as that of the residents, or the marginal contribution of
population to economic growth equals the average per capita annual
income of this region, i.e.$
#J[50]#D
*FͺΜ¦€Y*T/i@/ͺΜP*T/i@@=*F¦€Y*T/i@/P*T/i@@*B(21)@#C
#d&
where ¦€Y#-[i]/P#-[i]_ is the growth per capita of annual income of i-th region,
or$
#J[50]#D
*F¦€Y*T/i@/P*T/i@@=*Fd(PCI*T/i@)/dt@*B(22)@#C
#d&
Substitute (22), (21) into (20) and compare with (14), we find that
the constant ¦Λ in (20) is just the constant in (14). Then we complete
the proof that the pattern of population movement according to (14) is
the same pattern which maximizes the nation's total economic growth.
The prerequisite here is the equal pay between immigrants and
residents. The general implication is that free flow of population and
fare competition of labor lead to maximum economic growth.$
$
#T4NOTES#t
$
#P[80]#FS
##[D1J100P80]
_#+[1]_#FKStatistical Abstract of the U.S. 1992#FS, p.9.$
_#+[2]_Jackson et al, #FKRegional Diversity, Growth in the
U.S. 1960-1990#FS, pp.9, 118.$
_#+[3]_#FKIbid.#FS, p.2.$
_#+[4]_#FKThe United States Constitution#FS, Article 1,
section 9 and 10.$
_#+[5]_Smilnov, #FKLecture on Advanced Mathematics#FS,
No.1, Vol.4, p.425.$
_#+[6]_Paul A. Samuelson, #FKEconomics#FS, 11 edition,
p.86.$#E