#? #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$ Šand 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 Š(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 Šdiminishing 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 Šthat 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 Š*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 Šthe 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 Špopulation 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