- Email: [email protected]
The effects of city size and moving costs on public project benefits
JOURNAL
OP
URBANECONOMICS
9, 14% 164(1981)
The Effects of City Size and Moving Public Project Benefits’ WILLIAM
Costs on
GREER
Department of Economibs, Unitwsity of Penmyfwnai~ Philaa@hia, Penmyloania I9104 AND MICHELLE J. WHITE Graduate School of BusinessAdministration, New York Universi@,New York, New York loo06 Received February 6, 1979; revised November 22, 1979 This paper uses a simulation model to investigate the benefits of an urban public project when the size of the system of cities (or regions) in which migration occurs can vary and when moving costs impede migration. The model captures the range of possibilities between the perfectly open and closed city cases. We find that the total number of households migrating between cities declines sharply as moving costs rise. Also the level and distribution of project benefits between renters and landowners in the project city and other cities is sensitive to both the level of moving costs and the size of the urban system. The total benefits of the project fall by half or more when moving costs are introduced, even when moving costs themselves are netted out.
I. INTRODUCTION Economists have devoted much ink and effort over the last ten years to methods of measuring the benefits of large-scale urban public projects such as freeway construction or air quality improvements. On the one hand, Freeman [2], Polinsky and Shave11[6], and others have debated the theoretical question of whether and when the benefits of urban projects can be measured by their effects on land rents. Polinsky and Shave11 established that rents measure benefits correctly only in one special case: that of the perfectly open city.’ Here migration between the city where the project occurs and elsewhere (other cities and rural areas) is assumed to be costless and, in addition, the total population of the urban system is assumed to be large relative
to that of the project
city.3 On the other hand,
‘We are grateful to Richard Arnott, Malcolm Getz, and Ed Mills for their comments and suggestions. 2Polinsky and Shaveh [7] call this the “small open city” case. ‘This case also requires that the project itself be small, i.e., it must produce only a marginal improvement in air quality or the like. See White [lo] for a discussion. 149 0094-1190/81/020149-16$02.00/O CopyrightQ 1981 by Academic Rcsa, Inc. All lights
Of mpmduction
in any form
-cd.
150
GREER
AND
WHITE
Mills [4], Amott and MacKinnon [l], and others have developed increasingly detailed simulation models that explore the effects of urban projects on land rents, modal choice, residential location choice, job location, etc. These models isolate the city from the rest of the world by assuming another very special case: the closed city, whose population is fixed regardless what happens within it. This paper has two goals. First we synthesize the open and closed city strands of the literature. We present a simulation model of the effects of an urban public project in one city, but we examine carefully the role of the city’s relationship with the rest of the world. By varying the parameters of this relationship, we can determine how and how much the city’s links with the outside world affect the project’s benefits. We also determine the extent to which benefits and land rent changes are over- or underestimated by concentrating- as most economists have-on the conventional special cases of the perfectly open and perfectly closed cities.4 Second, we introduce explicit intercity moving costs into the model and examine their effects on households’ propensities to migrate and on project benefits, The perfectly open city case makes two special assumptions. First, it assumes that the cost of intercity migration is zero and, second, that the population elsewhere is large relative to the population of the single city. In our model these two assumptions are varied over a set of parameter values. We assume that migration may be costless or costly and that the population of the system of cities may be large or small relative to the city’s population. Thus four rather than two special cases are suggested. The “frictionless migration/many cities system” (F-M) is equivalent to the “perfectly open city.” The “costly migration/many cities system” (C-M) more realistically represents the U.S. situation, where migration occurs across many cities and large distances, but moving costs are nonzero. The “frictionless migration/few cities system” (F-F) and the “costly migration/few cities system” (C-F) are intended to represent countries such as England and Thailand, where the largest city is very large relative to the population of all other cities and towns put together. It also may represent the migration pattern of culturally distinct regions where households are more likely to move within the region than into or out of it.5 Intraregional migration may be seen as costless or as less costly than interregional migration.6 ‘Getz [3] suggests model@ a city’s relation with the outside world in terms of generalized elasticities of labor supply and product demand, but he solves only the perfectly open and perfectly closed city cases. ?he perfectly closed city is an extreme form of the “costly migration/many cities system” within which no migration occurs at all. 6The various models should not be seen as short-term versus long-term models of adjustment to the same project, however, since all assume full adjustment in housing markets. See White [ 10, 1 l] for a short-term adjustment model.
CITY
SIZE AND
MOVING
COSTS
151
Thus the simulation model proposed here provides an outside world context for the measurement of benefits from projects in particular cities and enables us to determine how important that context may be. Our results suggest that assumptions about the inside/outside relationship are important. We find that the level of intercity migration is sensitive to migration costs, that total benefits in particular cities vary greatly depending on the size of the project city relative to the system of cities and that even qualitative changes such as whether rents rise or fall in a particular location may depend on the context within which the project occurs. Thus it is important for policymakers to consider a wider range of models than the conventional two that economists are accustomed to using. II. THE MODEL We postulate a system of k + 1 identical monocentric cities.’ All households have identical utility functions
u = L(u)“X(u)’ -O,
(1)
where L(U) is household consumption of land at distance u from the central business district, X(U) is consumption of a numeraire composite good at distance u from the CBD, and a is the percentage of income spent on land. Households maximize utility subject to the budget constraint Y = R(u)L(u)
+ X(u) + tu,
(2)
where Y is daily income, R(u) is rent at distance u from the CBD and t is transportation.cost per two-mile distance. Thus tu is total daily commuting cost for a household residing at U. Demand functions for land and the composite good are L(u) =
a(Y - tu) R(u)
and X(u) = (1 - a)(Y - tu).
(4)
Equilibrium conditions for each city require, first, that all households within a single city achieve equal utility levels. This implies
-Wu) = -f/L(u). au ‘See Mills (51 for a discussion of the monocentric
urban model.
152
GREER AND WHITE
Second, aggregate land demand must equal aggregate land supply, or
N( u)L( u) = +u
for all u,
(6)
where N(u) is household population residing at u and + radians of a circle are occupied by residential land. Third, the total population of each city, N, must be
N=
“N(u)&, I0
where ii is the outer edge of the city. Fourth, rent at the edge of the city must equal rent on agricultural land, R, or R(G) = k.
(8)
Solving for the rent function and for total population, we find, R(u) =R
and
NC
+R a(Y - tuya
y - tu l/o Y-
[ 1
J0“u(Y -
tzp*
- “du.
Also solving for the indirect utility function for a household located at G, u = aq 1 - a)’ - “(Y - tG)/R”.
(11)
Equations (10) and (11) define the conventional closed and open city models. For the open city model, (10) and (11) are solved for ii and N, given a value of U. For the closed city model, they are solved for H and U, given a value of N. For the simulation, we use the following initial values in all cities. The initial population of each city in the system, N,, is 100,000households. The percent of income spent on land., (I, is 5%. Daily income, Y, is $40. Initial commuting cost per two miles, to, is $0.50. Using these values, we can solve (10) and (11) for the initial radius of each city, i& = 5.271 miles, and for the initial utility level, U, = 21.254. These values prevail initially in all cities. III. THE SIMULATION Assume that the public project is a transportation improvement which lowers the cost of commuting per two miles from to = $0.50 to 1, = $0.30 in one city only. Commuting cost remains the same in all k other cities.
CITY SIZE AND MOVING COSTS
153
The model must determine a new equilibrium following the project when two factors are varied separately: the size of the project city relative to the system of cities and the level of moving costs. Examining the first, we vary the size of the project city relative to the system of cities by varying k, the number of identically sized cities. If k is small, the project city’s population is large relative to the population of the system; if k is large, the opposite holds. However, regardless of the magnitude of k, all cities initially are identical in population. In the simulation, we vary k from 1 to 100. In the two frictionless migration cases, moving costs are zero and households move from any of the unimproved cities to the project city until utility levels in all k + 1 cities are equal. Thus
a=(1 ;y’ -O(y_t,q =aY1 if” _a(y_t,P*) or t,ii’ = t&i*,
(12)
where u’ is the radius at the new equilibrium of any of the k unimproved cities and U* is the radius of the project city. Also we assume that total population in the system of cities is constant at (k + l)N,, thus (k + l)N, = N* + kN’,
(13)
where N* and N’ are the populations of the project city and other cities respectively at the new equilibrium. The population levels N* and N’ also must satisfy Eq. (10) evaluated respectively at t = t,, P = U* and at t = t,, 24= U’. Thus for the frictionless migration/few cities and frictionless migration/many cities models (F-F and F-M), (12), (13) and the two population equations can be solved for U*, E’, N* and N’. Then the indirect utility function, (1 I), can be solved at C* and P’ for the new utility levels, U* and U’, and (9) can be solved for the new rent functions, R*(u) and R’(u). (By assumption U* = U’.) By varying the value of k and solving the same system of equations, we simulate the effects of frictionless migration within small, large or intermediate sized systems of cities. When moving costs are nonzero, the model becomes more complicated. We assume that the cost of moving between cities is either 10%or 20% of a year’s income. This includes both actual moving expenses and the cost of acquiring specific knowledge of job and housing markets in the new city. Assuming a IO-year time horizon and ignoring discounting factors, a migrant household faces a 1% of 2% reduction in income in the new city, from Y = $40 to Y’ = $39.60 or $39.20 per day.
GREER
154
AND
WHITE
Given moving costs, utility levels in the new equilibrium are no longer equal among all cities. If migration follows the transportation improvement, in equilibrium households remaining in an unimproved city and having Y = $40 must achieve the same utility level as households moving and having Y’ = $39.60 ($39.20). But after migration, the project city has two classes,a “poor” class with Y = $39.60 ($39.20) that initially resided in the unimproved cities and a “rich” class with Y = $40 that initially resided in the project city. The monocentric urban model implies that if two classeslive in the same city, the poor live in the center and the rich in the suburbs, with ub denoting the boundary between them.’ However, migration does not necessarily occur in the costly migration model. To determine whether it does, we solve for the rent function of the initial residents in the project city assuming no migration, and compare that with the rent function which the initial residents of unimproved cities can pay in the project city and still attain their initial utility levels (taking into account their reduced income). If the latter rent function is greater than the former at any point, migration will occur. Since the poor live in the center of the project city if migration occurs, we need only compare the two rent functions at the CBD.9 The initial rent function for the project city is found by solving (10) for ii*, where N = 100,000and t, = 0.3. Then substituting ii* and t, = 0.3 into (9), we get the rent function for the “rich.” The rent function for the poor is found by solving the indirect utility function (11) for U with t, = 0.3, Y = 0.99Y (0.98Y) and utility level U,. The resulting value of U is then substituted into the rent function (9), where again Y’ = $39.60 ($39.20) and t, = 0.3. If migration occurs, the new equilibrium must satisfy four conditions: First, utility levels for the poor in the project city and for all households in unimproved cities must be equal or Y-
t,ii R-a
=
Y’ - t,Ub R(d
where Y’ = 0.99Y or 0.98Y.
(14)
'
Second, the rich population in the project city must equal the initial population of the project city, N, or N=
+R a(Y - t,ii*y
c*u(Y - t,uya I Ub
- “du.
(15)
sin the model presented by Mills [S, Chap. 51, this requires that households have income elasticities of demand for housing at least equal to unity. QIf no migration occurs, then the system degenerates into the conventional two equation closed city model, (10) and (ll), which solves for the new utility level and new radius of the project city. Conditions do not change at all in other cities.
CITY SIZE AND MOVING COSTS
155
Third, the initial population of all unimproved cities, kN, must equal the final poor population in the project city plus the total remaining population in other cities, or kc#aR
kN=
a(Y - t,uy
U’ u( Y - t,uy” s0
- ‘)
I’& + +R(ud / ““,(r _ t,u)wOa(Y- t*up 0 Fourth, rent at the rich-poor border, R(u,), is R(+J=~
I I Y - t,u, Y-tu* I
‘/O .
(17)
Thus for the two costly migration models, C-F and C-M, the new equilibrium with migration is the solution of (14)-(17) for ii*, 2, ub and R(ub). The indirect utility function (10) can be solved for the new utility levels of both the rich and poor households, U,* and U,*. The three rent functions can be derived by evaluating (9). For the unimproved cities, (9) is evaluated for t = to. For the rich sector of the project city from u = u6 to ii*, (9) is evaluated t = t,, while for the poor sector from u = 0 to ub, (9) is evaluated for t = tl, Y’ and R(Q). The total population of each unimproved city, N’, can be calculated by evaluating (10) given u” and t = to. The population of the project city, N*, is
+
@if a(Y - t*u*y
JUb“‘u(Y -
t,u)l’a
- ’ du.
IV. MEASURING BENEFITS Two types of benefits result from the transportation improvement: increases in household welfare and changes in land rents. Either or both may occur in the project city and in the unimproved cities. In addition the project has a cost which must be subtracted from total benefits to obtain a measure of net total benefits. Looking first at land rents, the change in rents in a city equals the difference between aggregate land rents before versus after the project.
156
GREER AND WHITE
Aggregate land rent is found by integrating the rent function over u. For the project city, the change in aggregate rent is thus AR = I”‘[
R*(u)c#au] du - (J,“[
R’(u)cpu]
du + I”‘[
&
&w]
ch)
(19)
Here R’(u) and R*(u) are the initial and final rent functions, from (9). The first term in (19) is aggregate land rent after the project. Since the city expands as a result of the project, the initial level of aggregate rent has two parts, terms two and three in (19). These are the initial level of urban rents evaluated from u = 0 to i’&plus the level of rural rents, R, evaluated from 17 to U*. Thus in the area added to the city by the project, the increase in land rents is net of their previous value in agricultural use. Further, in the costly migration case, the final rent function, R*(u), is the upper envelope of rich households’ rent function from U* to ub and poor households’ rent function from ub to 0. Finally AR must be separately calculated for the project city and for the k unimproved cities and aggregated over all of them. Turning to project costs, we assume for simplicity that the project is financed externally, with no special tax payments levied either on residents of the project city or other cities to pay for it.” Turning to the effect of the project on household welfare, we must transform the increase in utility that accrues to households as a result of the project into a dollar measure of willingness-to-pay (WTP) or the amount of income a household can relinquish following the project and still achieve its original utility level. There are several ways to do this. The method used here is to solve for the new equilibrium and to measure households’ willingness-to-pay for the project after it occurs.” WTP is measured assuming that rents and taxes remain constant. For the Cobb-Douglas utility function, the elasticity of utility with respect to income net of transportation cost is unity. Consequently, WTP ‘OThis assumption is made partly to avoid the necessity of considering alternate tax schemes and partly because it is fairly realistic. Many large urban projects are primarily financed with Federal or State government revenues which are unrelated to the project itself. An alternate assumption might be that the project’s costs are small or negligible. Examples include levying parking fees where none existed before or timing the traffic lights on a major highway to speed the flow of traffic. “Amott and Mackinnon [l] use an alternative method. They assume that the project is accompanied by lumpsum taxes and transfers which keep households at the same utility level before and after the project. Then willingness-to-pay for the project is measured by the change in government revenues attributable to it. This method does not easily generalize to consideration of multiple city models with moving costs, however, since under their method all households must be taxed at the same level, while in the multicity case different households are in different situations. Various methoda of measuring willingness-to-pay give rise to different income effects across households and therefore to different levels of total willingness-to-pay and total benefits.
157
CITY SIZE AND MOVING COSTS TABLE 1 Simulation Results: Frictionless Migration Model Post-project k
P
P’
0 1 2 5 10 50 100
5.831 6.843 7.330 7.950 8.295 8.672 8.727
4.106 4.398 4.770 4.977 5.203 5.236
Migration per unimproved
Utility
city
21.757 21.585 21.502 21.396 21.338 21.273 21.264
0 47,093 37,251 22,853 13,917 3,371 1,731
Total households migrating
0 47,093 74,502 114J66 139,170 168,550 173,058
by a household at u is the precent increase in utility multiplied by net income. Since net income varies with u, WTP also varies with I(. For the two frictionless migration cases, the percent change in utility is equal for all households. Total WTP in each unimproved city is /“[ (%AU).(Y - t&V(u)] 0
du.
(20)
Total WTP in the project city is also (20), but evaluated at t = t, and from u = 0 to ii*. Aggregate WTP over all cities is the sum of WTP for the project city plus k times WTP for each unimproved city. For the two costly migration cases,WTP in the project city must be calculated separately for both the rich and poor households and summed over both classes.12 V. SIMULATION
RESULTS
A. Frictionless Migration
Simulation results for the frictionless migration model are shown in Table 1, for various values of k.13 The results show that as the number of cities in the urban system rises, more households in total migrate to the project city from other cities. The radius of the project city, P*, therefore rises. Also as k rises, fewer households migrate from each unimproved city, so U’ also rises. Finally while the project always causes utility to rise from its initial level of 21.254, larger k implies a smaller increase in utility since the benefits of the project are spread over more households. ‘*Note that WTP varies with location. Thus if a tax were levied on households equal to WTP, then post-project income levels would vary with distance. Households would relocate, resulting in a new equilibrium. 13Thek - 0 line gives results for the perfectly closed city case. We do not report results for k > 100, since as k gets very large the aggregate results are subject to considerable computational error.
CITY
SIZE AND
MOVING
COSTS
163
Finally the distribution of benefits from the project is also affected by moving costs. Landowners in the project city are big gainers when moving costs are zero and moderate gainers when moving costs are I%, but they are made worse off when moving costs are 2%. Policymakers thus need to be conscious of the possibility that landowners may lobby against public projects in their cities even when such projects are in the public interest. Households in the project city have the opposite interests: they are strong gainers when moving is costly and gain much less when it is free. These households also may have an incentive to lobby in favor of public projects that are not in the public interest. In our 2% moving cost case, their benefit exceeds the total benefit from the project, since other parties lose.22 VI. CONCLUSIONS This paper uses a simulation model to investigate the benefits of an urban project when the size of the system of cities and the level of intercity moving costs can vary. The model captures the range of casesbetween the perfectly open and perfectly closed city assumptions. We show the following results: (1) The amount of migration is very sensitive to the level of moving costs. In our model, moving costs equal to 2% of income cause the total number of households migrating to decline by 80% to 90% compared to the frictionless case, depending on the size of the urban system. (2) Moving costs act as a buffer which protect the initial residents of the project city from competition in the land market by in-migrants. When moving costs are zero, part of the project benefits are capitalized as an increase in land rents in the project city. As moving costs rise, the original residents of the project city receive a larger share of the benefits. When moving costs are 2% the original residents get more than 100% of the benefits (other effects are negative). Further, the benefits to original residents are almost unaffected by increases in the size of the urban system. (3) The total benefits from a project are larger if migration is free than if it is costly and the difference exceedsthe level of deadweight loss accounted for by moving costs. (4) The distribution of benefits between rent changes and willingness-topay and between the project city and other cities varies widely in the various casesas moving costs and the size of the urban system vary. (5) The level of total benefits for a particular project may fall by half or more when moving costs are introduced, even when moving costs themselves are netted out. The distortion increases as both the size of the urban UThese results may be sensitive to our assumption of a Cobb-Douglas utility function. It would be interesting to examine the effect of a more general utility function which allows the elasticity of substitution between land and other goods to vary.
CITY SIZE AND MOVING COSTS
159
total utility gain to households rises with the size of the urban system; the smaller gain accruing to each household is more than fully offset by the increased number of households sharing in it.i5 The size of the urban system thus affects the distribution of benefits. If the urban system is small, the improvement results in a significant utility gain for households in both improved and unimproved cities. Landowners in the project city receive only moderate capital gains on land, while landowners in other cities face substantial losses. If the urban system is large, households receive few benefits; landowners in the project city receive substantial capital gains, while landowners in other cities are indifferent.16 The last column in Table 2 shows total benefits of both types for the entire urban system. The level of total benefits increases with the size of the urban system. This suggests that we can define the notion of an optimal context for particular public projects. For our transportation project, the optimal location is the largest possible urban system, since the more households affected by the project, the higher the benefits. However, for other projects the opposite may be true.17 Policymakers should also be conscious of the distributional effects of locating projects in different contexts, since strong support or opposition may come from parties who fare differently depending on the size of the urban systern.I8 In our example, the level of total benefits (rent change plus WTP) in the project city and other cities both vary depending on the size of the urban system, with the gain in the project city rising and the loss in other cities falling as k rises. Thus the effect of a project on particular cities may be over- or underestimated if the wrong intercity migration assumptions are made. Finally, in our example landowners will favor projects in their cities and oppose projects in other cities. But their opposition to projects in other cities may run counter to the interests of society at large.” t5As k approaches infinity (the perfectly open city case), WTP in the project city and in each unimproved city must approach zero, since the utility gain to households approaches zero. Therefore total WTP might or might not approach zero depending on whether WTP in each unimproved city approaches zero faster or slower than k approaches infinity. In the simulations, even for very high values of k, total WTP always increased with k. t6Note that many of the landowners may be owner-occupiers: thus the same households may accrue both utility gains and capital gains or losses on land. “The level of total benefits depends on income effects generated by the method of measuring WTP. Other methods of measuring WIT may lead to different results. “Note that the results presented here, particularly those concerning the distribution of benefits between landlords and tenants, may depend on our assumption of Cobb-Douglas utility functions. Other utility functions may have different results. t9Note that as k approaches infinity, the rent change and WTP in each unimproved city, as well as WTP in the project city, all approach zero. Thus aa the model comes closer to the perfectly open city case, total benefits are more closely approximated by the change in rents in the project city.
160
GREER AND WHITE TABLE 3 Simulation Results: Costly Migration Model
k
1 2 5 10 50 100
ii’
ii’
ub
5.993 (6.501) 6.055 (6.771) 6.126 (7.067) 6.161 (7.245) 6.196 (7.431) 6.201 (7.458)
5.138 (4.676) 5.178 (4.840) 5.221 (5.020) 5.243 (5.127) 5.265 (5.239) 5.268 (5.255)
1.181 (2.440) 1.413 (2.994) 1.643 (3.706) 1.749 (3.998) 1.851 (4.269) 1.864 (4.262)
Migration Utility of rich per unimproved households city 21.730 (21.644) 21.720 (21.598) 21.708 (21.547) 21.702 (21.517) 21.6% (21.485) 21.695 (21.480)
(2gg (2:g) 2,497
WD44) (E) 314 (1SW 159 (805)
Total households migrating
Total moving costs
(2g?) 9,327 (4ON5) 12,483 (6OJl9) 14,092 (7OXn) 15,711
($gg 7,462 (16,018) 9,986 (24,088) 11274
(WW 12,569
(%385)
(32,154)
15,396 (80,539)
(32,216)
12,749
B. Cost& Migration Simulation results for the costly migration model are shown in Tables 3 and 4. Results for the 1% moving cost case are given in parentheses below the figures for the 2% moving cost case. The most skiking result is the dramatic decreasein the number of households migrating as moving costs rise. For k = 1, 47,092 households migrate when moving costs are zero, 26,818 when they are 1% and 6.600 when they are 2Yk20 Most other qualitative results are similar, however. Under costly migration as under frictionless migration, the radii of both the project city and other cities rise as the size of the urban system increases, but the changes are now smaller in magnitude. The original residents of the project city (the “rich” households) benefit from higher moving costs, since with fewer in-migrants, they face less competition for land. And conversely, the original residents of other cities are hurt by higher moving costs, since they reduce out-migration. However, as the urban system becomes larger, a given level of moving costs provides less protection for rich households and their utility level declines. The last column of Table 3 gives total moving costs, equal to $0.40 and $0.80 times the number of households migrating in the 1% and 2% cases, respectively. Table 4 reports the benefits of the project for the two levels of moving costs. Again the rent change in each unimproved city is negative, but the decrease is smaller in magnitude when moving is costly than when it is free, since fewer households out-migrate. Thus when a project occurs in 2oWhen migration costs exceed 2%, there is little or no migration.
- 6,620 ( - 26,326) - 4,737 (- 19,893) - 2,545 (- 12,130) - 1,435 (-7,134) - 313 (- 1,630) - 154 ( - 820)
(45,254)
- 14,248 (24,821) - 13,021 (33,775) - 11,771 (43,815) - 11,594
(W’87)
- 18,635 ( - 349) - 16,651
Rent change in the project city
4
(6::)
(1.3::)
(5.6
(7,876)
458 (9,021) 325
(8,356)
458
tiP by poor in project city*
Model
9 and 10 for discussion.)
(821)
312 (1,625) 153
cwm
6,407 (22,533) 4,629 (17,819) 2,514 (11,394) 1,425
w-win each unimproved city*
(See Footnotes
- 25,255 (-26,675) - 26,125 (-28,799) - 26,973 (- 35,829) - 27,371 (-37,565) - 27,421 ( - 37,685) - 26,994 (- 36,746)
rent change
TOtd
Note. AU figures in dollars *Approaches zero as k approaches infinity.
loo
SO
10
5
2
1
k
Rent change in each unimproved city*
TABLE Benefits in the Costly Migration
87,074 (73,034) 85,068 wsw 82,797 (52,781) 81,658 (47,377) 80,525 (42,027) 80,369 (41,795)
WTP by rich in project city
95,694 (124,568)
W4,W
93,939 (103,923) 94,784 (109,203) 95,692 (117,627) %,114 (121,494) %,175
Total WrP
58,733 (57,710 57,469 (55,847) 56,185 (54,777) 56,384 (55,606)
W36)
63,404 (66,521) 61,197
Total benefits net of moving costs
162
GREER
AND
WHITE
one city, landlords in other cities are better off if moving costs are high, since their capital losseson land are smaller. The rent change in the project city is negative in the 2% moving cost case for all k, since there is never enough in-migration to offset the decrease in rents at the center caused by the flattening of the rent function. (In the 1% cost case, however, more migration causes rents to rise in the project city for k > 1.) In both cases as k rises, the rent change in the project city becomes more positive.*’ The change in rents everywhere in the urban system is also negative, for all values of k. WTP in the unimproved cities is smaller with costly migration than with frictionless migration and again it declines as k increases. The pattern of WTP in the project city, however, varies with moving costs. Under frictionless migration, WTP declines strongly as the size of the urban system increases, but under 2% moving costs, WTP by rich households remains almost unchanged. WTP by the poor declines strongly with k, but is very small in all cases.Thus total WTP in the project city is about the same in absolute magnitude whether migration is costly or free when the urban system is small. But it declines sharply as k increases in the frictionless migration case while remaining relatively constant as k increases in the costly migration cases. In effect the existence of moving costs buffers the original residents of the project city from the effects of competition in the land market from in-migrants. With the two class situation, benefits to the original residents remain almost unaffected as the size of the urban system increases, because the number of in-migrating households increases only slightly. Thus while our results in the frictionless migration case suggest that the benefits to the original residents of the project city are nearly erased by competition from in-migrants, in the 2% moving cost case the original residents are nearly indifferent to the size of the urban system. The last column of Table 4 shows total benefits including both rent change and WTP, net of moving costs, for the entire urban system. For both levels of moving costs, benefits decline very gradually as the size of the urban system rises. Thus the optimal context for a public project differs depending on the costs of migration. When moving costs are zero, it is socially efficient to place the project in the largest possible urban system; with costly migration, society is more nearly indifferent to the size of the migration area. These results may reflect the particular assumptions of our model; it would be of interest to investigate others. 2’An increase in household utility levels in the monocentric urban model causes households to surburbanize and the rent function to fall at the center and rise in the suburbs. Thus the level of aggregate rents can either rise or fall. In-migration, however, always causes land demand to rise and aggregate rents to increase. Smce moving costs and the size of the urban system both affect the number of households migrating, the effect of the project on aggregate rents in the project city can be either positive or negative.
CITY
SIZE AND
MOVING
COSTS
163
Finally the distribution of benefits from the project is also affected by moving costs. Landowners in the project city are big gainers when moving costs are zero and moderate gainers when moving costs are I%, but they are made worse off when moving costs are 2%. Policymakers thus need to be conscious of the possibility that landowners may lobby against public projects in their cities even when such projects are in the public interest. Households in the project city have the opposite interests: they are strong gainers when moving is costly and gain much less when it is free. These households also may have an incentive to lobby in favor of public projects that are not in the public interest. In our 2% moving cost case, their benefit exceeds the total benefit from the project, since other parties lose.22 VI. CONCLUSIONS This paper uses a simulation model to investigate the benefits of an urban project when the size of the system of cities and the level of intercity moving costs can vary. The model captures the range of casesbetween the perfectly open and perfectly closed city assumptions. We show the following results: (1) The amount of migration is very sensitive to the level of moving costs. In our model, moving costs equal to 2% of income cause the total number of households migrating to decline by 80% to 90% compared to the frictionless case, depending on the size of the urban system. (2) Moving costs act as a buffer which protect the initial residents of the project city from competition in the land market by in-migrants. When moving costs are zero, part of the project benefits are capitalized as an increase in land rents in the project city. As moving costs rise, the original residents of the project city receive a larger share of the benefits. When moving costs are 2% the original residents get more than 100% of the benefits (other effects are negative). Further, the benefits to original residents are almost unaffected by increases in the size of the urban system. (3) The total benefits from a project are larger if migration is free than if it is costly and the difference exceedsthe level of deadweight loss accounted for by moving costs. (4) The distribution of benefits between rent changes and willingness-topay and between the project city and other cities varies widely in the various casesas moving costs and the size of the urban system vary. (5) The level of total benefits for a particular project may fall by half or more when moving costs are introduced, even when moving costs themselves are netted out. The distortion increases as both the size of the urban UThese results may be sensitive to our assumption of a Cobb-Douglas utility function. It would be interesting to examine the effect of a more general utility function which allows the elasticity of substitution between land and other goods to vary.
164
GREER AND WHITE
system and the level of moving costs increase. Thus neglecting the costs of moving when appraising the benefits of a project may cause the level of benefits to be sharply overestimated. Thus we have shown that the level and distribution of benefits generated by the same public project depend on the setting in which the project is executed. Planners making specific assumptions such as those of the perfectly closed or open city models may severely over- or underestimate both the level and distribution of benefits. Such variables as the size of the migration area around the project city and the level of moving costs may be very important in determining the level of benefits of public projects, who gets them and in what context they should be placed. REFERENCES 1. R. J. Amott, and J. Mackinnon, The effects of urban transportation changes: A general equilibrium simulation, J. Public &on. II, 19-38 (1977). 2. A. M. Freeman, III, Air pollution and property values: A methodological comment, Rev. Econ. Starist. 53,415-416 (1971). 3. M. Gets, A model of the impact of transportation investment on land rents, J. public Econ. 4, 57-74 (1975). 4. E. S. Mills, Planning and market processes in urban models in “Public and Urban Economics: Essays in Honor of Wiiam Vickrey” (R. E. Grieson, Ed.), Lexington Books, Cambridge (1976). 5. E. S. Mills, “Urban Economics,” Scott, Foresman, Glenview, Ill. (1972). 6. A. M. Polinsky, and D. L. Rubinfeld, Property values and the benefits of environmental improvements: Theory and measurement, in ‘Public Economics and the Quality of Lie” (L. Wingo and A. W. Evans, Eds.), Johns Hopkins Press, Baltimore, (1977). 7. A. M. Polinsb, and S. Shavell, The air pollution and property value debate, Reu. Econ. statist., 57, RIO- 104 (1975). 8. R. G. Ridker, and J. A. Henning, The determinants of residential property values with special reference to air pollution, Reu. hon. Statisr., 49, 246-257 (1967). 9. S. Rose-Ackerman, Cht the distribution of public program benefits between landlords and tenants, J. Emiron. Econ. Manage., 5, 167-184 (1977). 10. M. J. White, On the short-term effects of long-term change in cities: An efficient land markets model, J. Urban Con., 5,485-504 (1978). 11. M. J. White, Measuring the benefits of environmental and public policy changea in cities: Short-term and long-term considerations, 1. public Econ., 11, 247-260 (1979).
OP
URBANECONOMICS
9, 14% 164(1981)
The Effects of City Size and Moving Public Project Benefits’ WILLIAM
Costs on
GREER
Department of Economibs, Unitwsity of Penmyfwnai~ Philaa@hia, Penmyloania I9104 AND MICHELLE J. WHITE Graduate School of BusinessAdministration, New York Universi@,New York, New York loo06 Received February 6, 1979; revised November 22, 1979 This paper uses a simulation model to investigate the benefits of an urban public project when the size of the system of cities (or regions) in which migration occurs can vary and when moving costs impede migration. The model captures the range of possibilities between the perfectly open and closed city cases. We find that the total number of households migrating between cities declines sharply as moving costs rise. Also the level and distribution of project benefits between renters and landowners in the project city and other cities is sensitive to both the level of moving costs and the size of the urban system. The total benefits of the project fall by half or more when moving costs are introduced, even when moving costs themselves are netted out.
I. INTRODUCTION Economists have devoted much ink and effort over the last ten years to methods of measuring the benefits of large-scale urban public projects such as freeway construction or air quality improvements. On the one hand, Freeman [2], Polinsky and Shave11[6], and others have debated the theoretical question of whether and when the benefits of urban projects can be measured by their effects on land rents. Polinsky and Shave11 established that rents measure benefits correctly only in one special case: that of the perfectly open city.’ Here migration between the city where the project occurs and elsewhere (other cities and rural areas) is assumed to be costless and, in addition, the total population of the urban system is assumed to be large relative
to that of the project
city.3 On the other hand,
‘We are grateful to Richard Arnott, Malcolm Getz, and Ed Mills for their comments and suggestions. 2Polinsky and Shaveh [7] call this the “small open city” case. ‘This case also requires that the project itself be small, i.e., it must produce only a marginal improvement in air quality or the like. See White [lo] for a discussion. 149 0094-1190/81/020149-16$02.00/O CopyrightQ 1981 by Academic Rcsa, Inc. All lights
Of mpmduction
in any form
-cd.
150
GREER
AND
WHITE
Mills [4], Amott and MacKinnon [l], and others have developed increasingly detailed simulation models that explore the effects of urban projects on land rents, modal choice, residential location choice, job location, etc. These models isolate the city from the rest of the world by assuming another very special case: the closed city, whose population is fixed regardless what happens within it. This paper has two goals. First we synthesize the open and closed city strands of the literature. We present a simulation model of the effects of an urban public project in one city, but we examine carefully the role of the city’s relationship with the rest of the world. By varying the parameters of this relationship, we can determine how and how much the city’s links with the outside world affect the project’s benefits. We also determine the extent to which benefits and land rent changes are over- or underestimated by concentrating- as most economists have-on the conventional special cases of the perfectly open and perfectly closed cities.4 Second, we introduce explicit intercity moving costs into the model and examine their effects on households’ propensities to migrate and on project benefits, The perfectly open city case makes two special assumptions. First, it assumes that the cost of intercity migration is zero and, second, that the population elsewhere is large relative to the population of the single city. In our model these two assumptions are varied over a set of parameter values. We assume that migration may be costless or costly and that the population of the system of cities may be large or small relative to the city’s population. Thus four rather than two special cases are suggested. The “frictionless migration/many cities system” (F-M) is equivalent to the “perfectly open city.” The “costly migration/many cities system” (C-M) more realistically represents the U.S. situation, where migration occurs across many cities and large distances, but moving costs are nonzero. The “frictionless migration/few cities system” (F-F) and the “costly migration/few cities system” (C-F) are intended to represent countries such as England and Thailand, where the largest city is very large relative to the population of all other cities and towns put together. It also may represent the migration pattern of culturally distinct regions where households are more likely to move within the region than into or out of it.5 Intraregional migration may be seen as costless or as less costly than interregional migration.6 ‘Getz [3] suggests model@ a city’s relation with the outside world in terms of generalized elasticities of labor supply and product demand, but he solves only the perfectly open and perfectly closed city cases. ?he perfectly closed city is an extreme form of the “costly migration/many cities system” within which no migration occurs at all. 6The various models should not be seen as short-term versus long-term models of adjustment to the same project, however, since all assume full adjustment in housing markets. See White [ 10, 1 l] for a short-term adjustment model.
CITY
SIZE AND
MOVING
COSTS
151
Thus the simulation model proposed here provides an outside world context for the measurement of benefits from projects in particular cities and enables us to determine how important that context may be. Our results suggest that assumptions about the inside/outside relationship are important. We find that the level of intercity migration is sensitive to migration costs, that total benefits in particular cities vary greatly depending on the size of the project city relative to the system of cities and that even qualitative changes such as whether rents rise or fall in a particular location may depend on the context within which the project occurs. Thus it is important for policymakers to consider a wider range of models than the conventional two that economists are accustomed to using. II. THE MODEL We postulate a system of k + 1 identical monocentric cities.’ All households have identical utility functions
u = L(u)“X(u)’ -O,
(1)
where L(U) is household consumption of land at distance u from the central business district, X(U) is consumption of a numeraire composite good at distance u from the CBD, and a is the percentage of income spent on land. Households maximize utility subject to the budget constraint Y = R(u)L(u)
+ X(u) + tu,
(2)
where Y is daily income, R(u) is rent at distance u from the CBD and t is transportation.cost per two-mile distance. Thus tu is total daily commuting cost for a household residing at U. Demand functions for land and the composite good are L(u) =
a(Y - tu) R(u)
and X(u) = (1 - a)(Y - tu).
(4)
Equilibrium conditions for each city require, first, that all households within a single city achieve equal utility levels. This implies
-Wu) = -f/L(u). au ‘See Mills (51 for a discussion of the monocentric
urban model.
152
GREER AND WHITE
Second, aggregate land demand must equal aggregate land supply, or
N( u)L( u) = +u
for all u,
(6)
where N(u) is household population residing at u and + radians of a circle are occupied by residential land. Third, the total population of each city, N, must be
N=
“N(u)&, I0
where ii is the outer edge of the city. Fourth, rent at the edge of the city must equal rent on agricultural land, R, or R(G) = k.
(8)
Solving for the rent function and for total population, we find, R(u) =R
and
NC
+R a(Y - tuya
y - tu l/o Y-
[ 1
J0“u(Y -
tzp*
- “du.
Also solving for the indirect utility function for a household located at G, u = aq 1 - a)’ - “(Y - tG)/R”.
(11)
Equations (10) and (11) define the conventional closed and open city models. For the open city model, (10) and (11) are solved for ii and N, given a value of U. For the closed city model, they are solved for H and U, given a value of N. For the simulation, we use the following initial values in all cities. The initial population of each city in the system, N,, is 100,000households. The percent of income spent on land., (I, is 5%. Daily income, Y, is $40. Initial commuting cost per two miles, to, is $0.50. Using these values, we can solve (10) and (11) for the initial radius of each city, i& = 5.271 miles, and for the initial utility level, U, = 21.254. These values prevail initially in all cities. III. THE SIMULATION Assume that the public project is a transportation improvement which lowers the cost of commuting per two miles from to = $0.50 to 1, = $0.30 in one city only. Commuting cost remains the same in all k other cities.
CITY SIZE AND MOVING COSTS
153
The model must determine a new equilibrium following the project when two factors are varied separately: the size of the project city relative to the system of cities and the level of moving costs. Examining the first, we vary the size of the project city relative to the system of cities by varying k, the number of identically sized cities. If k is small, the project city’s population is large relative to the population of the system; if k is large, the opposite holds. However, regardless of the magnitude of k, all cities initially are identical in population. In the simulation, we vary k from 1 to 100. In the two frictionless migration cases, moving costs are zero and households move from any of the unimproved cities to the project city until utility levels in all k + 1 cities are equal. Thus
a=(1 ;y’ -O(y_t,q =aY1 if” _a(y_t,P*) or t,ii’ = t&i*,
(12)
where u’ is the radius at the new equilibrium of any of the k unimproved cities and U* is the radius of the project city. Also we assume that total population in the system of cities is constant at (k + l)N,, thus (k + l)N, = N* + kN’,
(13)
where N* and N’ are the populations of the project city and other cities respectively at the new equilibrium. The population levels N* and N’ also must satisfy Eq. (10) evaluated respectively at t = t,, P = U* and at t = t,, 24= U’. Thus for the frictionless migration/few cities and frictionless migration/many cities models (F-F and F-M), (12), (13) and the two population equations can be solved for U*, E’, N* and N’. Then the indirect utility function, (1 I), can be solved at C* and P’ for the new utility levels, U* and U’, and (9) can be solved for the new rent functions, R*(u) and R’(u). (By assumption U* = U’.) By varying the value of k and solving the same system of equations, we simulate the effects of frictionless migration within small, large or intermediate sized systems of cities. When moving costs are nonzero, the model becomes more complicated. We assume that the cost of moving between cities is either 10%or 20% of a year’s income. This includes both actual moving expenses and the cost of acquiring specific knowledge of job and housing markets in the new city. Assuming a IO-year time horizon and ignoring discounting factors, a migrant household faces a 1% of 2% reduction in income in the new city, from Y = $40 to Y’ = $39.60 or $39.20 per day.
GREER
154
AND
WHITE
Given moving costs, utility levels in the new equilibrium are no longer equal among all cities. If migration follows the transportation improvement, in equilibrium households remaining in an unimproved city and having Y = $40 must achieve the same utility level as households moving and having Y’ = $39.60 ($39.20). But after migration, the project city has two classes,a “poor” class with Y = $39.60 ($39.20) that initially resided in the unimproved cities and a “rich” class with Y = $40 that initially resided in the project city. The monocentric urban model implies that if two classeslive in the same city, the poor live in the center and the rich in the suburbs, with ub denoting the boundary between them.’ However, migration does not necessarily occur in the costly migration model. To determine whether it does, we solve for the rent function of the initial residents in the project city assuming no migration, and compare that with the rent function which the initial residents of unimproved cities can pay in the project city and still attain their initial utility levels (taking into account their reduced income). If the latter rent function is greater than the former at any point, migration will occur. Since the poor live in the center of the project city if migration occurs, we need only compare the two rent functions at the CBD.9 The initial rent function for the project city is found by solving (10) for ii*, where N = 100,000and t, = 0.3. Then substituting ii* and t, = 0.3 into (9), we get the rent function for the “rich.” The rent function for the poor is found by solving the indirect utility function (11) for U with t, = 0.3, Y = 0.99Y (0.98Y) and utility level U,. The resulting value of U is then substituted into the rent function (9), where again Y’ = $39.60 ($39.20) and t, = 0.3. If migration occurs, the new equilibrium must satisfy four conditions: First, utility levels for the poor in the project city and for all households in unimproved cities must be equal or Y-
t,ii R-a
=
Y’ - t,Ub R(d
where Y’ = 0.99Y or 0.98Y.
(14)
'
Second, the rich population in the project city must equal the initial population of the project city, N, or N=
+R a(Y - t,ii*y
c*u(Y - t,uya I Ub
- “du.
(15)
sin the model presented by Mills [S, Chap. 51, this requires that households have income elasticities of demand for housing at least equal to unity. QIf no migration occurs, then the system degenerates into the conventional two equation closed city model, (10) and (ll), which solves for the new utility level and new radius of the project city. Conditions do not change at all in other cities.
CITY SIZE AND MOVING COSTS
155
Third, the initial population of all unimproved cities, kN, must equal the final poor population in the project city plus the total remaining population in other cities, or kc#aR
kN=
a(Y - t,uy
U’ u( Y - t,uy” s0
- ‘)
I’& + +R(ud / ““,(r _ t,u)wOa(Y- t*up 0 Fourth, rent at the rich-poor border, R(u,), is R(+J=~
I I Y - t,u, Y-tu* I
‘/O .
(17)
Thus for the two costly migration models, C-F and C-M, the new equilibrium with migration is the solution of (14)-(17) for ii*, 2, ub and R(ub). The indirect utility function (10) can be solved for the new utility levels of both the rich and poor households, U,* and U,*. The three rent functions can be derived by evaluating (9). For the unimproved cities, (9) is evaluated for t = to. For the rich sector of the project city from u = u6 to ii*, (9) is evaluated t = t,, while for the poor sector from u = 0 to ub, (9) is evaluated for t = tl, Y’ and R(Q). The total population of each unimproved city, N’, can be calculated by evaluating (10) given u” and t = to. The population of the project city, N*, is
+
@if a(Y - t*u*y
JUb“‘u(Y -
t,u)l’a
- ’ du.
IV. MEASURING BENEFITS Two types of benefits result from the transportation improvement: increases in household welfare and changes in land rents. Either or both may occur in the project city and in the unimproved cities. In addition the project has a cost which must be subtracted from total benefits to obtain a measure of net total benefits. Looking first at land rents, the change in rents in a city equals the difference between aggregate land rents before versus after the project.
156
GREER AND WHITE
Aggregate land rent is found by integrating the rent function over u. For the project city, the change in aggregate rent is thus AR = I”‘[
R*(u)c#au] du - (J,“[
R’(u)cpu]
du + I”‘[
&
&w]
ch)
(19)
Here R’(u) and R*(u) are the initial and final rent functions, from (9). The first term in (19) is aggregate land rent after the project. Since the city expands as a result of the project, the initial level of aggregate rent has two parts, terms two and three in (19). These are the initial level of urban rents evaluated from u = 0 to i’&plus the level of rural rents, R, evaluated from 17 to U*. Thus in the area added to the city by the project, the increase in land rents is net of their previous value in agricultural use. Further, in the costly migration case, the final rent function, R*(u), is the upper envelope of rich households’ rent function from U* to ub and poor households’ rent function from ub to 0. Finally AR must be separately calculated for the project city and for the k unimproved cities and aggregated over all of them. Turning to project costs, we assume for simplicity that the project is financed externally, with no special tax payments levied either on residents of the project city or other cities to pay for it.” Turning to the effect of the project on household welfare, we must transform the increase in utility that accrues to households as a result of the project into a dollar measure of willingness-to-pay (WTP) or the amount of income a household can relinquish following the project and still achieve its original utility level. There are several ways to do this. The method used here is to solve for the new equilibrium and to measure households’ willingness-to-pay for the project after it occurs.” WTP is measured assuming that rents and taxes remain constant. For the Cobb-Douglas utility function, the elasticity of utility with respect to income net of transportation cost is unity. Consequently, WTP ‘OThis assumption is made partly to avoid the necessity of considering alternate tax schemes and partly because it is fairly realistic. Many large urban projects are primarily financed with Federal or State government revenues which are unrelated to the project itself. An alternate assumption might be that the project’s costs are small or negligible. Examples include levying parking fees where none existed before or timing the traffic lights on a major highway to speed the flow of traffic. “Amott and Mackinnon [l] use an alternative method. They assume that the project is accompanied by lumpsum taxes and transfers which keep households at the same utility level before and after the project. Then willingness-to-pay for the project is measured by the change in government revenues attributable to it. This method does not easily generalize to consideration of multiple city models with moving costs, however, since under their method all households must be taxed at the same level, while in the multicity case different households are in different situations. Various methoda of measuring willingness-to-pay give rise to different income effects across households and therefore to different levels of total willingness-to-pay and total benefits.
157
CITY SIZE AND MOVING COSTS TABLE 1 Simulation Results: Frictionless Migration Model Post-project k
P
P’
0 1 2 5 10 50 100
5.831 6.843 7.330 7.950 8.295 8.672 8.727
4.106 4.398 4.770 4.977 5.203 5.236
Migration per unimproved
Utility
city
21.757 21.585 21.502 21.396 21.338 21.273 21.264
0 47,093 37,251 22,853 13,917 3,371 1,731
Total households migrating
0 47,093 74,502 114J66 139,170 168,550 173,058
by a household at u is the precent increase in utility multiplied by net income. Since net income varies with u, WTP also varies with I(. For the two frictionless migration cases, the percent change in utility is equal for all households. Total WTP in each unimproved city is /“[ (%AU).(Y - t&V(u)] 0
du.
(20)
Total WTP in the project city is also (20), but evaluated at t = t, and from u = 0 to ii*. Aggregate WTP over all cities is the sum of WTP for the project city plus k times WTP for each unimproved city. For the two costly migration cases,WTP in the project city must be calculated separately for both the rich and poor households and summed over both classes.12 V. SIMULATION
RESULTS
A. Frictionless Migration
Simulation results for the frictionless migration model are shown in Table 1, for various values of k.13 The results show that as the number of cities in the urban system rises, more households in total migrate to the project city from other cities. The radius of the project city, P*, therefore rises. Also as k rises, fewer households migrate from each unimproved city, so U’ also rises. Finally while the project always causes utility to rise from its initial level of 21.254, larger k implies a smaller increase in utility since the benefits of the project are spread over more households. ‘*Note that WTP varies with location. Thus if a tax were levied on households equal to WTP, then post-project income levels would vary with distance. Households would relocate, resulting in a new equilibrium. 13Thek - 0 line gives results for the perfectly closed city case. We do not report results for k > 100, since as k gets very large the aggregate results are subject to considerable computational error.
CITY
SIZE AND
MOVING
COSTS
163
Finally the distribution of benefits from the project is also affected by moving costs. Landowners in the project city are big gainers when moving costs are zero and moderate gainers when moving costs are I%, but they are made worse off when moving costs are 2%. Policymakers thus need to be conscious of the possibility that landowners may lobby against public projects in their cities even when such projects are in the public interest. Households in the project city have the opposite interests: they are strong gainers when moving is costly and gain much less when it is free. These households also may have an incentive to lobby in favor of public projects that are not in the public interest. In our 2% moving cost case, their benefit exceeds the total benefit from the project, since other parties lose.22 VI. CONCLUSIONS This paper uses a simulation model to investigate the benefits of an urban project when the size of the system of cities and the level of intercity moving costs can vary. The model captures the range of casesbetween the perfectly open and perfectly closed city assumptions. We show the following results: (1) The amount of migration is very sensitive to the level of moving costs. In our model, moving costs equal to 2% of income cause the total number of households migrating to decline by 80% to 90% compared to the frictionless case, depending on the size of the urban system. (2) Moving costs act as a buffer which protect the initial residents of the project city from competition in the land market by in-migrants. When moving costs are zero, part of the project benefits are capitalized as an increase in land rents in the project city. As moving costs rise, the original residents of the project city receive a larger share of the benefits. When moving costs are 2% the original residents get more than 100% of the benefits (other effects are negative). Further, the benefits to original residents are almost unaffected by increases in the size of the urban system. (3) The total benefits from a project are larger if migration is free than if it is costly and the difference exceedsthe level of deadweight loss accounted for by moving costs. (4) The distribution of benefits between rent changes and willingness-topay and between the project city and other cities varies widely in the various casesas moving costs and the size of the urban system vary. (5) The level of total benefits for a particular project may fall by half or more when moving costs are introduced, even when moving costs themselves are netted out. The distortion increases as both the size of the urban UThese results may be sensitive to our assumption of a Cobb-Douglas utility function. It would be interesting to examine the effect of a more general utility function which allows the elasticity of substitution between land and other goods to vary.
CITY SIZE AND MOVING COSTS
159
total utility gain to households rises with the size of the urban system; the smaller gain accruing to each household is more than fully offset by the increased number of households sharing in it.i5 The size of the urban system thus affects the distribution of benefits. If the urban system is small, the improvement results in a significant utility gain for households in both improved and unimproved cities. Landowners in the project city receive only moderate capital gains on land, while landowners in other cities face substantial losses. If the urban system is large, households receive few benefits; landowners in the project city receive substantial capital gains, while landowners in other cities are indifferent.16 The last column in Table 2 shows total benefits of both types for the entire urban system. The level of total benefits increases with the size of the urban system. This suggests that we can define the notion of an optimal context for particular public projects. For our transportation project, the optimal location is the largest possible urban system, since the more households affected by the project, the higher the benefits. However, for other projects the opposite may be true.17 Policymakers should also be conscious of the distributional effects of locating projects in different contexts, since strong support or opposition may come from parties who fare differently depending on the size of the urban systern.I8 In our example, the level of total benefits (rent change plus WTP) in the project city and other cities both vary depending on the size of the urban system, with the gain in the project city rising and the loss in other cities falling as k rises. Thus the effect of a project on particular cities may be over- or underestimated if the wrong intercity migration assumptions are made. Finally, in our example landowners will favor projects in their cities and oppose projects in other cities. But their opposition to projects in other cities may run counter to the interests of society at large.” t5As k approaches infinity (the perfectly open city case), WTP in the project city and in each unimproved city must approach zero, since the utility gain to households approaches zero. Therefore total WTP might or might not approach zero depending on whether WTP in each unimproved city approaches zero faster or slower than k approaches infinity. In the simulations, even for very high values of k, total WTP always increased with k. t6Note that many of the landowners may be owner-occupiers: thus the same households may accrue both utility gains and capital gains or losses on land. “The level of total benefits depends on income effects generated by the method of measuring WTP. Other methods of measuring WIT may lead to different results. “Note that the results presented here, particularly those concerning the distribution of benefits between landlords and tenants, may depend on our assumption of Cobb-Douglas utility functions. Other utility functions may have different results. t9Note that as k approaches infinity, the rent change and WTP in each unimproved city, as well as WTP in the project city, all approach zero. Thus aa the model comes closer to the perfectly open city case, total benefits are more closely approximated by the change in rents in the project city.
160
GREER AND WHITE TABLE 3 Simulation Results: Costly Migration Model
k
1 2 5 10 50 100
ii’
ii’
ub
5.993 (6.501) 6.055 (6.771) 6.126 (7.067) 6.161 (7.245) 6.196 (7.431) 6.201 (7.458)
5.138 (4.676) 5.178 (4.840) 5.221 (5.020) 5.243 (5.127) 5.265 (5.239) 5.268 (5.255)
1.181 (2.440) 1.413 (2.994) 1.643 (3.706) 1.749 (3.998) 1.851 (4.269) 1.864 (4.262)
Migration Utility of rich per unimproved households city 21.730 (21.644) 21.720 (21.598) 21.708 (21.547) 21.702 (21.517) 21.6% (21.485) 21.695 (21.480)
(2gg (2:g) 2,497
WD44) (E) 314 (1SW 159 (805)
Total households migrating
Total moving costs
(2g?) 9,327 (4ON5) 12,483 (6OJl9) 14,092 (7OXn) 15,711
($gg 7,462 (16,018) 9,986 (24,088) 11274
(WW 12,569
(%385)
(32,154)
15,396 (80,539)
(32,216)
12,749
B. Cost& Migration Simulation results for the costly migration model are shown in Tables 3 and 4. Results for the 1% moving cost case are given in parentheses below the figures for the 2% moving cost case. The most skiking result is the dramatic decreasein the number of households migrating as moving costs rise. For k = 1, 47,092 households migrate when moving costs are zero, 26,818 when they are 1% and 6.600 when they are 2Yk20 Most other qualitative results are similar, however. Under costly migration as under frictionless migration, the radii of both the project city and other cities rise as the size of the urban system increases, but the changes are now smaller in magnitude. The original residents of the project city (the “rich” households) benefit from higher moving costs, since with fewer in-migrants, they face less competition for land. And conversely, the original residents of other cities are hurt by higher moving costs, since they reduce out-migration. However, as the urban system becomes larger, a given level of moving costs provides less protection for rich households and their utility level declines. The last column of Table 3 gives total moving costs, equal to $0.40 and $0.80 times the number of households migrating in the 1% and 2% cases, respectively. Table 4 reports the benefits of the project for the two levels of moving costs. Again the rent change in each unimproved city is negative, but the decrease is smaller in magnitude when moving is costly than when it is free, since fewer households out-migrate. Thus when a project occurs in 2oWhen migration costs exceed 2%, there is little or no migration.
- 6,620 ( - 26,326) - 4,737 (- 19,893) - 2,545 (- 12,130) - 1,435 (-7,134) - 313 (- 1,630) - 154 ( - 820)
(45,254)
- 14,248 (24,821) - 13,021 (33,775) - 11,771 (43,815) - 11,594
(W’87)
- 18,635 ( - 349) - 16,651
Rent change in the project city
4
(6::)
(1.3::)
(5.6
(7,876)
458 (9,021) 325
(8,356)
458
tiP by poor in project city*
Model
9 and 10 for discussion.)
(821)
312 (1,625) 153
cwm
6,407 (22,533) 4,629 (17,819) 2,514 (11,394) 1,425
w-win each unimproved city*
(See Footnotes
- 25,255 (-26,675) - 26,125 (-28,799) - 26,973 (- 35,829) - 27,371 (-37,565) - 27,421 ( - 37,685) - 26,994 (- 36,746)
rent change
TOtd
Note. AU figures in dollars *Approaches zero as k approaches infinity.
loo
SO
10
5
2
1
k
Rent change in each unimproved city*
TABLE Benefits in the Costly Migration
87,074 (73,034) 85,068 wsw 82,797 (52,781) 81,658 (47,377) 80,525 (42,027) 80,369 (41,795)
WTP by rich in project city
95,694 (124,568)
W4,W
93,939 (103,923) 94,784 (109,203) 95,692 (117,627) %,114 (121,494) %,175
Total WrP
58,733 (57,710 57,469 (55,847) 56,185 (54,777) 56,384 (55,606)
W36)
63,404 (66,521) 61,197
Total benefits net of moving costs
162
GREER
AND
WHITE
one city, landlords in other cities are better off if moving costs are high, since their capital losseson land are smaller. The rent change in the project city is negative in the 2% moving cost case for all k, since there is never enough in-migration to offset the decrease in rents at the center caused by the flattening of the rent function. (In the 1% cost case, however, more migration causes rents to rise in the project city for k > 1.) In both cases as k rises, the rent change in the project city becomes more positive.*’ The change in rents everywhere in the urban system is also negative, for all values of k. WTP in the unimproved cities is smaller with costly migration than with frictionless migration and again it declines as k increases. The pattern of WTP in the project city, however, varies with moving costs. Under frictionless migration, WTP declines strongly as the size of the urban system increases, but under 2% moving costs, WTP by rich households remains almost unchanged. WTP by the poor declines strongly with k, but is very small in all cases.Thus total WTP in the project city is about the same in absolute magnitude whether migration is costly or free when the urban system is small. But it declines sharply as k increases in the frictionless migration case while remaining relatively constant as k increases in the costly migration cases. In effect the existence of moving costs buffers the original residents of the project city from the effects of competition in the land market from in-migrants. With the two class situation, benefits to the original residents remain almost unaffected as the size of the urban system increases, because the number of in-migrating households increases only slightly. Thus while our results in the frictionless migration case suggest that the benefits to the original residents of the project city are nearly erased by competition from in-migrants, in the 2% moving cost case the original residents are nearly indifferent to the size of the urban system. The last column of Table 4 shows total benefits including both rent change and WTP, net of moving costs, for the entire urban system. For both levels of moving costs, benefits decline very gradually as the size of the urban system rises. Thus the optimal context for a public project differs depending on the costs of migration. When moving costs are zero, it is socially efficient to place the project in the largest possible urban system; with costly migration, society is more nearly indifferent to the size of the migration area. These results may reflect the particular assumptions of our model; it would be of interest to investigate others. 2’An increase in household utility levels in the monocentric urban model causes households to surburbanize and the rent function to fall at the center and rise in the suburbs. Thus the level of aggregate rents can either rise or fall. In-migration, however, always causes land demand to rise and aggregate rents to increase. Smce moving costs and the size of the urban system both affect the number of households migrating, the effect of the project on aggregate rents in the project city can be either positive or negative.
CITY
SIZE AND
MOVING
COSTS
163
Finally the distribution of benefits from the project is also affected by moving costs. Landowners in the project city are big gainers when moving costs are zero and moderate gainers when moving costs are I%, but they are made worse off when moving costs are 2%. Policymakers thus need to be conscious of the possibility that landowners may lobby against public projects in their cities even when such projects are in the public interest. Households in the project city have the opposite interests: they are strong gainers when moving is costly and gain much less when it is free. These households also may have an incentive to lobby in favor of public projects that are not in the public interest. In our 2% moving cost case, their benefit exceeds the total benefit from the project, since other parties lose.22 VI. CONCLUSIONS This paper uses a simulation model to investigate the benefits of an urban project when the size of the system of cities and the level of intercity moving costs can vary. The model captures the range of casesbetween the perfectly open and perfectly closed city assumptions. We show the following results: (1) The amount of migration is very sensitive to the level of moving costs. In our model, moving costs equal to 2% of income cause the total number of households migrating to decline by 80% to 90% compared to the frictionless case, depending on the size of the urban system. (2) Moving costs act as a buffer which protect the initial residents of the project city from competition in the land market by in-migrants. When moving costs are zero, part of the project benefits are capitalized as an increase in land rents in the project city. As moving costs rise, the original residents of the project city receive a larger share of the benefits. When moving costs are 2% the original residents get more than 100% of the benefits (other effects are negative). Further, the benefits to original residents are almost unaffected by increases in the size of the urban system. (3) The total benefits from a project are larger if migration is free than if it is costly and the difference exceedsthe level of deadweight loss accounted for by moving costs. (4) The distribution of benefits between rent changes and willingness-topay and between the project city and other cities varies widely in the various casesas moving costs and the size of the urban system vary. (5) The level of total benefits for a particular project may fall by half or more when moving costs are introduced, even when moving costs themselves are netted out. The distortion increases as both the size of the urban UThese results may be sensitive to our assumption of a Cobb-Douglas utility function. It would be interesting to examine the effect of a more general utility function which allows the elasticity of substitution between land and other goods to vary.
164
GREER AND WHITE
system and the level of moving costs increase. Thus neglecting the costs of moving when appraising the benefits of a project may cause the level of benefits to be sharply overestimated. Thus we have shown that the level and distribution of benefits generated by the same public project depend on the setting in which the project is executed. Planners making specific assumptions such as those of the perfectly closed or open city models may severely over- or underestimate both the level and distribution of benefits. Such variables as the size of the migration area around the project city and the level of moving costs may be very important in determining the level of benefits of public projects, who gets them and in what context they should be placed. REFERENCES 1. R. J. Amott, and J. Mackinnon, The effects of urban transportation changes: A general equilibrium simulation, J. Public &on. II, 19-38 (1977). 2. A. M. Freeman, III, Air pollution and property values: A methodological comment, Rev. Econ. Starist. 53,415-416 (1971). 3. M. Gets, A model of the impact of transportation investment on land rents, J. public Econ. 4, 57-74 (1975). 4. E. S. Mills, Planning and market processes in urban models in “Public and Urban Economics: Essays in Honor of Wiiam Vickrey” (R. E. Grieson, Ed.), Lexington Books, Cambridge (1976). 5. E. S. Mills, “Urban Economics,” Scott, Foresman, Glenview, Ill. (1972). 6. A. M. Polinsky, and D. L. Rubinfeld, Property values and the benefits of environmental improvements: Theory and measurement, in ‘Public Economics and the Quality of Lie” (L. Wingo and A. W. Evans, Eds.), Johns Hopkins Press, Baltimore, (1977). 7. A. M. Polinsb, and S. Shavell, The air pollution and property value debate, Reu. Econ. statist., 57, RIO- 104 (1975). 8. R. G. Ridker, and J. A. Henning, The determinants of residential property values with special reference to air pollution, Reu. hon. Statisr., 49, 246-257 (1967). 9. S. Rose-Ackerman, Cht the distribution of public program benefits between landlords and tenants, J. Emiron. Econ. Manage., 5, 167-184 (1977). 10. M. J. White, On the short-term effects of long-term change in cities: An efficient land markets model, J. Urban Con., 5,485-504 (1978). 11. M. J. White, Measuring the benefits of environmental and public policy changea in cities: Short-term and long-term considerations, 1. public Econ., 11, 247-260 (1979).