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Subsidies for intracity and intercity commuting
Journal of Urban Economics 66 (2009) 25–32
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Journal of Urban Economics www.elsevier.com/locate/jue
Subsidies for intracity and intercity commuting Rainald Borck a, Matthias Wrede b,* a b
University of Passau, Faculty of Economics, Innstrasse 27, 94034 Passau, Germany University of Marburg, Faculty of Business Administration and Economics, Am Plan 2, 35032 Marburg, Germany
a r t i c l e
i n f o
Article history: Received 21 May 2008 Revised 9 February 2009 Available online 31 March 2009
a b s t r a c t This paper analyzes subsidies for intracity and intercity commuting in an urban economics framework with two cities and agglomeration externalities, where workers may commute within and between cities. First, commuting subsidies serve to internalize agglomeration externalities: intracity commuting subsidies give incentives to move to the larger city and intercity commuting subsidies make residents of the periphery commute to the core. Second, if agglomeration rents are locally captured, commuting subsidies act as a welfare enhancing transfer from the core to the periphery. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Today, in most countries, the vast majority of workers commute to work, many over long distances. The proportion of long-distance commuters seems to have risen in many countries. For instance, in Germany in 2004, 47% of workers commuted more than 10 km to work, and 17% commuted more than 25 km, 5% even more than 50 km (Statistisches Bundesamt, 2005). According to Pisarski (2006), the average commute in the US was 12.2 miles in 2001, an increase of almost 14% since 1990. Moreover, about 9% of all workers worked outside their home metro area. Hence, commuting long distance seems to be an important phenomenon. It is also worthy to note that many countries subsidize commuting, and in some we find subsidy rates vary with distance. According to the American Public Transportation Association (2006), total fares in several thousand public transit authorities in the US accounted for only 33% of operating costs and 23% of total operating and capital costs, while for Europe the corresponding figures show fares cover about half of operating costs. Urban transit is obviously heavily subsidized through general tax revenue. Interestingly, automobile travel too, often does not cover the full cost of road construction and usage (let alone environmental and accident costs). Indeed, in the US, user fees (including gasoline taxes, license fees and other charges) accounted for only 60% of total highway expenditures (Brueckner, 2005). In Europe, higher petrol taxes suggest that subsidies to automobile travel should be significantly lower.1 In the US, there are no direct tax benefits for commuters, but commuting is subsidized through transit and highway construc-
* Corresponding author. Fax: +49 6421 2824852. E-mail addresses: [email protected] (R. uni-marburg.de (M. Wrede). 1 See Borck (2008) for more discussion and references.
Borck),
tion subsidies, as well as various fringe benefits such as cash benefits for parking or transit. In many European countries, by contrast, there are distinct provisions on the tax deductibility of commuting costs (see Bach, 2003). We briefly summarize these provisions with an eye on how deductibility varies with distance.2 Some countries including the US, Canada, the UK, Australia, Spain and Portugal do not allow deductibility of commuting costs. In some other countries, for instance, Finland and Norway, commuting costs are deductible regardless of distance, but only up to a certain maximum amount per year. In many other countries, however, deductibility varies with distance: In Germany, the tax code was changed in 2007 by eliminating deductibility of commuting distances of less than 20 km. For larger distances, deductibility remained. However, in 2008 the German Constitutional Court decided that this differentiated treatment was unconstitutional. In Sweden, transit costs are deductible, but commuting by automobile is deductible only above 5 km distance. Austria has a lump sum up to 20 km with higher deductibility for distances beyond 20 km. Denmark has deductibility only from 12 km with a lower rate for distances above 100 km. Some countries, such as Belgium and France limit deductibility to a maximum of 40 or 50 km. In general, the widespread subsidization of commuting provokes the question of why these subsidies are paid, and why subsidy rates should vary with distance. Possible explanations rest on politics: for instance, Borck and Wrede (2005, 2008) show how subsidies redistribute between rich and poor city dwellers and between renters and landowners. Another possibility would be that cheap commuting increases the demand for residential land and housing and requires extended road networks which benefits the construction industry. Firms in this industry would then have an incentive to lobby for commuting subsidies.3
wrede@wiwi.
0094-1190/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jue.2009.02.003
2 3
The provisions refer to the state of affairs as of 2003. This explanation was suggested to us by William Strange.
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In this paper, we analyze whether a welfare maximizing government would have incentives to subsidize commuting, and whether optimal subsidies vary with commuting distance. This is not to say that we believe that differentiated commuting subsidies exist because they are efficient. This obviously depends on the objectives of governments and whether these coincide with social welfare (which typically they do not). But in order to understand the political economy of some government intervention, it is useful to look at the outcome of welfare maximization as a benchmark. On the theoretical side, our paper also contributes to the literature by analyzing long-distance commuting between cities as well as short-distance commuting within cities.4 Commuting long distance may occur for a variety of reasons: (1) households with high housing demand may choose to locate far from employment centers to economize on housing costs, (2) households may want to move away from city centers to benefit from various rural amenities such as greenspace or from the absence of disamenities such as crime and pollution, (3) people may move to suburbs with better schools or other local public goods, and (4) dual-career couples may be forced to separate workplace and residence for at least one worker which may result in long commutes. To fix ideas, we mostly assume away these motivations and instead focus on households’ decision to commute to one of two employment centers: the one of the city they live in or the one of the other city. There are then two decisions households have to make: choice of residence and choice of workplace. For given residences, the choice of whether or not to commute long distance depends on the difference in wages and the cost of commuting long distance. Thus, workers will commute to the larger city if wages there are sufficiently higher due to agglomeration effects. The choice of residence is governed by within-city commuting and housing costs. Hence, households may choose to live in small cities in order to economize on these urban costs. Thus, our notion of long-distance commuting is not related to distance per se, but rather to the question of whether commuters commute to the employment center of the city they live in or to another, geographically distinct city. In our model, however, this will imply long distances since commuters to the other city first have to travel to their own city center. In reality, the distinction between what we take to be long- and short-distance commuting is somewhat blurred. This is necessarily so, given the abstractions of our model. Consider, for instance, workers in New York City. Whether commuters from, say, Newark, NJ or Philadelphia, or both are considered to commute long distance depends on their motives. We would term long-distance commuter someone who lives in Newark and works in New York if the same job were available in Newark, albeit at a lower wage.5 By contrast, if someone commutes from Newark to NYC because housing is cheaper in Newark, but the job they have can only be found in NYC (say, a typical Wall Street job), we would say this person is a short-distance commuter. The blurriness comes from the fact that job opportunities for people of different professions are not equally affected by agglomeration externalities, and the motives for commuting mentioned above are not the same for all commuters. Our model combines an Alonso type urban model with Marshallian agglomeration externalities à la Henderson (1974). These externalities might be due to benefits from input sharing, labor market pooling, and knowledge spillovers (see Duranton and Puga, 2004, for the microfoundations of agglomeration economies). Many studies have shown that localization economies and urbanization economies are present in cities (see, for a comprehensive overview, Rosenthal and Strange, 2004). Agglomeration effects generally imply that social and private returns to working in a city 4
Borck et al. (2007) also study long-distance commuting but do not model the spatial extension of cities and hence ignore commuting within cities. 5 An example may be the insurance industry.
do not coincide. In order to achieve first-best efficiency, workers should be confronted with the social effects of their commuting decision.6 When differentiated wage subsidies are not feasible, we show that differentiated subsidies to long- and short-distance commuting may achieve first-best efficiency. That is, our analysis can be viewed as a possible normative justification of why commuting should be subsidized and why long-distance commuting between cities and short-distance commuting within cities should be treated differently. The basic reason for subsidizing commuting is that workers do not receive their full marginal social product. Therefore, commuting subsidies help to ensure that workers commute to that employment center which yields the highest social output. Beyond that, however, (short-distance) commuting subsidies also serve to make workers choose the right place of residence. Hence, our paper extends the literature on the allocative effects of commuting subsidies by combining agglomeration externalities with land use. Within the standard urban model commutings subsidies are inefficient because they distort land use (see, e.g., Brueckner, 2005). Standard arguments for commuting subsidies rest on externalities, market imperfections, and pre-existing distortions (see Martin, 2001; Wrede, 2000, 2001, and Zenou, 2000). Most closely related to our paper are Verhoef and Nijkamp (2003) who discuss the trade-off between the internalization of pollution and agglomeration externalities and Arnott (2007), who studies the trade-off between congestion and agglomeration externalities. None of these papers explicitly combines commuting and residence decision, however, as we do. The paper proceeds as follows. The next section introduces the model. In Section 3, we study the social planner problem of how to allocate the population between two cities, where the choice variables relate to the resident and working populations. In Section 4, we focus on the market equilibrium, which is achieved by individuals choosing their place of residence and workplace. Section cas discusses the impact of subsidies when rents are only locally distributed, and the last section concludes. 2. The model We consider an economy with two cities, labeled 1 and 2. There is a total of N mobile workers. Workers simultaneously choose their place of residence and their place of work. That is, by commuting from one city to the other, workers may separate job and residence. Workers confer Marshallian urbanization externalities upon each other, that is, a worker is more productive the larger the number of other workers in the city is, perhaps because of knowledge spillovers. This is captured in a simplistic way by assuming that the wage rate earned by a worker working in city i; wi , is a function of the local labor force, Hi .7 Suppose that the production function of firm j in city i is yji ¼ EðHi ÞFðhji ; lji Þ, where hji is labor employed by firm j; Hi is aggregate labor in the city, and lji is an immobile input employed by firm j in city i. The immobile input could be either buildings or publicly provided infrastructure. Without loss of generality, we assume that city 1 is endowed with a stock of the immobile input at least as large as that in city 2: L1 P L2 , where Li is the aggregate amount of the immobile input in city i. We call FðÞ the ‘internal production function’ and we assume it is a linearly homogeneous function with Fð0; Þ ¼ 0, and positive but decreasing marginal products for both factors. The function EðHi Þ, with Eð0Þ P 0 and E0 ðHi Þ > 0, captures the positive agglomeration externality. When FðÞ fulfills the Inada condition, limHi !0 F H ðHi ; Þ ¼ 1, the ‘‘internal” marginal product of labor goes to infinity if the number of workers 6 We abstract from other efficiency reasons for intervening in commuting decisions such as congestion. 7 This approach follows among others Henderson (1974), Michel et al. (1996), and Abdel-Rahman and Anas (2004).
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goes to zero. Together with EðÞ > 0, the Inada condition precludes full agglomeration of labor (see Michel et al., 1996). Firms are assumed to maximize profits under perfect competition. Each firm takes aggregate labor in the city as given. Thus, even though there are increasing returns at the city level, each firm views itself as facing constant returns. Hence, workers are paid their marginal product, and the wage of a worker working in city i is
wi ¼ EðHi ÞF H ðHi ; Li Þ;
ð1Þ
where profits are zero. Each city is linear and extends from zero to r i ; i ¼ 1; 2, where the boundary ri is endogenously determined by individuals’ residence choices. All citizens live on plots of land of fixed size normalized to unity. Land rent at distance r will be denoted by pðrÞ. All workers working in a city commute to the central business district (CBD) to work; all those working in the other city first commute to the CBD and then from there to the other city. A natural way to think of this assumption is workers commuting by train, where the train stations are located in the city centers.8 Hence, regardless of where a worker works, if she lives at r km from the CBD, she incurs commuting costs of twr where w is the wage and t > 0 the inverse of travel speed (commuting costs are in terms of time). A worker who commutes long distance additionally incurs a cost of kw, with k > 0, to commute from CBD to CBD.9 An individual living at r km from the CBD where she works supplies 1 tr units of labor. If she lives r km from CBD i and commutes to CBD j, she supplies 1 k tr units of labor. We will assume that 1 tN k > 0. This implies that even if everyone were to live in, say, city 2, then if the outermost residents commute to city 1 they would still supply a positive amount of labor there (i.e., commuting does not take the entire workday). The number of inhabitants of city i working in the same city is denoted by N i , and the number of commuters from 2 to 1 by M 21 . Without loss of generality, the focus is on long-distance commuting from CBD 2 to CBD 1, i.e., M 21 P 0. Note that transport costs, together with homogeneity of workers, exclude two-way commuting. The regional government may subsidize commuting at rate ss within cities and at rate sl across cities. While subsidies differ between short-distance commuting and long-distance commuting, they are uniform across cities.10 Short-distance commuters in city i receive a subsidy of ss wi tr, long-distance commuters from 2 to 1 receive ss w1 tr þ sl w1 k. These subsidies may come in the form of subsidized public transport, tolls or other transport fees below total costs, or income tax deductions for commuting expenses. Subsidies are financed by a lump-sum tax T on all citizens. In order to obtain reasonable rent schedules, we assume throughout the analysis that shortdistance subsidies do not exceed costs. Hence, we assume ss 6 1. We assume that economic rents, namely land rents and the income of the immobile factor, are equally distributed among all citizens living in the entire metropolitan area. Aggregate land rent in city i is denoted by Ri , and immobile input income in city i is denoted by Q i ¼ EðHi ÞFðHi ; Li Þ EðHi ÞF H ðHi ; Li ÞHi . Per capita rents in city 1 and city 2 are 8
This assumption is a relatively accurate description for commuting by train, but when workers commute by car, things may get more complicated, since intercity roads generally do not connect the CBDs. It may then be that some workers living away from the CBD have shorter commutes to the other city than someone living at the same distance from their own city center but on the ‘far side’. Automobile travel may also raise issues of congestion. Including these effects would, however, greatly complicate the analysis. 9 Note that with some change in notation, our main results could also be obtained by assuming that part of the commuting cost is a monetary cost, i.e., independent of the wage. 10 Introducing differentiated subsidy rates would not change qualitatively our main results. Furthermore, we note that in many countries, while city governments have some discretion over transport policies, many instruments such as income tax policies or large infrastructure policies tend to be under the control of regional or even national governments. Hence, our assumption seems to be a reasonable approximation of reality.
I1 ¼ I2 ¼
R1 þ R2 þ Q 1 þ Q 2 : N
ð2Þ
Since our assumption on the distribution of rents affects the results regarding the necessity of subsidies, in Section 5 we also consider economic rents which are distributed only locally. Finally, we assume that utility, v, equals consumption, c.11 Suppose first that nobody commutes from the city under consideration to the other city. Then utility for a worker living in city i is given by
v i ¼ ci ¼ wi ½1 ð1 ss Þtr pi ðrÞ þ Ii T;
ð3Þ
which is simply the wage net of commuting costs, minus housing rents, plus economic rents minus lump-sum tax. Solving (3) for land rent as a function of commuting distance defines an individual’s bid rent function:
pi ðrÞ ¼ wi ½1 ð1 ss Þtr þ Ii T ci :
ð4Þ
As usual, we assume that competition for land implies that the highest bidder lives on any plot. Thus, which of two individuals lives closer to the CBD will depend on the slope of the bid rent function
@pi ðrÞ ¼ ð1 ss Þwi t: @r In particular, since commuting is time consuming, individuals with higher wages live closer to the CBD.12 At the city border, land rent must equal the agricultural land rent, which is normalized to zero. Since ri ¼ N i , we can solve pi ðN i Þ ¼ 0, from (4) to find an expression for utility:
ci ¼ wi ½1 ð1 ss ÞtNi þ Ii T: Using the definition of aggregate land rent, Ri ¼ sumption can be alternatively written as
ð5Þ R Ni 0
Ni Ri þ Ii T: ci ¼ wi 1 ð1 ss Þt 2 Ni
pi ðrÞ dr, con-
ð6Þ
Using the same procedure, we find the utility of residents in city 2 when all commute to city 1 by substituting M 21 for N i and taking account of long-distance commuting costs:
M21 R2 þ I2 c21 ¼ w1 1 ð1 sl Þk ð1 ss Þt T: 2 M 21
ð7Þ
Residents of city 2 will only commute to CBD 1 if their wage net of commuting costs is higher in city 1 than in city 2, i.e., if
w1 ½1 ð1 sl Þk ð1 ss Þtr > w2 ½1 ð1 ss Þtr:
ð8Þ
Provided that w1 > w2 it can easily be shown that there is a critical distance ~r such that all residents living at distance r > ~r work in CBD 2 and all residents living at r 6 ~r commute to CBD 1, where
~r ¼
½1 ð1 sl Þkw1 w2 : ð1 ss Þtðw1 w2 Þ
11 Hence, we also take leisure as fixed. Including leisure would clearly reduce the tractability of the model. Furthermore, if leisure time were variable, commuters would consume more leisure and work more than in our model. On the one hand, less commuting would be necessary to equalize marginal products of labor. On the other hand, commuting would be less harmful. However, we expect that the main results of our model would not be affected by the inclusion of leisure. 12 Of course, this assumption depends on equal housing consumption for all individuals. Otherwise, the slope of the bid rent function is given by the ratio of marginal commuting costs to housing consumption.
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Note that if some individuals commute from city 2 to city 1, the commuters will live closer to the CBD of city 2 if they have higher bid rents for locations close to the CBD. Hence, if the wage in city 1 is higher than the wage in city 2, long-distance commuters will tend to have steeper bid rents and will, therefore, prefer to live close to the CBD. With long-distance commuting the land rent in city 2 is
p2 ðrÞ ¼
w1 ½1 ð1 sl Þk ð1 ss Þtr c21 þ I2 T w2 ½1 ð1 ss Þtr c2 þ I2 T
for r < ~r
: otherwise ð9Þ
R2 ¼
M21
p2 ðrÞ dr þ
Z
N 2 þM 21
p2 ðrÞ dr M 21 M 21 ¼ w1 1 ð1 sl Þk ð1 ss Þt M 21 2 ðN2 þ M 21 Þ þ w2 ðN2 þ M 21 Þ 1 ð1 ss Þt 2 M 21 c2 N2 c21 M 21 w2 M 21 1 ð1 ss Þt 2 0
þ ðI2 TÞðN2 þ M 21 Þ:
ð10Þ
ð11Þ
ð12Þ
Using the pattern of locations, we can give a general formula for the government budget constraint:
Z N1 Z M21 þN2 Z M21 1 T¼ r dr þ w2 r dr þ w1 r dr ss t w1 N 0 M 21 0 Z M21 þ sl w1 k dr 0 ( " # 1 N21 ðN 2 þ M21 Þ2 M221 ¼ ss tw1 þ ss tw2 N 2 2 2 ) 2 M þ ss tw1 21 þ sl w1 kM 21 : 2
Z
Z M21 ð1 trÞ dr þ ð1 k trÞ dr 0 0 N1 M21 ¼ N1 1 t þ M21 1 k t ; 2 2 Z M21 þN2 N2 þ 2M 21 : ð1 trÞ dr ¼ N2 1 t H2 ¼ 2 M 21
ð16Þ
Substituting for H1 and H2 from (14) and (15), the Lagrangian can be written
GH ðH1 ; L1 Þð1 tN1 Þ k ¼ 0;
ð17Þ
GH ðH2 ; L2 Þ½1 tðN 2 þ M 21 Þ k þ l1 ¼ 0; GH ðH1 ; L1 Þð1 k tM21 Þ GH ðH2 ; L2 ÞtN2 k þ l2 ¼ 0:
ð18Þ ð19Þ
Due to the non-convexity of the optimization problem, solutions of the system of first-order conditions need not be global maxima. However, some characteristics of optima can be obtained by analyzing the first-order conditions. Case 1. N 2 ¼ M21 ¼ 0. This would imply full agglomeration of population ðN 1 ¼ NÞ. The complementary slackness conditions are l1 ; l2 P 0. Examining (17)–(19) shows that this implies
GH ð0; L2 Þ GH ðNð1 tN=2Þ; L1 Þð1 tNÞ 6 0;
ð20Þ
GH ðNð1 tN=2Þ; L1 ÞðtN kÞ 6 0:
ð21Þ
Since GH ðH; LÞ ¼ EðHÞF H ðH; LÞ þ E0 ðHÞFðH; LÞ, we have:
ð13Þ
Residents of city 1, workers of city 2, and workers of city 1 living in city 2 receive subsidies. Labor supply can now be written as
H1 ¼
N1 þ N 2 þ M21 ¼ N; N 2 P 0; M 21 P 0; ð14Þ and ð15Þ;
and the corresponding first-order conditions are
At the urban fringe in city 2; r 2 ¼ N N 1 , utility is
c2 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ þ I2 T:
s:t:
Y ¼ GðH1 ; L1 Þ þ GðH2 ; L2 Þ
þ l1 N2 þ l2 M 21 ;
There is then a kink of the land-rent schedule at the distance ~r ¼ M21 where a city 2 resident is just indifferent between commuting to city 1 or working in city 2. At this kink, we have
w1 ½1 ð1 sl Þk ð1 ss ÞtM 21 c21 ¼ w2 ½1 ð1 ss ÞtM 21 c2 :
max
N1 ;N2 ;M21
N1 M 21 L ¼ G N1 1 t þ M21 1 k t ; L1 2 2 N2 þ 2M 21 ; L2 þ kðN N 1 N2 M 21 Þ þ G N2 1 t 2
Hence, aggregate land rent in city 2 is given by
Z
simply maximizes total output: population is homogeneous, the lot size is fixed, leisure is not explicitly modeled, and non-time costs of commuting are neglected. Hence, denoting output in city i by GðHi ; Li Þ, a social planner who can choose residences and workplaces solvess
N1
ð14Þ ð15Þ
In city 1, each resident and each commuter from city 2 supplies one unit of time minus commuting time. In city 2, labor is only supplied by a fraction of residents. Here, we assume that subsidies increase income, but do not directly reduce travel time. 3. Optimum allocation of workers We now characterize the socially optimal allocation of workers across communities, both in terms of residences, and in terms of employment. Due to our assumptions, the welfare maximum
Proposition 1. Full agglomeration of population and employment, N 1 ¼ N, can be socially optimal if and only if k P tN and either (i) Eð0Þ > 0 and the Inada condition does not hold (limHi !0 F H ðHi ; Þ is finite) or (ii) Eð0Þ ¼ 0. Thus, less than full agglomeration of population can be optimal only if long-distance commuting is sufficiently cheap, in particular, if commuting from CBD to CBD costs less than commuting within a city would if all workers were to live there. Further, if the Inada conditions hold, full agglomeration of population cannot be optimal, since then the marginal product of shifting one worker from city 1 to city 2 must be positive. Case 2. N 2 ¼ 0; M 21 > 0. We now have (17)–(19) gives
l1 P 0; l2 ¼ 0. Combining
GH ð0; L2 Þð1 tðN N1 ÞÞ GH ðH1 ; L1 Þ½1 k tðN N1 Þ 6 0;
ð22Þ
GH ðH1 ; L1 Þðtð2N1 NÞ kÞ ¼ 0:
ð23Þ
Thus, we have the following result: Proposition 2. Full agglomeration of employment and dispersed population, i.e., N 2 ¼ 0 and M21 > 0, can be socially optimal only if k < tN and either (i) Eð0Þ > 0 and the Inada condition does not hold ðlimHi !0 F H ðHi ; Þ is finite) or (ii) Eð0Þ ¼ 0. Furthermore, the optimal allocation satisfies
N1 ¼
N k þ : 2 2t
R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
Case 3. N 2 > 0; M 21 ¼ 0. We now have (19) we find
l1 ¼ 0; l2 P 0. From (17)–
GH ðN 1 ð1 tN1 =2Þ; L1 Þð1 kÞ GH ðN2 ð1 tN2 =2Þ; L2 Þ 6 0;
ð24Þ
GH ðN 1 ð1 tN1 =2Þ; L1 Þð1 tN1 Þ GH ðN2 ð1 tN2 =2Þ; L2 Þ½1 tðN N 1 Þ ¼ 0:
ð25Þ
This shows that long-distance commuting is probably inefficient if it is very time consuming and if the differences in the stocks of immobile inputs are not too large. If L1 ¼ L2 , a symmetric allocation without long-distance commuting, i.e., N 1 ¼ N 2 ¼ N=2, possibly maximizes output. Indeed, as can readily be seen from (25), this will be the case if the production function is concave, i.e., GHH < 0. If L1 > L2 , an asymmetric allocation with N 1 > N 2 > 0 and M21 ¼ 0 may also be an optimum. Case 4. N 2 > 0; M 21 > 0. We now have l1 ; l2 > 0. Combining (17)– (19) gives
GH ðH1 ; L1 Þ½1 k tðN N1 N2 Þ ¼ GH ðH2 ; L2 Þ½1 tðN N1 N2 Þ; GH ðH1 ; L1 Þð1 tN1 Þ ¼ GH ðH2 ; L2 Þ½1 tðN N 1 Þ:
ð26Þ ð27Þ
According to the second condition (27), the social marginal product of labor should be equalized for the outermost residents of the two cities (note that the outermost city 2 resident works in city 2). The first condition (26) requires that the social marginal product of city 2 resident living at M21 who works in city 2 be equalized to the social marginal product of city 2 resident who commutes to city 1. These conditions imply
N2 ¼
½1 tðN N1 Þ½k þ tðN 2N1 Þ : ð2N1 NÞt 2
ð28Þ
4. Migration equilibria and subsidies We now study market equilibria, where individuals can freely migrate and commute, that is, they choose their place of residence and their place of work based on the utility obtained when living and working in city 1 versus living and working in city 2 or living in city 2 and working in 1. Comparing utility levels between cities, and between commuting and not commuting, we find interior equilibria with 0 < N 1 < N; 0 < M 21 < N N 1 if
c1 ðN1 ; M 21 Þ ¼ c2 ðN1 ; M 21 Þ ¼ c21 ðN1 ; M21 Þ;
where ci ðN 1 ; M21 Þ; i ¼ 1; 2 and c21 ðN 1 ; M21 Þ are determined using the definition of land rent in combination with the government budget constraint. By contrast, we get corner solutions with full agglomeration of population if c1 ðN; 0Þ P c2 ðN; 0Þ. Likewise, there would be no long-distance commuting if c21 ðN1 ; 0Þ 6 c2 ðN1 ; 0Þ while all city 2 residents would commute to city 1 if c21 ðN 1 ; N N 1 Þ P c2 ðN1 ; N N1 Þ.13 As is usual in the new economic geography literature, we assume a myopic adjustment process. That is, the change in population and in workforce is governed by the ad-hoc dynamics
N_ 1 ¼ ðc1 c2 ÞN1 N2 ; _ 21 ¼ ðc21 c2 ÞM 21 N2 ; M
ð29Þ
ð30Þ ð31Þ
if 0 < N 1 < N; 0 < M21 < N, and 0 < N 2 < N. Provided that L1 ¼ L2 , the symmetric allocation N 1 ¼ N 2 ¼ N=2 is obviously an equilibrium regardless of the size of subsidies, since utility is the same in both cities and due to equal wages no one can gain from long-distance commuting. However, this equilibrium might be unstable. An analysis of the first-order conditions of the central planner’s problem in cases 1–4 reveals that these conditions directly translate into the respective equilibrium conditions with zero subsidies ðss ¼ sl ¼ 0Þ if
E0 ðH1 ÞFðH1 ; L1 Þ þ EðH1 ÞF H ðH1 ; L1 Þ EðH1 ÞF H ðH1 ; L1 Þ ¼ ; E0 ðH2 ÞFðH2 ; L2 Þ þ EðH2 ÞF H ðH2 ; L2 Þ EðH2 ÞF H ðH2 ; L2 Þ
ð32Þ
where the left-hand side is the ratio of social marginal products of labor at the optimum, GH ðH1 ; L1 Þ=GH ðH2 ; L2 Þ, and the right-hand side is the ratio of internal marginal products of labor, i.e., the wage ratio. For instance, in case 4 where N 2 > 0 and M21 > 0, the equilibrium conditions derived from (29) read:
w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 N2 Þ ¼ w2 ½1 ð1 ss ÞtðN N1 N2 Þ;
Since we have assumed 1 > k þ tN and since the optimum with long-distance commuting requires N1 > N=2 and N 2 > 0, we conclude that N 1 < N=2 þ k=ð2tÞ holds. From closer inspection of the problem it becomes immediately clear that an asymmetric allocation with long-distance commuting, where N 1 ; N 2 , and M21 are all greater than zero, is only optimal when agglomeration externalities are sufficiently strong and long-distance commuting costs k are sufficiently low. Otherwise, a less asymmetric allocation without long-distance commuting maximizes output. Moreover, (26) directly implies GH ðH1 ; L1 Þ > GH ðH2 ; L2 Þ, i.e., workers commute to the CBD with the higher social marginal product of labor. Since working hours in CBD 1 are more valuable than in CBD 2, efficiency requires that in city 2 long-distance commuters live closer to the CBD than short-distance commuters. Furthermore, as an implication of (26), M21 þ N 2 < N=2 < N 1 . Long-distance commuting to CBD 1 is only optimal if already more than half the population live in city 1. Moreover, a very large difference between input endowments, L1 L2 also makes this type of optimum more likely. Finally, as already stated, N 2 > 0.
29
w1 ½1 ð1 ss ÞtN 1 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ;
ð33Þ ð34Þ
where we have used M21 ¼ N N 1 N 2 . The first condition ensures that at the kink of the land-rent schedule in city 2, residents are indifferent between long-distance commuting and working in the city of residence. The second condition equalizes utility of the outermost residents in both cities. Note that (32) implies that the ratio of the external marginal product to the internal marginal product has to be equalized across regions. If (32) were fulfilled, the first-order conditions would obviously coincide with the equilibrium conditions. Similar results could be obtained for cases 1–3, too. Our next proposition summarizes these findings: Proposition 3. If the production function is such that (32) holds, an efficient allocation can be sustained as a market equilibrium without commuting subsidies. A natural question is whether or not feasible production functions exist which always satisfy (32). Consider for example the a , Cobb-Douglas ‘‘internal” production function FðHi ; Li Þ ¼ Hai L1 i with 0 < a < 1. If EðHi Þ ¼ Hbi , an asymmetric equilibrium without subsidy would satisfy the necessary conditions of spatial efficiency. However, this implies Eð0Þ ¼ 0. Indeed, in order to fulfill (32) automatically, E0 ðHÞFðH; LÞ has to be strictly proportional to F H ðH; LÞEðHÞ. A necessary (but not sufficient) condition for this is Eð0Þ ¼ 0. For instance, while EðHÞ ¼ Hb fulfills the condition, logð1 þ HÞ does not. However, in general a migration equilibrium would not be spatially efficient with zero commuting subsidies. Due to agglomeration externalities workers do not accrue the full marginal benefit
13
The last two conditions are obviously equivalent to ~r 6 0 and ~r P N N 1 .
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
from moving residence and/or workplace to the core: wages are not equal to the marginal social product of labor. Only if (32) is fulfilled, explicit internalization of agglomeration benefits is not necessary. The reason is simply that wages in both cities deviate by the same fraction from the social marginal product of labor and that commuting costs are proportional to wages. But if (32) does not automatically hold, an explicit internalization mechanism is required. Commuting subsidies fulfill this task. In the remainder of this section, we derive necessary conditions for first-best optima with subsidies, where we assume that (32) does not hold. Case 1. N 2 ¼ M21 ¼ 0. Nobody will live in city 2 and, thus, will neither work in city 2 nor commute from city 2 to city 1 only if for N 2 ¼ M21 ¼ 0:
w2 6 w1 ½1 ð1 ss ÞtN;
ð35Þ
ð1 ss ÞtN 6 ð1 sl Þk:
ð36Þ
Condition (35) says that living in city 2 and not incurring commuting costs is inferior to living in city 1 with the rest of the population, and condition (36) says that living and working in city 1 is more attractive than living in city 2 and commute to city 1, even if one were the only resident of city 2. Although the social marginal product in city 1 exceeds the social marginal product in city 2, it is not ensured that the wage in the core is also sufficiently higher than in the periphery. Hence, using the definition of the social marginal product and the first-order conditions (20) and (21), we derive the following proposition.
Proof. Since full agglomeration of employment is only optimal if N 1 ¼ N=2 þ k=ð2tÞ; ss ¼ sl ¼ s follows immediately from (40). Using (40), (39) can be written as w2 ½1 ð1 s ÞtðN N 1 Þ 6 w1 ½1 ð1 s ÞtN 1 . Combining this with the definition of the social marginal product of labor, yields the lower bound on s . h A uniform subsidy rate ensures that the residence choice is not distorted. However, if the subsidy rate were lower than permitted by (41), some citizens would choose to work in the periphery. Case 3. N 2 > 0; M 21 ¼ 0. Dispersed production without any longdistance commuting is only a migration equilibrium if
w1 ½1 ð1 ss ÞtN1 ¼ w2 ½1 ð1 ss ÞtðN N 1 Þ;
ð42Þ
w1 ½1 ð1 sl Þk 6 w2 ;
ð43Þ
w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 Þ 6 w2 ½1 ð1 ss ÞtðN N1 Þ:
Condition (42) says that utility for the outermost residents of the two cities is equalized, while the second and the third condition exclude long-distance commuting. The next proposition assumes that at the optimum w1 > w2 is binding. A similar condition could be obtained, were w2 > w1 at the optimum. Proposition 6. Dispersion of production without long-distance comrequires for muting, i.e., N 2 > 0; M 21 ¼ 0, GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ > GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ss ¼ 1
GH ðH1 ;L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ; 0 0 t GH ðH1 ; L1 Þ E ðH1 ÞFðH1 ; L1 Þ N1 GH ðH2 ;L2 Þ þ E ðH2 ÞFðH2 ;L2 Þ ðN N 1 Þ
Proposition 4. Optimal full agglomeration of both residences and employment, i.e., N 2 ¼ M21 ¼ 0, require subsidies
ss
1 P1 tN
ð44Þ
sl
GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ GH ðH1 ;L1 Þ E0 ðH1 ÞFðH1 ;L1 Þ 61þ : 0 k GH ðH1 ;L1 Þ E ðH1 ÞFðH1 ; L1 Þ
ð45Þ ð46Þ
0
þ
GH ð0; L2 Þ E ð0ÞFð0;L2 Þ ; tN GH ðNð1 tN=2Þ; L1 Þ E0 ðNð1 tN=2ÞÞFðNð1 tN=2Þ; L1 Þ ð37Þ
sl 6 1
tNð1 ss Þ k
:
ð38Þ
A sufficiently high subsidy for short-distance commuting guarantees that even the resident at the urban fringe of city 1 would not benefit from living and working in city 2. If the subsidy for long-distance commuting were higher than given by (38), the outermost resident of city 1 would prefer to live in city 2 and to commute to city 1. Case 2. N 2 ¼ 0; M 21 > 0. All residents of city 2 work in city 1 only if
w2 ½1 ð1 ss ÞtðN N1 Þ 6 w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 Þ; ð39Þ ð1 ss ÞtN1 ¼ ð1 sl Þk þ ð1 ss ÞtðN N1 Þ:
ð40Þ
Again (39) states that the utility of the outermost city 2 resident if she works in city 2 is less than her utility from commuting to city 1. (40) states that the utility of the outermost city 1 resident equals that of the outermost city 2 residents (who commutes to city 1).
Proof. Taking into account the relationship between the wage and the social marginal product of labor, ss can be calculated from (25) and (42). (43) implies the upper bound on sl . h The subsidy for short-distance commuting serves as an instrument to achieve the optimal allocation of residences. By a sufficiently low subsidy sl long-distance commuting will be excluded. Case 4. N 2 > 0; M 21 > 0. An equilibrium with long-distance commuting where N 1 ; N 2 ; M21 > 0 is characterized by (33) and (34).14 In city 2, long-distance commuters only live closer to the CBD than short-distance commuters – as required by efficiency – if w1 ½1 ð1 sl Þk > w2 and w1 > w2 . The first condition implies that at CBD 2 long-distance commuters are willing to pay more for land than CBD 2 workers, the second condition ensures that the bid-rent curve of the former group is steeper than that of the latter group. Again, commuting subsidies support the first best as stated by the following proposition: Proposition 7. A combination of long-distance commuting subsidies and short-distance commuting subsidies can support the spatially efficient allocation, where in general sl –ss :
The following proposition characterizes the optimal set of subsidies: Proposition 5. Optimal full agglomeration of employment, i.e., N 2 ¼ 0; M 21 > 0, requires ss
¼ sl
¼s P1
GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ð0;L2 Þ E0 ð0ÞFð0;L2 Þ : t GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ N1 GH ð0;L2 Þ E0 ð0ÞFð0;L2 Þ ðN N 1 Þ
ð41Þ
14 Using the aggregate land rent formula (10), consumption at the urban fringe (12) and (29) yields
R2 ¼
h io ð1 ss Þt n w1 M 221 þ w2 ðM 21 þ N 2 Þ2 M 221 ; 2
which is equal to net aggregate commuting costs within city 2 (see Arnott and Stiglitz, 1981).
31
R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
c1 _ c2
c21 _ c2
1.5
5
1.0
10
15
M21
_ 0.5
0.5
_ 1.0
5
10
15
N1 _ 1.5
_ 0.5
Fig. 1. Stability of residence choice and commuting.
ss
¼1 GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ; 0 0 t GH ðH1 ; L1 Þ E ðH1 ÞFðH1 ; L1 Þ N1 GH ðH2 ; L2 Þ þ E ðH2 ÞFðH2 ; L2 Þ ðN N1 Þ
ð47Þ sl
¼ ss
½k þ tðN 2N1 Þð1 ss Þss þ : k½1 ð1 ss ÞtðN N1 Þ
ð48Þ
Proof. Taking into account the relationship between the wage and the social marginal product of labor, ss can be calculated from (27) and (34). Using (26), (27), and (33), sl can be obtained. h While short-distance commuting subsidies have a direct impact on residence choices [see (34)], long-distance commuting subsidies are targeted at the commuting decision. Short-distance commuting subsidies reduce the costs of living in the larger city, i.e., the core; long-distance commuting subsidies enhance the benefits of commuting from the periphery to the core. If short-distance commuting were not subsidized, i.e., if ss ¼ 0, long-distance commuting subsidies should be zero, too. Thus, either all commuters or no one is to be subsidized.15 Furthermore, the previous proposition has the following corollary: Corollary 1. If 0 < ss < 1; sl > ss . Proof. This follows immediately from N 1 < N=2 þ k=ð2tÞ. Hence, for ss < 1, both the numerator and denominator of the second term in (48) are positive. h Short-distance commuting subsidies lead workers to live in city 1, but also encourage inhabitants of city 2 to abstain from long-distance commuting. Thus, even stronger incentives to commute from the low-wage city to the high-wage city are useful: the long-distance subsidy rate should exceed the short-distance subsidy rate. Numerical example: To illustrate our findings, we now present a numerical example. We assume a Cobb-Douglas ‘‘internal” production function a FðHi ; Li Þ ¼ Hai L1 ; i
with 0 < a < 1, and an externality function
EðHi Þ ¼ ðH þ cÞb ; where c P 0 and 0 < b and focus on interior social optima. For the distribution of inputs, we assume N ¼ 20; L1 ¼ 20, and L2 ¼ 10. Furthermore, income shares, agglomeration externalities, and commuting costs are a ¼ 0:5; b ¼ 0:1; t ¼ 0:03, and k ¼ 0:1. Since our model has only one mobile input, a moderate income share of this 15
Note that both cities would voluntarily subsidize commuting and – if necessary – transfer resources to the other city if they maximize welfare of mobile citizens, since population is homogeneous and mobility is perfect. It follows immediately from (Myers, 1990) that the spatially efficient allocation of resources could be decentralized.
input seems to be reasonable.16 We have chosen a low b, since from the empirical literature on agglomeration externalities it is well known that elasticities indicating agglomeration externalities are rather small (see Rosenthal and Strange, 2004; Graham, 2007; Rappaport, 2008). Commuting costs consists of time costs, monetary costs, risk of accidents, and psychological costs. A large number of transport economics studies provide values and show that marginal costs of commuting are quite substantial (see Zax, 1991; Small et al., 2005; Van Ommeren and Fosgerau, 2009). Our cost parameters imply (gross) commuting costs ranging from 0% to roughly 35% of total wage income which seems to be in line with recent empirical findings. If c ¼ 0, the optimum is N1 ¼ 11:2941; N2 ¼ 7:0894, and M21 ¼ 1:61647 (case 4). Subsidies are superfluous, since c ¼ 0 implies that, for a fixed stock of L; E0 ðHÞFðH; LÞ is strictly proportional to F H ðH; LÞEðHÞ. With c ¼ 0:5, positive subsidies are necessary to achieve an efficient allocation of labor in a free migration equilibrium. The optimum is N1 ¼ 11:295; N 2 ¼ 7:06847, and M21 ¼ 1:63653. The core’s workers earn more than the periphery’s workers: w1 ¼ 0:866668 > 0:779856 ¼ w2 . The optimum subsidy rates are sl ¼ 0:0457541 > 0:0355369 ¼ ss : Long-distance subsidies exceed short-distance subsidies. To compensate for large commuting distances within the city, residents in the larger city also receive a positive (short-distance) subsidy. Local asymptotic stability of the equilibrium is also an issue. Calculating the trace and the determinant of the system defined by (30) and (31), local stability can be verified. For this numerical example, it can be easily shown that the difference in consumption of resident workers in both cities, c1 c2 , declines as N1 increases. The same holds true for the difference in consumption of commuters and local workers in city 2, namelyc21 c2 , as M21 increases (see Fig. 1).
5. Subsidies and locally captured rents Since we have assumed that rents are equally distributed across the entire metropolitan area, commuting subsidies do not redistribute rents, but only correct the agglomeration externality with respect to long-distance commuting. This assumes that the regional government can collect land rents (e.g., by taxation) and immobile factor incomes and redistribute them equally to all residents of either city in the region. In this section, we depart from this assumption. Following Abdel-Rahman and Anas (2004) and others, we now assume that rents are equally distributed within each city. Rents per capita in city i are Ii ¼ ðRi þ Q i Þ=N i . Thus, the major difference to the case of dissipated rents is that now rents play a role for the residence decision, since individuals earn different shares of rents depending on their residence. The equilibrium condition (34) becomes
w1 ½1 ð1 ss ÞtN 1 þ I1 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ þ I2 :
16
See, e.g., Rappaport (2008) for income shares in a three-factor model.
ð49Þ
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
By comparison of the first-order conditions of the social optimum and the market equilibrium conditions, i.e., on the one hand, (26) and (33), and, on the other hand, (27) and (49), it becomes clear that an interior migration equilibrium with N 1 ; N 2 ; M 21 > 0 would in general not be spatially efficient with zero commuting subsidies. Not only do wages differ from the marginal social product of labor, but also commuters from the periphery to the core do not participate in economic rents earned in the core. Again, it can be shown that subsidies may lead to first-best efficiency. We can solve (26), (27), (33), and (49) for the first-best subsidies ss and sl .17
we did – to focus on efficiency. Subsidy competition would bring about a further source of inefficiency if mobility were imperfect and the population heterogeneous, which is however, related to the organization of jurisdictions and not to the existence of correct instruments.
Proposition 8. If spatial efficiency requires N 1 ; N 2 ; M 21 > 0 and EðH1 ÞF H ðH1 ; L1 Þ > EðH2 ÞF H ðH2 ; L2 Þ, the efficient allocation can be supported by a combination of short-distance commuting subsidies and long-distance commuting subsidies, where in general, sl –ss .
References
6. Concluding remarks This paper has analyzed subsidies for intracity and intercity commuting when agglomeration externalities are present. In order to analyze commuting within and between cities, the monocentric city framework was extended to a two-city model where workers choose where to live and where to work. In contrast to Abdel-Rahman and Anas (2004) and many others, we have explicitly allowed for partial agglomeration of employment instead of full agglomeration. We have shown that commuting subsidies can help internalize externalities: intracity commuting subsidies give incentives to move to the larger city and intercity commuting subsidies make residents of the periphery commute to the core. Commuting subsidies would act as a welfare enhancing transfer from the core to the periphery, were rents locally captured. Note also that our analysis gives a rationale for differing treatment of short- and long-distance commuting, a policy that can be observed in some countries. One reason for this treatment may be that this way, government can redistribute rents from those working in the core to those working in the periphery. There are alternative policy instruments to achieve efficiency. Since workers’ wages do not compensate them for the agglomeration externalities they generate, an obvious corrective policy would be to subsidize wages. Indeed, location specific wage subsidies are an alternative instrument which may be used to achieve the firstbest outcome. We have not elaborated on this point in this paper, but note that wage subsidies would obviously have to be differentiated by workplace, which in many countries may be deemed unconstitutional. Another possibility would be to differentiate short-distance subsidies between cities. However, as we have shown, this is not necessary to achieve efficiency. There are several directions for further research. First, we have assumed that subsidies reduce monetary commuting costs but do not directly affect the time costs of commuting. It seems to be worthwhile to analyze the provision of traffic infrastructure which would directly reduce travel time and thus increase labor supply. Balancing speed improvement and infrastructure costs, the government would have a further incentive to subsidize commuting. Second, we have not considered locally financed commuting subsidies and have, therefore, excluded subsidy competition from our analysis. However, this choice seems natural if one wants – as
17
We omit the equations for sl and ss , since they are somewhat cumbersome.
Acknowledgments We thank the editor, William Strange, and two referees for helpful comments and suggestions.
Abdel-Rahman, H.M., Anas, A., 2004. Theories of systems of cities. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2293–2339. American Public Transportation Association, 2006. Public Transportation Factbook, APTA, Washington, DC. Arnott, R.J., 2007. Congestion tolling with agglomeration externalities. Journal of Urban Economics 62, 187–203. Arnott, R.J., Stiglitz, J.E., 1981. Aggregate land rents and aggregate transport costs. Economic Journal 91, 331–347. Bach, S., 2003. Entfernungspauschale: Kürzung gerechtfertigt. DIW Wochenbericht 70, 602–608. Borck, R., 2008. The political economy of urban transit. In: OECD (Ed.), ITF Round Table 141 Privatisation and Regulation of Urban Transit, OECD, Paris. Borck, R., Pflüger, M., Wrede, M., 2007. A Simple Theory of Industry Location and Residence Choice, IZA Discussion Paper 2862. Borck, R., Wrede, M., 2005. Political economy of commuting subsidies. Journal of Urban Economics 57, 478–499. Borck, R., Wrede, M., 2008. Commuting subsidies with two transport modes. Journal of Urban Economics 63, 841–848. Brueckner, J.K., 2005. Transport subsidies, system choice, and urban sprawl. Regional Science and Urban Economics 35, 715–733. Duranton, G., Puga, D., 2004. Micro-foundations of urban agglomeration economies. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2063–2117. Graham, D.J., 2007. Agglomeration Economies and Transport Investment, Working Paper, London. Henderson, J.V., 1974. The sizes and types of cities. American Economic Review 64, 640–656. Martin, R.W., 2001. Spatial mismatch and costly suburban commutes: can commuting subsidies help? Urban Studies 38, 1305–1318. Michel, P., Perrot, A., Thisse, J.F., 1996. Interregional equilibrium with heterogeneous labor. Journal of Population Economics 9, 95–113. Myers, G.M., 1990. Optimality, free mobility, and the regional authority in a federation. Journal of Public Economics 43, 107–121. Pisarski, A., 2006. Commuting in America III. The Third National Report on Commuting Patterns and Trends, Transportation Research Board, Washington, DC. Rappaport, J., 2008. A productivity model of city crowdedness. Journal of Urban Economics 63, 715–722. Rosenthal, S.S., Strange, W.C., 2004. Evidence on the nature and sources of agglomeration economies. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2119–2167. Small, K.A., Winston, C., Yan, J., 2005. Uncovering the distribution of motorists’ preferences for travel time and reliability. Econometrica 73, 1367–1382. Statistisches Bundesamt, 2005. Leben und Arbeiten in Deutschland: Ergebnisse des Mikrozensus 2004, Statistisches Bundesamt, Wiesbaden. Van Ommeren, J., Fosgerau, M., 2009. Workers’ marginal costs of commuting. Journal of Urban Economics 65, 38–47. Verhoef, E.T., Nijkamp, P., 2003. Externalities in the Urban Economy, Tinbergen Institute Discussion Paper 2003-078/3. Wrede, M., 2000. Tax deductibility of commuting expenses and leisure: On the tax treatment of time-saving expenditure. FinanzArchiv 57, 216–224. Wrede, M., 2001. Should commuting expenses be tax deductible? A welfare analysis. Journal of Urban Economics 49, 80–99. Zax, J.S., 1991. Compensation for commutes in labor and housing markets. Journal of Urban Economics 30, 192–207. Zenou, Y., 2000. Urban unemployment, agglomeration and transportation policies. Journal of Public Economics 77, 97–133.
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Journal of Urban Economics www.elsevier.com/locate/jue
Subsidies for intracity and intercity commuting Rainald Borck a, Matthias Wrede b,* a b
University of Passau, Faculty of Economics, Innstrasse 27, 94034 Passau, Germany University of Marburg, Faculty of Business Administration and Economics, Am Plan 2, 35032 Marburg, Germany
a r t i c l e
i n f o
Article history: Received 21 May 2008 Revised 9 February 2009 Available online 31 March 2009
a b s t r a c t This paper analyzes subsidies for intracity and intercity commuting in an urban economics framework with two cities and agglomeration externalities, where workers may commute within and between cities. First, commuting subsidies serve to internalize agglomeration externalities: intracity commuting subsidies give incentives to move to the larger city and intercity commuting subsidies make residents of the periphery commute to the core. Second, if agglomeration rents are locally captured, commuting subsidies act as a welfare enhancing transfer from the core to the periphery. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Today, in most countries, the vast majority of workers commute to work, many over long distances. The proportion of long-distance commuters seems to have risen in many countries. For instance, in Germany in 2004, 47% of workers commuted more than 10 km to work, and 17% commuted more than 25 km, 5% even more than 50 km (Statistisches Bundesamt, 2005). According to Pisarski (2006), the average commute in the US was 12.2 miles in 2001, an increase of almost 14% since 1990. Moreover, about 9% of all workers worked outside their home metro area. Hence, commuting long distance seems to be an important phenomenon. It is also worthy to note that many countries subsidize commuting, and in some we find subsidy rates vary with distance. According to the American Public Transportation Association (2006), total fares in several thousand public transit authorities in the US accounted for only 33% of operating costs and 23% of total operating and capital costs, while for Europe the corresponding figures show fares cover about half of operating costs. Urban transit is obviously heavily subsidized through general tax revenue. Interestingly, automobile travel too, often does not cover the full cost of road construction and usage (let alone environmental and accident costs). Indeed, in the US, user fees (including gasoline taxes, license fees and other charges) accounted for only 60% of total highway expenditures (Brueckner, 2005). In Europe, higher petrol taxes suggest that subsidies to automobile travel should be significantly lower.1 In the US, there are no direct tax benefits for commuters, but commuting is subsidized through transit and highway construc-
* Corresponding author. Fax: +49 6421 2824852. E-mail addresses: [email protected] (R. uni-marburg.de (M. Wrede). 1 See Borck (2008) for more discussion and references.
Borck),
tion subsidies, as well as various fringe benefits such as cash benefits for parking or transit. In many European countries, by contrast, there are distinct provisions on the tax deductibility of commuting costs (see Bach, 2003). We briefly summarize these provisions with an eye on how deductibility varies with distance.2 Some countries including the US, Canada, the UK, Australia, Spain and Portugal do not allow deductibility of commuting costs. In some other countries, for instance, Finland and Norway, commuting costs are deductible regardless of distance, but only up to a certain maximum amount per year. In many other countries, however, deductibility varies with distance: In Germany, the tax code was changed in 2007 by eliminating deductibility of commuting distances of less than 20 km. For larger distances, deductibility remained. However, in 2008 the German Constitutional Court decided that this differentiated treatment was unconstitutional. In Sweden, transit costs are deductible, but commuting by automobile is deductible only above 5 km distance. Austria has a lump sum up to 20 km with higher deductibility for distances beyond 20 km. Denmark has deductibility only from 12 km with a lower rate for distances above 100 km. Some countries, such as Belgium and France limit deductibility to a maximum of 40 or 50 km. In general, the widespread subsidization of commuting provokes the question of why these subsidies are paid, and why subsidy rates should vary with distance. Possible explanations rest on politics: for instance, Borck and Wrede (2005, 2008) show how subsidies redistribute between rich and poor city dwellers and between renters and landowners. Another possibility would be that cheap commuting increases the demand for residential land and housing and requires extended road networks which benefits the construction industry. Firms in this industry would then have an incentive to lobby for commuting subsidies.3
wrede@wiwi.
0094-1190/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jue.2009.02.003
2 3
The provisions refer to the state of affairs as of 2003. This explanation was suggested to us by William Strange.
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
In this paper, we analyze whether a welfare maximizing government would have incentives to subsidize commuting, and whether optimal subsidies vary with commuting distance. This is not to say that we believe that differentiated commuting subsidies exist because they are efficient. This obviously depends on the objectives of governments and whether these coincide with social welfare (which typically they do not). But in order to understand the political economy of some government intervention, it is useful to look at the outcome of welfare maximization as a benchmark. On the theoretical side, our paper also contributes to the literature by analyzing long-distance commuting between cities as well as short-distance commuting within cities.4 Commuting long distance may occur for a variety of reasons: (1) households with high housing demand may choose to locate far from employment centers to economize on housing costs, (2) households may want to move away from city centers to benefit from various rural amenities such as greenspace or from the absence of disamenities such as crime and pollution, (3) people may move to suburbs with better schools or other local public goods, and (4) dual-career couples may be forced to separate workplace and residence for at least one worker which may result in long commutes. To fix ideas, we mostly assume away these motivations and instead focus on households’ decision to commute to one of two employment centers: the one of the city they live in or the one of the other city. There are then two decisions households have to make: choice of residence and choice of workplace. For given residences, the choice of whether or not to commute long distance depends on the difference in wages and the cost of commuting long distance. Thus, workers will commute to the larger city if wages there are sufficiently higher due to agglomeration effects. The choice of residence is governed by within-city commuting and housing costs. Hence, households may choose to live in small cities in order to economize on these urban costs. Thus, our notion of long-distance commuting is not related to distance per se, but rather to the question of whether commuters commute to the employment center of the city they live in or to another, geographically distinct city. In our model, however, this will imply long distances since commuters to the other city first have to travel to their own city center. In reality, the distinction between what we take to be long- and short-distance commuting is somewhat blurred. This is necessarily so, given the abstractions of our model. Consider, for instance, workers in New York City. Whether commuters from, say, Newark, NJ or Philadelphia, or both are considered to commute long distance depends on their motives. We would term long-distance commuter someone who lives in Newark and works in New York if the same job were available in Newark, albeit at a lower wage.5 By contrast, if someone commutes from Newark to NYC because housing is cheaper in Newark, but the job they have can only be found in NYC (say, a typical Wall Street job), we would say this person is a short-distance commuter. The blurriness comes from the fact that job opportunities for people of different professions are not equally affected by agglomeration externalities, and the motives for commuting mentioned above are not the same for all commuters. Our model combines an Alonso type urban model with Marshallian agglomeration externalities à la Henderson (1974). These externalities might be due to benefits from input sharing, labor market pooling, and knowledge spillovers (see Duranton and Puga, 2004, for the microfoundations of agglomeration economies). Many studies have shown that localization economies and urbanization economies are present in cities (see, for a comprehensive overview, Rosenthal and Strange, 2004). Agglomeration effects generally imply that social and private returns to working in a city 4
Borck et al. (2007) also study long-distance commuting but do not model the spatial extension of cities and hence ignore commuting within cities. 5 An example may be the insurance industry.
do not coincide. In order to achieve first-best efficiency, workers should be confronted with the social effects of their commuting decision.6 When differentiated wage subsidies are not feasible, we show that differentiated subsidies to long- and short-distance commuting may achieve first-best efficiency. That is, our analysis can be viewed as a possible normative justification of why commuting should be subsidized and why long-distance commuting between cities and short-distance commuting within cities should be treated differently. The basic reason for subsidizing commuting is that workers do not receive their full marginal social product. Therefore, commuting subsidies help to ensure that workers commute to that employment center which yields the highest social output. Beyond that, however, (short-distance) commuting subsidies also serve to make workers choose the right place of residence. Hence, our paper extends the literature on the allocative effects of commuting subsidies by combining agglomeration externalities with land use. Within the standard urban model commutings subsidies are inefficient because they distort land use (see, e.g., Brueckner, 2005). Standard arguments for commuting subsidies rest on externalities, market imperfections, and pre-existing distortions (see Martin, 2001; Wrede, 2000, 2001, and Zenou, 2000). Most closely related to our paper are Verhoef and Nijkamp (2003) who discuss the trade-off between the internalization of pollution and agglomeration externalities and Arnott (2007), who studies the trade-off between congestion and agglomeration externalities. None of these papers explicitly combines commuting and residence decision, however, as we do. The paper proceeds as follows. The next section introduces the model. In Section 3, we study the social planner problem of how to allocate the population between two cities, where the choice variables relate to the resident and working populations. In Section 4, we focus on the market equilibrium, which is achieved by individuals choosing their place of residence and workplace. Section cas discusses the impact of subsidies when rents are only locally distributed, and the last section concludes. 2. The model We consider an economy with two cities, labeled 1 and 2. There is a total of N mobile workers. Workers simultaneously choose their place of residence and their place of work. That is, by commuting from one city to the other, workers may separate job and residence. Workers confer Marshallian urbanization externalities upon each other, that is, a worker is more productive the larger the number of other workers in the city is, perhaps because of knowledge spillovers. This is captured in a simplistic way by assuming that the wage rate earned by a worker working in city i; wi , is a function of the local labor force, Hi .7 Suppose that the production function of firm j in city i is yji ¼ EðHi ÞFðhji ; lji Þ, where hji is labor employed by firm j; Hi is aggregate labor in the city, and lji is an immobile input employed by firm j in city i. The immobile input could be either buildings or publicly provided infrastructure. Without loss of generality, we assume that city 1 is endowed with a stock of the immobile input at least as large as that in city 2: L1 P L2 , where Li is the aggregate amount of the immobile input in city i. We call FðÞ the ‘internal production function’ and we assume it is a linearly homogeneous function with Fð0; Þ ¼ 0, and positive but decreasing marginal products for both factors. The function EðHi Þ, with Eð0Þ P 0 and E0 ðHi Þ > 0, captures the positive agglomeration externality. When FðÞ fulfills the Inada condition, limHi !0 F H ðHi ; Þ ¼ 1, the ‘‘internal” marginal product of labor goes to infinity if the number of workers 6 We abstract from other efficiency reasons for intervening in commuting decisions such as congestion. 7 This approach follows among others Henderson (1974), Michel et al. (1996), and Abdel-Rahman and Anas (2004).
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
goes to zero. Together with EðÞ > 0, the Inada condition precludes full agglomeration of labor (see Michel et al., 1996). Firms are assumed to maximize profits under perfect competition. Each firm takes aggregate labor in the city as given. Thus, even though there are increasing returns at the city level, each firm views itself as facing constant returns. Hence, workers are paid their marginal product, and the wage of a worker working in city i is
wi ¼ EðHi ÞF H ðHi ; Li Þ;
ð1Þ
where profits are zero. Each city is linear and extends from zero to r i ; i ¼ 1; 2, where the boundary ri is endogenously determined by individuals’ residence choices. All citizens live on plots of land of fixed size normalized to unity. Land rent at distance r will be denoted by pðrÞ. All workers working in a city commute to the central business district (CBD) to work; all those working in the other city first commute to the CBD and then from there to the other city. A natural way to think of this assumption is workers commuting by train, where the train stations are located in the city centers.8 Hence, regardless of where a worker works, if she lives at r km from the CBD, she incurs commuting costs of twr where w is the wage and t > 0 the inverse of travel speed (commuting costs are in terms of time). A worker who commutes long distance additionally incurs a cost of kw, with k > 0, to commute from CBD to CBD.9 An individual living at r km from the CBD where she works supplies 1 tr units of labor. If she lives r km from CBD i and commutes to CBD j, she supplies 1 k tr units of labor. We will assume that 1 tN k > 0. This implies that even if everyone were to live in, say, city 2, then if the outermost residents commute to city 1 they would still supply a positive amount of labor there (i.e., commuting does not take the entire workday). The number of inhabitants of city i working in the same city is denoted by N i , and the number of commuters from 2 to 1 by M 21 . Without loss of generality, the focus is on long-distance commuting from CBD 2 to CBD 1, i.e., M 21 P 0. Note that transport costs, together with homogeneity of workers, exclude two-way commuting. The regional government may subsidize commuting at rate ss within cities and at rate sl across cities. While subsidies differ between short-distance commuting and long-distance commuting, they are uniform across cities.10 Short-distance commuters in city i receive a subsidy of ss wi tr, long-distance commuters from 2 to 1 receive ss w1 tr þ sl w1 k. These subsidies may come in the form of subsidized public transport, tolls or other transport fees below total costs, or income tax deductions for commuting expenses. Subsidies are financed by a lump-sum tax T on all citizens. In order to obtain reasonable rent schedules, we assume throughout the analysis that shortdistance subsidies do not exceed costs. Hence, we assume ss 6 1. We assume that economic rents, namely land rents and the income of the immobile factor, are equally distributed among all citizens living in the entire metropolitan area. Aggregate land rent in city i is denoted by Ri , and immobile input income in city i is denoted by Q i ¼ EðHi ÞFðHi ; Li Þ EðHi ÞF H ðHi ; Li ÞHi . Per capita rents in city 1 and city 2 are 8
This assumption is a relatively accurate description for commuting by train, but when workers commute by car, things may get more complicated, since intercity roads generally do not connect the CBDs. It may then be that some workers living away from the CBD have shorter commutes to the other city than someone living at the same distance from their own city center but on the ‘far side’. Automobile travel may also raise issues of congestion. Including these effects would, however, greatly complicate the analysis. 9 Note that with some change in notation, our main results could also be obtained by assuming that part of the commuting cost is a monetary cost, i.e., independent of the wage. 10 Introducing differentiated subsidy rates would not change qualitatively our main results. Furthermore, we note that in many countries, while city governments have some discretion over transport policies, many instruments such as income tax policies or large infrastructure policies tend to be under the control of regional or even national governments. Hence, our assumption seems to be a reasonable approximation of reality.
I1 ¼ I2 ¼
R1 þ R2 þ Q 1 þ Q 2 : N
ð2Þ
Since our assumption on the distribution of rents affects the results regarding the necessity of subsidies, in Section 5 we also consider economic rents which are distributed only locally. Finally, we assume that utility, v, equals consumption, c.11 Suppose first that nobody commutes from the city under consideration to the other city. Then utility for a worker living in city i is given by
v i ¼ ci ¼ wi ½1 ð1 ss Þtr pi ðrÞ þ Ii T;
ð3Þ
which is simply the wage net of commuting costs, minus housing rents, plus economic rents minus lump-sum tax. Solving (3) for land rent as a function of commuting distance defines an individual’s bid rent function:
pi ðrÞ ¼ wi ½1 ð1 ss Þtr þ Ii T ci :
ð4Þ
As usual, we assume that competition for land implies that the highest bidder lives on any plot. Thus, which of two individuals lives closer to the CBD will depend on the slope of the bid rent function
@pi ðrÞ ¼ ð1 ss Þwi t: @r In particular, since commuting is time consuming, individuals with higher wages live closer to the CBD.12 At the city border, land rent must equal the agricultural land rent, which is normalized to zero. Since ri ¼ N i , we can solve pi ðN i Þ ¼ 0, from (4) to find an expression for utility:
ci ¼ wi ½1 ð1 ss ÞtNi þ Ii T: Using the definition of aggregate land rent, Ri ¼ sumption can be alternatively written as
ð5Þ R Ni 0
Ni Ri þ Ii T: ci ¼ wi 1 ð1 ss Þt 2 Ni
pi ðrÞ dr, con-
ð6Þ
Using the same procedure, we find the utility of residents in city 2 when all commute to city 1 by substituting M 21 for N i and taking account of long-distance commuting costs:
M21 R2 þ I2 c21 ¼ w1 1 ð1 sl Þk ð1 ss Þt T: 2 M 21
ð7Þ
Residents of city 2 will only commute to CBD 1 if their wage net of commuting costs is higher in city 1 than in city 2, i.e., if
w1 ½1 ð1 sl Þk ð1 ss Þtr > w2 ½1 ð1 ss Þtr:
ð8Þ
Provided that w1 > w2 it can easily be shown that there is a critical distance ~r such that all residents living at distance r > ~r work in CBD 2 and all residents living at r 6 ~r commute to CBD 1, where
~r ¼
½1 ð1 sl Þkw1 w2 : ð1 ss Þtðw1 w2 Þ
11 Hence, we also take leisure as fixed. Including leisure would clearly reduce the tractability of the model. Furthermore, if leisure time were variable, commuters would consume more leisure and work more than in our model. On the one hand, less commuting would be necessary to equalize marginal products of labor. On the other hand, commuting would be less harmful. However, we expect that the main results of our model would not be affected by the inclusion of leisure. 12 Of course, this assumption depends on equal housing consumption for all individuals. Otherwise, the slope of the bid rent function is given by the ratio of marginal commuting costs to housing consumption.
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
Note that if some individuals commute from city 2 to city 1, the commuters will live closer to the CBD of city 2 if they have higher bid rents for locations close to the CBD. Hence, if the wage in city 1 is higher than the wage in city 2, long-distance commuters will tend to have steeper bid rents and will, therefore, prefer to live close to the CBD. With long-distance commuting the land rent in city 2 is
p2 ðrÞ ¼
w1 ½1 ð1 sl Þk ð1 ss Þtr c21 þ I2 T w2 ½1 ð1 ss Þtr c2 þ I2 T
for r < ~r
: otherwise ð9Þ
R2 ¼
M21
p2 ðrÞ dr þ
Z
N 2 þM 21
p2 ðrÞ dr M 21 M 21 ¼ w1 1 ð1 sl Þk ð1 ss Þt M 21 2 ðN2 þ M 21 Þ þ w2 ðN2 þ M 21 Þ 1 ð1 ss Þt 2 M 21 c2 N2 c21 M 21 w2 M 21 1 ð1 ss Þt 2 0
þ ðI2 TÞðN2 þ M 21 Þ:
ð10Þ
ð11Þ
ð12Þ
Using the pattern of locations, we can give a general formula for the government budget constraint:
Z N1 Z M21 þN2 Z M21 1 T¼ r dr þ w2 r dr þ w1 r dr ss t w1 N 0 M 21 0 Z M21 þ sl w1 k dr 0 ( " # 1 N21 ðN 2 þ M21 Þ2 M221 ¼ ss tw1 þ ss tw2 N 2 2 2 ) 2 M þ ss tw1 21 þ sl w1 kM 21 : 2
Z
Z M21 ð1 trÞ dr þ ð1 k trÞ dr 0 0 N1 M21 ¼ N1 1 t þ M21 1 k t ; 2 2 Z M21 þN2 N2 þ 2M 21 : ð1 trÞ dr ¼ N2 1 t H2 ¼ 2 M 21
ð16Þ
Substituting for H1 and H2 from (14) and (15), the Lagrangian can be written
GH ðH1 ; L1 Þð1 tN1 Þ k ¼ 0;
ð17Þ
GH ðH2 ; L2 Þ½1 tðN 2 þ M 21 Þ k þ l1 ¼ 0; GH ðH1 ; L1 Þð1 k tM21 Þ GH ðH2 ; L2 ÞtN2 k þ l2 ¼ 0:
ð18Þ ð19Þ
Due to the non-convexity of the optimization problem, solutions of the system of first-order conditions need not be global maxima. However, some characteristics of optima can be obtained by analyzing the first-order conditions. Case 1. N 2 ¼ M21 ¼ 0. This would imply full agglomeration of population ðN 1 ¼ NÞ. The complementary slackness conditions are l1 ; l2 P 0. Examining (17)–(19) shows that this implies
GH ð0; L2 Þ GH ðNð1 tN=2Þ; L1 Þð1 tNÞ 6 0;
ð20Þ
GH ðNð1 tN=2Þ; L1 ÞðtN kÞ 6 0:
ð21Þ
Since GH ðH; LÞ ¼ EðHÞF H ðH; LÞ þ E0 ðHÞFðH; LÞ, we have:
ð13Þ
Residents of city 1, workers of city 2, and workers of city 1 living in city 2 receive subsidies. Labor supply can now be written as
H1 ¼
N1 þ N 2 þ M21 ¼ N; N 2 P 0; M 21 P 0; ð14Þ and ð15Þ;
and the corresponding first-order conditions are
At the urban fringe in city 2; r 2 ¼ N N 1 , utility is
c2 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ þ I2 T:
s:t:
Y ¼ GðH1 ; L1 Þ þ GðH2 ; L2 Þ
þ l1 N2 þ l2 M 21 ;
There is then a kink of the land-rent schedule at the distance ~r ¼ M21 where a city 2 resident is just indifferent between commuting to city 1 or working in city 2. At this kink, we have
w1 ½1 ð1 sl Þk ð1 ss ÞtM 21 c21 ¼ w2 ½1 ð1 ss ÞtM 21 c2 :
max
N1 ;N2 ;M21
N1 M 21 L ¼ G N1 1 t þ M21 1 k t ; L1 2 2 N2 þ 2M 21 ; L2 þ kðN N 1 N2 M 21 Þ þ G N2 1 t 2
Hence, aggregate land rent in city 2 is given by
Z
simply maximizes total output: population is homogeneous, the lot size is fixed, leisure is not explicitly modeled, and non-time costs of commuting are neglected. Hence, denoting output in city i by GðHi ; Li Þ, a social planner who can choose residences and workplaces solvess
N1
ð14Þ ð15Þ
In city 1, each resident and each commuter from city 2 supplies one unit of time minus commuting time. In city 2, labor is only supplied by a fraction of residents. Here, we assume that subsidies increase income, but do not directly reduce travel time. 3. Optimum allocation of workers We now characterize the socially optimal allocation of workers across communities, both in terms of residences, and in terms of employment. Due to our assumptions, the welfare maximum
Proposition 1. Full agglomeration of population and employment, N 1 ¼ N, can be socially optimal if and only if k P tN and either (i) Eð0Þ > 0 and the Inada condition does not hold (limHi !0 F H ðHi ; Þ is finite) or (ii) Eð0Þ ¼ 0. Thus, less than full agglomeration of population can be optimal only if long-distance commuting is sufficiently cheap, in particular, if commuting from CBD to CBD costs less than commuting within a city would if all workers were to live there. Further, if the Inada conditions hold, full agglomeration of population cannot be optimal, since then the marginal product of shifting one worker from city 1 to city 2 must be positive. Case 2. N 2 ¼ 0; M 21 > 0. We now have (17)–(19) gives
l1 P 0; l2 ¼ 0. Combining
GH ð0; L2 Þð1 tðN N1 ÞÞ GH ðH1 ; L1 Þ½1 k tðN N1 Þ 6 0;
ð22Þ
GH ðH1 ; L1 Þðtð2N1 NÞ kÞ ¼ 0:
ð23Þ
Thus, we have the following result: Proposition 2. Full agglomeration of employment and dispersed population, i.e., N 2 ¼ 0 and M21 > 0, can be socially optimal only if k < tN and either (i) Eð0Þ > 0 and the Inada condition does not hold ðlimHi !0 F H ðHi ; Þ is finite) or (ii) Eð0Þ ¼ 0. Furthermore, the optimal allocation satisfies
N1 ¼
N k þ : 2 2t
R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
Case 3. N 2 > 0; M 21 ¼ 0. We now have (19) we find
l1 ¼ 0; l2 P 0. From (17)–
GH ðN 1 ð1 tN1 =2Þ; L1 Þð1 kÞ GH ðN2 ð1 tN2 =2Þ; L2 Þ 6 0;
ð24Þ
GH ðN 1 ð1 tN1 =2Þ; L1 Þð1 tN1 Þ GH ðN2 ð1 tN2 =2Þ; L2 Þ½1 tðN N 1 Þ ¼ 0:
ð25Þ
This shows that long-distance commuting is probably inefficient if it is very time consuming and if the differences in the stocks of immobile inputs are not too large. If L1 ¼ L2 , a symmetric allocation without long-distance commuting, i.e., N 1 ¼ N 2 ¼ N=2, possibly maximizes output. Indeed, as can readily be seen from (25), this will be the case if the production function is concave, i.e., GHH < 0. If L1 > L2 , an asymmetric allocation with N 1 > N 2 > 0 and M21 ¼ 0 may also be an optimum. Case 4. N 2 > 0; M 21 > 0. We now have l1 ; l2 > 0. Combining (17)– (19) gives
GH ðH1 ; L1 Þ½1 k tðN N1 N2 Þ ¼ GH ðH2 ; L2 Þ½1 tðN N1 N2 Þ; GH ðH1 ; L1 Þð1 tN1 Þ ¼ GH ðH2 ; L2 Þ½1 tðN N 1 Þ:
ð26Þ ð27Þ
According to the second condition (27), the social marginal product of labor should be equalized for the outermost residents of the two cities (note that the outermost city 2 resident works in city 2). The first condition (26) requires that the social marginal product of city 2 resident living at M21 who works in city 2 be equalized to the social marginal product of city 2 resident who commutes to city 1. These conditions imply
N2 ¼
½1 tðN N1 Þ½k þ tðN 2N1 Þ : ð2N1 NÞt 2
ð28Þ
4. Migration equilibria and subsidies We now study market equilibria, where individuals can freely migrate and commute, that is, they choose their place of residence and their place of work based on the utility obtained when living and working in city 1 versus living and working in city 2 or living in city 2 and working in 1. Comparing utility levels between cities, and between commuting and not commuting, we find interior equilibria with 0 < N 1 < N; 0 < M 21 < N N 1 if
c1 ðN1 ; M 21 Þ ¼ c2 ðN1 ; M 21 Þ ¼ c21 ðN1 ; M21 Þ;
where ci ðN 1 ; M21 Þ; i ¼ 1; 2 and c21 ðN 1 ; M21 Þ are determined using the definition of land rent in combination with the government budget constraint. By contrast, we get corner solutions with full agglomeration of population if c1 ðN; 0Þ P c2 ðN; 0Þ. Likewise, there would be no long-distance commuting if c21 ðN1 ; 0Þ 6 c2 ðN1 ; 0Þ while all city 2 residents would commute to city 1 if c21 ðN 1 ; N N 1 Þ P c2 ðN1 ; N N1 Þ.13 As is usual in the new economic geography literature, we assume a myopic adjustment process. That is, the change in population and in workforce is governed by the ad-hoc dynamics
N_ 1 ¼ ðc1 c2 ÞN1 N2 ; _ 21 ¼ ðc21 c2 ÞM 21 N2 ; M
ð29Þ
ð30Þ ð31Þ
if 0 < N 1 < N; 0 < M21 < N, and 0 < N 2 < N. Provided that L1 ¼ L2 , the symmetric allocation N 1 ¼ N 2 ¼ N=2 is obviously an equilibrium regardless of the size of subsidies, since utility is the same in both cities and due to equal wages no one can gain from long-distance commuting. However, this equilibrium might be unstable. An analysis of the first-order conditions of the central planner’s problem in cases 1–4 reveals that these conditions directly translate into the respective equilibrium conditions with zero subsidies ðss ¼ sl ¼ 0Þ if
E0 ðH1 ÞFðH1 ; L1 Þ þ EðH1 ÞF H ðH1 ; L1 Þ EðH1 ÞF H ðH1 ; L1 Þ ¼ ; E0 ðH2 ÞFðH2 ; L2 Þ þ EðH2 ÞF H ðH2 ; L2 Þ EðH2 ÞF H ðH2 ; L2 Þ
ð32Þ
where the left-hand side is the ratio of social marginal products of labor at the optimum, GH ðH1 ; L1 Þ=GH ðH2 ; L2 Þ, and the right-hand side is the ratio of internal marginal products of labor, i.e., the wage ratio. For instance, in case 4 where N 2 > 0 and M21 > 0, the equilibrium conditions derived from (29) read:
w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 N2 Þ ¼ w2 ½1 ð1 ss ÞtðN N1 N2 Þ;
Since we have assumed 1 > k þ tN and since the optimum with long-distance commuting requires N1 > N=2 and N 2 > 0, we conclude that N 1 < N=2 þ k=ð2tÞ holds. From closer inspection of the problem it becomes immediately clear that an asymmetric allocation with long-distance commuting, where N 1 ; N 2 , and M21 are all greater than zero, is only optimal when agglomeration externalities are sufficiently strong and long-distance commuting costs k are sufficiently low. Otherwise, a less asymmetric allocation without long-distance commuting maximizes output. Moreover, (26) directly implies GH ðH1 ; L1 Þ > GH ðH2 ; L2 Þ, i.e., workers commute to the CBD with the higher social marginal product of labor. Since working hours in CBD 1 are more valuable than in CBD 2, efficiency requires that in city 2 long-distance commuters live closer to the CBD than short-distance commuters. Furthermore, as an implication of (26), M21 þ N 2 < N=2 < N 1 . Long-distance commuting to CBD 1 is only optimal if already more than half the population live in city 1. Moreover, a very large difference between input endowments, L1 L2 also makes this type of optimum more likely. Finally, as already stated, N 2 > 0.
29
w1 ½1 ð1 ss ÞtN 1 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ;
ð33Þ ð34Þ
where we have used M21 ¼ N N 1 N 2 . The first condition ensures that at the kink of the land-rent schedule in city 2, residents are indifferent between long-distance commuting and working in the city of residence. The second condition equalizes utility of the outermost residents in both cities. Note that (32) implies that the ratio of the external marginal product to the internal marginal product has to be equalized across regions. If (32) were fulfilled, the first-order conditions would obviously coincide with the equilibrium conditions. Similar results could be obtained for cases 1–3, too. Our next proposition summarizes these findings: Proposition 3. If the production function is such that (32) holds, an efficient allocation can be sustained as a market equilibrium without commuting subsidies. A natural question is whether or not feasible production functions exist which always satisfy (32). Consider for example the a , Cobb-Douglas ‘‘internal” production function FðHi ; Li Þ ¼ Hai L1 i with 0 < a < 1. If EðHi Þ ¼ Hbi , an asymmetric equilibrium without subsidy would satisfy the necessary conditions of spatial efficiency. However, this implies Eð0Þ ¼ 0. Indeed, in order to fulfill (32) automatically, E0 ðHÞFðH; LÞ has to be strictly proportional to F H ðH; LÞEðHÞ. A necessary (but not sufficient) condition for this is Eð0Þ ¼ 0. For instance, while EðHÞ ¼ Hb fulfills the condition, logð1 þ HÞ does not. However, in general a migration equilibrium would not be spatially efficient with zero commuting subsidies. Due to agglomeration externalities workers do not accrue the full marginal benefit
13
The last two conditions are obviously equivalent to ~r 6 0 and ~r P N N 1 .
30
R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
from moving residence and/or workplace to the core: wages are not equal to the marginal social product of labor. Only if (32) is fulfilled, explicit internalization of agglomeration benefits is not necessary. The reason is simply that wages in both cities deviate by the same fraction from the social marginal product of labor and that commuting costs are proportional to wages. But if (32) does not automatically hold, an explicit internalization mechanism is required. Commuting subsidies fulfill this task. In the remainder of this section, we derive necessary conditions for first-best optima with subsidies, where we assume that (32) does not hold. Case 1. N 2 ¼ M21 ¼ 0. Nobody will live in city 2 and, thus, will neither work in city 2 nor commute from city 2 to city 1 only if for N 2 ¼ M21 ¼ 0:
w2 6 w1 ½1 ð1 ss ÞtN;
ð35Þ
ð1 ss ÞtN 6 ð1 sl Þk:
ð36Þ
Condition (35) says that living in city 2 and not incurring commuting costs is inferior to living in city 1 with the rest of the population, and condition (36) says that living and working in city 1 is more attractive than living in city 2 and commute to city 1, even if one were the only resident of city 2. Although the social marginal product in city 1 exceeds the social marginal product in city 2, it is not ensured that the wage in the core is also sufficiently higher than in the periphery. Hence, using the definition of the social marginal product and the first-order conditions (20) and (21), we derive the following proposition.
Proof. Since full agglomeration of employment is only optimal if N 1 ¼ N=2 þ k=ð2tÞ; ss ¼ sl ¼ s follows immediately from (40). Using (40), (39) can be written as w2 ½1 ð1 s ÞtðN N 1 Þ 6 w1 ½1 ð1 s ÞtN 1 . Combining this with the definition of the social marginal product of labor, yields the lower bound on s . h A uniform subsidy rate ensures that the residence choice is not distorted. However, if the subsidy rate were lower than permitted by (41), some citizens would choose to work in the periphery. Case 3. N 2 > 0; M 21 ¼ 0. Dispersed production without any longdistance commuting is only a migration equilibrium if
w1 ½1 ð1 ss ÞtN1 ¼ w2 ½1 ð1 ss ÞtðN N 1 Þ;
ð42Þ
w1 ½1 ð1 sl Þk 6 w2 ;
ð43Þ
w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 Þ 6 w2 ½1 ð1 ss ÞtðN N1 Þ:
Condition (42) says that utility for the outermost residents of the two cities is equalized, while the second and the third condition exclude long-distance commuting. The next proposition assumes that at the optimum w1 > w2 is binding. A similar condition could be obtained, were w2 > w1 at the optimum. Proposition 6. Dispersion of production without long-distance comrequires for muting, i.e., N 2 > 0; M 21 ¼ 0, GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ > GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ss ¼ 1
GH ðH1 ;L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ; 0 0 t GH ðH1 ; L1 Þ E ðH1 ÞFðH1 ; L1 Þ N1 GH ðH2 ;L2 Þ þ E ðH2 ÞFðH2 ;L2 Þ ðN N 1 Þ
Proposition 4. Optimal full agglomeration of both residences and employment, i.e., N 2 ¼ M21 ¼ 0, require subsidies
ss
1 P1 tN
ð44Þ
sl
GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ GH ðH1 ;L1 Þ E0 ðH1 ÞFðH1 ;L1 Þ 61þ : 0 k GH ðH1 ;L1 Þ E ðH1 ÞFðH1 ; L1 Þ
ð45Þ ð46Þ
0
þ
GH ð0; L2 Þ E ð0ÞFð0;L2 Þ ; tN GH ðNð1 tN=2Þ; L1 Þ E0 ðNð1 tN=2ÞÞFðNð1 tN=2Þ; L1 Þ ð37Þ
sl 6 1
tNð1 ss Þ k
:
ð38Þ
A sufficiently high subsidy for short-distance commuting guarantees that even the resident at the urban fringe of city 1 would not benefit from living and working in city 2. If the subsidy for long-distance commuting were higher than given by (38), the outermost resident of city 1 would prefer to live in city 2 and to commute to city 1. Case 2. N 2 ¼ 0; M 21 > 0. All residents of city 2 work in city 1 only if
w2 ½1 ð1 ss ÞtðN N1 Þ 6 w1 ½1 ð1 sl Þk ð1 ss ÞtðN N1 Þ; ð39Þ ð1 ss ÞtN1 ¼ ð1 sl Þk þ ð1 ss ÞtðN N1 Þ:
ð40Þ
Again (39) states that the utility of the outermost city 2 resident if she works in city 2 is less than her utility from commuting to city 1. (40) states that the utility of the outermost city 1 resident equals that of the outermost city 2 residents (who commutes to city 1).
Proof. Taking into account the relationship between the wage and the social marginal product of labor, ss can be calculated from (25) and (42). (43) implies the upper bound on sl . h The subsidy for short-distance commuting serves as an instrument to achieve the optimal allocation of residences. By a sufficiently low subsidy sl long-distance commuting will be excluded. Case 4. N 2 > 0; M 21 > 0. An equilibrium with long-distance commuting where N 1 ; N 2 ; M21 > 0 is characterized by (33) and (34).14 In city 2, long-distance commuters only live closer to the CBD than short-distance commuters – as required by efficiency – if w1 ½1 ð1 sl Þk > w2 and w1 > w2 . The first condition implies that at CBD 2 long-distance commuters are willing to pay more for land than CBD 2 workers, the second condition ensures that the bid-rent curve of the former group is steeper than that of the latter group. Again, commuting subsidies support the first best as stated by the following proposition: Proposition 7. A combination of long-distance commuting subsidies and short-distance commuting subsidies can support the spatially efficient allocation, where in general sl –ss :
The following proposition characterizes the optimal set of subsidies: Proposition 5. Optimal full agglomeration of employment, i.e., N 2 ¼ 0; M 21 > 0, requires ss
¼ sl
¼s P1
GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ð0;L2 Þ E0 ð0ÞFð0;L2 Þ : t GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ N1 GH ð0;L2 Þ E0 ð0ÞFð0;L2 Þ ðN N 1 Þ
ð41Þ
14 Using the aggregate land rent formula (10), consumption at the urban fringe (12) and (29) yields
R2 ¼
h io ð1 ss Þt n w1 M 221 þ w2 ðM 21 þ N 2 Þ2 M 221 ; 2
which is equal to net aggregate commuting costs within city 2 (see Arnott and Stiglitz, 1981).
31
R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
c1 _ c2
c21 _ c2
1.5
5
1.0
10
15
M21
_ 0.5
0.5
_ 1.0
5
10
15
N1 _ 1.5
_ 0.5
Fig. 1. Stability of residence choice and commuting.
ss
¼1 GH ðH1 ; L1 Þ E0 ðH1 ÞFðH1 ; L1 Þ GH ðH2 ; L2 Þ E0 ðH2 ÞFðH2 ; L2 Þ ; 0 0 t GH ðH1 ; L1 Þ E ðH1 ÞFðH1 ; L1 Þ N1 GH ðH2 ; L2 Þ þ E ðH2 ÞFðH2 ; L2 Þ ðN N1 Þ
ð47Þ sl
¼ ss
½k þ tðN 2N1 Þð1 ss Þss þ : k½1 ð1 ss ÞtðN N1 Þ
ð48Þ
Proof. Taking into account the relationship between the wage and the social marginal product of labor, ss can be calculated from (27) and (34). Using (26), (27), and (33), sl can be obtained. h While short-distance commuting subsidies have a direct impact on residence choices [see (34)], long-distance commuting subsidies are targeted at the commuting decision. Short-distance commuting subsidies reduce the costs of living in the larger city, i.e., the core; long-distance commuting subsidies enhance the benefits of commuting from the periphery to the core. If short-distance commuting were not subsidized, i.e., if ss ¼ 0, long-distance commuting subsidies should be zero, too. Thus, either all commuters or no one is to be subsidized.15 Furthermore, the previous proposition has the following corollary: Corollary 1. If 0 < ss < 1; sl > ss . Proof. This follows immediately from N 1 < N=2 þ k=ð2tÞ. Hence, for ss < 1, both the numerator and denominator of the second term in (48) are positive. h Short-distance commuting subsidies lead workers to live in city 1, but also encourage inhabitants of city 2 to abstain from long-distance commuting. Thus, even stronger incentives to commute from the low-wage city to the high-wage city are useful: the long-distance subsidy rate should exceed the short-distance subsidy rate. Numerical example: To illustrate our findings, we now present a numerical example. We assume a Cobb-Douglas ‘‘internal” production function a FðHi ; Li Þ ¼ Hai L1 ; i
with 0 < a < 1, and an externality function
EðHi Þ ¼ ðH þ cÞb ; where c P 0 and 0 < b and focus on interior social optima. For the distribution of inputs, we assume N ¼ 20; L1 ¼ 20, and L2 ¼ 10. Furthermore, income shares, agglomeration externalities, and commuting costs are a ¼ 0:5; b ¼ 0:1; t ¼ 0:03, and k ¼ 0:1. Since our model has only one mobile input, a moderate income share of this 15
Note that both cities would voluntarily subsidize commuting and – if necessary – transfer resources to the other city if they maximize welfare of mobile citizens, since population is homogeneous and mobility is perfect. It follows immediately from (Myers, 1990) that the spatially efficient allocation of resources could be decentralized.
input seems to be reasonable.16 We have chosen a low b, since from the empirical literature on agglomeration externalities it is well known that elasticities indicating agglomeration externalities are rather small (see Rosenthal and Strange, 2004; Graham, 2007; Rappaport, 2008). Commuting costs consists of time costs, monetary costs, risk of accidents, and psychological costs. A large number of transport economics studies provide values and show that marginal costs of commuting are quite substantial (see Zax, 1991; Small et al., 2005; Van Ommeren and Fosgerau, 2009). Our cost parameters imply (gross) commuting costs ranging from 0% to roughly 35% of total wage income which seems to be in line with recent empirical findings. If c ¼ 0, the optimum is N1 ¼ 11:2941; N2 ¼ 7:0894, and M21 ¼ 1:61647 (case 4). Subsidies are superfluous, since c ¼ 0 implies that, for a fixed stock of L; E0 ðHÞFðH; LÞ is strictly proportional to F H ðH; LÞEðHÞ. With c ¼ 0:5, positive subsidies are necessary to achieve an efficient allocation of labor in a free migration equilibrium. The optimum is N1 ¼ 11:295; N 2 ¼ 7:06847, and M21 ¼ 1:63653. The core’s workers earn more than the periphery’s workers: w1 ¼ 0:866668 > 0:779856 ¼ w2 . The optimum subsidy rates are sl ¼ 0:0457541 > 0:0355369 ¼ ss : Long-distance subsidies exceed short-distance subsidies. To compensate for large commuting distances within the city, residents in the larger city also receive a positive (short-distance) subsidy. Local asymptotic stability of the equilibrium is also an issue. Calculating the trace and the determinant of the system defined by (30) and (31), local stability can be verified. For this numerical example, it can be easily shown that the difference in consumption of resident workers in both cities, c1 c2 , declines as N1 increases. The same holds true for the difference in consumption of commuters and local workers in city 2, namelyc21 c2 , as M21 increases (see Fig. 1).
5. Subsidies and locally captured rents Since we have assumed that rents are equally distributed across the entire metropolitan area, commuting subsidies do not redistribute rents, but only correct the agglomeration externality with respect to long-distance commuting. This assumes that the regional government can collect land rents (e.g., by taxation) and immobile factor incomes and redistribute them equally to all residents of either city in the region. In this section, we depart from this assumption. Following Abdel-Rahman and Anas (2004) and others, we now assume that rents are equally distributed within each city. Rents per capita in city i are Ii ¼ ðRi þ Q i Þ=N i . Thus, the major difference to the case of dissipated rents is that now rents play a role for the residence decision, since individuals earn different shares of rents depending on their residence. The equilibrium condition (34) becomes
w1 ½1 ð1 ss ÞtN 1 þ I1 ¼ w2 ½1 ð1 ss ÞtðN N1 Þ þ I2 :
16
See, e.g., Rappaport (2008) for income shares in a three-factor model.
ð49Þ
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R. Borck, M. Wrede / Journal of Urban Economics 66 (2009) 25–32
By comparison of the first-order conditions of the social optimum and the market equilibrium conditions, i.e., on the one hand, (26) and (33), and, on the other hand, (27) and (49), it becomes clear that an interior migration equilibrium with N 1 ; N 2 ; M 21 > 0 would in general not be spatially efficient with zero commuting subsidies. Not only do wages differ from the marginal social product of labor, but also commuters from the periphery to the core do not participate in economic rents earned in the core. Again, it can be shown that subsidies may lead to first-best efficiency. We can solve (26), (27), (33), and (49) for the first-best subsidies ss and sl .17
we did – to focus on efficiency. Subsidy competition would bring about a further source of inefficiency if mobility were imperfect and the population heterogeneous, which is however, related to the organization of jurisdictions and not to the existence of correct instruments.
Proposition 8. If spatial efficiency requires N 1 ; N 2 ; M 21 > 0 and EðH1 ÞF H ðH1 ; L1 Þ > EðH2 ÞF H ðH2 ; L2 Þ, the efficient allocation can be supported by a combination of short-distance commuting subsidies and long-distance commuting subsidies, where in general, sl –ss .
References
6. Concluding remarks This paper has analyzed subsidies for intracity and intercity commuting when agglomeration externalities are present. In order to analyze commuting within and between cities, the monocentric city framework was extended to a two-city model where workers choose where to live and where to work. In contrast to Abdel-Rahman and Anas (2004) and many others, we have explicitly allowed for partial agglomeration of employment instead of full agglomeration. We have shown that commuting subsidies can help internalize externalities: intracity commuting subsidies give incentives to move to the larger city and intercity commuting subsidies make residents of the periphery commute to the core. Commuting subsidies would act as a welfare enhancing transfer from the core to the periphery, were rents locally captured. Note also that our analysis gives a rationale for differing treatment of short- and long-distance commuting, a policy that can be observed in some countries. One reason for this treatment may be that this way, government can redistribute rents from those working in the core to those working in the periphery. There are alternative policy instruments to achieve efficiency. Since workers’ wages do not compensate them for the agglomeration externalities they generate, an obvious corrective policy would be to subsidize wages. Indeed, location specific wage subsidies are an alternative instrument which may be used to achieve the firstbest outcome. We have not elaborated on this point in this paper, but note that wage subsidies would obviously have to be differentiated by workplace, which in many countries may be deemed unconstitutional. Another possibility would be to differentiate short-distance subsidies between cities. However, as we have shown, this is not necessary to achieve efficiency. There are several directions for further research. First, we have assumed that subsidies reduce monetary commuting costs but do not directly affect the time costs of commuting. It seems to be worthwhile to analyze the provision of traffic infrastructure which would directly reduce travel time and thus increase labor supply. Balancing speed improvement and infrastructure costs, the government would have a further incentive to subsidize commuting. Second, we have not considered locally financed commuting subsidies and have, therefore, excluded subsidy competition from our analysis. However, this choice seems natural if one wants – as
17
We omit the equations for sl and ss , since they are somewhat cumbersome.
Acknowledgments We thank the editor, William Strange, and two referees for helpful comments and suggestions.
Abdel-Rahman, H.M., Anas, A., 2004. Theories of systems of cities. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2293–2339. American Public Transportation Association, 2006. Public Transportation Factbook, APTA, Washington, DC. Arnott, R.J., 2007. Congestion tolling with agglomeration externalities. Journal of Urban Economics 62, 187–203. Arnott, R.J., Stiglitz, J.E., 1981. Aggregate land rents and aggregate transport costs. Economic Journal 91, 331–347. Bach, S., 2003. Entfernungspauschale: Kürzung gerechtfertigt. DIW Wochenbericht 70, 602–608. Borck, R., 2008. The political economy of urban transit. In: OECD (Ed.), ITF Round Table 141 Privatisation and Regulation of Urban Transit, OECD, Paris. Borck, R., Pflüger, M., Wrede, M., 2007. A Simple Theory of Industry Location and Residence Choice, IZA Discussion Paper 2862. Borck, R., Wrede, M., 2005. Political economy of commuting subsidies. Journal of Urban Economics 57, 478–499. Borck, R., Wrede, M., 2008. Commuting subsidies with two transport modes. Journal of Urban Economics 63, 841–848. Brueckner, J.K., 2005. Transport subsidies, system choice, and urban sprawl. Regional Science and Urban Economics 35, 715–733. Duranton, G., Puga, D., 2004. Micro-foundations of urban agglomeration economies. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2063–2117. Graham, D.J., 2007. Agglomeration Economies and Transport Investment, Working Paper, London. Henderson, J.V., 1974. The sizes and types of cities. American Economic Review 64, 640–656. Martin, R.W., 2001. Spatial mismatch and costly suburban commutes: can commuting subsidies help? Urban Studies 38, 1305–1318. Michel, P., Perrot, A., Thisse, J.F., 1996. Interregional equilibrium with heterogeneous labor. Journal of Population Economics 9, 95–113. Myers, G.M., 1990. Optimality, free mobility, and the regional authority in a federation. Journal of Public Economics 43, 107–121. Pisarski, A., 2006. Commuting in America III. The Third National Report on Commuting Patterns and Trends, Transportation Research Board, Washington, DC. Rappaport, J., 2008. A productivity model of city crowdedness. Journal of Urban Economics 63, 715–722. Rosenthal, S.S., Strange, W.C., 2004. Evidence on the nature and sources of agglomeration economies. In: Henderson, J.V., Thisse, J.F. (Eds.), Handbook of Regional and Urban Economics, vol. 4. Elsevier, New York, pp. 2119–2167. Small, K.A., Winston, C., Yan, J., 2005. Uncovering the distribution of motorists’ preferences for travel time and reliability. Econometrica 73, 1367–1382. Statistisches Bundesamt, 2005. Leben und Arbeiten in Deutschland: Ergebnisse des Mikrozensus 2004, Statistisches Bundesamt, Wiesbaden. Van Ommeren, J., Fosgerau, M., 2009. Workers’ marginal costs of commuting. Journal of Urban Economics 65, 38–47. Verhoef, E.T., Nijkamp, P., 2003. Externalities in the Urban Economy, Tinbergen Institute Discussion Paper 2003-078/3. Wrede, M., 2000. Tax deductibility of commuting expenses and leisure: On the tax treatment of time-saving expenditure. FinanzArchiv 57, 216–224. Wrede, M., 2001. Should commuting expenses be tax deductible? A welfare analysis. Journal of Urban Economics 49, 80–99. Zax, J.S., 1991. Compensation for commutes in labor and housing markets. Journal of Urban Economics 30, 192–207. Zenou, Y., 2000. Urban unemployment, agglomeration and transportation policies. Journal of Public Economics 77, 97–133.