BARRIERS TO MIGRATION IN A SYSTEM OF CITIES NATHAN SEEGERT

Abstract. This paper creates a model of city creation and growth to analyze how different barriers to migration affect the resulting system of cities. How many and which set of cities are created is shown to differ based on the barriers to migration that exist in the system of cities. In this model individuals create and move across cities maximizing their utility. Barriers to migration affect the fundamental wedge that is created when uncoordinated individuals disregard the impact of their migration decision on people in the city they choose to live in. When barriers to migration are high or do not exist the population distribution and the number and set of cities created is suboptimal. However, when barriers to migration are capitalized as fees charged to migrants the population is efficiently distributed across the optimal number and set of cities.

The mobility of individuals in a country affects whether urban population is concentrated in a few cities or dispersed among many cities. The distribution of population fundamentally affects the economic growth in a country. Hence, creating the efficient level of mobility within a country is crucial for economic growth especially if in the next 40 years urban population increases by 2.8 billion, or 80 percent, as the United Nations projects. This paper considers the efficient level of mobility within a system of cities and contrasts the resulting distribution of population with systems of cities with different levels of mobility. This paper considers the distribution of population across cities (the intensive margin) as well as the number and set of heterogeneous cities a system of cities creates (the extensive margin). Barriers to migration limit the ability of individuals to move across cities. The barriers may be moving costs, information, or explicit policies such as zoning laws. These barriers may be beneficial in limiting the over-population of cities which occurs when individuals do not internalize the externalities they cause on current residents. In contrast, these barriers may be costly if they allow clubs to monopolize the heterogeneous amenities cities offer. This paper characterizes the barriers to migration a benevolent social planner sets to efficiently distribute population within a 1

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system of cities and explores mechanisms that can create this distribution within a competitive equilibrium. This paper proposes a two-stage model of city formation and population distribution where individuals create cities in the first stage and move across cities in the second stage. The model compares the distribution of population with four levels of barriers to migration. The first distribution of population is chosen by a social planner that maximize the total benefit produced in the system of cities. The second distribution results from a competitive equilibrium with free mobility across cities. The third distribution of population results from a competitive equilibrium with cities able to set population limits. The fourth distribution of population results from a competitive equilibrium where cities are able to charge migrants a fee to enter the city. In the second stage individuals move across cities to maximize their individual benefit, disregarding any externalities their choices may have. Each individual is assumed to receive the average benefit produced within the city in which they reside. When there is free mobility across cities individuals move to equalize the average benefit across cities causing cities to become inefficiently over-populated. The intuition is the same as in the two-road example proposed by Pigou [15], where one road is slow but provides a constant speed independent of the number of drivers on it and the other is fast (if empty) but congestible. Efficiency requires that the marginal benefit of an additional car on each of these two roads to be equal. However, uncoordinated individuals equalize the average benefits, causing too much traffic on the congestible road, and hence the need for a “Pigouvian tax”. If migrants are charged a fee set by a revenue-maximizing city planner the distribution of population across cities is efficient. This result is identical to Knight’s 1924 [12] response to Pigou in which he demonstrates the toll set by a revenue-maximizing toll-setter causes the distribution of cars across the two roads to be efficient. This paper extends this intuition of the intensive

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margin to the extensive margin to determine whether revenue-maximizing city planners create the efficient number and set of heterogeneous cities. Section 1 defines the model and the four levels of barriers to migration within a system of cities. Section 2 solves for and compares the distribution of population across cities in the four cases. Similarly, sections 3 and 4 solve for and compare the number and set of cities created. Section 5 extends the analysis to the case when there are spill-overs across cities.

1. Model 1.1. Foundations of the Model. Much of the current urban research is built upon Henderson’s seminal paper on city sizes and types. The model presented in Henderson’s 1974 paper has been extended to systems of cities by Henderson (1986) [7], Ioannides (1979) [11], and Henderson and Ioannides (1981) [8]. The goal of these extensions is to determine what causes some systems to have growth concentrated in one major city while in other systems growth is dispersed by the creation of new cities. While these models have provided a variety of interesting results they have proven cumbersome. In these models new cities are created by land developers that cap city size. Each city’s population is capped at the population that maximizes the per resident utility. Cities accommodate additional population only when the total population can not be divided into cities with their utility-maximizing populations. New cities are created as soon as the total population is large enough to populate all cities with their utility-maximizing populations.1 The main result from these models is that cities will be created with time intervals that become shorter as total population increases. Further developments in modeling city growth and formation attempted to compare laissezfaire and planned city creation [3]. These extensions found that a system of cities characterized 1

This constraint on creating a new city is an equilibrium stability condition.

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by laissez-faire policies would create cities at a slower rate than a system of cities organized by a planner. In addition, the laissez-faire system of cities would be characterized by “panicmigrations,” as Anas stated in his 1992 paper. These cycles of booms and busts are caused by a coordination failure amongst individuals who are unwilling to create a new city unilaterally until current cities are grossly-overpopulated. Given the dire patterns of laissez-faire city creation this model requires periodic government intervention to improve welfare. Given the undesirable aspects of city growth inherent in laissez-faire systems of cities, subsequent literature focused on “large agents” such as land developers to create cities. Helsley and Strange in their 1994 paper [6] introduced the idea that durable capital could be used by land developers to solve the coordination problem amongst individuals. Later dynamic models by Cuberes [5], Venables[17], and Henderson [9] used this idea from Helsley and Strange’s static model to solve the coordination problem. Despite the tractability of using land developers Krugman in his 1996 paper [13] emphasizes the desirability of creating urban models solely as a result of individuals’ choices. Following this emphasis Seegert’s 2011 paper [16] creates a model of forward-looking individuals that tradeoff benefits that exist in established cities with opportunities that exist in new cities, which when aggregated characterizes the dynamic growth of cities. In the following model individuals create cities and move across them in a two stage game. The dynamic choices of individuals aggregate to characterize which cities are created and the population that resides within them.

1.2. Setup of the Model. Cities combine positive and negative externalities through higher wages and higher costs of living. These forces characterize a total benefit function, T Bj , for each city j, that is increasing in population, convex for small populations, and concave for large populations with only one inflection point. Therefore, the average benefit is single peaked with respect to population and is maximized at the point the marginal benefit intersects the average. The difference between the average and the marginal benefit is defined as the within-city wedge,

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W CW , which is zero at the peak of the average benefit curve, positive to the right, and negative to the left. The within city wedge represents the canonical wedge in economics that exists when individuals consider the average instead of the marginal effects. (1)

W CWj (Nj ) = SABj (Nj ) − SM Bj (Nj )

At the outset, there is an empty plane of J potential city sites and a hinterland indexed by 0 which provides a constant payoff, regardless of the number of individuals inhabiting it. There is an exogenous amount of population, N tot , that is assumed to be homogeneous with no preference for a given city site. Population in any given city is assumed to be any nonnegative real number, abstracting from indivisibility problems.2 Population is distributed across an endogenous number K ⊂ J of inhabited cities, which may include the hinterland. This model is a two-stage game with complete information with sequential moves in each stage.3 Individuals know the order of moves and observe the choices made by everyone else. The equilibrium concept of this two-stage model is a trembling-hand perfect equilibrium and is solved by backwards induction. The trembling-hand perfect equilibrium concept is a refinement of subgame perfect equilibrium which in this context excludes unstable equilibria. In the first stage cities and barriers to migration are created. The model analyzes four cases with different possible barriers to migration; a tax set by the social planner, free mobility, a city specific population cap, and a fee set by the city creator. In the second stage individuals move across cities to maximize their payoff. An individual’s payoff is the average benefit produced in the city they reside in minus any taxes or fees they must pay for living in the city. In the first case a total benefit maximizing social planner creates the efficient distribution of population. The planner does this by creating cities and setting city specific taxes in the first 2This

assumption is justified when populations are large. order in which individual’s move is random between the two stages but known to everyone before the game begins. 3The

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stage. In the second stage individuals sequentially decide which city to move to after observing the set of cities and taxes created by the planner in the first stage. In the second case there is free mobility because individuals lack the ability to create barriers to migration. In the first stage individuals choose sequentially which, if any, city to create. Individuals that create a city in the first stage are obliged to live in the city they create in the second stage. All other individuals in the second stage sequentially decide which city to reside in. The third case models a system of cities created by individuals with a quantity mechanism which allows individuals that create cities to set the maximum population in their city. In the first stage individuals choose sequentially which, if any, city to create and the maximum population. In the second stage individuals sequentially decide which city to reside in. In the fourth case individuals create cities and are able to set a fee all other individuals must pay to reside in the city. In the first stage individuals choose sequentially which, if any, city to create and the fee others must pay to enter. In the second stage individuals sequentially decide which city to reside in knowing the menu of fees charged by each city. The following sections determine, by backwards induction, the distribution of population (section 2), the number of cities created (section 3), and which cities are created (section 4) for each of the four cases.

2. Stage Two Analysis: Distribution of Population In this section the distribution of population across cities is determined assuming the number and set of cities inhabited is exogenous and that the exogenous total population is large enough that the hinterland is populated in equilibrium and in all cases. The full equilibrium problem is

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solved by backwards induction starting with the distribution of population in stage-two. Therefore this section, which determines the population distribution across cities, is the first step in solving the full problem.

2.1. Case One: Planner Optimization. A benevolent social planner maximizes total benefit created by setting a city specific lump-sum tax, τj , which is paid by all individuals residing in the city.4 The full planner’s problem, given in equation (2), is a mixed integer problem where the tax rate τj is any real number and xj ∈ {0, 1} is equal to one if the city is inhabited and zero otherwise. In this section the planner takes as given the set of cities K that are inhabited. After observing the set of cities created and their associated taxes individuals move across cities equalizing the payoff they receive, the planner’s mobility condition given in equation (3). The planner is able to determine the population in each city by setting the city specific tax rates and is constrained to ensure that the sum of populations across inhabited cities is equal to the total P population, Jj=0 xj Nj = N tot . (2)

maxτj

J X

xj Nj (τj )SABj (Nj (τj ))

j=0

(3)

SABj (Nj ) − τj = SABk (Nk ) − τk

∀j, k ∈ K

The first order conditions for all inhabited cities with respect to the tax rates are SM Bj (Nj,1 ) = λ, where λ is the lagrangian multiplier, j indexes city, and 1 indexes population in case 1. Therefore, the planner sets the tax rates to equalize the marginal benefits in each inhabited city.5

4Tax

revenue is redistributed evenly to all resident. first order conditions for all inhabited cities with respect to the tax rates are xj (∂Nj /∂τj )SABj + xj (∂Nj /∂τj )Nj (∂SABj /∂Nj ) = λxj (∂Nj /∂τj ) which can be reduced to SABj + Nj (∂SABj /∂Nj ) = λ. To get the first order condition given in the text note that SABj + Nj (∂SABj /∂Nj ) = SM Bj . 5The

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2.2. Case Two: No Mechanism-Free Mobility. In the system of cities with with no mechanism to limit migration there is free mobility in the second stage. In equilibrium all individuals must be content with their choice of residence which in this case implies that all individuals must receive the same payoff. If this were not the case some individuals would have an incentive, ex post, to move to a different city. Therefore in equilibrium with free mobility all inhabited cities produce the same average benefit according to the free-mobility mobility condition given in equation (4) where j, k index cities, K represents the set of inhabited cities, and 2 indexes the equilibrium population for case 2. (4)

SABj (Nj,2 ) = SABk (Nk,2 )

∀j, k ∈ K

2.3. Case Three: Quantity Mechanism-Limited Mobility. City creators in the first stage with the ability to cap population do so to maximize their benefit which is the per-resident benefit produced within the city. The per-resident benefit is maximized when the average benefit is equal to the marginal benefit. In equilibrium these limits are binding for all cities causing the average benefit produced in each city to be heterogeneous. Individuals may want to migrate to a different city if the benefit they receive in their current city is less than that in another city. However, the population cap for the given city restricts additional migrants from moving into the city.6 (5)

SABj (Nj,3 ) = SM Bj (Nj,3 )

∀j ∈ K

2.4. Case Four: Price Mechanism-Intermediate Mobility. In this case individuals are able to move across cities freely but must pay a fee to enter the city. In equilibrium all individuals must be content with their choice of residence implying the average benefit produced in a city minus the fee they must pay to enter the city must be equal across inhabited cities, the price-mechanism

6If

the maximum per-resident benefit produced in a city is less than the benefit produced in the hinterland then the city will be empty in equilibrium.

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mobility condition given in equation (6). (6)

SABj (Nj,4 ) − fj = SABk (Nk,4 ) − fk

∀j, k ∈ K

The city creator sets the fee in the first stage to maximize the profit from the fees.7 The equilibrium fee is found by substituting the condition in equation (6) into the city creator’s objective function in equation (7) and taking the first order condition with respect to the fee noting that the population Nj,4 (fj ) is a function of the fee.8 (7)

maxfj

fj Nj

The fee charged in equilibrium for all inhabited cities is the within-city wedge, fj = W CWj . The condition in equation (8) is the equilibrium condition for a system of cities with a price mechanism and is found by substituting the equilibrium fee into the price mechanism mobility condition given in equation (6). This condition states that the fees set by decentralized profit maximizing individuals cause individuals in the second stage to move across cities in a way that equalizes the marginal benefit across all inhabited cities. (8)

SM Bj (Nj,4 ) = SM Bj (Nj,4 )

∀j, k ∈ K

2.5. Population Distribution Analysis. Figure 1 graphs the average and marginal benefit of a given city with respect to population with the equilibrium populations in the four cases 7Maximizing

the profit from fees is the correct objective function for the city creator because population is assumed to be a real number. To see this consider the more general objective function Ω(ω)SABj (Nj,4 (f )) + f [Nj,4 (f ) − Ω(ω)] where Ω is the density function of population and Ω(ω) represents the density at point ω which represents the city creator. In the case where population is an integer Ω(ω) = 1 representing a unit mass for each individual. However, in the case where population is a real number Ω(ω) = 0. Substituting Ω(ω) = 0 into the more general objective function we note that it reduces to maximizing the profit from the fees. 8The new objective function is (SAB − SAB + f )N . The first order condition is (∂SAB /∂N )(∂N /∂f )N + j k k j j j j j j (∂Nj /∂fj )(SABj − SABk + fk ) = 0. Rearranging (∂SABj /∂Nj )Nj + SABj = SABk − fk which is fk = SABk − SM Bj . Substituting fk and fj from the first order conditions into condition 6 gives SABj (Nj,4 ) − (SABj − SM Bk ) = SABk (Nk,4 ) − (SABk − SM Bj ) which gives the condition SM Bk = SM Bj in the text. The fee is found by noting the first order condition with the condition that the marginal benefits be equal imply that SM Bj = SABj − fj .

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marked. Across the four cases the population residing in the hinterland differs such that the population in all cities for a given case can be less than the equilibrium populations in a different case. For example, Figure 1 demonstrates a city able to cap its population chooses a population level less than the other cases, and this holds for all cities. Therefore, there is more population living in the hinterland in the system of cities able to cap city populations. In contrast, when there is free mobility across cities every city has its largest equilibrium population. Finally, the equilibrium population resulting from city creators setting fees is the efficient population set by social planner. Result 1: For a given number of inhabited cities and a large population such that the hinterland is inhabited; a system of cities with free mobility has cities that are all over-populated, a system of cities with the quantity mechanism has cities that are all under-populated, and a system of cities with fees has cities that are all efficiently populated. Result 1 demonstrates the ability of the price mechanism to solve the inefficiency in the allocation of population across cities that occurs when the system of cities have access to either a quantity mechanism or no mechanism to limit migration. This result is surprising and encouraging because it implies that if cities use zoning optimally to set a fee for migrants, the population distribution may be efficient on the intensive margin.9 The fact that a system of cities with free mobility has cities that are over-populated follows from the equilibrium conditions.

9There

is a special case where the population in each city is the same across all cases. In this special case all cities are homogeneous, the total population is divisible by the shared population that maximizes average product in a city, and the hinterland is uninhabited. In this special case each city has the capped population, the average benefits are equal, the marginal benefits are equal, and the fee in all cities in equilibrium is zero.

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Proof SABj (Nj,2 ) = SAB0

by case 2 equilibrium condition

= SM B0

by definition hinterland

= SM Bj (Nj,1 ) = SABj (Nj,1 ) + Nj,1 ⇒

by planner equilibrium condition ∂SABj (Nj,1 ) ∂N

by definition SM B and SAB

SABj (Nj,2 ) ≤ SABj (Nj,1 ) ⇒

Nj,2 ≥ Nj,1

The result that the system of cities with free mobility over-populates cities relative to other cases is well documented in the literature. Arnott in his 1979 paper [4] proposes that “no stable equilibrium exists in which some cities are less than optimal size.”10 The result that the population distribution created by the planner can also be created in a competitive equilibrium where fees are charged by city creators is similar to Knight’s optimal highway toll. Knight demonstrated that the efficient distribution of traffic between a slow uncongestible road and a fast congestible road could be achieved by allowing the owner of the congestible road to charge a toll. The toll the profit-maximizing entrepreneur charges is exactly the tax Pigou suggested to align private and social incentives. Similar intuition holds in this paper as well, the planner sets the distribution across cities to equalize the marginal benefits by setting a tax, and this distribution can be decentralized by allowing city creators, similar to

10However,

recently Albouy and Seegert in their 2010 paper [2] loosen Arnott’s assumptions that cities are homogeneous and that the total benefit produced within a city is consumed within a city to demonstrate that cities can be inefficiently small even with free mobility. The assumption that the total benefit produced within a city is consumed within a city is maintained in this section but loosened in section 5.

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Knight’s entrepreneur’s, to charge a fee. The model extends this intuition in the following sections by investigating the distribution of population when the number and set of cities inhabited are endogenous.

3. Extensive Margin: How Many Cities to Create Individuals in the first stage make a binary decision between creating a city or not based on their expected utilities in either case. The order in which cities are created is taken as given in this section and is determined in section 4. An individual’s expected utility of creating or not creating a city depends on the second stage outcomes, hence the number of cities created can be heterogeneous across cases because the second stage outcomes are heterogeneous across cases.

3.1. Case One: Planner Optimization. The planner creates K cities when the total benefit produced by efficiently allocating the population across K cities is larger than the total benefit produced by efficiently allocating the population across K − 1 cities or K + 1 cities.

3.2. Case Two: No Mechanism-Free Mobility. In this case the second stage ensures the per-resident benefit each individual receives is equal across all inhabited cities. Individuals decide to create a city if by doing so increases the equilibrium per-resident benefit. Therefore, this case creates the number of cities that maximize the shared per-resident benefit.

3.3. Case Three: Quantity Mechanism-Limited Mobility. In this case individuals compare the per-resident benefit they would receive in the second stage with the maximum perresident benefit in the city they would create. In this case there are two subcases, either the hinterland is inhabited or it is not. If the hinterland is not inhabited the maximum number of cities is created, given the exogenous total population. Each city, with the exception of one, has population that maximizes its per-resident benefit. The possible exception is for the last city created which could have a population less than its per-resident benefit maximum if it is

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able to produce a per-resident benefit greater than the hinterland. If the hinterland is inhabited either the population living in the hinterland is less than the population that maximizes the per-resident benefit in the next possible city or the maximum per-resident benefit in the next city is less than the hinterland benefit.

3.4. Case Four: Price Mechanism-Intermediate Mobility. Individuals in a system of cities with a price mechanism have an incentive to create cities in the first stage as long as the equilibrium fee they would be able to charge is nonnegative. The equilibrium fee is the within-city wedge which is zero at the capped population and positive for larger populations.

3.5. How Many Cities Analysis. Intuitively, the third case with the ability to cap populations creates the most cities and the second case with free mobility creates the fewest cities because in equilibrium the third case under-populates its cities and the second case over-populates its cities. However, whether the price mechanism in case four is able to create the efficient number of cities is not obvious. In the second stage allowing city creators to set a fee aligned the social and private incentives but in the first stage the public and private incentives do not seem to be aligned. However, to be able to create an additional city and charge a positive fee necessitates an increase in total benefit produced within the system of cities. Therefore, the price mechanism is able to create the efficient number of cities. Result 2: For a given ordering of city creation; a system of cities with free mobility creates the fewest cities, a system of cities with the quantity mechanism creates the most cities, and a system of cities with the price mechanism creates the efficient number of cities. The intuition for result 2 is formalized in the appendix (section 7) and demonstrated below by simulating the number of cities each case produces.

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3.6. System of Cities Simulation. Result 2 is demonstrated by simulating the number of cities that each case produces as total population increases.11 The calibrated model is a general function representing the economies and diseconomies of scale given in equation (9). Cities are heterogeneous in the level of Ai , a multiplicative factor on the economies of scale, and in Qi , an additively separable factor. Following Albouy and Seegert (2010), the multiplicative factor represents production amenities and the additively separable factor represents consumption amenities. A wide range of parameter values for this functional form are used in the simulations, only constrained such that the average benefit is a single-peaked function. The simulation is performed using an algorithm similar to the add-routine algorithm described in Kuehn and Hamburger’s 1963 paper [14]; details are provided in the appendix (section 8) (9)

SABi (Ni ) = Ai Niα − BNiβ + Qi

Figure 2 plots the number of cities created as total population increases in the cases with the quantity mechanism, free mobility, and the social planner. For small levels of total population, the difference in the number of cities among cases is small. However, as the total population increases the difference in cities created diverges. In the special case where all cities are homogenous each case creates the same number of cities. However, as cities become more heterogeneous the difference between the number of cities created in each case increases.

4. Extensive Margin: Which Cities to Create This section determines the order cities are inhabited.12 City sites are modeled to be heterogeneous in their production amenities Aj and quality of life amenities Qj according to equation (9). The benefit that a city or system of cities receives from these amenity levels depends on the 11The

simulations are done for 10, 000 different values of total population and 10, 000 different parameter values. In addition the simulation is run with the microfounded calibrated model in Albouy and Seegert (2010). 12In the static model presented here, it could be welfare improving to create multiple cities lower on the priority list instead of a city higher on the priority list. However, these equilibria are eliminated because they are not robust to dynamic models where the total population in the city is increasing with the assumption that once a city is created it cannot be uninhabited or to do so would incur a large cost.

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ability of the city to limit migration. Therefore, individuals value the production amenities and quality of life amenities based on their ability to limit migration. Cities in this model are heterogeneous in two-dimensions, but to order the cities, the twodimensional space must be projected into a one-dimensional effective benefit space. This section produces two mappings from Aj x Qj space to A˜j space that are similar to the equivalent and compensating variation introduced by John Hicks in 1939 [10]. The equivalent variation maps points in price-wealth space onto a fixed price line. Similarly, the first mapping in this section maps production and quality-of-life-amenity space onto a fixed quality of life line, holding population fixed. Therefore, this mapping gives the amount of change in quality of life amenities needed to offset a change in production amenities such that the individuals would receive the same benefit with the same level of population. The projection in this mapping is the indifference curve between production and quality of life amenities, given below in equation (10).13 (10)

β α Qj = C¯ − Aj Nj,i + BNj,i

α which, by result 1, implies that the indifference curve The slope of the indifference curve is Nj,i

is steepest for the system of cities with free mobility and flattest in the system with the quantity mechanism. The indifference curves, drawn in Figure 3, demonstrate that barriers to migration cause individuals and the social planner to value amenities differently. Result 3: Systems of cities with the quantity mechanism over-value quality of life amenities and systems of cities with free mobility over-value production amenities. The second mapping compares the equilibrium level of benefit for two different production amenity levels allowing equilibrium populations to differ. This is similar to the compensating variation that compares utility levels, allowing individuals to choose different bundles for different 13The

equivalent variation and the indifference curve projection are well suited for individual comparisons. However, changes in amenity levels change the fee the city creator is able to charge in case four, which is not an individual level comparison but a city wide comparison as shown in the compensating variation.

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relative prices. The compensating and equivalent variation benefits for systems of cities with free mobility and the system with the quantity mechanism are graphed in Figures 4 and 5. The compensating variation for the system of cities with free mobility is zero because population in equilibrium perfectly compensates for differences in amenity levels. In contrast, the compensating variation is larger than the equivalent variation for systems of cities with the quantity mechanism because individuals capitalize the full benefit of the additional amenities by adjusting the cap on population. The compensating variation for the system with the price mechanism compares the difference in the fees collected by the city creator. The fees collected increase with the level of production amenities because both the fee and the number of migrants paying the fee increases. The increase in fees collected is given by the difference in the two rectangles depicted in Figure 8. The compensating variation for the social planner can be conceptualized for a single person or for all individuals. For comparison with systems of cities with free mobility and the quantity mechanism, the single person compensating variation is used and depicted in Figure 6. When the social planner is considering the next city to create, the objective is to maximize the total benefit and the relevant comparison is across all individuals. When a city with more production amenities is created, it provides a higher level of average benefit for more individuals, resulting in a total benefit that is represented graphically as two rectangles in Figure 7. Figure 8 demonstrates that the compensating variations for the system of cities with the price mechanism and the social planner are the same, which implies that although they have different objectives, they value production and quality of life amenities in the same way. Result 4: Systems of cities with the price mechanism value amenity levels in the same way as the social planner and produce the same ranking of cities. The compensating and equivalent variation projections provide an ordering of cities given estimates of the quality of life and production amenities that characterize each city. Table 1 lists

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the ordering using the equivalent variation projection for the system of cities with the planner, free mobility, and the quantity mechanism. The population of the city in each case is given in columns 1 and 2. Population is estimated using the calibrated model from Albouy and Seegert (2010) [2] and the amenity levels from Albouy (2009) [1] . The cities in Table 1 are ordered by which cities case 2 (the quantity mechanism) most over-values (relative to the social planner). The first city in Table 1 is Portland, Maine, which is the most over-valued city in a system with the quantity mechanism relative to the social planner. The effective benefit, given in dollars, of each city is given in columns 3, 5, and 7. The ranks of each city for each case are given in columns 4, 6, and 8 and the differences in ranks among the cases are given in columns 9, 10, and 11. Columns 1 and 2 demonstrate result 3 that the cities over-valued by systems with the quantity mechanism are those with relatively large quality of life amenities relative to their production amenities. For example, Houston, Texas is valued most highly by systems of cities with free mobility, but is valued least by systems with the quantity mechanism. The social planner values Houston at an intermediate level. This ranking reflects the fact that Houston has relatively more production amenities than quality of life amenities. In contrast, Portland, Oregon is ranked higher by the system of cities with the quantity mechanism than the system of cities with free mobility. This ranking holds even though Portland, Oregon has an estimated level of production amenities that is higher than its quality of life amenities. However, relative to the indifference curve, Portland has higher quality of life amenities. These two cities are of particular interest because they are often used as examples of the extremes in land-use policies; Houston has very few zoning laws and Portland is noted for its strong land-use planning. The relative rankings in Table 1 suggest that the strength of these cities’ zoning laws may not be a coincidence but a result of the relative preference of individuals.

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Figure 3 plots a few cities and their amenity levels and possible indifference curves for the social planner, system of cities with free mobility and system of cities with the quantity mechanism all going through San Luis Obispo, CA. San Luis Obispo, CA has slightly more production amenities than Denver, CO and significantly more quality of life amenities. Therefore in all three cases San Luis Obispo, CA is preferred over Denver, CO. In contrast, Seattle, WA and San Diego, CA are preferred over San Luis Obispo, CA even though they have less quality of life amenities (but make up for it with significantly more production amenities). Honolulu, HI has more quality of life amenities than San Luis Obispo, CA but less production amenities. The social planner prefers San Luis Obsipo, CA over Honolulu, HI but systems of cities with the quantity mechanism prefer Honolulu, HI. The quantity mechanism prefers Honolulu, HI because it over-values (relative to the social planner) the quality of life amenities that Honolulu, HI offers. Similarly, Stockton, CA is preferred over San Luis Obispo, CA by systems of cities with free mobility but not by the social planner because systems of cities with free mobility over-value (relative to the social planner) production amenities.

5. Across-City Wedge This section allows for across-city externalities by relaxing the assumption (maintained in the previous sections) the benefits produced within a city remain in the city. There are many realworld examples where benefits produced within cities are combined into a common pool from which cities receive benefits; examples include federal income taxation and land rents if land owners do not live in the city where they own land. The transfer of benefits from some cities to others is defined as the across-city wedge (ACW ). This section demonstrates that if across-city wedges exist then the price mechanism is unable to solve the inefficiencies in a system of cities. The across-city wedge is defined in equation (11) where P AB is the average benefit in the city net the across-city wedge and is assumed to be single-peaked. The across-city wedge is allowed to differ with population and across cities; therefore, there are cities that are net beneficiaries

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

19

and cities that are net providers. Let cities that are net providers be in the set S = {1, 2, 3...I} and cities that are net beneficiaries be in the set S c = {I + 1, I + 2, I + 3...J}. P ABj (Nj ) = SABj (Nj ) − ACWj (Nj )

(11)

Result 5: For a given number of inhabited cities and a total population large enough such that the hinterland is inhabited in all cases, if the across-city wedge is positive for city j then the social planner allocates more population to city j than the system of cities with the price mechanism, and if the across-city wedge is negative for city j then the social planner allocates less population to city j than the system of cities with the price mechanism. Proof: P M Bj (Nj,4 ) = SAB0

Equilibrium condition for case 4.

= SM Bj (Nj,p )

Equilibrium condition for planner.

SM Bj (Nj,4 ) − ACWj = SM Bj (Nj,p )

Definition across-city wedge

SM Bj (Nj,4 ) > SM Bj (Nj,p )

Given j ∈ S

⇒ Nj,4 < Nj,p SM Bj (Nj,4 ) < SM Bj (Nj,p )

Given j ∈ S c

⇒ Nj,4 > Nj,p

When across-city wedges exist the price mechanism is no longer able to efficiently distribute population across a given number of cities. Similarly, when the across-city wedge exists individuals with the price mechanism value cities differently than the social planner. As a result individuals may create a different set of cities both in number and type. For example, the system of cities the with price mechanism values cities that create large surpluses to individuals and have small

20

NATHAN SEEGERT

across-city wedges. Therefore, the across-city wedge differentiates how individuals with the price mechanism value cities in comparison to the social planner.14 If the across-city wedge is federal taxation, this implies that the system of cities with the price mechanism over-values cities with large quality of life amenities and under-value cities with large production amenities relative to the social planner.

5.1. Second Best World. If across-city wedges encompass fundamental aspects of society, such as federal taxation and land rents, then limiting the across-city wedge may not be possible. The across-city wedge decreases the equilibrium population of net provider cities and increase the equilibrium population of net beneficiary cities. Therefore, if across-city wedges cannot be limited then cities within a system should be given different abilities to limit migration to counter the effects of the across-city wedge. Specifically, cities that are net providers should be constrained in their ability to limit migration and cities that are net beneficiaries should be encouraged to limit migration. Result 6: For a given number of inhabited cities and a total population large enough such that the hinterland is inhabited in all cases, the equilibrium population levels Nj,2 , Nj,3 , and Nj,4 are nonincreasing functions of the across-city wedge. Result 6 follows directly from the equilibrium conditions and the definition of the across-city wedge. Implication 1: Cities that are net providers should have a restricted set of zoning policy tools to allow more mobility to these cities to counter the effects of the across-city wedge. Implication 2: Cities that are net beneficiaries should be allowed a wide range of zoning policy tools to create more restrictive mobility to these cities to counter the effects of the across-city wedge. 14The

social planner does not consider across-city wedges because they are transfers across cities and do not change the total production in the system of cities.

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

21

6. Conclusion This paper proposes a tractable strategic urban model in which individuals endogenously create and move among cities to maximize their own benefit. The model emphasizes the importance of barriers to migration on individual’s incentives. If able, individuals limit the population of the city in which they live. Doing so can maximize the per-resident benefit to individuals within the city, but not the total benefit across cities, causing cities to be inefficiently small in equilibrium. In contrast, if individuals are unable to limit migration, cities become inefficiently large. The efficient population for a city can be achieved by a decentralized system of individuals able to charge a fee to migrants entering the city. Conceptually, residents of a city may charge a fee to migrants by artificially limiting housing supply with land-use policies. These results describing how to efficiently distribute population for a given number of cities corresponds to the economic intuition from Pigou and Knight and are similar to results in the urban literature (Arnott, 1979 and Anas, 1992). This model extends the results in the urban literature, producing solutions to how many and which cities should be created. Different barriers to migration cause different numbers of cities and different sets of cities to be created. When there are large barriers to migration, individuals produce too many cities. When there are no barriers to migration individuals produce too few cities. When the barriers to migration are capitalized in fees charged to migrants, the efficient number of cities are produced. This result is both surprising and encouraging because it suggests that self-interested individuals with the ability to create barriers to migration through a price mechanism do so efficiently. The model also demonstrates that the value of production and quality of life amenities are valued differently depending on the type of barriers to migration that exist. Quality of life amenities are valued highly by systems of cities that have large barriers to migration, whereas production amenities are valued highly by systems of cities that have no barriers to migration.

22

NATHAN SEEGERT

When barriers to migration consist of fees to migrants, quality of life and production amenities are valued in the same was as the benevolent social planner. These results imply that landuse policies may act as a market for migrants among cities, causing population to be optimally distributed across the optimal number and set of cities. This model provides a framework for further research on the extensive margin of city formation. For example, the model is built using homogeneous agents but could be extended to heterogeneous agents. In addition, this paper (and most migration models) focus on wages and cost of living as the sole determinants of migration. However, this model could be extended to allow the proximity of individuals to different cities both geographically and in preference-space to enter the model. Geographic proximity can be an important factor in structuring migration patterns among cities. For instance, Chicago is a productive city which offers high wages and a reasonable cost of living, which should encourage in-migration from across the entire United States but receives migrants disproportionately from the immediately adjacent states. This regionalism which is unaddressed in most models may have important ramifications for city creation and growth.

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

23

References [1] D. Albouy. What are cities worth? land rents, local productivity, and the capitalization of amenity values, 2009. [2] D. Albouy and N. Seegert. Optimal city size and the private-social wedge. Ann Arbor, 1001:48109–1220. [3] A. Anas. On the birth and growth of cities: Laissez-faire and planning compared. Regional Science and Urban Economics, 22(2):243–258, 1992. [4] R. Arnott. Optimal city size in a spatial economy. Journal of Urban Economics, 6(1):65–89, 1979. [5] D. Cuberes. The rise and decline of cities. University of Chicago, Dept. of Economics, 2005. [6] R.W. Helsley and W.C. Strange. Private government. Journal of Public Economics, 69(2):281–304, 1998. [7] J.V. Henderson. Efficiency of resource usage and city size. Journal of Urban Economics, 19(1):47–70, 1986. [8] J.V. Henderson and Y.M. Ioannides. Aspects of growth in a system of cities. Journal of Urban Economics, 10(1):117–139, 1981. [9] J.V. Henderson and A.J. Venables. The dynamics of city formation. Review of Economic Dynamics, 12(2):233–254, 2009. [10] J.R. Hicks. Value and capital, volume 2. Clarendon press Oxford, 1974. [11] Y.M. Ioannides. Temporal risks and the tenure decision in housing markets* 1. Economics letters, 4(3):293– 297, 1979. [12] F.H. Knight. Some fallacies in the interpretation of social cost. The Quarterly Journal of Economics, 38(4):582–606, 1924. [13] P. Krugman. Urban concentration: the role of increasing returns and transport costs. International Regional Science Review, 19(1-2):5, 1996. [14] A.A. Kuehn and M.J. Hamburger. A heuristic program for locating warehouses. Management science, pages 643–666, 1963. [15] A.C. Pigou. The economics of welfare. Transaction Publishers, 1952. [16] N. Seegert. Dynamic city formation. June 2011. [17] A.J. Venables. Spatial disparities in developing countries: cities, regions, and international trade. Journal of Economic Geography, 5(1):3, 2005.

24

NATHAN SEEGERT

Figure 1. Equilibrium Populations

Figure 2. Simulated Number of Cities Created

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

25

Table 1. Free Mobility and Social Planner Ranking of Cities

City Portland, ME Norfolk–Virginia Beach -Newport News, VA–NC non-metropolitan areas, OR Sarasota–Bradenton, FL Provo–Orem, UT Honolulu, HI non-metropolitan areas, HI non-metropolitan areas, VT Bellingham, WA non-metropolitan areas, AZ Savannah, GA non-metropolitan areas, MT Albuquerque, NM Tucson, AZ Charlottesville, VA Flagstaff, AZ–UT Bloomington, IN non-metropolitan areas, AK Santa Fe, NM non-metropolitan areas, NH non-metropolitan areas, CO Salt Lake City–Ogden, UT Fort Collins–Loveland, CO Tampa-St. Petersburg -Clearwater, FL Colorado Springs, CO Iowa City, IA Fort Pierce–Port St. Lucie, FL San Diego, CA Barnstable–Yarmouth, MA West Palm BeachBoca Raton, FL Naples, FL Milwaukee–Racine, WI C Madison, WI non-metropolitan areas, CA non-metropolitan areas, WA Des Moines, IA Rochester, NY non-metropolitan areas, UT non-metropolitan areas, FL non-metropolitan areas, ME non-metropolitan areas, ID non-metropolitan areas, WY non-metropolitan areas, VA Visalia–Tulare–Porterville, CA non-metropolitan areas, NM Greensboro–Winston -Salem–High Point, NC Chico–Paradise, CA Greenville–SpartanburgAnderson, SC Yuba City, CA Redding, CA Eugene–Springfield, OR Medford–Ashland, OR Tulsa, OK Wilmington, NC Tallahassee, FL Evansville–Henderson, IN–KY Yuma, AZ Tuscaloosa, AL Melbourne–TitusvillePalm Bay, FL Bryan–College Station, TX Knoxville, TN Montgomery, AL Rocky Mount, Lincoln, NE Gainesville, FL Hickory–Morganton– Lenoir, NC Punta Gorda, FL Grand Junction, CO Amarillo, TX

243537 1569541

Amenity Estimate QOL Production (1) (2) 0.058 -0.056 0.03 -0.092

Quantity Mech. Value Rank (3) (4) 9174 114 8485 175

Social Planner Value Rank (5) (6) 9072 121 8413 180

Free Market Value Rank (7) (8) 9037 122 8382 180

1194699 589959 368536 876156 335651 608387 166814 942343 293000 774080 712738 843746 159576 122366 120563 367124 147635 1011597 924086 1333914 251494 2395997

0.057 0.073 0.013 0.165 0.111 0.064 0.063 0.035 0.021 0.059 0.048 0.054 0.053 0.085 0.026 0.011 0.115 0.018 0.088 0.019 0.064 0.013

-0.036 -0.045 -0.047 0.049 -0.016 -0.041 -0.045 -0.041 -0.053 -0.062 -0.064 -0.089 -0.089 -0.105 -0.105 0.007 -0.02 -0.006 -0.024 -0.016 -0.03 -0.051

9498 9408 9163 11269 10016 9441 9372 9339 9092 9079 9008 8620 8616 8469 8258 10042 9964 9853 9804 9693 9621 9097

77 88 116 14 42 84 92 96 125 127 135 165 166 180 197 39 45 50 53 63 69 124

9390 9293 9080 11080 9873 9331 9263 9243 9007 8979 8914 8534 8531 8378 8196 9947 9820 9757 9676 9598 9507 9015

81 92 120 17 45 87 95 99 128 130 138 168 169 183 200 41 47 52 55 65 71 126

9355 9257 9051 11032 9831 9295 9228 9211 8977 8944 8880 8500 8497 8339 8165 9919 9778 9728 9637 9569 9472 8986

83 93 120 17 45 89 97 99 128 131 139 168 169 184 201 41 48 52 56 65 71 127

-4 -4 -4 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -2 -2 -2 -2 -2 -2 -2

2 1 0 0 0 2 2 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1

-6 -5 -4 -3 -3 -5 -5 -3 -3 -4 -4 -3 -3 -4 -4 -2 -3 -2 -3 -2 -2 -3

516929 111006 319426 2813833 162582 1131184

0.053 0.027 0.022 0.108 0.086 0.03

-0.062 -0.072 -0.076 0.096 0.04 0.044

9058 8802 8719 11840 10846 10716

130 151 160 9 22 26

8961 8721 8642 11679 10705 10605

132 153 162 10 23 27

8926 8691 8612 11638 10667 10575

134 153 162 10 23 27

-2 -2 -2 -1 -1 -1

2 0 0 0 0 0

-4 -2 -2 -1 -1 -1

251377 1689572 426526 1249739 1063531 456022 1098201 531967 1222532 1033664 863855 493849 1640567 368021 783050 1251509

0.098 -0.007 0.05 0.044 0.034 -0.011 -0.026 0.014 0.018 0.021 0.008 0.012 -0.028 -0.024 0.006 -0.018

0.024 0.039 -0.018 -0.017 -0.022 -0.023 -0.021 -0.032 -0.039 -0.041 -0.04 -0.042 -0.035 -0.038 -0.047 -0.049

10625 10505 9769 9764 9647 9472 9452 9412 9312 9289 9260 9241 9215 9180 9138 9020

28 31 56 57 67 79 82 87 101 102 104 106 108 112 122 133

10480 10414 9659 9657 9546 9394 9381 9324 9224 9201 9177 9157 9148 9112 9058 8952

29 32 57 58 68 80 83 88 102 103 105 107 109 113 123 134

10441 10389 9625 9624 9515 9369 9358 9296 9194 9171 9149 9128 9125 9088 9030 8928

29 32 57 59 70 80 82 87 102 104 105 108 109 114 123 133

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

0 0 0 1 2 0 -1 -1 0 1 0 1 0 1 0 -1

-1 -1 -1 -2 -3 -1 0 0 -1 -2 -1 -2 -1 -2 -1 0

203171 962441

0.047 -0.019

-0.07 -0.062

8906 8803

145 150

8815 8740

146 151

8781 8715

146 151

-1 -1

0 0

-1 -1

139149 163256 322959 181269 803235 233450 284539 296195 160026 164875 476230

-0.001 0.039 0.08 0.09 -0.014 0.071 0.028 -0.026 0.004 -0.009 0.005

-0.067 -0.077 -0.086 -0.099 -0.082 -0.102 -0.095 -0.091 -0.098 -0.096 -0.101

8785 8763 8761 8585 8493 8468 8429 8302 8294 8281 8249

153 156 157 170 174 181 184 191 193 194 200

8715 8679 8661 8487 8435 8382 8359 8254 8237 8228 8193

154 157 158 171 175 182 185 192 194 195 201

8688 8646 8623 8448 8410 8345 8328 8230 8210 8202 8165

154 157 158 171 175 183 185 192 194 195 200

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

0 0 0 0 0 1 0 0 0 0 -1

-1 -1 -1 -1 -1 -2 -1 -1 -1 -1 0

152415 687249 333055 143026 250291 217955 341851

0.033 -0.007 0.001 -0.024 0.021 0.035 -0.004

-0.116 -0.111 -0.114 -0.109 -0.12 -0.129 -0.121

8102 8042 8021 8014 7994 7897 7888

205 210 213 214 216 225 226

8046 7999 7977 7976 7947 7854 7854

206 211 214 215 217 226 227

8015 7973 7951 7953 7918 7823 7828

206 211 215 214 217 229 228

-1 -1 -1 -1 -1 -1 -1

0 0 1 -1 0 3 1

-1 -1 -2 0 -1 -4 -2

141627 116255 217858

0.058 0.07 -0.001

-0.142 -0.148 -0.137

7768 7712 7637

237 243 248

7732 7682 7626

238 244 249

7700 7650 7605

238 246 248

-1 -1 -1

0 2 -1

-1 -3 0

Population

Difference Rank QM- SP SP - FM QM - FM (9) (10) (11) -7 1 -8 -5 0 -5

26

NATHAN SEEGERT

Table 1. Free Mobility and Social Planner Ranking of Cities

City San Francisco– Oakland–San Jose, CA New York–Northern New Jersey–Long Island Santa Barbara-Santa Maria–Lompoc, CA Los Angeles–Riverside –Orange County, CA Boston-Worcester-Lawrence MA–NH–ME–CT Salinas-Monterey-Carmel, CA Chicago–Gary-Kenosha, IL–IN–WI Hartford, CT Detroit–Ann Arbor –Flint, MI Seattle–Tacoma-Bremerton, WA Philadelphia–WilmingtonAtlantic City, PA-NJ-DE-MD San Luis Obispo– Atascadero–Paso Robles, CA Sacramento–Yolo, CA Las Vegas, NV–AZ Minneapolis-St. Paul, MN-WI Denver–Boulder-Greeley, CO C Portland-Salem, OR-WA Reno, NV Phoenix–Mesa, AZ Austin–San Marcos, TX Raleigh–Durham-Chapel Hill, NC non-metropolitan areas, CT Cincinnati-Hamilton, OH-KY-IN Miami–Fort Lauderdale, FL non-metropolitan areas, RI Columbus, OH non-metropolitan areas, NV non-metropolitan areas, MA Allentown-Bethlehem -Easton, PA non-metropolitan areas, MD Kansas City, MO–KS Richmond–Petersburg, VA St. Louis, MO–IL Albany-SchenectadyTroy, NY non-metropolitan areas, DE Lancaster, PA Fresno, CA Merced, CA Green Bay, WI non-metropolitan areas, SC Yakima, WA Orlando, FL non-metropolitan areas, NY non-metropolitan areas, NC non-metropolitan areas, WV Toledo, OH Louisville, KY–IN Baton Rouge, LA non-metropolitan areas, OK non-metropolitan areas, ND non-metropolitan areas, NE non-metropolitan areas, SD Lexington, KY New Orleans, LA Charleston-North Charleston, SC Jacksonville, FL Omaha, NE–IA Fort Myers–Cape Coral, FL Boise City, ID Cedar Rapids, IA Springfield, IL Benton Harbor, MI Dover, DE Canton–Massillon, OH

7039362

Amenity Estimate QOL Production (1) (2) 0.114 0.285

Quantity Mech. Value Rank (3) (4) 14966 1

Social Planner Value Rank (5) (6) 14784 1

Free Market Value Rank (7) (8) 14745 1

21199864

0.033

0.21

13456

2

13328

2

13298

2

0

0

0

399347

0.158

0.156

12997

3

12799

3

12753

3

0

0

0

16373645

0.065

0.143

12464

4

12322

4

12288

4

0

0

0

5819100

0.045

0.145

12428

5

12297

5

12266

5

0

0

0

401762 9157540

0.126 0.004

0.125 0.131

12378 12056

6 7

12203 11950

6 7

12160 11924

6 7

0 0

0 0

0 0

1183110 5456428

-0.029 -0.037

0.134 0.115

11992 11652

8 11

11904 11569

8 11

11882 11548

8 11

0 0

0 0

0 0

3554760

0.049

0.094

11604

12

11476

12

11443

12

0

0

0

6188463

-0.036

0.1

11408

13

11326

13

11305

13

0

0

0

246681

0.115

0.058

11242

15

11081

15

11040

16

0

1

-1

1796857 1563282 2968806 2581506

0.025 -0.023 -0.023 0.045

0.072 0.077 0.075 0.058

11159 11075 11042 10998

18 19 20 21

11047 10988 10955 10877

18 19 20 21

11017 10964 10932 10845

18 19 20 21

0 0 0 0

0 0 0 0

0 0 0 0

2265223 339486 3251876 1249763 1187941

0.041 0.05 0.018 0.029 0.01

0.044 0.042 0.035 0.026 0.019

10754 10753 10527 10417 10236

24 25 30 33 34

10638 10632 10423 10309 10139

24 25 30 33 34

10606 10598 10394 10279 10111

25 26 31 33 34

0 0 0 0 0

1 1 1 0 0

-1 -1 -1 0 0

1350818 1979202 3876380 258023 1540157 285196 569691 637958

-0.013 -0.039 0.046 0.035 -0.027 -0.011 0.021 -0.029

0.022 0.026 0.007 0.006 0.015 0.003 -0.005 -0.004

10205 10180 10165 10110 10040 9899 9880 9721

35 36 37 38 40 48 49 59

10119 10107 10051 10003 9963 9816 9782 9648

35 36 37 38 40 48 49 59

10094 10085 10019 9971 9940 9791 9753 9625

35 36 37 39 40 47 50 58

0 0 0 0 0 0 0 0

0 0 0 1 0 -1 1 -1

0 0 0 -1 0 1 -1 1

666998 1776062 996512 2603607 875583

-0.03 -0.03 -0.031 -0.031 -0.021

-0.005 -0.005 -0.005 -0.006 -0.01

9701 9701 9698 9681 9651

60 61 62 64 66

9629 9629 9626 9610 9575

60 61 62 64 66

9606 9606 9603 9587 9551

60 61 62 64 66

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

158149 470658 922516 210554 226778 1616255 222581 1644561 1744930 2632956 1809034 618203 1025598 602894 1862951 521239 878760 629811 479198 1337726 549033 1100491 716998 440888 432345 191701 201437 162453 126697 406934

0.001 -0.018 -0.012 -0.018 -0.009 -0.02 -0.01 0.012 -0.021 -0.005 -0.056 -0.038 -0.02 -0.026 -0.033 -0.035 -0.018 -0.001 -0.002 0.016 0.05 0.008 -0.007 0.058 0.008 0 -0.029 -0.027 -0.013 -0.029

-0.019 -0.017 -0.019 -0.018 -0.021 -0.026 -0.03 -0.035 -0.03 -0.034 -0.031 -0.035 -0.046 -0.05 -0.05 -0.05 -0.054 -0.058 -0.059 -0.065 -0.073 -0.066 -0.067 -0.082 -0.077 -0.076 -0.078 -0.079 -0.083 -0.08

9580 9546 9534 9530 9512 9391 9361 9356 9322 9313 9182 9180 9063 8975 8951 8943 8938 8933 8913 8878 8867 8833 8763 8749 8653 8641 8505 8495 8480 8472

72 73 74 75 76 90 93 94 98 100 110 111 129 139 140 141 142 143 144 147 148 149 155 159 163 164 172 173 177 178

9495 9470 9456 9454 9433 9318 9284 9270 9250 9234 9127 9117 8994 8912 8890 8884 8872 8860 8840 8800 8776 8759 8697 8657 8583 8574 8452 8442 8422 8420

72 73 74 75 76 90 93 94 98 100 110 111 129 139 140 141 142 143 144 147 148 149 155 159 163 164 172 173 177 178

9468 9446 9431 9429 9407 9294 9258 9241 9226 9208 9107 9095 8970 8888 8867 8861 8847 8833 8814 8770 8742 8731 8670 8622 8555 8547 8429 8419 8397 8397

72 73 74 75 76 90 92 94 98 101 110 111 129 138 140 141 142 143 144 147 148 149 155 159 163 165 172 173 178 177

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 -1 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1

0 0 0 0 0 0 1 0 0 -1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 1

Population

Difference Rank QM- SP SP - FM QM - FM (9) (10) (11) 0 0 0

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

27

Table 1. Free Mobility and Social Planner Ranking of Cities

City Davenport–Moline-Rock Island, IA–IL Spokane, WA Lake Charles, LA Augusta–Aiken, GA–SC San Antonio, TX Wausau, WI Little Rock-North Little Jackson, TN State College, PA Tyler, TX Roanoke, VA Glens Falls, NY Lafayette, LA Scranton-WilkesBarre-Hazleton, PA Athens, GA Sioux Falls, SD La Crosse, WI–MN Asheville, NC Erie, PA LakelandWinter Haven, FL Oklahoma City, OK Mansfield, OH St. Cloud, MN ShreveportBossier City, LA Muncie, IN Columbus, GA–AL Mobile, AL Panama City, FL Eau Claire, WI Binghamton, NY FayettevilleSpringdale-Rogers, AR Auburn–Opelika, AL Monroe, LA Sioux City, IA–NE Williamsport, PA Waterloo–Cedar Falls, IA Myrtle Beach, SC Longview-Marshall, TX Pensacola, FL Topeka, KS Lynchburg, VA Odessa–Midland, TX Terre Haute, IN El Paso, TX Florence, AL Pueblo, CO Lubbock, TX Fort Walton Beach, FL Fargo-Moorhead, ND-MN Sharon, PA Columbia, MO Johnson City-KingsportBristol, TN–VA Ocala, FL Fayetteville, NC Gadsden, AL Billings, MT Altoona, PA Jamestown, NY St. Joseph, MO Alexandria, LA Danville, VA Springfield, MO Goldsboro, NC Fort Smith, AR–OK

359062

Amenity Estimate QOL Production (1) (2) -0.02 -0.082

Quantity Mech. Value Rank (3) (4) 8471 179

Social Planner Value Rank (5) (6) 8416 179

Free Market Value Rank (7) (8) 8392 179

417939 183577 477441 1592383 125834 583845 107377 135758 174706 235932 124345 385647 624776

0.002 -0.06 -0.044 -0.016 -0.046 -0.003 -0.048 0.04 -0.013 -0.015 -0.015 -0.03 -0.031

-0.091 -0.079 -0.083 -0.091 -0.086 -0.098 -0.092 -0.113 -0.103 -0.107 -0.108 -0.105 -0.107

8402 8378 8370 8337 8313 8269 8207 8177 8151 8079 8062 8058 8022

186 187 188 189 190 196 202 203 204 207 208 209 212

8342 8338 8325 8285 8270 8215 8168 8115 8105 8036 8020 8020 7985

186 187 188 189 190 196 202 203 204 207 208 209 212

8314 8319 8304 8260 8250 8189 8148 8082 8080 8011 7996 7997 7963

187 186 188 189 190 196 202 203 204 207 209 208 213

0 0 0 0 0 0 0 0 0 0 0 0 0

1 -1 0 0 0 0 0 0 0 0 1 -1 1

-1 1 0 0 0 0 0 0 0 0 -1 1 -1

153444 172412 126838 225965 280843 483924

0.019 0.007 -0.003 0.055 -0.036 -0.014

-0.12 -0.118 -0.116 -0.13 -0.112 -0.117

7987 7977 7974 7953 7922 7919

218 219 220 221 222 223

7941 7934 7934 7900 7891 7884

218 219 220 221 222 223

7912 7907 7908 7867 7870 7859

218 220 219 222 221 223

0 0 0 0 0 0

0 1 -1 1 -1 0

0 -1 1 -1 1 0

1083346 175818 167392 392302

-0.003 -0.049 -0.049 -0.029

-0.12 -0.112 -0.112 -0.118

7908 7875 7875 7848

224 228 229 230

7872 7850 7850 7821

224 228 229 230

7847 7830 7830 7799

224 226 227 230

0 0 0 0

0 -2 -2 0

0 2 2 0

118769 274624 540258 148217 148337 252320 311121

-0.035 -0.008 -0.009 0.031 -0.025 -0.047 0.005

-0.117 -0.123 -0.123 -0.133 -0.122 -0.118 -0.132

7843 7841 7837 7817 7797 7784 7741

231 232 233 234 235 236 239

7817 7810 7807 7782 7772 7763 7716

231 232 233 234 235 236 239

7796 7786 7783 7752 7750 7745 7692

231 232 233 234 235 236 241

0 0 0 0 0 0 0

0 0 0 0 0 0 2

0 0 0 0 0 0 -2

115092 147250 124130 120044 128012 196629 208780 412153 169871 214911 237132 149192 679622 142950 141472 242628 170498 174367 120293 135454 480091

-0.019 -0.029 -0.025 -0.035 -0.019 0.042 -0.036 0.014 -0.018 -0.031 -0.049 -0.06 -0.032 -0.04 0 0.003 0.067 -0.008 -0.034 0.025 -0.022

-0.127 -0.125 -0.126 -0.126 -0.13 -0.146 -0.132 -0.144 -0.139 -0.138 -0.135 -0.134 -0.141 -0.143 -0.152 -0.153 -0.171 -0.156 -0.151 -0.165 -0.156

7736 7733 7731 7695 7687 7644 7593 7577 7543 7513 7497 7474 7460 7399 7395 7389 7326 7301 7289 7272 7250

240 241 242 245 246 247 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264

7716 7714 7712 7680 7671 7628 7590 7574 7543 7513 7497 7474 7460 7399 7395 7389 7326 7301 7289 7272 7250

240 241 242 245 246 247 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264

7694 7694 7691 7662 7651 7603 7579 7565 7543 7513 7497 7474 7460 7399 7395 7389 7326 7301 7289 7272 7250

239 240 242 243 245 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-1 -1 0 -2 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 2 1 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

258916 302963 103459 129352 129144 139750 102490 126337 110156 325721 113329 207290

-0.003 0.03 -0.072 0.011 -0.05 -0.063 -0.026 -0.025 -0.056 0.002 -0.003 -0.018

-0.162 -0.171 -0.149 -0.168 -0.155 -0.155 -0.165 -0.168 -0.162 -0.176 -0.175 -0.175

7221 7192 7185 7173 7166 7119 7088 7043 7029 7010 7008 6954

265 266 267 268 269 270 271 272 273 274 275 276

7221 7192 7185 7173 7166 7119 7088 7043 7029 7010 7008 6954

265 266 267 268 269 270 271 272 273 274 275 276

7221 7192 7185 7173 7166 7119 7088 7043 7029 7010 7008 6954

265 266 267 268 269 270 271 272 273 274 275 276

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

Population

Difference Rank QM- SP SP-FM QM - FM (9) (10) (11) 0 0 0

28

NATHAN SEEGERT

Table 1. Free Mobility and Social Planner Ranking of Cities

City Hattiesburg, MS Sumter, SC Las Cruces, ClarksvilleHopkinsville, TN-KY Dothan, AL Killeen–Temple, TX Anniston, AL Laredo, TX Johnstown, PA Wichita Falls, TX Abilene, TX Brownsville–Harlingen-San Benito, TX McAllen–Edinburg–Mission, TX Joplin, MO Washington–Baltimore, DC–MD–VA–WV C Stockton–Lodi, CA Atlanta, GA Dallas–Fort Worth, TX C Modesto, CA Houston–Galveston-Brazoria, TX C Providence–Fall River-Warwick, RI–MA Charlotte–Gastonia-Rock Hill, NC–SC Cleveland–Akron, OH C Memphis, TN–AR–MS Indianapolis, IN Bakersfield, CA Bloomington–Normal, IL Nashville, TN Lansing–East Lansing, MI Grand Rapids–Muskegon-Holland, MI Harrisburg–Lebanon-Carlisle, PA Janesville–Beloit, WI Reading, PA non-metropolitan areas, IN non-metropolitan areas, GA non-metropolitan areas, LA non-metropolitan areas, IL non-metropolitan areas, KY non-metropolitan areas, IA non-metropolitan areas, TN non-metropolitan areas, MN York, PA non-metropolitan areas, TX non-metropolitan areas, AR Peoria–Pekin, IL Appleton–Oshkosh-Neenah, WI non-metropolitan areas, KS Pittsburgh, PA non-metropolitan areas, MO Columbia, SC Sheboygan, WI Beaumont– Port Arthur, TX Greenville, NC Decatur, AL Jackson, MS Chattanooga, TN–GA Albany, GA Duluth-Superior, MN-WI Houma, LA Biloxi-GulfportPascagoula, MS

111674 104646 174682 207033

Amenity Estimate QOL Production (1) (2) -0.024 -0.176 -0.026 -0.177 0.027 -0.19 0.012 -0.192

Quantity Mech. Value Rank (3) (4) 6915 277 6891 278 6871 279 6784 280

Social Planner Value Rank (5) (6) 6915 277 6891 278 6871 279 6784 280

Free Market Value Rank (7) (8) 6915 277 6891 278 6871 279 6784 280

137916 312952 112249 193117 232621 140518 126555 335227

-0.033 0.04 -0.048 0.009 -0.064 0.012 0.014 -0.041

-0.183 -0.208 -0.189 -0.207 -0.193 -0.212 -0.221 -0.227

6768 6626 6615 6529 6491 6457 6318 6018

281 282 283 284 285 286 287 288

6768 6626 6615 6529 6491 6457 6318 6018

281 282 283 284 285 286 287 288

6768 6626 6615 6529 6491 6457 6318 6018

281 282 283 284 285 286 287 288

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

569463 157322 7608070

-0.069 -0.01 -0.012

-0.229 -0.246 0.12

5882 5821 11820

289 290 10

5882 5821 11724

289 290 9

5882 5821 11699

289 290 9

0 0 1

0 0 0

0 0 1

563598 4112198 5221801 446997 4669571

-0.008 -0.032 -0.033 -0.016 -0.06

0.08 0.063 0.057 0.047 0.049

11176 10813 10711 10605 10485

17 23 27 29 32

11081 10732 10631 10518 10420

16 22 26 28 31

11056 10710 10609 10494 10402

15 22 24 28 30

1 1 1 1 1

-1 0 -2 0 -1

2 1 3 1 2

1188613

-0.008

0.009

10008

43

9923

42

9897

42

1

0

1

1499293

-0.009

0.009

10005

44

9920

43

9894

43

1

0

1

2945831 1135614 1607486 661645 150433 1231311 447728 1088514

-0.017 -0.044 -0.038 -0.058 -0.064 -0.001 -0.043 -0.044

0.005 0.007 0.005 0.006 0.007 -0.015 -0.008 -0.008

9911 9850 9838 9784 9780 9639 9606 9603

47 51 52 54 55 68 70 71

9831 9782 9767 9723 9721 9554 9541 9537

46 50 51 53 54 67 69 70

9806 9761 9746 9704 9703 9527 9519 9517

46 49 51 53 54 67 68 69

1 1 1 1 1 1 1 1

0 -1 0 0 0 0 -1 -1

1 2 1 1 1 1 2 2

629401

-0.033

-0.018

9477

78

9408

77

9386

77

1

0

1

152307 373638 1791003 2744802 1415540 2202549 2828647 1863270 2123330 1565030 381751 4030376 1607993 347387 358365

-0.045 -0.052 -0.055 -0.039 -0.061 -0.056 -0.063 -0.029 -0.036 -0.043 -0.041 -0.039 -0.027 -0.063 -0.02

-0.017 -0.017 -0.017 -0.026 -0.026 -0.029 -0.028 -0.037 -0.036 -0.035 -0.036 -0.038 -0.046 -0.04 -0.05

9451 9427 9416 9324 9247 9215 9207 9178 9170 9162 9153 9127 9038 9009 8996

83 85 86 97 105 107 109 113 115 117 119 123 132 134 136

9388 9367 9358 9260 9193 9159 9154 9112 9107 9102 9092 9065 8973 8959 8930

82 84 85 96 104 106 108 112 114 116 118 122 131 133 135

9367 9347 9338 9238 9174 9140 9135 9089 9085 9081 9070 9044 8950 8940 8906

81 84 85 95 103 106 107 112 115 116 118 121 130 132 135

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-1 0 0 -1 -1 0 -1 0 1 0 0 -1 -1 -1 0

2 1 1 2 2 1 2 1 0 1 1 2 2 2 1

1366517 2358695 1798819 536691 112646 385090

-0.02 -0.038 -0.021 -0.003 -0.024 -0.093

-0.05 -0.047 -0.056 -0.071 -0.067 -0.063

8996 8982 8894 8712 8703 8525

137 138 146 161 162 171

8930 8923 8830 8645 8644 8492

136 137 145 160 161 170

8906 8901 8806 8618 8620 8477

136 137 145 161 160 170

1 1 1 1 1 1

0 0 0 1 -1 0

1 1 1 0 2 1

133798 145867 440801 465161 120822 243815 194477 363988

-0.024 -0.069 -0.02 -0.021 -0.06 -0.065 -0.048 -0.008

-0.084 -0.082 -0.095 -0.095 -0.097 -0.099 -0.105 -0.128

8424 8297 8258 8254 8082 8032 7994 7759

185 192 198 199 206 211 217 238

8372 8261 8209 8206 8051 8003 7962 7735

184 191 197 198 205 210 216 237

8348 8244 8185 8182 8032 7985 7943 7712

182 191 197 199 205 210 216 237

1 1 1 1 1 1 1 1

-2 0 0 1 0 0 0 0

3 1 1 0 1 1 1 1

Population

Difference Rank QM- SP SP-FM QM-FM (9) (10) (11) 0 0 0 0 0 0 0 0 0 0 0 0

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

29

Table 1. Free Mobility and Social Planner Ranking of Cities

City Daytona Beach, FL Utica–Rome, NY Anchorage, AK Kokomo, IN Richland–Kennewick-Pasco, WA Springfield, MA Rochester, MN non-metropolitan areas, MI Birmingham, AL non-metropolitan areas, WI Dayton–Springfield, OH non-metropolitan areas, PA Kalamazoo–Battle Creek, MI Buffalo–Niagara Falls, NY Huntsville, AL Syracuse, NY Lafayette, IN Fort Wayne, IN South Bend, IN Champaign–Urbana, IL Wichita, KS Corpus Christi, TX Decatur, IL Youngstown–Warren, OH Lima, OH Waco, TX non-metropolitan areas, OH non-metropolitan areas, AL Saginaw–Bay City-Midland, MI non-metropolitan areas, MS Rockford, IL Jackson, MI Macon, GA

493175 299896 260283 101541 191822

Amenity Estimate QOL Production (1) (2) 0.032 -0.14 -0.057 -0.125 0.024 0.075 -0.111 0.032 -0.049 0.015

Quantity Mech. Value Rank (3) (4) 7706 244 7633 249 11205 16 10027 41 9964 46

Social Planner Value Rank (5) (6) 7683 243 7626 248 11093 14 9988 39 9897 44

Free Market Value Rank (7) (8) 7655 244 7612 247 11063 14 9976 38 9877 44

591932 124277 2178963 921106 1866585 950558 2023193 452851 1170111 342376 732117 182821 502141 265559 179669 545220 380783 114706 594746 155084 213517 2548986 1504381 403070

-0.007 -0.06 -0.05 -0.032 -0.025 -0.031 -0.054 -0.053 -0.045 -0.055 -0.061 -0.009 -0.059 -0.042 -0.006 -0.044 -0.019 -0.086 -0.052 -0.066 -0.037 -0.057 -0.068 -0.066

-0.007 0 -0.015 -0.019 -0.025 -0.028 -0.025 -0.034 -0.039 -0.055 -0.055 -0.067 -0.065 -0.069 -0.077 -0.076 -0.083 -0.08 -0.089 -0.1 -0.114 -0.018 -0.03 -0.031

9749 9678 9467 9464 9390 9320 9288 9143 9090 8791 8769 8756 8612 8607 8603 8485 8458 8269 8242 8012 7885 9393 9156 9147

58 65 80 81 91 99 103 121 126 152 154 158 167 168 169 176 183 195 201 215 227 89 118 120

9666 9619 9405 9395 9319 9252 9230 9087 9031 8741 8722 8690 8567 8556 8540 8437 8403 8239 8204 7984 7857 9335 9106 9096

56 63 78 79 89 97 101 119 124 150 152 156 165 166 167 174 181 193 199 213 225 86 115 117

9640 9600 9385 9373 9296 9229 9211 9067 9010 8721 8703 8664 8548 8534 8513 8416 8379 8224 8184 7966 7836 9316 9088 9078

55 63 78 79 88 96 100 119 124 150 152 156 164 166 167 174 181 193 198 212 225 86 113 117

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3

-1 0 0 0 -1 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 -1 0 0 -2 0

3 2 2 2 3 3 3 2 2 2 2 2 3 2 2 2 2 2 3 3 2 3 5 3

1869256 371236 158422 322549

-0.062 -0.069 -0.068 -0.058

-0.036 -0.018 -0.036 -0.074

9079 9350 9058 8467

128 95 131 182

9027 9298 9008 8425

125 91 127 176

9008 9281 8991 8406

125 91 126 176

3 4 4 6

0 0 -1 0

3 4 5 6

Population

Difference Rank QM- SP SP-FM QM-FM (9) (10) (11) 1 1 0 1 -1 2 2 0 2 2 -1 3 2 0 2

30

NATHAN SEEGERT

Figure 3. Equivalent Variation Indifference Curves

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

Figure 4. Free Mobility: Equivalent and Compensating Variation

31

32

NATHAN SEEGERT

Figure 5. Quantity Mechanism: Equivalent and Compensating Variation

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

Figure 6. Social Planner: Equivalent and Compensating Variation

33

34

NATHAN SEEGERT

Figure 7. Social Planner Total Population: Compensating Variation

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

(a) Market Mechanism Fees Collected Pre- (b) Market Mechanism Fees Collected Post Shock Shock

(c) Market Mechanism Compensating Variation (d) Social Planner Compensating Variation Total Population

Figure 8. Compensating Variation Social Planner and Market Mechanism

35

36

NATHAN SEEGERT

7. APPENDIX: Result 2 Result 2: For a given ordering of city creation; a system of cities with free mobility creates the fewest cities, a system of cities with the quantity mechanism creates the most cities, and a system of cities with the price mechanism creates the efficient number of cities. Result 2 follows from the following arguments which demonstrate the planner produces at least as many cities as the system of cities with free mobility and creates the same number of cities as the system of cities with the price mechanism. The planner produces at least as many cities as the system of cities with free mobility. Assume toward contradiction that the planner produces K − n cities and the system of cities with free mobility produces K cities. In equilibrium if K cities are created in the system with free mobility then the per-resident benefit with K cities must be larger than with K − n cities. This implies the total benefit produced in the system of cities with K cities is greater than the system with K − n a contradiction. Therefore, the planner produces at least as many cities as the system of cities with free mobility. The system of cities with the price mechanism creates the efficient number of cities. First, the distribution of population across a given number of cities is the same for the price mechanism and the social planner. Second, assume toward contradiction the case where social planner creates more cities than the system with the price mechanism. In this case the equilibrium fee that would be charged in the system with the price mechanism by definition is negative. The equilibrium fee is negative when the population in the city is less than the capped population. In the second stage there is no trembling hand perfect equilibria where the population of a city is positive and less than the capped population, a contradiction. Third, consider toward contradiction the case where the social planner creates fewer cities than the system with the price mechanism. In this case the equilibrium fee that is charged in the additional cities is positive by definition. However, if the fee is positive there is excess total benefit that is not being realized when the social planner

BARRIERS TO MIGRATION IN A SYSTEM OF CITIES

37

creates the cities, which is contradiction of the social planner’s objective. Therefore the system of cities with the price mechanism creates the efficient number of cities.15

8. Appendix: Simulation Algorithm The simulation uses an algorithm that solves the integer problem of how many cities to create by reframing the intensive margin in a way that ensures a solution in a fixed number of steps. The first step solves the population in each city for a given shared level of benefit. The population in each city is calculated for all values of the shared benefit equal to the discrete benefits calculated for city 1. This step produces a matrix with rows representing each value of the shared benefit and each column representing a different city. The second step constrains this matrix such that the sum of the populations across all created cities is less than or equal to the total population. This constricts the matrix down to a vector with rows representing each level of shared benefit and each entry giving the number of cities that would be created. The third step performs the extensive margin optimization for each case. The social planner’s objective is to maximize the total benefit produced. Therefore the algorithm chooses the marginal benefit level from the constrained set that produces the largest total benefit. Individuals with free mobility create the number of cities in the first stage that maximizes the shared average benefit from the constrained set. Individuals that are able to cap city size produces the maximum number of cities in the constrained set. Individuals that are able to set fees create the number of cities in the first stage such that the equilibrium fee charged in all cities is nonnegative. E-mail address: [email protected] 15In

two special cases the number of cities created is the same in all cases. The first case is when all cities are homogeneous. The number of cities created is the same because the population in the cities is the same for all cases except for the system able to cap city sizes. The second case is when the total population is large enough such that the maximum average benefit in the K + 1 city is less than the hinterland benefit. In this case the hinterland and K cities are inhabited but the population distribution across cities differ.

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