2.1. Spatial organization, costs and preferences

12 We consider a standard urban monocentric model with linear transportation costs. All activities and public facilities take place at the center of the city, or central business district (CBD). The transportation costs incurred by resident households are directly measured by their distance (l) from the CBD. Available space at distance l is represented by the function Φ (l), and the consumption of space by a household at this location is denoted by x (l). In other words, 1/x (l) defines urban residential densities. Beyond the (endogenous) limit of the city (L), land rents equal agricultural rents (a).

13 Our aim is to analyze the management and funding of a public facility which simultaneously provides the residents of the city a pure public good (z) and multiple “private” services. The public good can take different levels (z) and we denote by Y = (y_i ) the vector of services provided to one household (each y_i representing the consumption of a type of service, with i ∈ I the set of supplied services, such as transport services, shops, restaurants etc.)

14 Household preferences are quasi-linear and separable into different subgroups of goods and services. This hypothesis will make the vector Y of services purchased by residents always identical for all of them. The willingness u (x) to pay for a surface x of land is identical for all potential residents and satisfies usual properties: u (0) = 0; u′ > 0; u″ < 0. Hence a plot of size x at distance l is (socially) valued (u (x) − l) for each of them.

15 Let σ (z, Y, h) denote the value of urban services provided (z) or purchased (Y) by a certain resident, h representing the cost of construction of his housing (a higher h being supposed to provide him a higher well-being). We will write:

∂σ ∂σ
σ ′ (z, Y, h) = (z, Y, h), or simply σ ′ =
y_{i ∂y}i y_{i ∂y}i

16 The level of h will be decided by landowners, who are assumed to be directly the developers of their land. For the sake of simplicity, we assume σ separable between (z, Y) and (h). This separability allows us to define a simple underlying demand for quality of housing (h). If we denote (p_h) its price, this demand H (p_h ) is such that , and we denote . H′ (p_h) its price-elasticity.

17 For example, if the project is a new metro network such like Grand Paris Express, (z) would represent the offered quality of accessibility to the CBD, the number of trips being described by a component of the vector (Y). So, there is no a priori distinction between services directly bound up with the nature of the facility (here transportation services), and what could be described as additional services. The distinction is only between pure public good services (z) and services (Y) which induce direct costs and can be priced. However, the complementarity or potential substitutability between the different services are embedded in the willingness to pay for these different services (σ).

18 Potential residents (indexed by n) differ only by the specific value (b (n)) that they attribute to residing in this city. For perfectly mobile residents between cities, this value is zero, but it may be high for weakly mobile ones. We assume potential residents ranked by increasing ability to leave the city, so that:

b ′ (n) ≤ 0 and lim b (n) = 0
n→∞

19 This specification encompasses all possible situations between the two extreme cases: that of the “open city” which underpins the standard economic doctrine, with all potential residents perfectly mobile to other cities (b (n) = 0, ∀n); and that of the “closed city” (fixed population, vertical function b (n)).

20 We assume a simple production function for the public facility, with infrastructure costs (f (z), f ′ > 0) depending on the level of z, and constant marginal costs (c_i) for each of the services (or, globally, vector C). Hence, if the population of the city is N, the total cost for the operator will be:

f (z) + N.C · Y with C · Y = Σ (c_i .y_i)

21 Let V denote the social value of the amenities and services provided by the public facilities to a resident, net of the direct costs associated with their provision, and net of the construction cost of his housing. With our assumptions on the utility functions, this will be a common value at the city level.

V = σ (z, Y, h) − C · Y − h [1]

22 Thus, the social value provided by the location of agent “n” in the city, at distance t from its CBD with an area of land x, equals (by reference to his location outside the city):

V + u (x) − l + b (n) − a.x

23 This value includes three terms: the common net value of services offered by the infrastructure (V); plus the specific amenity value (b (n)); less the “urban costs” (l + a.x − u (x)). In the same manner, we denote by ν the corresponding (common, net) private value of services, if the vector of prices p_i for the different services y_i is (P), and if a residence-based tax on household τ is applied:

ν = σ (z, Y, h) − P · Y − τ [2]

24 The net private benefit of this location is then:

ν + u (x) − l − r.x + b (n)

25 r being the level of rents (by unit of consumption of space) paid to the landowner.

26 Since V and ν will be common values for all residents, the definition of the assignment of funding instruments can distinguish four steps: first, we will consider the optimal organization of land in the city for a given level of its population; then, we will introduce tools to generically characterize city size efficiency; the analysis of distortions associated with the urban equilibrium follows; and provides the basis for the derivation of optimal management, pricing and fiscal policy rules.

2.2. Aggregate urban costs

27 The optimal internal organization of the city faces the usual trade-off between densities and transportation costs. Since the population is homogeneous at this level, the only variables to be considered to characterize it are the limit of the city and the structure of urban densities.

28 Let A (N) denote the minimum overall urban costs, for any given N. This is the result of the following program (3), in which the constraint reflects the assumption of a given population size (N) and the objective incorporates the different urban costs mentioned above (eviction of agricultural land and transportation costs, net of housing amenities).

{_{∫( ∫}^L₎^u(x(l))−l
Min_{( L, x ( l ) )} ⌊ () ⎥
Φ (l).dl = A (N)
a−
0^x(l)
[3]
L Φ (l)
(μ (N)) s.t. .dl = N
0^x(l)

29 The corresponding Lagrange multiplier is denoted by (μ (N)), which is to be interpreted as the marginal social cost of locating an additional resident in the city:

A ′ (N) = μ (N) μ ′ (N) > 0 [4]

30 The necessary conditions of this program express the indifference between the alternative possibilities to increase the population of the city at the margin, by increasing densities at any distance l or by enlarging the area of urbanized land. They can be written:

∀l ≤ L, u′ (x (l)).x (l) = μ (N) + u (x (l)) − l
{_∫(^{μ(N) +}₎^{u(x(L )) − L}
a = = u′ (x (L))
x (L) [5]
L Φ (l)
.dl = N
0^x(l)

2.3. Efficient city size function

31 The total surplus provided by the city is the sum of common and specific residents’ benefits less urban costs. The optimal city size would maximize this surplus. But fiscal distortions will impact this size. For evaluating city size inefficiency induced by fiscal distortions it is useful to generally introduce the following function of two parameters (U, N):

⌊∫^N ⎥
∀ (U, N), S (U, N) = U.N + b (n).dn − A (N) [6]
0

32 The argument U will refer to a value of common amenities provided to the residents. This formulation allows for considering social or market values. In particular, if U = V (the social value of the amenities and services provided by the public facility to a resident, net of the direct costs associated with its provision, and net of the construction cost of his housing), S (V, N) represents the social (net) benefits of the city.

33 Let N̆ (U) denote the population size which maximizes S (U, N) for a given value of U, and S̆ (U) the corresponding outcome of the following program:

maxS (U, N) = S̆ (U)
N

34 This optimum is characterized by the following condition:

U + b (N̆ (U)) = μ (N̆ (U)) [7]

35 Combining with (5), this optimal size N = N̆ (U) and the associated efficient internal organization of the city verify:

∀l ≤ L, u′ (x (l)).x_N (l) − u (x (l)) + l = U + b (N)
{^U_∫^{+ b}₍^{(N) + u}₎^{(x(L )) − L}
a = = u′ (x (L))
x (L) [8]
L Φ (l)
with .dl = N
0^x(l)

36 A marginal variation of U implies a modification of the efficient population size such that:

N̆′ (U) = 1/ (μ ′ (N̆ (U)) − b ′ (N̆ (U))) [9]

37 From the envelope theorem, the associated impact on S equals:

S̆′ (U) = N̆ (U) (≥ 0) [10]

38 We will denote by ε_N the elasticity of this optimal population as a function of U:

ε_N = U.N̆′ (U)/N̆ (U)

39 And we define:

k (N) = μ ′ (N)/ (μ ′ (N) − b ′ (N))

40 This parameter equals 1 for the “open city”; is zero if residents are immobile ( “closed city”); and is intermediate between these extreme values in the general case. We also introduce the function:

R̆ (U) = (U + b (N̆ (U))). N̆ (U) − A (N̆ (U))
∫^{N̆ ( U)}
=S̆ (U) − [b (n) −b (N̆ (U))].dn [11]
0

41 which verifies:

R̆′ (U) = k (N̆ (U)). N̆ (U) [12]

42 This means that in the case of the open city, an improvement of U will pass fully into R̆ (U); it will not in the case of the closed city.

43 Since S (V, N) is the gross social benefit brought by the city, its “first-best” size (for a given value of V) would be N̆ (V). Then (7) reflects the equality between the marginal cost (μ (N)) induced by the “last” resident in the city and the social benefits of his location in the city, when the city size is optimal. Given the marginal cost-function μ (N) and b (n), this determination of an optimal population size can be described in a standard partial equilibrium framework (cf. figure 1, the shaded area depicting R̆ (V)).

Figure 1

First-best city size representation

First-best city size representation

2.4. Competitive equilibrium of land markets

44 Let us now consider the urban equilibrium. If we consider the “private” value of city services for the residents (ν = σ (z, Y, h) − P · Y − τ) as a given, r (l) and x (l) being respectively the level of rents and size of plots at distance l from the CBD, the equilibrium of land markets requires: the residents to be indifferent between locations inside the city; and the ‘marginal’ resident (n = N) to be indifferent with regard to locating inside it or not. This leads to the following conditions:

}^{∀l ≤ L, ν + u ( x ( l ) ) − l − r ( l ).x ( l ) = cst}
[13]
∀l ≤ L, ν + u (x (l)) − l − r (l).x (l) + b (N) = 0

45 If we consider that the property revenues above the level of the agricultural rents are taxed at rate t, profits obtained by landowners-developers at distance l from the CBD are:

π (l) = (1−t). (r (l) −a− h) [14]
(1 − t).x (l)

46 Since (13), the maximal level of rents that they can obtain by unit of space verifies:

r (l) = (ν + b (N) + u (x (l)) − l)/x (l) [15]

47 with ν = σ (z, Y, h) − P · Y − τ. Landowners choose h and x (l) so as to maximize their profits under this constraint. For h, this leads to the following first order condition, which reflects the distortions in quality induced by this imperfect land value tax:

σ ′_h (h) = 1/ (1 − t) or h = H (₁¹_−t)

48 Thus, the quality of housing is determined by t only. Furthermore, the equilibrium densities and land prices satisfy:

}^{∀l ≤ L, u′ ( x ( l ) ).x ( l ) = ν + b ( N ) + u ( x ( l ) ) − l − h/ ( 1 − t)}
h [16]
r (L) − = a = u′ (x (L))
(1 − t).x (L)

49 Comparing with (8), we observe that this spatial organization is similar to what should be the optimal one as regards to S (U, N) for :

N = N̆
(₁^h_−t)
[17]
ν−

50 This size satisfies:

51 Proposition 1: competitive land markets ensure an efficient internal spatial organization of the city but its size and the quality of buildings are distorted.

52 We will denote by w the (net) value:

53 The deviation between V and w corresponds to a tax wedge, which equals:

V − w = ((P − C) · Y + τ) + th/ (1 − t)

54 V − w represents the “global” taxation on the residential market, which determines the distortion in population size:

N = N̆ (w) with w + b (N̆ (w)) = μ (N̆ (w))

55 Furthermore, the global amount of differential land rents is:

∫^L
R = (r (l) − a). Φ (l).dl = R̆
(^h)(^h)(^h
ν− + . N̆ ν−
)
[18]
1−t 1−t 1−t
0

56 The corresponding amount of aggregated profits is thus:

1Π− t
= R̆ ν −
(₁^h_−t)
[19]

57 Figure 2 below illustrates this distortion in city size resulting from a deviation between w and V and shows the associated level of landowners’ profits:

Figure 2

Size of the city at the urban competitive equilibrium

Size of the city at the urban competitive equilibrium

3.1. Problem of the public Authority

58 Let us consider now the overall problem of management and funding of the platform, in a realistic world where capitalization is not perfect and where the public Authority has only two imperfect fiscal instruments at its disposal: household tax τ and linear property tax t. She can also use pricing policy (P) for the different services offered by the facility to comply with its budgetary constraint B ≥ 0. Indifferently, we can consider that the public Authority directly controls (Y) and then applies (P) such that: .

59 In this model, all available instruments for funding the public facility have distortionary impacts: the property tax modifies landowners’ choices regarding building quality; the household tax makes the city less attractive and has a direct negative impact on the city’s population size; markups above the direct costs of merchant services provided by the facility distort the demand for its services. The funding scheme must minimize the burden of all these distortions, the optimal second-best policy maximizing social welfare under market and fiscal constraints. This welfare objective, which balances social benefits S (V, N) and infrastructure costs f (z), can be written:

W = S (V, N) − f (z) [20]

60 with V = σ (z, Y, h) − C · Y − h and N = N̆ (w).

61 Hence W = S̆ (w) + (V − w). N̆ (w) − f (z), or:

W = S̆ (w) + (σ − C · Y − h − w). N̆ (w) − f (z) [21]

62 The budgetary balance of the public Authority is:

B = t. R + (P · Y + τ − C · Y). N − f (z)

63 Having in mind that:

(
1
)
th
; V−w= ((P−C)·Y+τ) +
h=H
1
1−t
−t

64 B can also be written:

B = t. [R̆ (w) + (₁^h_−t ). N̆ (w)] + (V − t. (₁^h_−t ) − w). N̆ (w) − f (z)
B = t. R̆ (w) + (σ − C · Y − h − w). N̆ (w) − f (z); B ≥ 0 [22]

3.2. Optimal policy

65 The public Authority maximizes W under the constraint B ≥ 0, with the instruments at its disposal. We note as λ the corresponding Lagrange multiplier and thus:

L=L (τ, t, Y, z, λ) = W + λB

66 This Lagrangian function can be written as follows, the impact of the household tax only passing through w:

L= S̆ (w) + tλR̆ (w) + (1 + λ). [(σ (z, Y, h) − C · Y − h − w) N̆ (w) − f (z)] [23]

67 The first-order necessary conditions of this program are thus (with simplified notations for better visibility, and using the condition on (τ) to simplify the other ones):

λ
(τ) = ()
()
1
V−w
. (1 − k (N̆ (w)).t).
1+λ
ε_N
w
⌊^{R̆ ( w )} ⎥
{^{(t) t=(1+λλ). (1−t)2.N̆}
h . (1/ε^h ) [24]
(w).
(Y) ∀i ∈ I, σ ′_y = c_i or p_i = c_i
i
(z) f ′ (z) = N̆ (w). (∂σ/∂z)

68 Proposition 2: Despite imperfect fiscal instruments, marginal cost pricing and Bowen-Lindhal-Samuelson rule prevail for pricing policy and for the selection of the level of public good.

69 This result comes from conditions (Y) which define the optimum levels for the different public services or retailing activities. They show that marginal cost pricing (∀i ∈ I, p_i = c_i) should apply for all of the services provided by the public facility: quite surprisingly, the optimal pricing policy remains “marginal cost pricing”, despite the constraints on available fiscal instruments.

70 Similarly, the level of public good (z) should satisfy the standard Bowen-Lindhal-Samuelson (BLS) rule, for the (endogenous) population of the city. Indeed, it was clear from (21) and (22), or (23), that optimal (Y) and (z) maximize [(σ (z, Y, h) − C · Y) N̆ (w) − f (z)], since w, and by these means the population of the city, can be controlled by (τ).

71 Thus, the consumers of the services (Y) should only bear their direct costs, coverage of the infrastructure costs of the facility being obtained exclusively from local taxes.

3.3. Structure of local taxes

72 The two first conditions (labelled (τ) and (t)) define the optimal tax rates and, thus, the optimal sharing between household taxes and property taxes. These reflect the potential distortions generated by the different taxes, under the minimization of associated deadweight losses. They are of Ramsey-Boiteux type, with markups () inversely proportional to the elasticities of demands, respectively for city size and quality of housing. Indeed, it appears a direct link between: property tax and quality of housing; and between (w), which is controlled by the household tax (τ), and the size of the city. Taking together the two equations, we obtain:

V−w
w = [(1 − kt)/ (1 − t) 2]. [N̆h/R̆]. εh [25]
t⁽ε_N̄⁾

73 If property taxes do not distort investments by landowners, they dominate the housing tax. Similarly, when the population of the city is not mobile, the household tax does not generate distortions, and is thus to be preferred. More generally, this markups ratio defines an ‘index’ of relative share of household tax to property tax.

74 Proposition 3: Conditions for (t) and (τ) reflect the trade-offs between raising revenues and distortions generated by these two variables. Ramsey-Boiteux rules apply at this level, with optimal “tolls” inversely proportional to the corresponding elasticities.

3.4. Assignment of instruments: comments

75 The condition on (τ) can also be written:

λ
(τ) τ + (P − C) · Y + h = w. ()
()
1
t
. (1 − k (N̆ (w)).t).
1+λ
1−t
ε_N

76 This formula helps to understand why pricing rules remain the first-best ones: the distortionary impact on the population size of the city coming from possible mark-ups in the pricing policy are the same as for the housing tax; but they would add Malthusian allocative inefficiencies on the level of corresponding activities. So, they are detrimental: in this context, the development of additional activities should not be seen as a source of revenues, but primarily a means to increase the social value created by the facility.

77 The rationale under this result is not that property taxes are perfectly neutral. They are not here. Moreover, the funding of the infrastructure costs f (z) is associated with a positive “Social Cost of Public Funds”, λ > 0.

78 The argument reflects a hierarchy of instruments: the household tax is a better tool than mark-ups for this funding. Indeed, this result is not surprising as the household tax and what should be a fixed part in the pricing scheme of services are strictly equivalent in this simple model.

79 This can also be seen in the logic of Jules Dupuits’s views (1849) that the best pricing system would be that which would “charge people according to how much each would be willing to pay”, to avoid usage discouragement.

4.1. Urban externalities

81 Among economic specificities of urban areas, we have focused on amenities provided by public facilities, which are among the rationales for concentration of activities in cities. Other factors are urban externalities, for example at the level of local job markets, or knowledge spillovers. Public facilities services interact with these externalities.

82 More generally, we must bear in mind that urban facilities provide services to the different sides of urban markets that they connect: residents and developers; retail; workers and firms etc. Moreover, these economic agents are users of public services, but (often) also contribute to their quality [2]. Thus, all direct and indirect externalities between these different sides should be incorporated into the analysis to define appropriate pricing rules, first of all those associated with the size of the city.

83 Indeed, modern urban economics (Combes and Lafourcade [2012]) emphasizes the role of agglomeration externalities for labor productivity of cities. Other types of externalities must also be considered, such as network externalities, which could be associated with the global level of the different services provided by the facility, or architectural externalities. In our model, the first ones could be measured, for example, by the mean value , and the latter ones, for example, by the mean value of construction quality [3] .

84 Hence, we can imagine that a more relevant specification for the value of urban amenities for each resident should be: . By these means [4], we incorporate agglomeration externalities (N), network externalities and architectural externalities . Necessary conditions (24) become:

{_⌊⎥⁽⁾
(Y) ∀i ∈ I, p_i = c_i − σ ′_y
i
(z) f ′ (z) = N̆ (w). (∂σ/∂z)
σ−C·Y−h+σ′ −w
N
λ
()
1
. (1 − k (N̆ (w)).t).
(τ) =
1+λ
ε_N
w
R̆ (w)
tλ
(t) =−σ′
h (
)
. (1 − t).
. (1/ε)
+
λ N̆ (w).
h
1−t
1+
h

85 Pricing rules turn to “social marginal cost” pricing, including external marginal costs of damages (or benefits) . The structure of local taxes is also modified. The condition on τ means that household taxes should be (relatively) reduced since locating in the city should be “subsidized” to mobilize agglomeration externalities ( . Similarly, property taxes should be lowered when urban quality impacts ( need to be taken account for. These latter effects act as indirect externalities between landowners and residents. Then, the results about the fiscal structure, between property and housing taxes, echo multi-sided markets formulas.

86 Proposition 4: The overall assignment of instruments remains valid with externalities: prices to guide demand for the services provided by the urban facility; local taxes for financing its infrastructure costs.

87 Of course, the model could be enriched to incorporate all other club and common quality effects and externalities which characterize urban activities. Integrating all – direct or cross – urban externalities is thus crucial for efficient pricing and fiscal structures.

4.2. Cases for Ramsey-Boiteux pricing

88 What arguments could explain or justify the fact that, instead above results, retailing activities are often seen as a source of revenues to finance fixed costs of local public facilities? Redistributive issues and political economy considerations seem the most natural explanations. However, it must be underlined that this is relevant in specific contexts only, in line with Dupuit’s reasoning.

89 For example, let us relax the assumption of a benevolent social planner maximizing a utilitarian welfare function and consider that urban choices result from political economy processes. The simplest alternative in this perspective is to consider that the policy is decided by a pivotal agent, this leading to deviations of the choices, according to his characteristics: type of resident in the b (n) dimension; relative magnitude of his wealth in city land.

90 However, the unique possible redistribution dimension with our set of taxes is between residents and landowners. In particular, our fiscal instruments cannot implement redistribution within residents, depending on b (n). Moreover, the net surplus of residents equals: [b (n) −b (N̆ (w))] .dn, and aggregate profits are (1 − t). R̆ (w). These two figures depend globally of w. Thus, the superiority of fiscal instruments for financing fixed costs remains, with marginal cost rules for pricing the services, whoever the pivotal agent is.

91 Nevertheless, when the city is composed of heterogeneous social groups, another questionable assumption concerns the ability to implement a uniform household tax: it is well-known that real local taxes incorporate a lot of political economy or redistributive constraints, as controversies about poll taxes or capitation have shown. In this perspective, the pricing of public services, and beyond, the development of the supply of such services, may be useful to target certain groups for redistributive purposes. For example, household taxes may be unfair for groups who do not benefit from the facility.

92 In this case, a trade-off remains between the redistributive objective which tends to favor markups, to alleviate the burden on such population, and the general funding objective which requires using the widest tax-base. In this perspective, markups are a narrower base than the household tax and, thus, they are more distorsive on the size of the city.

93 This suggests that the switch to Ramsey-Boiteux rules is primarily justified for services which are not local ones, for example services to non-residents. Indeed, it is obvious that, if we consider the case of a public facility which offers different services for residents (Y) and for non-residents (Y^X), whatever the weight that the social planner gives to the surplus of these consumers in his objective function, necessary conditions for the optimal prices of these latter services lead to standard monopoly or Ramsey-Boiteux multi-product markup structures.

94 If the services are similar, the operator being able to distinguish between consumers, this result can be interpreted as a legitimate cause [5] to discriminate: since the only way to make non-residents contribute to the coverage of fixed costs is to put markups on service prices, these are fair; residents contribute otherwise, by paying local taxes, which generate fewer distortions.

4.3. Regulatory issues

95 Although the question of the incentives of the operator to select welfare-improving projects is central for the economic recommendation in favor of property taxes, the above analysis has only been carried out from the fiscal incidence point of view, under the assumption that the social planner had perfect information.

96 But, again, regulatory constraints do not automatically change the general recommendation in favor of local taxes for funding facilities. Of course, the introduction of information asymmetries about the marginal cost of retailing activities would add informational costs reflecting the cost of extracting information about the operator’s performance by the regulator. But the basis for pricing would remain the perfect information case, here marginal cost, not Ramsey-Boiteux rules.

97 Traditionally, the most convincing argument in favor of average-cost pricing applies when the social planner does not have sufficient information on the value of the services provided by the facility. By constraining the operator to fund it only by markups, he ensures that fixed costs are justified: since consumers buy the services, their surplus is positive; the profit of the operator being zero, the global social surplus is positive. This reasoning transposes to our context, characterized by the following surpluses:

98

• the consumer surplus
• the profits of landowners Π = (1 − t). R̆ (w)
• the public budget surplus B = t. R + (P · Y + τ − C · Y). N̆ (w) − f (z), with R satisfying (18) and
• the overall surplus being W = Σ + Π + B.

99 Let us consider the case of a new investment improving the level of public service from z₀ to z, with no “private” service offered in the initial situation (Y = (0)). If the facility can be funded by markups on new additional services only (without higher tax rates), its implementation is socially justified because: Δw > 0 since Δw = [σ (z, Y) − P · Y − σ (z, (0))] + [σ (z, (0)) − σ (z₀, (0))], the first term being positive because these services are bought by residents and the second one because the level of z is higher. Thus Σ and Π increase, B also being improved. Such a project improves social welfare.

100 This is the exact transposition of the traditional argument in favor of average-cost pricing. Here, its limit is that this constraint is very restrictive because too small a share of the benefits of the project can be captured by these means if only additional services can be priced. But the local context offers opportunities to establish more appropriate constraints to check that the project is justified.

101 Firstly, it must be noticed that if the budget of the operator is just balanced, the public budget clears a surplus. Fiscal revenues induced by the project can be earmarked to fund it. In practice, the impact of the facility on city size and thus on receipts from the household tax is difficult to assess. But the impact on property tax revenues may be easier to observe. In that case, the above reasoning remains valid when the budget constraint on the project includes these induced revenues. This is exactly what is done by a “Tax Increment Financing” mechanism, which earmarks the additional property taxes tΔR generated in the neighborhood of a project to its funding. If these are measurable, we can imagine further that the social planner is able to evaluate land rents created by the facility. Their capture to finance the project is thus justified under the condition that the profits of landowners are at least maintained.

102 If residential mobility is high, most of the benefits of the project capitalize into land rents and by observing the latter, the social planner indirectly observes the benefits of the project. Imposing such a constraint on the level of increased property taxes which may be earmarked to the operator budget allows the decision-maker to sort out good projects and fund them without excessive distortions. This approach tries to earmark a part of additional Π into B for that purpose. The alternative one would consist of integrating B into Π. This can be implemented by means of “Joint Property Development” tools, whereby a private developer finances public urban facilities within a real estate project. Moreover, this also avoids the distortions on construction choices.

103 However, this mechanism encounters limits when residential mobility is low or land ownership is dispersed. In the former case, the value of the project does not capitalize if land markets are competitive. Moreover, if land ownership is concentrated, the private developer may try to use the project to exert monopoly power against residents. In the latter case, the surplus subject to appropriation by the private developer will remain partial, since he owns only a fraction of urban land.

4.4. Summary of the results

104 The case of perfect mobility of residents between cities is useful for summarizing our results, full capitalization appearing as a benchmark. It corresponds to the pure “open city”, which is associated with:

∀n, b (n) = 0

105 In this case, the functions R̆ and S̆ verify:

R̆ = S̆; R̆′ = S̆′ = N̆

106 Land rents completely capitalize the value created by the public facility, because the level of utility for the residents is strictly equal to what they would obtain if they locate outside. Then, the optimal policy for its management is to maximize the aggregate land rents. If we add the assumption of fixed construction quality (h exogenous, ε_h = 0), these observations support the traditional economic views on the optimality of land value taxes. A contrario, this emphasizes that this doctrine follows three assumptions:

107

• the considered property tax is non-distortionary. In our framework, it is not the case with a linear tax if the tax base does not contain deductibles for construction costs. Population size is modified, and, what is a bigger issue, so is construction quality;
• perfect mobility of residents. Otherwise, the value created by the facility is shared between landowners and the less mobile residents. In the extreme case of immobile residents (i.e. the closed city), all the value is captured by them, and the property tax is no longer able to finance the development of the type of facilities. Their financing by these means will also prove difficult as soon as this mobility is low in the short-term;
• a perfect discipline of public managers to maximize land rents.

108 The following table summarizes how the standard doctrine is impacted when these different assumptions are challenged.

Table

Summary of results.

	Relative share of property taxes vs household taxes index  (*)	Markups over appropriate marginal cost (**)
Benchmark: perfect mobility; non-distortionary property tax (4.4)	++ +	0
Imperfect capitalization (3.2/3)	↓	0
Distortionary property tax (3.2/3)	↓↓	0
Agglomeration externalities (4.1)	↑	0
Architectural externalities (4.1)	↓	0
Bias in favor of landowners (4.2)	↓	0
Heterogeneous preferences (4.2)	-	↑ or?
Non-resident beneficiaries (4.2)	-	↑
Fixed costs selection, open city (4.3)	↑↑	↑
Fixed costs selection, closed city (4.3)	-	↑↑

(*) t/ (V − w/w), cf. 3.3. In this table, arrows reflect direct (all things equal otherwise) impacts, for example of demand elasticities in equation (25).

(**) P − C

Summary of results.