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Dynamic Henry George Theorem and Optimal City Sizes

Shihe Fu, 

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Shihe Fu

Economics and Management School & The Center for Economic Development Research, Wuhan University, Wuhan, China

Correspondence: Shihe Fu ([email protected])

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First published: 21 July 2025
https://doi.org/10.1002/ise3.70013

ABSTRACT

The Henry George Theorem (HGT) in static models states that when a city has an optimal population size, aggregate urban differential land rents exactly cover costs of pure public goods. This paper extends the static HGT to dynamic settings. Through a series of dynamic models, the paper tentatively concludes that the HGT holds in dynamic settings in terms of present value—the present value of urban differential land rents equals the present value of public goods expenditure. In urban economies with congestion externalities or production externalities, the dynamic HGT still holds if externalities are priced correctly.

1 Introduction

Cities are concentrations of population. The social interactions of a high-density population generate all types of social benefits and social costs. On the one hand, as city population increases, people benefit more from learning from each other (knowledge spillover effects); firms benefit more from being located together (business agglomeration economies); consumers save more traveling costs from multi-purpose trips (shopping externalities); and, the average cost of providing urban public goods may decrease. These are examples of social benefits of population concentration. On the other hand, as city sizes become larger, traffic congestion, commuting distance or time, air pollution, and crime also increase. These are examples of social costs associated with population concentration. There must exist an optimal city size where the marginal social benefit of adding one more resident to the city equals its marginal social cost. What characterizes the optimal city size? The Henry George Theorem (HGT) states that when a city has an optimal population size, aggregate urban differential land rents (DLR) exactly cover costs of pure public goods. This theorem has been applied to test empirically whether or not the city of Tokyo is too large (Kanemoto et al. 1996).

Although Henry George invented the Single Tax Theory, urban and public finance economists discovered the HGT in the 1970s. The idea of the HGT dates back to at least as far as Hotelling (1938) as well as George. Hotelling discovered that marginal cost pricing is the correct way to maximize general welfare and proposed to use non-distortionary taxes to finance the fixed costs of decreasing-cost industries. Such taxes are imposed on land values or inheritance. Bos (1965) discussed the relationship between aggregate land rents (ALR), aggregate transportation costs, and total fixed costs of production in a linear city. Serck-Hanssen (1969) also studied a linear city and found that the DLR equal the production deficit from marginal cost pricing when the number of factories is optimal. Vickrey (1977) treated the city as the concentration of increasing-returns-to-scale industries and demonstrated that aggregate DLR can exactly cover the fixed costs when the amount of output is optimal. He dubbed his findings the “GHV Theorem” (George–Hotelling–Vickrey Theorem). Flatters et al. (1974) and Starrett (1974) also studied the HGT independently.

The HGT has been generalized to congestion externalities (Arnott 1979), external economy in production (Arnott 1979; Kanemoto 1980), heterogeneous agents (Stiglitz 1983), multiple regions (Hartwick 1980), uncertainty (Lee 2005), and monopolistic competition (Behrens and Murata 2009). In fact, the HGT holds in any interior Pareto optimal allocations in large, spatial economies with properly priced locations (Arnott 2004a; Schweizer 1996b). All of the above results are derived from static models.

The static HGT cannot, however, answer the following questions on optimal city sizes and financing of public goods in dynamic contexts:
  • 1.

    If public goods are durable, does the static HGT have to hold for every period? This question is of practical relevance. For example, the development of a new city requires huge investments in public goods, mainly infrastructures. Since infrastructures are durable, in later periods only maintenance and capacity expansion are needed. As shown later, an intertemporal optimum requires that the present value of DLR equal the present value of expenditure on pure public goods; it does not necessarily require that the HGT hold for every period.

  • 2.

    Can we apply the static HGT to test optimal city sizes? In the real world, population in many cities keep growing. In dynamic environments, we expect an optimal time path for city size. Applying the static HGT to test optimal city size may be misleading, unless a city is in a steady state and has a constant population size.

A dynamic version of the HGT is needed to answer the above questions. Furthermore, the dynamic HGT could shed light on policy issues, such as land pricing policy in industrial parks, infrastructure financing policy in gated communities or growing cities, and land value taxation. However, there is no study on the dynamic HGT. A very closely related paper is by Arnott and Kraus (1998). They extended the static model of self-financing of congestible facilities to dynamic models and found that the results of static models still hold in dynamic settings in present value terms. Black and Henderson (1999) mentioned the HGT in their endogenous urban growth model, but, in their model, the HGT is essentially static since it holds in every period without any intertemporal linkage.

The purpose and also the contribution of this paper are to extend the static HGT to dynamic models. Through a series of different dynamic models, the paper demonstrates that the HGT holds in terms of present value in dynamic contexts; that is, the present value of urban DLR over all time periods equals the present value of public goods expenditure over all time periods. In urban economies with congestion externalities or production externalities, the dynamic HGT still holds if externalities are priced correctly.

The rest of the paper is organized as follows: Section 2 recasts the static HGT model in a game-theoretic framework and summarizes the interpretations of the HGT; Section 3 treats pure public goods as durable and depreciable, and shows that the HGT holds in terms of present value. Section 4 discusses how technology improvement in the transport sector affects the steady state of the model; Section 5 takes into account congestion externalities in public goods and the transport sector, and finds that if congestion externalities are priced correctly, the dynamic HGT still holds. Section 6 discusses the production externalities case; Section 7 presents a simple open city model; and Section 8 concludes.

2 HGT in the Static Model

To set the stage, we first derive the HGT from a static model based on Fujita and Thisse (2002). They used a one-dimensional linear city model. Here we employ the traditional monocentric, circular city model with a city planner and pure public goods. We will also review different interpretations of the HGT.

Assume that there is a point central business district (CBD) surrounded by a circular residential area in a featureless plain. Consider the following simple two-stage Stackelberg game: In the first stage, a benevolent city planner, whose goal is to maximize the utility of a representative resident, rents agricultural land from an absentee landlord at rent, R A ${R}_{{\rm{A}}}$ , as urban land. She decides to provide pure public goods, P $P$ , and selects N identical people to immigrate to the city. To finance the public goods, she needs to design a transfer scheme τ ( x ) $\tau (x)$ (x denotes location or distance away from the CBD). τ ( x ) $\tau (x)$ could be a tax or a subsidy, and it is location-specific since residents are different only in terms of locations. In the second stage, given the level of public goods P and the specified transfer scheme τ ( x ) $\tau (x)$ , the N identical immigrants in this closed city compete with each other for locations in the residential area to maximize their utility. Residents commute to the CBD to work, each has the same, exogenous, general-purpose output Y, which can be used as composite goods consumption C, transport services, land rents R ( x ) $R(x)$ , and taxes, if applicable. To simplify the model, we assume a fixed lot size L, constant commuting cost (round trip) per unit distance T, and the absence of traffic congestion.

Obviously, the first stage is a planner's problem and an optimization problem; the second stage is a decentralization problem and an equilibrium problem. We solve the model by backward induction. From the point of view of residents at location x, each has resource Y plus transfer τ ( x ) $\tau (x)$ , which will be spent on consumption C, land rents R ( x ) L $R(x)L$ , and transport costs Tx. Residents consume free pure public goods since the marginal cost of pure public goods is zero. The equal utility constraint implies that each resident consumes the same amount of consumption goods C $C$ .

Assume that a representative resident's utility function is U ( C , P , L ) $U(C,{P},{L})$ , with all the neoclassical properties. The representative resident's decision problem can be modeled as follows:
Max C , x U ( C , P , L ) , $\mathop{\mathrm{Max}}\limits_{C,x}U(C,P,L),$
s . t . C + R ( x ) L + Tx = Y − τ ( x ) . ${\rm{s}}.{\rm{t}}.\,C+R(x)L+{Tx}=Y-\tau (x).$
Combining the first-order conditions, we can derive the spatial equilibrium condition: R ′ ( x ) = − T + τ ′ ( x ) L ${R}^{^{\prime} }(x)=-\frac{T+\tau ^{\prime} (x)}{L}$ . Together with the boundary condition R ( b ) = R A $R(b)={R}_{{\rm{A}}}$ , where b is the city boundary and b = NL π 1 / 2 $b={\left(\frac{{NL}}{\pi }\right)}^{1/2}$ , we can derive the bid rent function:
R ( x ) = R A + Tb − Tx + τ ( b ) − τ ( x ) L . $R(x)={R}_{{\rm{A}}}+\frac{{Tb}-{Tx}+\tau (b)-\tau (x)}{L}.$ ()
From Equation (1), we can see that not only are all transport cost savings fully reflected in land rents, but also the tax benefits or costs are fully capitalized into land rents. Denoting the maximum consumption of composite goods as C * ${C}^{* }$ , we can rewrite the bid rent function as
R ( x ) = Y − C * − Tx − τ ( x ) L . $R(x)=\frac{Y-{C}^{* }-{Tx}-\tau (x)}{L}.$ ()
From the budget constraint, C * ${C}^{* }$ depends on two parameters N and τ ( b ) $\tau (b)$ that are exogenous to residents but are endogenous to the city planner:
C * [ N , τ ( b ) ] = Y − R A L − 3 2 A N 1 / 2 − τ ( b ) , ${C}^{* }[N,\tau (b)]=Y-{R}_{{\rm{A}}}L-\frac{3}{2}A{N}^{1/2}-\tau (b),$ ()
where A ≡ 2 3 T L π 1 / 2 $A\equiv \frac{2}{3}T{\left(\frac{L}{\pi }\right)}^{1/2}$ a constant. Equation (3) can be considered the optimal reaction function of a resident.
The ALR, which residents pay to the city planner, are
ALR = ∫ 0 b 2 π xR ( x ) dx = NL R A + 1 2 A N 3 / 2 + ∫ 0 b 2 π x τ ( b ) − τ ( x ) L dx . $\mathrm{ALR}={\int }_{0}^{b}2\pi {xR}(x){dx}={NL}{R}_{{\rm{A}}}+\frac{1}{2}A{N}^{3/2}+{\int }_{0}^{b}2\pi x\left[\frac{\tau (b)-\tau (x)}{L}\right]{dx}.$ ()
Define urban DLR as DLR = ∫ 0 b 2 π x [ R ( x ) − R A ] dx $\mathrm{DLR}={\int }_{0}^{b}2\pi x[R(x)-{R}_{{\rm{A}}}]{dx}$ . Therefore,
DLR = 1 2 A N 3 / 2 + ∫ 0 b 2 π x τ ( b ) − τ ( x ) L dx . $\mathrm{DLR}=\frac{1}{2}A{N}^{3/2}+{\int }_{0}^{b}2\pi x\left[\frac{\tau (b)-\tau (x)}{L}\right]{dx}.$ ()
The aggregate transport costs (ATC) are
ATC = ∫ 0 b 2 π x L Txdx = A N 3 / 2 . $\mathrm{ATC}={\int }_{0}^{b}\frac{2\pi x}{L}{Txdx}=A{N}^{3/2}.$ ()

Obviously, aggregate DLR are half the ATC in the absence of transfer scheme.

The city planner's goal is to maximize the representative resident's utility by choosing an optimal city size N, public goods expenditure P, and a location-based transfer scheme τ ( x ) $\tau (x)$ , subject to an aggregate resource constraint. The planner knows that residents will behave competitively and optimally to allocate their resources among consumption, land rents, and commuting, to maximize their utility; that is, the planner can solve residents' optimal reaction C * ${C}^{* }$ . The planner's problem is described as follows:
Max P , N , τ ( x ) U [ C * ( N , τ ( b ) ) , P , L ] , $\mathop{\mathrm{Max}}\limits_{P,N,\tau (x)}U[{C}^{* }(N,\tau (b)),P,L],$
s . t . ∫ 0 b 2 π x [ R ( x ) − R A ] dx + ∫ 0 b 2 π x L τ ( x ) dx ≥ P . ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{b}2\pi x[R(x)-{R}_{{\rm{A}}}]{dx}+{\int }_{0}^{b}\frac{2\pi x}{L}\tau (x){dx}\ge P.$ ()

Equation (7) is the planner's budget constraint: The revenue from urban land rents net of aggregate transfers must cover the public goods expenditure and the opportunity costs of urban land.

Plugging Equation (2) into (7), we can rewrite the aggregate resource constraint (7) as
N C * + A N 3 / 2 + NL R A + P ≤ NY . $N{C}^{* }+A{N}^{3/2}+{NL}{R}_{{\rm{A}}}+P\le {NY}.$ ()
Inspecting Equation (7′), we see that the maximum consumption does not depend on the transfer scheme. This is because the effects of the transfer scheme are fully reflected in land rents. Since τ ( x ) $\tau (x)$ does not affect the solution to the resident's problem, without loss of generality, we set τ ( x ) = τ $\tau (x)=\tau $ in the ensuing analysis and restate the city planner's problem as follows:
Max P , N , τ U [ C ( N , τ ) , P , L ] , $\mathop{\mathrm{Max}}\limits_{P,N,\tau }U[C(N,\tau ),P,L],$
s . t . 1 2 A N 3 / 2 + N τ ≥ P . ${\rm{s}}.{\rm{t}}.\,\frac{1}{2}A{N}^{3/2}+N\tau \ge P.$ ()
Let λ > 0 $\lambda \gt 0$ denote the Lagrange multiplier for budget constraint (8). The first-order conditions for N, P, and τ $\tau $ are as follows:
N : U C C N = − λ 3 4 A N 1 / 2 + τ , $N:{U}_{C}{C}_{N}=-\lambda \left(\frac{3}{4}A{N}^{1/2}+\tau \right),$ ()
P : U P = λ , $P:{U}_{P}=\lambda ,$ ()
τ : U C C τ = − λ N . $\tau :{U}_{C}{C}_{\tau }=-\lambda N.$ ()
Inserting Equation (11) into (9), and using C τ = − 1 ${C}_{\tau }=-1$ and C N = − 3 4 A N − 1 / 2 ${C}_{N}=-\frac{3}{4}A{N}^{-1/2}$ (from Equation 3), we see τ = 0 $\tau =0$ . The budget constraint (8) now becomes
1 2 A N 3 / 2 = P = DLR . $\frac{1}{2}A{N}^{3/2}=P={DLR}.$ ()

Equation (12) means that for any level of pure public goods, when the city population size is optimal, the optimal transfer is 0, and the DLR equal the pure public goods expenditure. Since this single confiscatory tax on urban DLR can exactly finance urban public goods, this rule is dubbed the HGT in urban public finance literature, based on the Single Tax Theory of Henry George (1879).

Note that the average cost of public goods is P N $\frac{P}{N}$ , the average commuting cost per person is A N 1 / 2 $A{N}^{1/2}$ . The marginal benefit of adding one more resident to this city is the decrease in average cost of public goods: d P N d N = − P N 2 $\frac{d\frac{P}{N}}{dN}{=}{-}\frac{P}{{N}^{{2}}}$ ; the marginal cost is the increase in average commuting cost 1 2 A N − 1 / 2 $\frac{{1}}{{2}}A{N}^{{-}{1}{/}{2}}$ . When the population size is optimal, the marginal cost of adding one more resident to the city equals its marginal benefit P N 2 = 1 2 A N − 1 / 2 $\frac{P}{{N}^{{2}}}{=}\frac{{1}}{{2}}A{N}^{{-}{1}{/}{2}}$ , which is the same as Equation (12). This is the intuition mentioned in the beginning.

The spatial equilibrium in the second stage is a competitive equilibrium, which is Pareto efficient according to the first welfare theorem. This is a simple case of Schweizer's conclusion: The Pareto efficiency of spatial allocation implies the HGT.

There are other various ways to interpret the HGT:
  • 1.

    Suppose there exists a perfectly competitive market for “cities”—one developer in each city. The costs of development are land opportunity costs plus pure public goods expenditure, and the only revenue is urban land rents collected from residents. Competition will drive profits to zero, so in the competitive equilibrium of a city development market, land rents collected equal land opportunity costs plus pure public goods expenditure. Thus, the HGT holds (Fujita and Thisse 2002). Section 7 presents a simple open city model that demonstrates this interpretation.

  • 2.

    There exist two opposing forces in cities as population increases: forces of increasing returns to scale such as public goods or external scale economies in production, and forces of decreasing returns to scale such as aggregate (average) transport costs, land rents, and congestion externalities. The optimal city size occurs in the locally constant returns to scale, where the forces of increasing returns to scale are offset by the forces of decreasing returns to scale (Arnott 2004b; Vickrey 1977). In the simplest case, if pure public goods are the only force of increasing returns to scale, and land rents (or transport costs) are the only force of decreasing returns to scale, then, under marginal cost pricing, the profits from the decreasing-returns-to-scale activity just equal the loss from increasing-returns-to-scale activity (Arnott and Stiglitz 1979). That is, aggregate DLR equal pure public goods expenditure.

Further interpretations of the HGT include the efficiency of marginal cost pricing (Hotelling 1938), Edgeworth's principle (Schweizer 1983), and city as a club or a community (Berglas and Pines 1981). Regardless of the various perspectives of interpretations, the HGT is essentially an efficiency rule for resource allocation across space, and it holds in any interior Pareto efficient spatial economy (Arnott 2004a). Since the city planner's optimization problem is about the efficiency of resource allocation in her city, we will focus on only the city planner's problem in the rest of the paper.

3 A Simple Dynamic HGT Model

In this section, we treat pure public goods as durable and depreciable and construct a simple dynamic model that shows that the HGT holds in terms of present value. Our methodology is based mainly on Arnott and Kraus (1998).

3.1 Model Setup

We use most of the assumptions of the monocentric, circular city model in Section 2, but add some new assumptions related to dynamic environments.
  • 1.

    All residents are identical, and each resident consumes fixed lot size L. So, if at time t, there are N ( t ) $N(t)$ residents in the city, the city boundary b ( t ) $b(t)$ is N ( t ) L π 1 / 2 ${\left[\frac{N(t)L}{\pi }\right]}^{1/2}$ .

  • 2.

    Commuting cost per unit distance is T, a constant across different locations. The ATC at time t are

ATC ( t ) = ∫ 0 b ( t ) 2 π x L Txdx = A N ( t ) 3 / 2 . $\mathrm{ATC}(t)={\int }_{0}^{b(t)}\frac{2\pi x}{L}{Txdx}=A{N(t)}^{3/2}.$
From the spatial equilibrium problem in Section 2, we know that urban residential land rents reflect only the savings of transport costs. If agricultural land rents per unit area is R A ${R}_{{\rm{A}}}$ , then aggregate urban land rents payment at time t are
∫ 0 b ( t ) 2 π xR ( x , t ) dx = 1 2 A N ( t ) 3 / 2 + N ( t ) L R A . ${\int }_{0}^{b(t)}2\pi {xR}(x,t){dx}=\frac{1}{2}A{N(t)}^{3/2}+N(t)L{R}_{{\rm{A}}}.$
The aggregate DLR at time t are
DLR ( t ) = 1 2 A N ( t ) 3 / 2 . $\mathrm{DLR}(t)=\frac{1}{2}A{N(t)}^{3/2}.$ ()
  • 1.

    The production at the CBD is exogenous, so is the income of each resident, Y ( t ) $Y(t)$ . The general-purpose output goods can be used as consumption C ( t ) $C(t)$ , commuting services, pure local public goods investment I ( t ) $I(t)$ ; and a head tax τ ( t ) $\tau (t)$ imposed by the city planner: Our new, crucial assumption is that public goods are durable—once they are built, they can be used for many periods. The city planner has to decide which is the expenditure on public goods at time 0, P ( 0 ) $P(0)$ , and which is the path of public goods investment, I ( t ) $I(t)$ , so as to cover depreciation and maintenance.

  • 2.

    The utility function of a representative resident living at location x is U ( x , t ) = U [ C ( x , t ) , P ( x , t ) , L ( x , t ) ] $U(x,{t})=U[C(x,t),{P}(x,t),{L}(x,t)]$ . The spatial-intertemporal optimization requires spatial equilibrium at each point of time; otherwise, some residents will have an incentive to switch locations at some points of time. With fixed lot size and pure public goods, the utility function reduces U ( x , t ) = U ( t ) = U [ C ( t ) , P ( t ) , L ] $U(x,{t})=U(t)=U[C(t),{P}(t),{L}]$ .

  • 3.

    Contrary to the closed city and open city dichotomy found in urban literature, we assume an open city with growth control. Imagine a developing country under urbanization, where the smaller urban area is surrounded by a larger rural area. The individual utility level in the urban area is higher than that in the rural area, so the rural population will keep moving into the urban area if migration is free and costless, until the utility level is equalized. Here, however, we assume that the city planner has the power to control urban population growth through lot size control, residence administration, or other similar ways, so that a large utility gap between urban and rural areas can be maintained for a long time. Under this assumption, the dynamic model is essentially a planning problem.

The city planner has to choose an optimal population path N ( t ) $N(t)$ , an initial public goods expenditure P ( 0 ) $P(0)$ , and an optimal path of public goods investment I ( t ) $I(t)$ , so as to maximize the present value of individual life-time utility (assume infinite horizon). After the planner's problem is solved, at any time t, the N ( t ) $N(t)$ residents in the city compete with each other for locations to achieve the same maximum utility level U ( t ) $U(t)$ across locations. By backward induction, the planner knows the optimal reactions of residents, so, in the ensuing analysis, we restrict our analysis to only the planner's problem.

Let r denote the subjective discount rate. The present value of the life-time utility of an individual who moves into the city at time T is ∫ T ∞ e − rt U ( t ) dt ${\int }_{T}^{\infty }{e}^{-{rt}}U(t){dt}$ . Correspondingly, ∫ 0 ∞ e − rt U ( t ) dt ${\int }_{0}^{\infty }{e}^{-{rt}}U(t){dt}$ is the present value of the life-time utility of an individual who moves into the city at time 0. Suppose that the optimal life path for an immigrant who moves into the city at time 0 is found, say, ∫ 0 ∞ e − rt U * ( t ) dt ${\int }_{0}^{\infty }{e}^{-{rt}}{U}^{* }(t){dt}$ is a maximum over the period [ 0 , ∞ ] $[0,\infty ]$ . According to the Pontryagin Maximum Principle, a part of the optimal life path starting from any point of time T would bring the maximum present value of utility, ∫ T ∞ e − rt U * ( t ) dt ${\int }_{T}^{\infty }{e}^{-{rt}}{U}^{* }(t){dt}$ over the subperiod [ T , ∞ ] $[T,\infty ]$ ; that is, the optimal life path of an immigrant at time 0 implies the optimal life path (sub-path) for her starting from any later time T. Since all generations are assumed to be identical and live forever, the optimal life path of an immigrant at T is exactly the same part of the optimal path of an immigrant at time 0 starting from time T. Therefore, the city planner's problem reduces to just finding the optimal life path for an immigrant at time 0.

Since the city planner's problem boils down to maximizing the present value of life-time utility of individual immigrants at time 0, subject to the planner's resource constraint, it can be described as follows:
Max N ( t ) , τ ( t ) , I ( t ) ∫ 0 ∞ e − rt U [ C ( t ) , P ( t ) , L ] dt , $\mathop{\mathrm{Max}}\limits_{N(t),\tau (t),I(t)}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U[C(t),P(t),L]{dt},$
s . t . ∫ 0 ∞ e − rt N ( t ) τ ( t ) + 1 2 A N ( t ) 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt , ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N(t)\tau (t)+\frac{1}{2}A{N(t)}^{3/2}\right]{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt},$ ()
P ′ ( t ) = I ( t ) − δ P ( t ) , ${P}^{^{\prime} }(t)=I(t)-\delta P(t),$ ()
C [ N ( t ) , Y ( t ) , τ ( t ) ] = Y ( t ) − R A L − 3 2 A N ( t ) 1 / 2 − τ ( t ) , $C[N(t),Y(t),\tau (t)]=Y(t)-{R}_{{\rm{A}}}L-\frac{3}{2}A{N(t)}^{1/2}-\tau (t),$ ()
P ( 0 ) , P ( ∞ ) free $P(0),P({\rm{\infty }})\,\mathrm{free}$
0 ≤ I ( t ) ≤ Y ( t ) , $0\le I(t)\le Y(t),$
where I ( t ) $I(t)$ is the public goods investment at time t, δ $\delta $ is a positive constant denoting the depreciation and maintenance rate of public goods, r is the discount rate. Equation (14) is the aggregate resource constraint: the present value of revenues (tax revenues and land rents) should equal the present value of public good investment across all periods. Note that Equation (14) is an isoperimetric constraint, and Equation (16) is directly from Equation (3). We will drop the argument t later for simplicity in some equations. The current-value Hamiltonian is
H = U ( C , P , L ) + λ N τ + 1 2 A N 3 / 2 − I + μ ( I − δ P ) , $H=U(C,P,L)+\lambda \left(N\tau +\frac{1}{2}A{N}^{3/2}-I\right)+\mu (I-\delta P),$
where λ ( t ) $\lambda (t)$ and μ ( t ) $\mu (t)$ are the associated marginal valuations with respect to constraints (14) and (15) at time t. The first-order conditions for τ $\tau $ and N are
τ : U C − λ N = 0 , $\tau :{U}_{C}-\lambda N=0,$ ()
and
N : U C C N + λ ( τ + 3 4 A N 1 / 2 ) = 0 , $N:{U}_{C}{C}_{N}+\lambda (\tau +\frac{3}{4}A{N}^{1/2})=0,$ ()
respectively. Combining Equations (17) and (18), we have
τ = 0 . $\tau =0.$ ()
Substitute Equation (19) into the budget constraint (14), we see
∫ 0 ∞ e − rt 1 2 A N ( t ) 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\frac{1}{2}A{N(t)}^{3/2}{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$ ()
Using Equation (13), Equation (20) is the same as Equation (20′):
∫ 0 ∞ e − rt DLR ( t ) dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\,\mathrm{DLR}(t){dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$ ()

Equation (20′) is the dynamic version of the HGT in a simple case: For any given path of public goods expenditure, when the optimal population path is chosen, the present value of urban DLR over all time periods (i.e., the urban differential land value at time 0) equals the initial public goods expenditure P ( 0 ) $P(0)$ plus the discounted value of public goods investment over all the time periods. It goes without saying that the dynamic HGT does not require that DLR equal public goods expenditure at every period.

We therefore have the following proposition:

Proposition 1.In a monocentric, circular city with fixed lot size and exogenous production, if the city planner's objective is to maximize the present value of the representative resident's life-time utility, then for any given time path of public goods expenditure, when the population path is optimal, the present value of urban DLR over all time periods equals the present value of expenditure (investment) on pure public goods over all the time periods. That is, the HGT holds in terms of present value.

The first-order conditions for I and P are
I : λ = μ , $I:\lambda =\mu ,$ ()
and
P : − ∂ H ∂ P = − ( U P − μ δ ) = μ ′ − r μ , $P:-\frac{\partial H}{\partial P}=-({U}_{P}-\mu \delta )={\mu }^{^{\prime} }-r\mu ,$ ()
respectively. Note that using Equations (17), (21), and (22), we can derive the dynamic version of the Samuelson condition for pure public goods—the sum of marginal rates of substitution between public goods and private consumption equals the marginal rate of transformation between them:
N U P U C = r + δ − μ ′ μ , $N\frac{{U}_{P}}{{U}_{C}}=r+\delta -\frac{\mu ^{\prime} }{\mu },$ ()
where μ ′ μ $\frac{\mu ^{\prime} }{\mu }$ can be considered the appreciation of the marginal value of public goods.
The transversality condition for P(t) is lim t → ∞ μ ( t ) P ( t ) e − rt = 0 $\mathop{\mathrm{lim}}\limits_{t\to \infty }{\mu (t)P(t)e}^{-{rt}}=0$ . The transversality condition for P ( 0 ) $P(0)$ is
λ = μ ( 0 ) . $\lambda =\mu (0).$ ()

λ $\lambda $ is the marginal value of one dollar of income or consumption in terms of utility, and μ $\mu $ is the marginal value of one dollar of public goods stock in terms of utility. At time 0, the marginal value of allocating one dollar to consumption and to public goods should be the same.

3.2 Steady State

In this subsection, we discuss the steady state of the dynamic model and properties of the optimal paths of population, consumption, and public goods stock.

Total differentiating (17) and (22) with respect to t and rewriting the results in matrix form, we have
U CC C N − λ U CP U PC C N U PP N ′ P ′ = − U CC C Y Y ′ − U PC C Y Y ′ . $\left(\begin{array}{cc}{U}_{{CC}}{C}_{N}-\lambda & {U}_{{CP}}\\ {U}_{{PC}}{C}_{N} & {U}_{{PP}}\end{array}\right)\left(\begin{array}{c}N^{\prime} \\ P^{\prime} \end{array}\right)=\left(\begin{array}{c}-{U}_{{CC}}{C}_{Y}Y^{\prime} \\ -{U}_{{PC}}{C}_{Y}Y^{\prime} \end{array}\right).$ ()
Define
D = U CC C N − λ U CP U PC C N U PP , $D=\left(\begin{array}{cc}{U}_{{CC}}{C}_{N}-\lambda & {U}_{{CP}}\\ {U}_{{PC}}{C}_{N} & {U}_{{PP}}\end{array}\right),$
so the determinant of matrix D is | D | = C N ( U CC U PP − U CP 2 ) − λ U PP $|D|={C}_{N}({U}_{{CC}}{U}_{{PP}}-{U}_{{CP}}^{2})-\lambda {U}_{{PP}}$ . If the utility function is strictly concave, then the Hessian matrix is negative definite, which means that U CC U PP − U CP 2 > 0 ${U}_{{CC}}{U}_{{PP}}-{U}_{{CP}}^{2}\gt 0$ , U CC < 0 ${U}_{{CC}}\lt 0$ , and U PP < 0 ${U}_{{PP}}\lt 0$ . Since C N < 0 ${C}_{N}\lt 0$ , without concrete specification of the utility function, | D | $|D|$ could be positive, negative, or zero. Possible (exclusive but not exhaustive) cases are as follows:

Case 1: If | D | ≠ 0 $|D|\ne 0$ , and the exogenous income Y ( t ) = Y $Y(t)=Y$ , then the solution is P ′ ( t ) = 0 ${P}^{^{\prime} }(t)=0$ and N ′ ( t ) = 0 ${N}^{^{\prime} }(t)=0$ , therefore C ′ ( t ) = 0 ${C}^{^{\prime} }(t)=0$ . The steady state means C ′ C = N ′ N = P ′ P = 0 $\frac{C^{\prime} }{C}=\frac{N^{\prime} }{N}=\frac{P^{\prime} }{P}=0$ . There exists a unique, stationary, optimal city size, N * ${N}^{* }$ , although this city size may not be optimal for some periods from the static point of view. If the city planner has perfect foresight, with infinite horizon, she would simply choose the optimal population N * ${N}^{* }$ and the optimal public goods P * = P ( 0 ) ${P}^{* }=P(0)$ at the initial period and keep these values the same over time. There is no transitional dynamics. Investment at every period covers just the depreciation and maintenance of public goods stock; the DLR collected through all the periods cover the initial public goods expenditure and the later expenditure on depreciation and maintenance.

Note that integrating both sides of Equation (15), we have
∫ 0 ∞ e − rt I ( t ) dt = ∫ 0 ∞ e − rt [ P ′ ( t ) + δ P ( t ) ] dt = ( r + δ ) ∫ 0 ∞ e − rt P ( t ) dt − P ( 0 ) . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}={\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}[{P}^{^{\prime} }(t)+\delta P(t)]{dt}=(r+\delta ){\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}P(t){dt}-P(0).$
Combining Equation (20′) and the steady-state properties, we have
∫ 0 ∞ e − rt DLR ( t ) dt = ( r + δ ) P ( 0 ) r . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\,\mathrm{DLR}(t){dt}=\frac{(r+\delta )P(0)}{r}.$ ()

Equation (26) states that the urban differential land value at time 0 equals the value of amortized public goods expenditure. Also, note that since population is constant, so urban DLR are constant over time. But, the flow of public goods expenditure is not constant, with a huge fixed cost of P ( 0 ) $P(0)$ at time 0, and constant investment δ P ( 0 ) $\delta P(0)$ thereafter. Thus, the dynamic version of the HGT does not require that the static HGT hold period by period.

Since ( C * , P * , N * ) ${(C}^{* },{P}^{* },{N}^{* })$ consists of a steady state, it is easy to solve for these variables, if the utility function is known. For example, from Equation (26), we can solve for the optimal city size
N * = 2 ( r + δ ) P * A 2 / 3 . ${N}^{* }={\left[\frac{2(r+\delta ){P}^{* }}{A}\right]}^{2/3}.$ ()
If the utility function has a Cobb–Douglas form: U ( t ) = C ( t ) α P ( t ) β L 1 − α − β $U(t)={C(t)}^{\alpha }{P(t)}^{\beta }{L}^{1-\alpha -\beta }$ where 0 < α < 1 , 0 < β < 1 $0\lt \alpha \lt 1,0\lt \beta \lt 1$ and α + β < 1 $\alpha +\beta \lt 1$ , then the optimal city size is
N * = Y − R A L A 3 2 + α 2 β 2 . ${N}^{* }={\left[\frac{Y-{R}_{{\rm{A}}}L}{A\left(\frac{3}{2}+\frac{\alpha }{2\beta }\right)}\right]}^{2}.$ ()

(The proof is in Appendix A.)

Equation (28) implies that the optimal city size depends on income level, opportunity costs of land, and transport costs. A numerical example is helpful to see how large this city size is. Suppose a representative resident's annual income Y = $40,000; lot size is 0.12 acre. Since land value usually comprises one-third of the housing value, we adjust lot size to be 0.36 acre, since we have not included structures in the model. So, L ≈ 0.00056 $L\approx 0.00056$ square miles (1 square miles = 640 acres). Let R A = ${R}_{A}=$ $15,000 per acre; let T = $300 per mile per year. Housing expenditure usually takes one-third of household income, so we set 1 − α − β = 0.3 $1-\alpha -\beta =0.3$ ; public goods expenditure is approximately equal to the tax payment, say β = 0.1 $\beta =0.1$ , thus α = 0.6 $\alpha =0.6$ . Given these parameter values, N * ≈ ${N}^{* }\,\approx $ 8,254,000. This is a city as big as New York City now.

Case 2: If | D | < 0 $|D|\lt 0$ , Y ′ > 0 ${Y}^{^{\prime} }\gt 0$ , and U PC < 0 ${U}_{{PC}}\lt 0$ , then N ′ > 0 ${N}^{^{\prime} }\gt 0$ , P ′ > 0 ${P}^{^{\prime} }\gt 0$ . However, U PC < 0 ${U}_{{PC}}\lt 0$ is rather strange, and we rule out this scenario. If | D | < 0 $|D|\lt 0$ , Y ′ > 0 ${Y}^{^{\prime} }\gt 0$ , and U PC = 0 ${U}_{{PC}}=0$ , then N ′ > 0 ${N}^{^{\prime} }\gt 0$ , P ′ = 0 ${P}^{^{\prime} }=0$ . It is reasonable to assume U PC = 0 ${U}_{{PC}}=0$ . For example, U = α ln C + β ln P $U=\alpha \,\mathrm{ln}\,C+\beta \,\mathrm{ln}\,P$ . Case 2 implies that persistent growth in income or production can drive cities to expand persistently, which is consistent with empirical findings in the urban endogenous growth literature (Black and Henderson 1999; Eaton and Eckstein 1997). Case 2 also implies that it is problematic to test an optimal city size using the static HGT when city population grows persistently.

Case 3: If | D | = 0 $|D|=0$ , and the exogenous income Y ( t ) = Y $Y(t)=Y$ , then we will have multiple solutions for N ′ $N^{\prime} $ and P ′ $P^{\prime} $ . We also rule out this case since we focus on the unique equilibrium.

Case 1 of the above model resembles a rental market of pure public goods, since the only intertemporal linkage is the durability of the public goods, and the constant investment exactly covers the depreciation and maintenance costs. However, it does generate a dynamic version of the HGT: In dynamic environments, the HGT does not need to hold in each period; instead, the HGT holds in present value terms.

4 Comparative Dynamics: Technology Progress in the Transport Sector

The dynamic HGT in Section 3 considers both spatial equilibrium and intertemporal optimization. Transitional dynamics and comparative dynamic analysis could be conducted for exogenous shocks, such as technological progress in the transport sector or the production sector. This subsection takes transport technology improvement as an example and performs comparative dynamic analysis.

Assume that the commuting cost per unit distance is T θ ( t ) $\frac{T}{\theta (t)}$ , where θ ( t ) > 0 $\theta (t)\gt 0$ is a technology shift indicating that technology improvement in the transport sector decreases commuting cost per unit distance. For simplicity, we further assume that commuting cost per unit distance decreases at a constant rate of g; that is, θ ′ ( t ) θ ( t ) = g $\frac{\theta ^{\prime} (t)}{\theta (t)}=g$ . Without loss of generality, define θ ( t ) = ϑ e gt $\theta (t)={\vartheta e}^{{gt}}$ , where ϑ $\vartheta $ is a constant. Then, the commuting cost per unit distance at time t is T ϑ e − gt $\frac{T}{\vartheta }{e}^{-{gt}}$ . With this modification, the model can be formalized as follows:
Max N , τ , I ∫ 0 ∞ e − rt U [ C , P , L ] dt , $\mathop{\mathrm{Max}}\limits_{N,\tau ,I}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U[C,P,L]{dt},$
s . t . ∫ 0 ∞ e − rt N τ + 1 2 A ϑ e − gt N 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt Idt , ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N\tau +\frac{1}{2}\frac{A}{\vartheta }{{e}^{-{gt}}N}^{3/2}\right]{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}{Idt},$ ()
P ′ = I − δ P , ${P}^{^{\prime} }=I-\delta P,$
C [ N , Y , τ ] = Y − R A L − 3 2 A ϑ e − gt N 1 / 2 − τ . $C[N,Y,\tau ]=Y-{R}_{{\rm{A}}}L-\frac{3}{2}\frac{A}{\vartheta }{e}^{-{gt}}{N}^{1/2}-\tau .$ ()
Combining the first-order conditions of N and τ $\tau $ , and the budget constraint (29), we have
∫ 0 ∞ e − rt 1 2 A ϑ e − gt N ( t ) 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt , ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\frac{1}{2}\frac{A}{\vartheta }{{e}^{-{gt}}N(t)}^{3/2}{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt},$ ()
or
∫ 0 ∞ e − rt DLR ( t ) dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\,\mathrm{DLR}(t){dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$

The dynamic HGT still holds, except that commuting costs are modified.

To analyze the optimal city population path, again, we assume an exogenous income path Y ( t ) = Y $Y(t)=Y$ . A steady-state consumption path implies that the growth rate of city size is 2g: To be specific, total differentiate (30) with respect to t and set C ′ ( t ) = 0 ${C}^{^{\prime} }(t)=0$ , and we find
N ′ N = 2 g . $\frac{N^{\prime} }{N}=2g.$ ()
If U CP > 0 ${U}_{{CP}}\gt 0$ , then, P ′ = 2 g U C U CP ${P}^{^{\prime} }=2g\frac{{U}_{C}}{{U}_{{CP}}}$ , and the growth rate of public goods stock is positive. Growth in city size expands the city boundary, generating more DLR, which can be used to expand the public goods stock. If the utility function is also assumed to be U ( T ) = C ( t ) α P ( t ) β L 1 − α − β $U(T)={C(t)}^{\alpha }{P(t)}^{\beta }{L}^{1-\alpha -\beta }$ , then P ′ P = 2 g β $\frac{P^{\prime} }{P}=\frac{2g}{\beta }$ . If we write N ( t ) = N ( 0 ) e 2 gt $N(t)=N(0){e}^{2{gt}}$ and P ( t ) = P ( 0 ) e 2 g β t $P(t)=P(0){e}^{\frac{2g}{\beta }t}$ , insert them into the dynamic HGT Formula (31), and if 2 g β < r $\frac{2g}{\beta }\lt r$ , then we find the following equation for the initial city size and public goods stock:
N ( 0 ) = 2 ϑ ( r − 2 g ) ( r + δ ) P ( 0 ) A ( r − 2 g β ) 2 / 3 . $N(0)={\left[\frac{2\vartheta (r-2g)(r+\delta )P(0)}{A(r-\frac{2g}{\beta })}\right]}^{2/3}.$ ()

Obviously, if ϑ = 1 $\vartheta =1$ and g = 0 (no technology improvement in the transport sector), then N * = 2 ( r + δ ) P * A 2 / 3 ${N}^{* }={\left[\frac{2(r+\delta ){P}^{* }}{A}\right]}^{2/3}$ , which is exactly the same as Equation (27). By combining the spatial equilibrium condition at time 0, we can completely solve for the optimal trajectories for consumption, city size, and public goods stock.

We can see from Equation (32) that an anticipated constant technology improvement in the transport sector could be the driving force of persistent city growth. Since ∂ N ( 0 ) ∂ g > 0 $\frac{\partial N(0)}{\partial g}\gt 0$ (from Equation 33), a larger anticipated technology improvement in the transport sector not only leads to a higher growth rate of city population but also leads to a larger initial city size.

5 Congestion Externalities

In the previous models, there could be two types of congestion: congestion in public goods and traffic congestion. We will analyze both in turn.

5.1 Congestible Public Goods

If public goods are congestible, then individual utility from public goods is affected by the total population. We specify individual utility function as
U [ C , F ( P , N ) , L ] , $U[C,F(P,N),L],$
where F P > 0 , F N < 0 , U F > 0 ${F}_{P}\gt 0,{F}_{N}\lt 0,{U}_{F}\gt 0$ . All other aspects remain the same as in Section 3. Therefore, the planner's objective function is
Max N , τ , I ∫ 0 ∞ e − rt U [ C ( N , τ ) , F ( P , N ) , L ] dt , $\mathop{\mathrm{Max}}\limits_{N,\tau ,I}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U[C(N,\tau ),F(P,N),L]{dt},$
and the constraints (14)–(16) still hold. Combining the first-order conditions of N and τ $\tau $ , we have
τ = − 1 λ U F F N . $\tau =-\frac{1}{\lambda }{U}_{F}{F}_{N}.$ ()
Substitute Equation (34) into the budget constraint (14), and we see the dynamic HGT in the case of congestible public goods:
∫ 0 ∞ e − rt DLR ( t ) − N ( t ) 1 λ U F F N dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[\mathrm{DLR}(t)-N(t)\frac{1}{\lambda }{U}_{F}{F}_{N}\right]{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$ ()

Obviously, U F F N ${U}_{F}{F}_{N}$ is the negative congestion externality, or disutility, imposed on another resident by adding one more resident to the city. λ $\lambda $ is the marginal valuation of adding one more unit of public goods stock, in terms of utility to a representative resident. Therefore, 1 λ U F F N $\frac{1}{\lambda }{U}_{F}{F}_{N}$ is the material value, in units of the general-purpose goods, of the disutility imposed on another resident by adding one more resident to the city; and, N 1 λ U F F N $N\frac{1}{\lambda }{U}_{F}{F}_{N}$ is the total disutility (in terms of material value) imposed on all residents, or the total congestion externalities, generated by a marginal resident. If congestion externalities are not priced, then the present value of market DLR is less than the present value of public goods expenditure. If a toll is implemented and is set to equal the total congestion externalities imposed by a marginal resident, then the present value of DLR and toll revenue equals the present value of public goods expenditure. That is, the dynamic HGT holds in terms of shadow DLR and present value.

Proposition 2.In a monocentric, circular city with congestible public goods, if a toll is set to equal the total congestion externalities generated by a marginal resident, then the present value of DLR and toll revenue over all time periods equals the present value of public goods expenditure over all time periods.

5.2 Traffic Congestion

In this section, we model traffic congestion based on Arnott (1979), Kanemoto (1980), and Mills and Ferranti (1971), and find that if traffic congestion externalities are priced correctly, then the present value of DLR and toll revenue equals the present value of public goods expenditure.

Assume at any residential ring, there is a fixed proportion of land s used for residential lots, and 1 − s $1-s$ share for roads. To make things simple, we set a fixed lot size of L = 1 $L=1$ . The private transport cost of moving a resident at x to the CBD is ∫ 0 x G [ W ( z ) , Q ( z ) ] dz ${\int }_{0}^{x}G[W(z),{Q}(z)]{dz}$ , G W < 0 ${G}_{W}\lt 0$ , G Q > 0 ${G}_{Q}\gt 0$ , where W ( z ) $W(z)$ is the road width (the amount of land allocated to road use) at location z, and Q ( z ) ${Q}(z)$ is the accumulated traffic flow from the city boundary to location z:
Q ( z ) = ∫ z b 2 π ysdy = π s ( b 2 − z 2 ) . $Q(z)={\int }_{z}^{b}2\pi {ysdy}=\pi s({b}^{2}-{z}^{2}).$
The congestion externalities generated by a person at location x is ∫ 0 x G Q Q ( z ) dz ${\int }_{0}^{x}{G}_{Q}Q(z){dz}$ ; the aggregate congestion externalities (ACE) in the city are
ACE = ∫ 0 b 2 π xs ∫ 0 x G Q Q ( z ) dz dx . $\mathrm{ACE}={\int }_{0}^{b}2\pi {xs}\left[{\int }_{0}^{x}{G}_{Q}Q(z){dz}\right]{dx}.$ ()
If congestion is unpriced, the aggregate (private) transport costs are
ATC = ∫ 0 b 2 π xs ∫ 0 x G ( W , Q ) dz dx . $\mathrm{ATC}={\int }_{0}^{b}2\pi {xs}\left[{\int }_{0}^{x}G(W,Q){dz}\right]{dx}.$
To compute the ALR, we have to consider the spatial equilibrium problem. Suppose the planner imposes the same head tax τ $\tau $ to all residents. τ $\tau $ can be interpreted as a uniform toll, which will be clear in the end of this section. If congestion is unpriced, residents will use market land rents as the price signal to compete for locations. Therefore, the spatial equilibrium problem is modeled as follows:
Max C , x U ( C , P ) , $\mathop{\mathrm{Max}}\limits_{C,x}U(C,P),$
s . t . C + R ( x ) + ∫ 0 x G [ W ( z ) , Q ( z ) ] dz = Y − τ . ${\rm{s}}.{\rm{t}}.\,C+R(x)+{\int }_{0}^{x}G[W(z),Q(z)]{dz}=Y-\tau .$
The spatial equilibrium condition is R ′ ( x ) = − G ( x ) ${R}^{^{\prime} }(x)=-G(x)$ . The maximum consumption is
C * ( N , τ ) = Y − τ − R A − ∫ 0 b G [ W ( z ) , Q ( z ) ] dz . ${C}^{* }(N,\tau )=Y-\tau -{R}_{{\rm{A}}}-{\int }_{0}^{b}G[W(z),Q(z)]{dz}.$ ()
The bid-rent function becomes R ( x ) = R A + ∫ 0 b G ( z ) d z − ∫ 0 x G ( z ) d z $R(x)={R}_{{\rm{A}}}+{\int }_{0}^{b}G(z)dz-{\int }_{0}^{x}G(z)dz$ . Therefore, the ALR are
∫ 0 b 2 π xsR ( x ) dx = sN R A + π s ∫ 0 b x 2 G ( x ) dx . ${\int }_{0}^{b}2\pi {xsR}(x){dx}={sN}{R}_{A}+\pi s{\int }_{0}^{b}{x}^{2}G(x){dx}.$
And the urban DLR are
DLR = π s ∫ 0 b x 2 G ( x ) dx . $\mathrm{DLR}=\pi s{\int }_{0}^{b}{x}^{2}G(x){dx}.$
Now come back to the dynamic model:
Max N , τ , I ∫ 0 ∞ e − rt U [ C ( N , τ ) , P ] dt , $\mathop{\mathrm{Max}}\limits_{N,\tau ,I}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U[C(N,\tau ),P]{dt},$
s . t . ∫ 0 ∞ e − rt N τ + π s ∫ 0 b x 2 G ( x ) dx dt = P ( 0 ) + ∫ 0 ∞ e − rt Idt . ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N\tau +\pi s{\int }_{0}^{b}{x}^{2}G(x){dx}\right]{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}{Idt}.$ ()
P ′ = I − δ P , ${P}^{^{\prime} }=I-\delta P,$
and the constraint (37).
The first-order conditions for τ $\tau $ and N are
τ : U C = λ N , $\tau :\,{U}_{C}=\lambda N,$ ()
and
N : U C C N + λ τ + db dN π s b 2 G ( b ) + π s ∫ 0 b ∂ G ( x ) ∂ N x 2 dx = 0 . $N:{U}_{C}{C}_{N}+\lambda \left[\tau +\frac{{db}}{{dN}}\pi s{b}^{2}G(b)+\pi s{\int }_{0}^{b}\frac{\partial G(x)}{\partial N}{x}^{2}{dx}\right]=0.$ ()
Note that from Equation (37)
C N = − db dN G ( b ) − ∫ 0 b ∂ G ( x ) ∂ N dx . ${C}_{N}=-\frac{{db}}{{dN}}G(b)-{\int }_{0}^{b}\frac{\partial G(x)}{\partial N}{dx}.$ ()
Combining Equations (39) and (41), we can simplify Equation (40):
τ = ∫ 0 b N ∂ G ( x ) ∂ N dx − π s ∫ 0 b ∂ G ( x ) ∂ N x 2 dx $\tau ={\int }_{0}^{b}N\frac{\partial G(x)}{\partial N}{dx}-\pi s{\int }_{0}^{b}\frac{\partial G(x)}{\partial N}{x}^{2}{dx}$
= ∫ 0 b π s ( b 2 − x 2 ) ∂ G ( x ) ∂ N dx = ∫ 0 b G Q Q ( x ) dx . $={\int }_{0}^{b}\pi s({b}^{2}-{x}^{2})\frac{\partial G(x)}{\partial N}{dx}={\int }_{0}^{b}{G}_{Q}Q(x){dx}.$ ()
So, when the planner sets the optimal population size, the optimal tax equals the congestion externalities generated by adding one more resident at the city boundary. Insert Equation (42) into the budget constraint (38), and we find the dynamic HGT in a congested city:
∫ 0 ∞ e − rt N ∫ 0 b G Q Q ( x ) dx + DLR ( t ) dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N{\int }_{0}^{b}{G}_{Q}Q(x){dx}+\mathrm{DLR}(t)\right]{dt}\,=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$

Again, if traffic congestion externalities are priced correctly, the present value of market DLR plus toll revenue equals the present value of public goods expenditure across time.

Proposition 3.In a monocentric, circular city with traffic congestion, if a toll is set to equal the total congestion externalities generated by a marginal driver, then the present value of DLR and toll revenue over all time periods equals the present value of public goods expenditure over all time periods.

It is worth noting that the aggregate toll revenue is not equal to the ACE (36) because of marginal cost pricing in space. The difference between the aggregate toll revenue and the ACE is exactly the difference between the aggregate shadow land rents (or social land rents) and the aggregate market land rents (the proof is in Appendix B).

This section shows that in an urban economy with traffic congestion externalities, the dynamic HGT still holds if congestion externalities are priced properly. The generality of the HGT in distorted economies with unpriced congestion externalities in a static setting was discussed in Arnott (2004b).

6 Production Externalities

If production in a city delivers positive externalities or scale economies external to individual firms, then it should be subsidized to achieve the efficient scale. In this case, the static HGT says that aggregate DLR exactly cover public goods expenditure and subsidies for positive externalities. In a dynamic setting, the HGT still holds in present value terms. This section will demonstrate this result.

Assume each resident is also a producer with identical productivity Y ( t ) $Y(t)$ , but individual production benefits from the concentration of producers in the CBD. Let f ( N ) > 1 $f(N)\gt 1$ be a multiplicative productivity shift or external economies of scale, then the actual output of a resident is f ( N ) Y ( t ) $f(N)Y(t)$ . In our model, f ( N ) $f(N)$ could be explained as localization economies or urbanization economies since we assume only one industry in the city. The aggregate output in the city at time t is Nf ( N ) Y ${Nf}(N)Y$ . The marginal externalities generated by an additional resident are Nf ′ ( N ) Y ${Nf}^{\prime} (N)Y$ ; thus, the aggregate externalities in the city are N 2 f ′ ( N ) Y ${N}^{2}f^{\prime} (N)Y$ .

The dynamic model is set as follows:
Max N , τ , I ∫ 0 ∞ e − rt U ( C , P , L ) dt , $\mathop{\mathrm{Max}}\limits_{N,\tau ,I}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U(C,P,L){dt},$
s . t . ∫ 0 ∞ e − rt N τ + 1 2 A N 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt Idt , ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left(N\tau +\frac{1}{2}A{N}^{3/2}\right){dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}{Idt},$ ()
P ′ = I − δ P , ${P}^{^{\prime} }=I-\delta P,$
C = f ( N ) Y − R A L − 3 2 A N 1 / 2 − τ . $C=f(N)Y-{R}_{{\rm{A}}}L-\frac{3}{2}A{N}^{1/2}-\tau .$
Combine the first-order conditions for τ $\tau $ and N, and we have
τ = − N f ′ ( N ) Y . $\tau =-N{f}^{^{\prime} }(N)Y.$ ()

The optimal tax is the Pigouvian subsidy that equals the production externalities generated by an additional resident.

Insert Equation (44) into the budget constraint (43), and we obtain
∫ 0 ∞ e − rt DLR   dt = P ( 0 ) + ∫ 0 ∞ e − rt [ I + N 2 f ′ ( N ) Y ] dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\,\mathrm{DLR}{\ }{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}[I+{N}^{2}{f}^{^{\prime} }(N)Y]{dt}.$ ()

Equation (45) states that the present value of market DLR equals the present value of Pigouvian subsidies to the production externalities and expenditure on public goods over all time periods.

Proposition 4.In a monocentric, circular city with positive production externalities, if a subsidy provided to firms is set to equal the total production externalities generated by a marginal firm, then the present value of DLR over all time periods equals the present value of public goods expenditure plus the Pigouvian subsidies over all time periods.

This conclusion is of practical importance as far as land pricing policies in industrial parks are concerned, where production has strong agglomeration economies, and firms share high-quality infrastructure.

7 An Open City Model

To demonstrate further the dynamic HGT, this section presents a simple dynamic model in an open city. Assume there is a system of cities of different sizes. Cities are open in terms of free and costless intercity migration. Each city has a developer. Each developer provides free pure public goods to establish a new city, and charges each resident a head tax τ $\tau $ and urban land rents. To ensure residents live in her city, each developer must guarantee residents in her city at least the utility level u ¯ $\mathop{u}\limits^{&#773;}$ , which is prevailing in other cities and can be considered exogenous. A developer's goal is to maximize the present value of profit flow. The profits at time t > 0 $t\gt 0$ are the sum of urban DLR and tax revenue minus investment on public goods. If the city development markets are perfectly competitive, then, in equilibrium, each developer gets zero discounted profits. Thus, the dynamic model for an open city can be summarized as follows:
Max N , τ , I ∫ 0 ∞ e − rt N τ + 1 2 A N 3 / 2 − I dt − P ( 0 ) , $\mathop{\mathrm{Max}}\limits_{N,\tau ,I}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N\tau +\frac{1}{2}A{N}^{3/2}-I\right]{dt}-P(0),$
s . t . ∫ 0 ∞ e − rt U ( C , P , L ) dt = u ¯ , ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}U(C,P,L){dt}=\bar{u},$
P ′ = I − δ P , ${P}^{^{\prime} }=I-\delta P,$
plus the constraint (16). Since we know that competitive equilibrium implies that the present value of profit flow is zero, to test the dynamic HGT, we need only to check whether the optimal tax is zero as well. The combination of the first-order conditions for N and τ $\tau $ does imply that the optimal tax τ $\tau $ equals zero; thus, the HGT also holds in the open city case.

8 Conclusions

We have shown that the HGT holds in terms of present value in various dynamic contexts. These contexts can be combined together to one scenario with pure public goods and multiple urban externalities. However, a general proof of the dynamic HGT is very difficult, since it is hard to characterize a general spatial economy. In addition, a general dynamic spatial model involves optimal control problems governed by first-order partial differential equations, hardly tractable and rarely employed in economics. Possible extensions of the current paper could be taking into account convex adjustment costs of investment, variable lot size, heterogeneous residents, and so forth. For example, adding convex adjustment costs of investment is fairly tractable. The reason is straightforward: Suppose the convex function of investment adjustment costs is κ I P $\kappa \left(\frac{I}{P}\right)$ , adding ∫ 0 ∞ e − rt κ I P dt ${\int }_{0}^{\infty }{e}^{-{rt}}\kappa \left(\frac{I}{P}\right){dt}$ to the right-hand side of Equation (14) does not change the first-order conditions (17) and (18) from which the dynamic HGT is derived.

The dynamic HGT we have tentatively concluded could have important policy insight. For example, firms are colocated in industrial parks to enjoy agglomeration economies, but initially, firms have little incentive to be the first movers since there is little agglomeration in a new industrial park. Park developers incur huge sunk costs at the beginning since they need to provide all the infrastructure facilities before firms move in. Park developers charge first movers lower land rents and later movers higher rents. This discriminatory pricing of sites over time could partially overcome the inertia of firms' relocation (Rauch 1993). When the number of firms is optimal, can the present value of land rents cover sunk costs and first-period subsidies to first movers? Another example is impact fees. When a city keeps growing, new infrastructure needs to be constructed to provide public services for new residents. Nowadays, one typical way to finance the incremental infrastructure is to charge impact fees: Developers have to pay for the incremental infrastructure when developing a new site. Brueckner (1997) proved that a developer would not develop a site until the urban DLR at the location equal the amortized costs of the infrastructure investments. If a city planner provides new infrastructure as the city grows, will the present value of incremental DLR equal the present value of incremental infrastructure investments? The dynamic HGT can be applied to study these questions. However, detailed modeling would be worth separate papers.

Acknowledgments

I am deeply indebted to Professor Richard Arnott for his generous help in model specification and proof. I also thank Professors Marvin Kraus, Stephen L. Ross, Tony Yezer, Hideo Konishi, Patrick Bayer, Junfu Zhang, and two anonymous referees for very helpful comments. Financial support from the Robert Schalkenbach Foundation and the Lincoln Institute of Land Policy is gratefully acknowledged.

    Ethics Statement

    The author has nothing to report.

    Conflicts of Interest

    The author declares no conflicts of interest.

    Endnotes

  1. 1 Urban differential land rent refers to the difference between urban land rents and the opportunity costs of urban land. The rents of agricultural land at the urban boundary can be considered the opportunity costs of urban land.
  2. 2 In a separate paper, Arnott and Fu (2005) also extended the HGT to dynamic contexts, but focused on the self-financing of congestible transport facility in cities.
  3. 3 One of the annoying problems in spatial models is that maximizing social welfare (except in the Rawlsian social welfare function) results in equal people having unequal utility levels due to non-convexity (Mirrlees 1972). In this section, since we assume people are identical and consume the same fixed lot size, the spatial equilibrium results in equal utility across locations, so the planner's objective reduces to maximizing the utility level of the representative resident.
  4. 4 The problem of maximizing per head utility with respect to private consumption, public consumption, and population size is called club efficient by Schweizer (1996a). The conventional maximization problem with respect to private and public consumption is called efficient, whereas with respect to private consumption and population, it is called population efficient. In our specification, from an individual resident's problem, the optimal consumption depends only on N and τ ( x ) $\tau (x)$ , so the planner's choice variables reduce to P, N, and τ ( x ) $\tau (x)$ . This is equivalent to the club efficiency.
  5. 5 The Kuhn–Tucker Theorem implies that the resource constraint (8) is binding if λ > 0 $\lambda \gt 0$ .
  6. 6 This is happening in China now. One problem related to growth control is that the boundary condition may not hold, but we ignore this issue in this paper. A complete open city can achieve an equilibrium size but not necessarily an optimal size. An open city with growth control is a simplified scenario to derive an optimal city size path.
  7. 7 We need to assume that the planner has access to the capital market but residents do not; otherwise, residents will choose an optimal time path of locations based on their life-time credit constraints; this will involve partial differential equations, which makes the problem very complicated. For the models with individual, location-specific capital, see Boucekkine et al. (2009) and Camacho and Zou (2004).
  8. 8 The subjective rate of discounting utility should be different from the discount rate for the capital market, but this does not generate new insight. So, to simplify the model, we choose the same discount rate for both utility and capital.
  9. 9 The second-order necessary conditions H τ τ ≤ 0 ${H}_{\tau \tau }\le 0$ and H NN ≤ 0 ${H}_{{NN}}\le 0$ are satisfied.
  10. 10 The neoclassical properties of the utility function imply C ( t ) > 0 $C(t)\gt 0$ and thus I ( t ) < Y ( t ) $I(t)\lt Y(t)$ ; δ > 0 $\delta \gt 0$ implies I ( t ) > 0 $I(t)\gt 0$ ; therefore, even though the Hamiltonian is linear in I ( t ) $I(t)$ , the bang–bang control case for I ( t ) $I(t)$ is excluded. Note that the last term of Equation (22) used Equation (21).
  11. 11 Many studies have estimated the share of land value in home value, and the estimates vary across cities and time, ranging between 30% and 70% (see e.g., Albouy and Ehrlich 2018; Davis and Heathcote 2007).
  12. 12 If lot size is variable, then unpriced congestion externalities will generate a distortion in the lot size choice. The assumption of fixed lot size does not result in any distortion.
  13. 13 Apply integration by parts and the Liebnitz's rule.

    14. Appendix A

    Assume the utility function of a representative resident living at location x is
    U ( x , t ) = C ( x , t ) α P ( x , t ) β L ( x , t ) 1 − α − β . $U(x,t)={C(x,t)}^{\alpha }{P(x,t)}^{\beta }{L(x,t)}^{1-\alpha -\beta }.$
    Since we assumed fixed lot size and pure public goods with equal utility constraint, the utility function can be reduced to
    U ( x , t ) = U ( t ) = C ( t ) α P ( t ) β L ( t ) 1 − α − β . $U(x,t)={U(t)=C(t)}^{\alpha }{P(t)}^{\beta }{L(t)}^{1-\alpha -\beta }.$ ()
    The city planner's problem is
    Max N ( t ) , τ , I ( t ) ∫ 0 ∞ e − rt C ( t ) α P ( t ) β L ( t ) 1 − α − β dt , $\mathop{\mathrm{Max}}\limits_{N(t),\tau ,I(t)}{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}{C(t)}^{\alpha }{P(t)}^{\beta }{L(t)}^{1-\alpha -\beta }{dt},$
    s . t . ∫ 0 ∞ e − rt N ( t ) τ + 1 2 A N ( t ) 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt , ${\rm{s}}.{\rm{t}}.\,{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\left[N(t)\tau +\frac{1}{2}A{N(t)}^{3/2}\right]{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt},$ ()
    P ′ ( t ) = I ( t ) − δ P ( t ) , ${P}^{^{\prime} }(t)=I(t)-\delta P(t),$ ()
    C ( N , Y , τ ) = Y ( t ) − R A L − 3 2 A N ( t ) 1 / 2 − τ . $C(N,Y,\tau )=Y(t)-{R}_{{\rm{A}}}L-\frac{3}{2}A{N(t)}^{1/2}-\tau .$ ()
    The current-value Hamiltonian is
    H = C α P β L 1 − α − β + λ N τ + 1 2 A N 3 / 2 − I + μ ( I − δ P ) , ${H=C}^{\alpha }{P}^{\beta }{L}^{1-\alpha -\beta }+\lambda \left(N\tau +\frac{1}{2}A{N}^{3/2}-I\right)+\mu (I-\delta P),$
    where λ $\lambda $ and μ ( t ) $\mu (t)$ are the associated marginal valuations with respect to constraints (A2) and (A3). The first-order conditions for τ $\tau $ , N, I, and P are
    τ : α C α − 1 P β L 1 − α − β = λ N , $\tau :\,\alpha {C}^{\alpha -1}{P}^{\beta }{L}^{1-\alpha -\beta }=\lambda N,$ ()
    N : − α C α − 1 3 4 A N − 1 / 2 P β L 1 − α − β + λ τ + 3 4 A N 1 / 2 = 0 , $N:-\alpha {C}^{\alpha -1}\left(\frac{3}{4}A{N}^{-1/2}\right){P}^{\beta }{L}^{1-\alpha -\beta }+\lambda \left(\tau +\frac{3}{4}A{N}^{1/2}\right)=0,$ ()
    I : λ = μ ( t ) , $I:\,\lambda =\mu (t),$ ()
    and
    P : − ∂ H ∂ P = − ( β C α P β − 1 L 1 − α − β − μ δ ) = μ ′ − r μ . $P:-\frac{\partial H}{\partial P}=-(\beta {C}^{\alpha }{P}^{\beta -1}{L}^{1-\alpha -\beta }-\mu \delta )={\mu }^{^{\prime} }-r\mu .$ ()
    Substituting Equation (A5) into Equation (A6), we obtain
    τ = 0 . $\tau =0.$
    Therefore, Equation (A2) becomes
    ∫ 0 ∞ e − rt 1 2 A N ( t ) 3 / 2 dt = P ( 0 ) + ∫ 0 ∞ e − rt I ( t ) dt . ${\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}\frac{1}{2}A{N(t)}^{3/2}{dt}=P(0)+{\int }_{0}^{{\rm{\infty }}}{e}^{-{rt}}I(t){dt}.$ ()
    From Equations (A5) and (A8), we can derive Equations (A10) and (A11) as
    ( α − 1 ) C ′ C + β P ′ P = N ′ N , $(\alpha -1)\frac{{C}^{^{\prime} }}{C}+\beta \frac{P^{\prime} }{P}=\frac{N^{\prime} }{N},$ ()
    and
    μ ′ μ − r − δ = − β α NC P . $\frac{\mu ^{\prime} }{\mu }-r-\delta =-\frac{\beta }{\alpha }\frac{{NC}}{P}.$ ()
    In steady state, P ′ P = μ ′ μ = 0 $\frac{P^{\prime} }{P}=\frac{\mu ^{\prime} }{\mu }=0$ , which implies P ( t ) = P * $P(t)={P}^{* }$ , I ( t ) = δ P * $I(t)=\delta {P}^{* }$ . By using Equations (A10) and (A11), we see C ′ C = N ′ N = 0 $\frac{{C}^{^{\prime} }}{C}=\frac{N^{\prime} }{N}=0$ . This steady state is unique. Therefore, the steady state means that the city planner will choose optimal population N * ${N}^{* }$ and the optimal public goods P * ${P}^{* }$ at the initial period and keep these levels of value over time. From Equation (A9) we have
    N * 3 / 2 = 2 ( r + δ ) P * A . ${{N}^{* }}^{3/2}=\frac{2(r+\delta ){P}^{* }}{A}.$
    Combine Equations (A11) and (A4), solve for optimal city size
    N * = Y − R A L A 3 2 + α 2 β 2 . ${N}^{* }={\left[\frac{Y-{R}_{{\rm{A}}}L}{A\left(\frac{3}{2}+\frac{\alpha }{2\beta }\right)}\right]}^{2}.$

    Appendix B

    The difference between aggregate toll revenue and aggregate congestion externalities is
    N τ − ACE = N ∫ 0 b G Q Q ( x ) dx − ∫ 0 b 2 π xs ∫ 0 x G Q Q ( z ) dz dx $N\tau -\mathrm{ACE}=N{\int }_{0}^{b}{G}_{Q}Q(x){dx}-{\int }_{0}^{b}2\pi {xs}\left[{\int }_{0}^{x}{G}_{Q}Q(z){dz}\right]{dx}$
    = N ∫ 0 b G Q Q ( x ) dx − ∫ 0 b 2 π xs ∫ 0 b G Q Q ( z ) dz − ∫ x b G Q Q ( z ) dz dx = ∫ 0 b 2 π xs ∫ x b G Q Q ( z ) dz dx . $=N{\int }_{0}^{b}{G}_{Q}Q(x){dx}-{\int }_{0}^{b}2\pi {xs}\left[{\int }_{0}^{b}{G}_{Q}Q(z){dz}-{\int }_{x}^{b}{G}_{Q}Q(z){dz}\right]{dx}={\int }_{0}^{b}2\pi {xs}\left[{\int }_{x}^{b}{G}_{Q}Q(z){dz}\right]{dx}.$
    If congestion is unpriced, the market rents gradient is
    R ′ ( x ) = − G . ${R}^{^{\prime} }(x)=-G.$
    The shadow rents S ( x ) $S(x)$ gradient is
    S ′ ( x ) = − G − G Q Q . ${S}^{^{\prime} }(x)=-G-{G}_{Q}Q.$
    At location x, the difference between shadow rents S ( x ) $S(x)$ and market rents R ( x ) $R(x)$ is
    S ( x ) − R ( x ) = − ∫ 0 b [ S ′ ( z ) − R ′ ( z ) ] dz = ∫ x b G Q Qdz , $S(x)-R(x)=-{\int }_{0}^{b}[{S}^{^{\prime} }(z)-{R}^{^{\prime} }(z)]{dz}={\int }_{x}^{b}{G}_{Q}{Qdz},$
    since in a closed city, S ( b ) − R ( b ) = R A $S(b)-R(b)={R}_{{\rm{A}}}$ . Therefore, the difference between the aggregate shadow rents and the aggregate market rents are
    ∫ 0 b 2 π xs [ S ( x ) − R ( x ) ] dx = ∫ 0 b 2 π xs ∫ x b G Q Qdz dx , ${\int }_{0}^{b}2\pi {xs}[S(x)-R(x)]{dx}={\int }_{0}^{b}2\pi {xs}\left[{\int }_{x}^{b}{G}_{Q}{Qdz}\right]{dx},$
    which equals the difference between the aggregate toll revenue and the aggregate congestion externalities.

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