Abstract
Urbanization is going on in China. This paper considers the optimal process of urbanization in a theoretic framework. We already have the Henry George theorem, which asserts that the optimal amount of the public goods should equal the total amount of land rent for the optimal distribution of the population over regions in a country, or optimal population size of each region. This theorem was originally developed in a static framework. We extend this theorem to a dynamic one using overlapping generations model. As the result, we derived that this theorem still holds even in a transient state as well as in the steady state under some types of the utility the production functions.
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Notes
See Ebara (2013).
The term region in this paper implies the area where people live together and jointly consume the service of the same public goods.
In China, people in agricultural sector are moving to urban sector. About 51% of the total population live in urban area now. However, there are 16% of the non-registered residents in urban regions, and they cannot receive enough public services from the governments. This is an important urbanization problem in China. See Ebara (2013), Kaniwa (2013). However, we cannot discuss this kind of problem and we limit our discussion in purely economic field in this paper.
Especially, Chapter 1 of his dissertation, “Dynamic Henry George Theorem and Optimal City Sizes”.
Fu assumes the circular city, where the residents commute to CBD. Letting \(N\) be the population of the city, we derive that the aggregate commuting cost is given by a function of the population, AN \(^{3/2}\), where \(A\) is positive constant. The residents’ utility levels are all equal regardless of the location at equilibrium, hence a resident who lives far from CBD pays high commuting cost pays small differential land rent. Differential land rent plus commuting cost is equal for all residents in the city. Therefore, aggregate commuting costs and the aggregate differential land rents in the city are the same amount. Then the budget constraint for the representative individual in static framework is given by
$$\begin{aligned} y=c+2AN^{1/2}+I/N, \end{aligned}$$(f-1)where \(y\) is the income per capita, \(I\) the investment for the public good, \(c\) is the consumption. The utility is given by \(U(c\),\(I)\). In order to maximize the utility, the consumption \(c\) must be maximized w.r.t. \(N\), given \(y\) and \(I\) in (f-1). Hence, as the result, we have \(I\) = AN \(^{3/2}\), which shows that the public investment expenditure
equals the differential land rent at each period. This is the Henry George theorem. However in the dynamic model, the budget constraint is given by
$$\begin{aligned} \int \limits _0^\infty {e^{-rt}y_t dt} =\int \limits _0^\infty {e^{-rt}\left( {c_t +2AN_t ^{1/2}+I_t /N_t }\right) dt}, \end{aligned}$$(f-2)where subscript \(t\) denotes period, and \(r\) is the interest rate. Then the static budget constraint (f-1) need not hold at each period and Henry George theorem does not hold at each period. However, in the sense of discounted sum, it still holds.
Fujita (1989) and others derived the physical size of the city endogenously. We assume the physical size is fixed for the sake of simplicity. This assumption does not affect the result crucially.
Without the introduction of the bliss, we cannot solve this maximization problem because the value of the objective function diverges to an infinity. We did not introduce the discount rate because we want to compare the Henry George theorem in dynamics and one in statics. Under the existence of a discount rate, the comparison of the result between dynamics and the statics becomes unclear. See Ramsey (1928) for the reference of bliss.
The number of the region is given by \(n_t =1/{N_t}\), hence, \(N_t \le 1\) must hold. In the followings, we assume that this constraint is always satisfied.
If we assume symmetric utility function \(U_t =\beta _1 \ln C_t^1 +\beta _2 \ln C_{r+1}^2 +\beta _3 \ln G_{t+1}\), then we cannot derive the steady-state equilibrium.
This is derived from \(C_t^{1^{*} } =\beta w_t ,C_{t+1}^{2^{*} } =(1-\beta )w_t ( {1+r_{t+1} }) \text{ and } w_t =\alpha N_t^{\alpha -1}.\)
The characteristic roots of Jacobian matrix around \(E^S\)are real numbers. One of them is negative and other is positive. While those at \(E^o\)are complex numbers.
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Kawano, M. Optimal process of urbanization in a developing country dynamic Henry George theorem. Lett Spat Resour Sci 7, 195–204 (2014). https://doi.org/10.1007/s12076-013-0111-x
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DOI: https://doi.org/10.1007/s12076-013-0111-x