Does the Henry George Theorem provide a practical guide to optimal city size?

American Journal of Economics and Sociology, The,  Nov, 2004  by Richard Arnott

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The HGT is derived on the assumption that land is homogeneous, but in reality locations differ in terms of fertility, natural amenities such as visual beauty and climate, and natural accessibility such as access to the sea or a navigable river. How do these Ricardian differences in land affect the Theorem qualitatively, and how important are they quantitatively? To my knowledge, this question has not been investigated in the literature.

Thus far, we have considered only static economies. Does the Theorem extend to intertemporal economies? In the absence of any state dependence, the answer is obviously affirmative, since the economy can be optimized anew every period. The most obviously important source of state dependence is durable capital, and in the urban context the most important forms of durable capital are buildings and public infrastructure such as roads. Arnott and Kraus (1998) have shown that the Mohring-Harwicz-Strotz self-financing results for congestible facilities, which were originally established for static economies, generalize to dynamic economies, but with the results in discounted terms. In light of the close similarity between the generalized Henry George Theorem and the Mohring-Harwicz-Strotz results, which is explored in Berglas and Pines (1981), it is natural to conjecture that the generalized Henry George Theorem extends to intertemporal settings. Consider, for example, extending the basic static model of the previous section to allow for durable housing and durable pure local public goods (such as a lighthouse). For the static model, the result was that in a city of optimal population size, aggregate land rents equal expenditure on the pure. local public good. The analog in a dynamic model with durable housing would be that, in cities whose population size follows the optimal trajectory, aggregate land values equal the discounted present value of expenditure on the pure, local public good. (4)

The final generalization to be considered in this section is distortions. As indicated earlier, the generalized Henry George Theorem applies for any Pareto optimal allocation in a large spatial economy with a nontrivial pattern of agglomeration. Furthermore, when a Pareto optimal allocation in such an economy is decentralized as a quasi-competitive equilibrium, market prices equal the corresponding shadow prices, so that the Theorem holds when aggregates are evaluated at market prices. The presence of unalterable distortions, however, generally precludes decentralization of Pareto optimal allocations and causes shadow prices to deviate from market prices. It might appear therefore that the generalized Henry George Theorem does not hold in distorted economies; if it does not, the practical relevance of the Theorem is dubious since uninternalized production externalities, interaction externalities, and congestion externalities are very important features of all real-world cities. Fortunately, the Henry George Theorem can be adapted to cover distorted economies. The intuition is that a constrained Pareto optimal allocation will still be an average cost minimum when evaluated at shadow prices. The presence of distortions introduces additional constraints, so that the planning problem becomes one of minimizing resource costs per capita of providing the exogenous vector of utilities to the various groups, holding fixed the proportion of individuals in each group, as before, but now subject to the additional constraints imposed by the distortions. This corresponds to minimizing average costs measured in terms of shadow prices. Thus, in distorted urbanized economies the generalized Henry George Theorem continues to hold when the aggregate magnitudes are valued at shadow prices.