A Thermodynamic Formulation of Home Prices in a Monocentric City

Abstract

In this work, we proposed a theoretical framework inspired by physical thermodynamics to explain the housing price distributions in monocentric cities. In the same spirit as the Alonso–Muth–Mills (AMM) model, we assume that the disposable income C=W−R(x)−Q(x) after renting a home a distance x from the center of a city is determined by the wage W generated at the point-like Central Business District (CBD), the rent R(x), and the transportation cost Q(x). Unlike in the AMM model, where the scaling exponents are phenomenological, we admitted only physically reasonable exponents for the scaling of various quantities with distance x from the CBD. We then determine the equilibrium rent R(x) by requiring dU∕dx=0, where we assumed for simplicity the utility function U=lnC (representing the demand side) has diminishing return in C. In the simplest model, the equilibrium rent is given by R(x)=R0−Q(x), i.e., the scaling of R(x) with x is entirely determined by Q(x). We then introduce additional home availability S(x) (representing the supply side) into the simple theory in the form of an entropic correction, F=U−TS. The equilibrium rent then becomes R(x)=R0−Q(x)+C0[1−exp(TS(x))]. This allows us to treat additional availability due to the two-dimensional nature of cities, as well as that due to high-rise buildings on equal footing. Finally, we compare the equilibrium theory against urban data in Singapore, London and Philadelphia. For Singapore, we find quantitative agreement between theory and data. For London, we find only qualitative agreement between theory and data because the transportation cost is zone based. For Philadelphia, the home price distribution is very different from Singapore and London, and shows clear signs of economic segregation, which is difficult to treat in our equilibrium theory.

Keywords

• thermodynamics
• free energy
• optimization
• urban economics
• home prices