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General Equilibrium Theories of Spatial Agglomeration  

Marcus Berliant and Ping Wang

General equilibrium theories of spatial agglomeration are closed models of agent location that explain the formation and growth of cities. There are several types of such theories: conventional Arrow-Debreu competitive equilibrium models and monopolistic competition models, as well as game theoretic models including search and matching setups. Three types of spatial agglomeration forces often come into play: trade, production, and knowledge transmission, under which cities are formed in equilibrium as marketplaces, factory towns, and idea laboratories, respectively. Agglomeration dynamics are linked to urban growth in the long run.


General Equilibrium Theory of Land  

Masahisa Fujita

Land is everywhere: the substratum of our existence. In addition, land is intimately linked to the dual concept of location in human activity. Together, land and location are essential ingredients for the lives of individuals as well as for national economies. In the early 21st century, there exist two different approaches to incorporating land and location into a general equilibrium theory. Dating from the classic work of von Thünen (1826), a rich variety of land-location density models have been developed. In a density model, a continuum of agents is distributed over a continuous location space. Given that simple calculus can be used in the analysis, these density models continue to be the “workhorse” of urban economics and location theory. However, the behavioral meaning of each agent occupying an infinitesimal “density of land” has long been in question. Given this situation, a radically new approach, called the σ -field approach, was developed in the mid-1980s for modeling land in a general equilibrium framework. In this approach: (1) the totality of land, L , is specified as a subset of ℝ 2 , (2) all possible land parcels in L are given by the σ -field of Lebesgue measurable subsets of L , and (3) each of a finite number of agents is postulated to choose one such parcel. Starting with Berliant (1985), increasingly more sophisticated σ -field models of land have been developed. Given these two different approaches to modeling land within a general equilibrium framework, several attempts have thus far been proposed for bridging the gap between them. But while a systematic study of the relationship between density models and σ -field models remains to be completed, the clarification of this relationship could open a new horizon toward a general equilibrium theory of land.


Q-Factors and Investment CAPM  

Lu Zhang

The Hou–Xue–Zhang q-factor model says that the expected return of an asset in excess of the risk-free rate is described by its sensitivities to the market factor, a size factor, an investment factor, and a return on equity (ROE) factor. Empirically, the q-factor model shows strong explanatory power and largely summarizes the cross-section of average stock returns. Most important, it fully subsumes the Fama–French 6-factor model in head-to-head spanning tests. The q-factor model is an empirical implementation of the investment-based capital asset pricing model (the Investment CAPM). The basic philosophy is to price risky assets from the perspective of their suppliers (firms), as opposed to their buyers (investors). Mathematically, the investment CAPM is a restatement of the net present value (NPV) rule in corporate finance. Intuitively, high investment relative to low expected profitability must imply low costs of capital, and low investment relative to high expected profitability must imply high costs of capital. In a multiperiod framework, if investment is high next period, the present value of cash flows from next period onward must be high. Consisting mostly of this next period present value, the benefits to investment this period must also be high. As such, high investment next period relative to current investment (high expected investment growth) must imply high costs of capital (to keep current investment low). As a disruptive innovation, the investment CAPM has broad-ranging implications for academic finance and asset management practice. First, the consumption CAPM, of which the classic Sharpe–Lintner CAPM is a special case, is conceptually incomplete. The crux is that it blindly focuses on the demand of risky assets, while abstracting from the supply altogether. Alas, anomalies are primarily relations between firm characteristics and expected returns. By focusing on the supply, the investment CAPM is the missing piece of equilibrium asset pricing. Second, the investment CAPM retains efficient markets, with cross-sectionally varying expected returns, depending on firms’ investment, profitability, and expected growth. As such, capital markets follow standard economic principles, in sharp contrast to the teachings of behavioral finance. Finally, the investment CAPM validates Graham and Dodd’s security analysis on equilibrium grounds, within efficient markets.