The Hotelling rule in non-renewable resource economics: A reassessment

Harold Hotelling's 1931 contribution is known for providing a basic principle—the Hotelling rule—to the economics of non-renewable resources. Nearly 90 years later, empirical tests conclude the rule lacks empirical validity, requiring strong amendments to describe the long-term, aggregate behaviour of its target object. On the basis of Hotelling's unpublished archival material, this paper revisits the place given to the Hotelling rule in non-renewable resource economics. Our reconstruction shows that Hotelling's 1931 paper has been misinterpreted: from the outset, the Hotelling rule was not valid for mineral resources. In contrast, the consideration of two inherent geological constraints, alongside exhaustibility, offered the opportunity for an alternative basic framework, capable to generate bell-shaped and U-shaped equilibrium trajectories for supplies and prices, respectively. Inspired by this unknown aspect of Hotelling's work brought to light by our archival investigation, we sketch this alternative basic model, enabling non-renewable resource economics to circumvent the empirical shortfalls of the Hotelling rule.     

Price equilibrium in address models of product differentiation: unit-elastic demand

Summary. I present a class of address models of product differentiation with unit-elastic individual demand and show the existence of Nash equilibrium in prices under assumptions on utility functions and the taste and income heterogeneity across consumers. This paper complements the work by Caplin and Nalebuff (1991, Econometrica), who analyze unit demand models of product differentiation. Received: December 28, 1998; revised version: September 5, 2001     

Competition and confidentiality: Signaling quality in a duopoly when there is universal private information

We model non-cooperative signaling by two firms that compete over a continuum of consumers, assuming each consumer has private information about the intensity of her preferences for the firms' respective products and each firm has private information about its own product's quality. We characterize a symmetric separating equilibrium in which each firm's price reveals its respective product quality. We show that the equilibrium prices, the difference between those prices, the associated outputs, and profits are all increasing functions of the ex ante probability of high safety. If horizontal product differentiation is sufficiently great then equilibrium prices and profits are higher under incomplete information about quality than if quality were commonly known. Thus, while signaling imposes a distortionary loss on a monopolist using price to signal quality, duopolists may benefit from the distortion as it can reduce competition. Finally, average quality is lower since signaling quality redistributes demand towards low-quality firms.     

Non‐Homothetic Multisector Growth Models

Multisector growth (MSG) models are dynamic versions of computable general equilibrium (CGE) models. Non‐homothetic preference (utility) functions are required for the evolution of factor allocations and industrial structures in accordance with consumption expenditure patterns implied by the non‐unitary income elasticities observed in all budget data since Engel in the 1850s. But comparative static general equilibrium solutions and particularly solving the dynamics of MSG models require explicit specifications of all demand and cost (price) functions. On the demand side, the constant differences of elasticity of substitution (CDES) non‐homothetic indirect utility functions and Roy's identity provide the explicit Marshallian demand functions and budget shares. Sectorial constant elasticity of substitution (CES) cost functions and Shephard's lemma provide the explicit relative commodity price functions and the sectorial cost shares and capital–labor ratios. Walrasian equilibria are given by one equation and the multisector dynamics by three differential equations. Benchmark solutions are given for three cost regimes of a 10‐sector MSG model. History patterns of industrial/allocational evolutions are recognized.     

The existence and computation of competitive equilibria in markets with an indivisible commodity

This paper considers an exchange economy called a generalized assignment market, in which sellers and buyers trade one indivisible commodity possibly with product differentiation for a perfectly divisible commodity. The existence of a competitive equilibrium in this economy is proved using Kakutani's fixed point theorem. This existence theorem is applied to a production economy in which sellers are formulated as producers with convex cost functions. Two examples of housing markets are provided and their competitive equilibria are numerically calculated.     

Niche products, generic products, and consumer search

We endogenize product design in a model of sequential search with random firm-consumer match value à la Wolinsky (Quart J Econ 96:493–511, 1986) and Anderson and Renault (RAND J Econ 30:719–735, 1999). We focus on a product design choice by which a firm can control the dispersion of consumer valuations for its product; we interpret low-dispersion products as “generic” and high-dispersion products as “nichy.” Equilibrium product design depends on a feedback loop: when reservation utility is high (low), the marginal customer’s match improves (worsens) with more nichy products, encouraging high (low) differentiation by firms. In turn, when firms offer more nichy products, this induces more intense search; depending on search costs, this could raise or lower consumers’ reservation utility. Remarkably, when the match distribution satisfies a hazard rate condition, firm and consumer interests align: equilibrium product design always adjusts to the level that maximizes utility. When this condition is not met, either multiple equilibria (one nichy, the other generic) or one asymmetric equilibrium (generic and nichy firms coexist) can arise; we argue that the former is more likely for common specifications of consumer preferences.     

WELFARE GAINS AND LOSSES IN SUNSPOT EQUILIBRIA*

We study a standard two period exchange economy with one nominal asset. As is well known, there is a continuum of sunspot equilibria around each efficient equilibrium. A sunspot equilibrium is inefficient but some households may gain in sunspot equilibria relative to the efficient equilibrium. We show that a household's equilibrium utility level is either locally maximized or locally minimized at the efficient equilibrium, and derive a condition which identifies whether or not a household's utility is locally minimized or maximized.     

Dumping,antidumping duties,and price undertakings

This paper examines which types of firms, from a developed country (DC) or a less developed country (LDC), tend to practice dumping, using a two-market equilibrium analysis of trade in similar products. Specifically, we present a vertical product differentiation model of duopolistic competition between a DC firm and an LDC firm under free trade to show that the DC firm sells a higher-quality product without dumping. In contrast, the LDC firm sells a lower-quality product and practices dumping in the DC market by charging a price lower than the product's price in the LDC's local market. In response to the LDC dumping, the DC government's use of an optimal antidumping duty increases its domestic welfare. The LDC's social welfare may increase if its exporting firm accepts price undertaking rather than dumping. From the perspective of world welfare, defined by aggregating the welfare of the trading countries (DC and LDC), the trade damage measure through imposing antidumping fines on LDC dumping is Pareto-improving compared to free trade (under which dumping takes place) and price undertakings.     

Are unit export values correct measures of the exports’ quality?

It has become common to measure the quality of exports using their unit export value (UEV). Applications of this method include studies of intra-industry trade (IIT) and analyses of industrial ‘competitiveness.’ This literature seems to assume that export quality and export price (the most natural interpretation of UEV) are not merely correlated but that they follow each other one-for-one. We put this assumption under scrutiny from both a theoretical and empirical point of view. In terms of theory, we formalize this assumption as a hypothesis of the proportionality of equilibrium prices and equilibrium qualities. We discuss several cases for which this hypothesis is theoretically doubtful (nonlinear utility and cost functions; strong and asymmetric horizontal product differentiation). We also suggest a method of verifying the hypothesis for cases in which it cannot be easily rejected theoretically. This method is then applied to German imports in the period of 1994–2009. We find that the implications of the proportionality hypothesis are largely contradicted by the data.     

ENTRY AND QUALITY SIGNALLING WHEN ONLY SOME CONSUMERS ARE INFORMED OF THE ENTRANT'S QUALITY

This paper focuses on competition between an incumbent and an entrant when only the entrant's quality is unknown to (some) consumers. The incumbent may or may not know the entrant's quality. The model reveals a separating equilibrium where the entrant's high price signals its high quality when the proportion of informed consumers is at some intermediate value. The case in which the incumbent knows the entrant's quality generates two additional equilibria. When the proportion of informed consumers is large enough, firms choose their prices as in the complete information case. The entrant's high price in combination with the incumbent's low price signals the entrant's high quality. When the proportion of informed consumers is at some intermediate value, the incumbent's high price signals the entrant's low quality, while its low price signals the entrant's high quality. Interestingly, we find that entry may be facilitated with informational product differentiation.     

On Hotelling's competition with non-uniform customer distributions

Hotelling's competition in a two-stage (location, price) game is examined with increasing densities of consumers towards the centre. It is shown that there is a range of, rather even, distributions for which firms locate at opposite ends of the market. As the distribution becomes more concentrated, duopolists will tend to move inside the market.     

The complexion of dynamic duopoly game with horizontal differentiated products

In this paper, firms are considered on the hypothesis of having incomplete rationality expectation and incomplete information of the market to get the dynamic development of price competition behavior in the Hotelling model (Hotelling, 1929). Under the assumption of the heterogeneous expectations of two firms, we have observed that the Nash equilibrium price can be a dynamic equilibrium to realize when the speed of price adjustment is lower. However, the numerical simulation shows that the system may present a periodic and chaotic status when the speed of price adjustment is higher. The effect of the degree of horizontal differentiation on the stability of Nash equilibrium of the system is also discussed. We have a different conclusion from Lucino Fanti and Luca Gori's (2012); that is the greater the degree of product horizontal differentiation is, the more stable the Nash equilibrium of the system is.     

Drèze equilibria and welfare maxima

We present an example of a production economy with incomplete markets, von Neumann–Morgenstern utility functions, and a unique Drèze equilibrium in order to illustrate and explain the following phenomenon. There exists a transfer scheme such that every shareholder’s utility after transfers and share adjustments increases the more the firm deviates from the Drèze equilibrium. However, shareholders’ welfare decreases the further the firm departs from the Drèze equilibrium. Shareholders’ welfare is defined as the sum of their utilities where every utility function is normalized such that the marginal utility of today’s consumption equals 1 at the Drèze equilibrium.     

RECURSIVE EQUILIBRIA IN AN AIYAGARI‐STYLE ECONOMY WITH PERMANENT INCOME SHOCKS

We prove existence of a recursive competitive equilibrium (RCE) for an Aiyagari‐style economy with permanent income shocks and derive important economic implications. We show that there exist equilibria where borrowing constraints are never binding and establish a nontrivial lower bound on the equilibrium interest rate. These results imply distinct consumption dynamics compared to existing studies. We present a new approach to solve the agent's problem that uses lattices of consumption functions to deal with permanent income shocks and an unbounded utility function. The approach provides a theoretical foundation for convergence of the time iteration algorithm widely used in applied work.     

Optimal Trade Policy with Horizontal Differentiation and Asymmetric Costs

This paper examines the optimal export policy under Bertrand competition when the products exhibit horizontal differentiation and production costs are asymmetric. The focus of this paper is on the product‐differentiation effect in the determination of the optimal export policy. We show that given that the equilibrium characteristic of a foreign firm's product R&D lies to the left‐hand side of its initial level , since the foreign firm has a unit cost advantage and the efficiency of its R&D technology is sufficiently low, a rise in the export subsidy of the domestic country increases a domestic firm's profits and then welfare by extending the degree of horizontal differentiation between the two products. Thus, the optimal export policy under Bertrand competition may turn out to be an export subsidy rather than an export tax. This result is in sharp contrast to that of Eaton and Grossman (1986 ).     

EXISTENCE AND UNIQUENESS OF EQUILIBRIUM FOR A SPATIAL MODEL OF SOCIAL INTERACTIONS*

We extend Beckmann's spatial model of social interactions to the case of a two‐dimensional spatial economy with a large class of utility functions, accessing costs, and space‐dependent amenities. We show that spatial equilibria derive from a potential functional. By proving the existence of a minimizer of the functional, we obtain that of spatial equilibrium. Under mild conditions on the primitives of the economy, the functional is shown to satisfy displacement convexity. Moreover, the strict displacement convexity of the functional ensures the uniqueness of equilibrium. Also, the spatial symmetry of equilibrium is derived from that of the primitives of the economy.     

Adaptive behavior and the existence of Weizsäcker's long-run indifference curves

Suppose that past consumptions of the first r commodities (r < n) influence present consumption. Then, the long-run demand function to which demand converges maximizes the equilibrium short-run utility function only under very restrictive conditions. The long-run demand functions can be rationalized by a utility function, different from the equilibrium short-run utility function, if and only if the short-run utility function is such that past consumptions of any good that experience, learning, or taste changes is separable from all other goods. The class of such utility functions has been found.     

Nonpaternalistic intergenerational altruism

The paper develops a concept of equilibrium behaviour and establishes its existence in a model of nonpaternalistic intergenerational altruism. Each generation derives utility from its own consumption and the utilities of its successors. Equilibrium capital stocks are time-monotone and therefore converge to a steady state. Finally. when each generation's utility depends on that of at least two successors, equilibria may be inefficient. This is shown by an example, where feasible programs exist providing greater consumption (compared to the equilibrium path) for every generation.     

The impact of brand selection on price competition - a double-edged sword

Brand selection - the number of alternative brands of a particular product - is a double-edged sword in that on the one hand it increases rivalry and so stimulates price competition but on the other hand it causes consumers to be poorly informed and thus vulnerable to exploitation and so dampens price competition. This phenomenon is demonstrated analytically and empirically, with analytical work based on Hotelling's unit market line. Using price dispersion as a measure of the state of price competition, it is shown that a widening in price dispersion, that is, a lessening of price competition, arises from an increase in brand selection. Any increase in the number of alternative brands on offer (brand selection) causes an increase in ‘DIF-ness’ (Distortion in Information Function) which in turn causes an increase in price dispersion (a lessening of price competition). The phenomenon is demonstrated empirically through a small survey of household beverages.     

Equilibrium prices in the Bertrand and Cournot oligopolies

The equilibrium prices for the Bertrand and Cournot oligopolies with product differentiation are compared. If all firms have linear demand and cost functions, and if, in addition, the Jacobian matrix of the demand functions has a dominant negative diagonal, the Cournot equilibrium prices are not lower than the Bertrand ones. The general condition for the comparison of the Bertrand and Cournot equilibrium prices can be derived even if the nonlinearity is involved in the cost and/or demand functions.