Quadratic Transportation Costs Research Articles

Location, Location, Quality:The Fixed Differentiation Principle

AbstractThis work revisits the classic spatial duopoly location-then-prices competition with quadratic transportation costs, in the presence of vertical differentiation. We show that contrary to the substitutability between horizontal and vertical differentiation that has appeared thus far in the literature, higher-quality asymmetry does not affect locational differentiation in equilibrium. While the better-quality seller indeed approaches the city center, the worse-quality seller is simultaneously driven further away from the city, and ultimately out of the market when quality differences become large enough. Interestingly, although vertical differentiation weakens competition, total welfare increases – even when the better-quality seller ultimately monopolizes the market.

Stochastic evolutionary p-Laplace equation: Large deviation principles and transportation cost inequality

We establish a large deviation principle for the strong solution of the evolutionary p-Laplace equation driven by small multiplicative Brownian noise, where the weak convergence approach plays a key role. Moreover, by using the Girsanov transformation along with the L1-contraction approach, we show the quadratic transportation cost inequality for the strong solution to the underlying problem.

A transportation inequality for reflected SPDEs on infinite spatial domain

For the stochastic partial differential equation with reflection on the infinite spatial domain, we establish a quadratic transportation cost inequality under the weighted sup-norm. The proof is based on a new moment inequality under the weighted sup-norm for the stochastic convolution with respect to the space–time white noise.

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

Quadratic transportation cost in the conditional central limit theorem for dependent sequences

R&D investments with spillovers and endogenous horizontal differentiation

We investigate the implications of cost-reducing R&D activities with spillovers in a Hotelling model with endogenous product differentiation driven by quadratic transportation costs. We consider two different three-stage games: in the first, firms choose locations, then R&D efforts and finally prices; in the second, R&D strategies are decided at the first stage and locations are chosen at the second, with price competition taking place at the third. We identify the conditions whereby process innovation brings about a reduction of product differentiation in equilibrium, consequently implying lower profits. Equilibrium product differentiation and profits are increasing in R&D spillovers. Then, transforming the structure into a twofold version of a four-stage game, we allow firms to commit not to invest in R&D, to show the presence of an underlying prisoners’ dilemma affecting R&D decisions. Finally, we illustrate R&D cartels’ ability to eliminate wasteful sunk costs, and outline policy prescriptions.

Transportation cost inequalities for SDEs with irregular drifts

In this paper, the quadratic transportation cost inequality for SDEs with Dini continuous drift and the W1-transportation cost inequality for SDEs with singular coefficients are established via the stability of the Wasserstein distance and relative entropy of measures under the homeomorphism induced by Zvonkin’s transformation.

Talagrand’s quadratic transportation cost inequalities for SPDEs driven by fractional noises with two reflection walls

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the laws of the solutions of stochastic partial differential equations (SPDEs) with two reflection walls under the uniform norm on the continuous functions space. These equations are driven by fractional noises.

The Nature of Equilibria Under Noncollusive Product Design and Collusive Pricing

The basic framework is Hotelling’s model of product choice with quadratic transportation cost. Duopolists choose locations in the initial period and compete in prices in subsequent infinite periods. The firms share profits on the profit possibility frontier. It is shown that under very general conditions, both the firms locating at the center is an equilibrium. It is not necessarily unique and multiple symmetric equilibria can exist. Thus, the products are not necessarily minimally differentiated. How the profits are shared when the firms are located together off the center has a critical bearing on the nature of equilibria. If the firms share profits equally at those locations, then all the equilibria are symmetric. Otherwise, asymmetric equilibria can appear. The equilibria can be classified into three types: a unique equilibrium at the center of the market, multiple symmetric equilibria and multiple asymmetric agglomerated equilibria. The second case entails nonminimal product differentiation. Sufficient conditions for each of these equilibria are given. Necessary conditions for multiple symmetric equilibria off the center are also obtained.

Transportation cost inequality for backward stochastic differential equations with mean reflection

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the law of the solution of backward stochastic differential equations with mean reflection, under the uniform norm and L2-norm. These equations are driven by a Brownian motion.

Quadratic transportation cost inequality for scalar stochastic conservation laws

In this paper, we established a quadratic transportation cost inequality for scalar stochastic conservation laws driven by multiplicative noise. The doubling variables method plays an important role.

Spatial pricing and collusion

AbstractThe stability of collusion is compared assuming quadratic transport costs under two spatial pricing schemes, uniform mill pricing and spatial price discrimination. This comparison uniquely allows fully endogenous location choice and examines when two firms collude, when two of three firms collude and when three firms collude. Spatial price discrimination facilitates collusion more than uniform pricing in the first two of the three examinations but not in the third. This heterogeneity argues that the presence of price discrimination by itself does not inform antitrust authorities about the likelihood of collusion.

On the insufficiency of some conditions for minimal product differentiation

The basic framework is Hotelling’s model of product choice with quadratic transportation cost. Duopolists choose locations in the initial period and compete in prices in subsequent infinite periods. The firms share profits on the profit possibility frontier. Friedman and Thisse (Rand J Econ 24:631–645, 1993) provides a set of sufficient conditions for a unique equilibrium and minimal product differentiation in this setting. This paper reexamines those conditions. In the presence of some mild continuity requirements, there is exactly one profit sharing rule which satisfies those conditions. Furthermore, given any discount factor(s), the corresponding profits cannot be the outcome of a subgame perfect Nash equilibrium at every pair of locations. This brings out an inconsistency in the conditions. A slight weakening of the conditions, to allow for a wider class of profit sharing rules, can result in multiple equilibria and minimal product differentiation need not obtain. Two examples demonstrate this. Thus, neither those conditions nor their weaker variants can be used to characterize a unique equilibrium.

Mixed duopoly in a Hotelling framework with cubic transportation costs

We examine a two-stage location-price model of a mixed duopoly where a private profit-maximizing firm competes with a public welfare-maximizing firm in a Hotelling-type framework. A noteworthy result in this model is that, with quadratic transportation costs, which has become the usual assumption in the literature, the socially optimal locations are obtained in equilibrium. We show here that under the alternative assumption of cubic transportation costs this result no longer holds: equilibrium locations are socially suboptimal. We find that just as in the case of linear transportation costs, previously studied in the literature, for some locations there is price equilibrium in the second stage of the game and for other locations there is not. But, in contrast with such a case, there is a location pair for which there is price equilibrium in the second stage of the game and neither firm has an incentive to marginally change its location. We also find that, in contrast with the case of quadratic transportation costs, this location pair is not socially optimal.

Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space–time white noise

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for laws of the solutions of reflected SPDEs under the uniform norm on the continuous path space. These equations are driven by space–time white noise with deterministic or random initial values. The proofs are heavily based on the methods of Shang and Zhang (2019) and Wu and Zhang (2014).

Optimal sediment transport for morphodynamic model validation

Although commonly used for the validation of morphological predictions, point-wise accuracy metrics, such as the root-mean-squared error (RMSE), are not well suited to demonstrate the quality of a high-variability prediction; in the presence of (often inevitable) location errors, the comparison of depth values per grid point tends to favour predictions that underestimate variability. In order to overcome this limitation, this paper presents a novel diagnostic tool that defines the distance between predicted and observed morphological fields in terms of an optimal sediment transport field, which moves the misplaced sediment from the predicted to the observed morphology. This optimal corrective transport field has the “cheapest” quadratic transportation cost and is relatively easily found through a parameter-free and symmetric solution procedure solving an elliptic partial differential equation. Our method, which we named effective transport difference (ETD), is a variation to a partial differential equation approach to the Monge–Kantorovich L2 optimal transport problem. As a new error metric, we propose the root-mean-squared transport error (RMSTE) as the root-mean-squared value of the optimal transport field. We illustrate the advantages of the RMSTE for simple 1D and 2D cases as well as for more realistic morphological fields, generated with Delft3D, for an idealized case of a tidal inlet developing from an initially highly schematized geometry. The results show that by accounting for the spatial structure of morphological fields, the RMSTE, as opposed to the RMSE, is able to discriminate between predictions that differ in the misplacement distance of predicted morphological features, and avoids the consistent favouring of the underprediction of morphological variability that the RMSE is prone to.

Transportation cost inequality for backward stochastic differential equations

We prove that probability laws of a backward stochastic differential equation, satisfy a quadratic transportation cost inequality under the uniform metric. That is, a comparison of the Wasserstein distance from the law of the solution of the equation to any other absolutely continuous measure with finite relative entropy. From this we derive concentration properties of Lipschitz functions of the solution.

On spatial competition with quadratic transport costs and one online firm

Guo and Lai (Ann Reg Sci 52(1):309–324, 2014) argue that, under quadratic transport costs, when two offline firms compete with one online firm, the two offline firms locate in such a way that they occupy unconnected regions in the market. However, we offer a counterexample to show that their provided condition is not sufficient to support existence of subgame perfect equilibria, because location deviations to different market structures are not taken into account.

Talagrand concentration inequalities for stochastic heat-type equations under uniform distance

In this paper, we established a quadratic transportation cost inequality under the uniform/maximum norm for solutions of stochastic heat equations driven by multiplicative space-time white noise. The proof is based on a new inequality we obtained for the moments of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

On a mixture of Brenier and Strassen Theorems

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem [14].

How market interactions shape the city structure

We develop a spatial monopolistic competition model in which city structure formation is entirely driven by market interactions. When preferences and transport costs are described by real analytic functions, equilibrium land-use patterns are segregated. We completely solve the case of quasilinear quadratic preferences and quadratic transport costs. The city is monocentric when firms are few, duocentric when they are neither too few nor too many, and involves a residential central area bordered by two commercial clusters when firms are many. In the long-run equilibrium, the city size and its spatial structure may change swiftly in response to tiny variations in the opportunity cost of land. Our model captures spatial price dispersion without involving any search frictions.