Optimal Timing of Cartel Formation Under Uncertainty

Understanding how business cartels form and expand is foundational for developing sound deterrence strategies. Past work (i.e. Connor, 2005) has relied on net present value (NPV) methods to evaluate the streams of costs and benefits of forming or joining a cartel. While NPV adequately measure the expected value of future streams of benefits and costs, higher moments of the distribution are also important in understanding agent behavior. Thus, in the presence of uncertainty about future streams and litigation costs, NPV may miss important dimensions that shape the issue. The decision to form or join a cartel is, at least, partially irreversible, because it exposes the firm or its involved managers to litigation on all previous returns and even after the cartel is dissolved. In this study, we rely on the aforementioned irreversible and uncertain nature of cartel participation and returns to develop a real-options framework that examines the optimal decision rules regarding the timing of cartel formation. This leads to suggestions for improved policy tools for antitrust agencies. In our model, all firms outside of a cartel essentially hold the option to form or join a cartel at some point in the future. The option is exercised the day the cartel is formed and has no cash value before that. The payoffs that firms give up by not immediately forming a cartel are weighed against uncertain and partially irreversible forming decision nature. Under the assumption of stochastic market demand, we find a threshold level of demand beyond which the cartel is formed. This threshold is analytically calculated as a function of a number of parameters. We then illustrate the conditions that determine the optimal timing decision of cartel formation by conducting comparative dynamics analysis. The timing of cartel formation is analyzed in both domestic and international settings. The qualitative results are obtained by comparative dynamics analysis and the quantitative results by numerical analysis. The results obtained in this study will assist in the development and improvement of guidelines to deter the formation of cartels by antitrust agencies in both developed and developing nations. In the first scenario of domestic cartel formation, at the beginning of each period, firms will choose to compete when the sunk costs related to a cartel operation are too high, current period demand is too low, and/or the expected duration of collusion is too short. Obviously, the possibilities of cartel formation increase when sunk costs go down, demand increases, or relationships with firms improve the prospects for a longer cartel arrangement. The value of waiting increases with a) increased uncertainty of market demand, b) increase irreversibility, and c) increased number of firms and d) a higher discount rate. We study the effect of demand uncertainty on the expected social welfare of cartel formation. The simulation results suggest that the expected social welfare under uncertainty could be higher than that under certainty. In a second scenario, we study the formation of international cartels. The model considers two markets with demand uncertainty that is either correlated or uncorrelated. The demand shocks in each market are assumed to follow geometric Brownian motion. We calculate the threshold value of demand faced by the cartel and obtain the rules guiding a firm's decision to form an international cartel. The comparative dynamics results obtained in the previous domestic scenario still apply. The simulation results suggest that cartel formation is most likely to occur between firms that come from countries with highly correlated markets and similar expected demand growth.Cartel Joining Behavior, Real Options Theory, International Cartels, Industrial Organization, K21, L00, L12,

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.Comment: 25 pages, 1 figur

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $\tau=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $\tau=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1511.0119

Oligopsony Power: Evidence from the U.S. Beef Packing Industry

Replaced with revised version of paper 08/24/09.Margin, Beef Packing, Fed Cattle Prices, Markov Regime Switching, Industrial Organization,

Research on the Utilization of Foreign Direct Investment in Xinjiang Uygur Autonomous Region of China

With the implement of economic reform and opening-up policy over 30 years, the utilization of foreign direct investment (FDI) in Xinjiang has been developing fast. FDI increased obviously in scale with great achievements of outstanding effects, however, there are still some problems in the utilization of FDI in Xinjiang. The paper summarizes the basic situation about using FDI in Xinjiang and proposes strategies and suggestions to optimize using FDI, extend the source of FDI, improve the scale and quality of foreign investment with perfect supporting system of Xinjiang government

It’s All About Produce: Flexing the Muscles of Western U.S. Organic Spinach Consumption

Over the past few decades, consumers have become more concerned about health and nutrition, which is displayed by an increased demand for organic foods. Once considered a niche product, organic food has become more affordable for consumers through its availability in conventional supermarkets. In 2010, organic food and beverages showed a 7.7% increase in annual sales compared to 2009 sales, with the highest growth rate of 11.8 % in organic produce (Organic Trade Association (OTA), 2011)

The Impact of Food Environment on Private Label versus Branded Produce Choice

Over the past two decades, retailers are providing more Private label foods (PL), which are directly competing with the National brand (NB) products. For years, PLs competed as generic and cheaper versions with their high-priced NB substitute. However, modern PLs have improved in product quality relative to NBs and are available in the premium, organic, and even produce sections with the goal to distinguish themselves from their competitors’ product lines. One of the fastest growing segments in the produce industry consists of triple-washed cello-packed spinach. Using a two-step Heckman model, we determine the impact of household purchase information, demographics, and food environment on PL spinach purchasing behavior. Given its regional dominance with regard to spinach production and fresh spinach consumption, we focus on households residing in the U.S. West. Results show that food environment is the main driver for PL spinach purchases. We determine that specialty stores might be traditional channels for purchasing organic PL spinach, while supercenters might take the role of the main outlet for conventional NBs. An understanding of what factors might encourage increased consumption of healthful foods is important to producers and marketers for developing effective strategies in order to reach beyond the traditional consumer base