Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Culture and Development: An Analytical Framework

This paper develops a framework which analyzes how a population's culture affects the decisions of rational profit maximizing firms, while simultaneously exploring how the actions of these firms in turn affect the population's culture.By endogenizing culture as well as the more usual economic variables, it shows how an economically valuablebehavioural trait can be sustained as part of a competitive equilibrium.It is shown that, for given primitives, an economy can be in either a 'good' steady state, in which the valuable cultural trait is present, or a welfare dominated 'bad' one in which the valuable cultural trait disappears.Starting from the 'good' steady state and implementing productivity improvements raises welfare, but if changes are too rapid this steady state will not be reached from the old one.Instead, the unique trajectory is to the bad steady state where welfare is reduced.culture;development;inequality;technological change

Trust, Social Capital and Economic Development

Many argue that elements of a society s norms, culture or social capital are central to understanding its development.However, these notions have been difficult to capture in economic models.Here we argue that trustworthiness is the economically relevant component of a society s culture and hence comprises its social capital.Individuals are trustworthy when they perform actions they have promised, even if these do not maximize their payoffs.The usual focus on incentive structures in motivating behaviour plays no role here.Instead, we emphasize more deep-seated modes of behaviour and consider that trustworthy agents are socialized to act as they do.To model this socialization, we borrow from a relatively new process of preference evolution pioneered by Bisin and Verdier (2001).The model developed endogenously accounts for social capital and explores its role in the process of economic development.It captures in a simple, formal way the interaction between social capital and the economy s productive process.The results obtained caution against rapid reform, provide an explanation for why late developing countries cannot easily transplant the modes of production that have proved useful in the West, and suggest an explanation for the pattern of reform experiences in ex-communist countries.technological change;public finance;economic development

Contracting Productivity Growth

In this paper, we analyze the interactions between growth and the contracting environment in the production sector.Allowing incompleteness in contracting implies that viable production relationships for firms and workers, and therefore the profitability of industries, depend on the rates of innovation and growth.The speed at which new innovations arrive in turn depends on the profitability of production, for the usual reasons examined in the endogenous growth literature.We show that these interactions can have important implications which are consistent with observed phenomena in both the micro and macro environment.In particular, we demonstrate that a technological shock (increasing productivity of research) can, through this interaction, lead to a productivity slowdown and a shift in labor market contracts away from firms providing implicit guarantees of lifetime employment and towards shorter-term "contractor" type arrangements.We show the consistency of an increase in the proportion of the labor force under short term employment, increased relative returns of workers in high productivity sectors, and increased income inequality, with a productivity slowdown of finite duration.economic growth;contracts;productivity

Characterization of the size and position of electron-hole puddles at a graphene p-n junction

The effect of an electron-hole puddle on the electrical transport when governed by snake states in a bipolar graphene structure is investigated. Using numerical simulations we show that information on the size and position of the electron-hole puddle can be obtained using the dependence of the conductance on magnetic field and electron density of the gated region. The presence of the scatterer disrupts snake state transport which alters the conduction pattern. We obtain a simple analytical formula that connects the position of the electron-hole puddle with features observed in the conductance. Size of the electron-hole puddle is estimated from the magnetic field and gate potential that maximizes the effect of the puddle on the electrical transport.Comment: This is an author-created, un-copyedited version of an article published in Nanotechnology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi:10.1088/0957-4484/27/10/10520

Gender Wage Differentials in a Competitive Labor Market: The Household Interaction Effect

We present a theoretical explanation of the gender wage gap which turns on the interaction between men and women in households.In equilibria where men are over-represented in full-time work, we show that firms rationally choose to hire women only at strictly lower wages than men.The model developed predicts a gap even controlling for education, occupation and industry of workers and does so in a competitive labor market where there exist no inherent gender differences. We test our theory using CPS data over the period 1979-98 and find it is strongly supported by the data.gender discrimination;household models;wage gap

Animal Spirits Meets Creative Destruction

We show how a Schumpeterian process of creative destruction can induce coordination in the timing of entrepreneurial activities across diverse sectors of the economy.Consequently, a multi-sector economy, in which sector-specific, productivity improvements are made by independent, profit-seeking entrepreneurs, can exhibit regular booms, slowdowns and downturns as an inherent part of the long-run growth process.The cyclical equilibrium that we study has a higher long-run growth rate but lower welfare than the corresponding acyclical one.We find that the cycles generated by our model share some features of actual business cycles, and that across cycling economies, a negative relationship emerges between volatility and growth.economic growth;entrepreneurship;innovation;business cycles

The relationship between noise and annoyance around Orly

The extent to which annoyance estimated by an isopsophic index is a good forecaster for annoyance perceived near airport approaches was investigated. An index of sensed annoyance is constructed, and the relationship between the annoyance index and the isopsophic index is studied

Co-movement, Capital and Contracts: 'Normal' Cycles Through Creative Destruction

We develop a unified theory of endogenous business cycles in which expansions are neoclassical growth periods driven by productivity improvements and capital accumulation, while downturns are the result of Keynesian contractions in aggregate demand below potential output. Recessions allow skilled labor to be reallocated to growth promoting activities which fuel subsequent expansions. However, rigidities in production and contractual limitations, inherent to the process of creative destruction, leave capital severely underutilized. A key feature of our equilibrium is the endogenous emergence of long term supply contracts between capitalist owners and producers.Long-term contracting;investment irreversibility;putty-clay technology;asset- specificity;Endogenous cycles and growth