The Lagrange multipliers and existence of competitive equilibrium in an intertemporal model with endogenous leisure

This paper proves the existence of competitive equilibrium in a single sector dynamic economy with elastic labor supply. The method of proof relies on some recent results (see Le Van and Saglam [2004]) concerning the existence of Lagrange multipliers in infinite dimensional spaces and their representation as a summable sequence.Optimal growth model, Lagrange multipliers, competitive equilibrium, elastic labor supply.

Growth and convergence in a model with renewable and non-renewable resources: existence, transitional dynamics, and empirical evidence

This paper studies an optimal endogenous growth model using physical capital, labor and two kinds of natural resources in the final goods sector and employing labor to accumulate knowledge. Based on results in calculus of variations, a direct proof of existence of optimal solution is provided. Analytical solutions for the planner case and the balanced growth paths are found for a specific CRRA utility and Cobb-Douglas production function. Transitional dynamics to the steady state from the theoretical model are used to derive three convergence equations of output intensity growth rate, exhaustible resource growth rate and renewable growth rate, which are tested based on data on production and energy consumption in 27 OECD countries.Optimal growth, existence of equilibrium, transitional dynamics, energy, renewable resource, non-renewable resource.

Existence of competitive equilibrium in a single-sector growth model with heterogeneous agents and endogenous leisure

We prove the existence of competitive equilibrium in a single-sector dynamic economy with heterogeneous agents and elastic labor supply. The method of proof relies on exploiting the existence of Lagrange multipliers in infinite dimensional spaces and the link between Pareto-optima and competitive equilibria.Optimal growth model, Lagrange multipliers, single-sector growth model, competitive equilibrium, elastic labor supply.

On alternative theorems and necessary conditions for efficiency

In this paper, we establish theorems of the alternative for a system described by inequalities, equalities and an inclusion, which are generalizations of Tucker's classical theorem of the alternative, and develop Kuhn-Tucker necessary conditions for efficiency to mathematical programs in normed spaces involving inequality, equality and set constraints with positive Lagrange multipliers of all the components of objective functions.Theorem of the alternative, Kuhn-Tucker necessary condition, constraint qualification.

ON NECESSARY CONDITIONS FOR EFFICIENCY IN DIRECTIONALLY DIFFERENTIABLE OPTIMIZATION PROBLEMS

This paper deals with multiobjective programming problems with in- equality, equality and set constraints involving Dini or Hadamard differentiable func- tions. A theorem of the alternative of Tucker type is established, and from which Kuhn-Tucker necessary conditions for local Pareto minima with positive Lagrange multipliers associated with all the components of objective functions are derived.Theorem of the alternative, Kuhn-Tucker necessary conditions, direc- tionally differentiable functions.

On constraint qualifications with generalized convexity and optimality conditions

This paper deals with a multiobjective programming problem involving both equality constraints in infinite dimensional spaces. It is shown that some constraint qualifications together with a condition of interior points are sufficient conditions for the invexity of constraint maps with respect to the same scale map. Under a new constraint qualification which involves an invexity condition and a generalized Slater condition a Kuhn-Tucker necessary condition is established.Invexity, scale, constraint qualification, nearly S-convelike mapping.

The role of poly-β-hydroxybutyrate as protection against Vibrio infections in blue mussel larvae

The blue mussel (Mytilus edulis) is an important species in Europe, not only as a model for environmental studies but also as aquaculture species. This PhD research is the first study to evaluate the use of poly-β-hydroxybutyrate (PHB) as an innovative production strategy to improve the performance of bivalve larvae. The following three approaches were used: First, the application of PHB in mussel larvae cultures in crystalline form (i.e. extracted from the bacterial cells, PHB-C) versus amorphous form (i.e. still contained in the bacterial cells, PHB-A) was compared. PHB was supplied at concentrations of 0.1 mg L-1, 1.0 mg L-1, and 10.0 mg L-1 to the culture water of the larvae, starting 2 days after fertilization; secondly, the possible antimicrobial effect of PHB-A and its monomer β-hydroxybutyrate was investigated since only the latter is biologically active. Their putative protective role against two pathogenic bacteria Vibrio splendidus and Vibrio coralliilyticus was studied both in vitro and in vivo. For the in vitro tests, the impact of β-hydroxybutyrate against different virulence factors was determined at pH7 and pH8; (3) finally, the impact of PHB-A (1mg L-1) on gene expression of the antimicrobial peptides mytimycin, mytilinB, defensin and the hydrolytic enzyme lysozyme was looked into during in vivo challenge tests with V. coralliilyticus (105 CFU mL-1). Phenoloxidase activity in the challenged mussel larvae was also evaluated. In conclusion, the work proved that the biological compound PHB-A supplied as freeze-dried Ralstonia eutropha can bring added value to blue mussel larvae culture and possibly to bivalve larval culture in general by its ability to interfere with some virulence factors of the pathogens and with the immune gene expression of AMPs and by stimulating phenoloxidase activity in larvae. Use of PHB-A as an energy source by the larvae is unlikely but needs further confirmation