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A gas is said to be in local equilibrium if it satisfies this equation. [4] The assumption of local equilibrium leads directly to the Euler equations, which describe fluids without dissipation, i.e. with thermal conductivity and viscosity equal to . The primary goal of Chapman–Enskog theory is to systematically obtain generalizations of the ...
In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence), there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.
There is a close connection between linear programs, eigenequations, John von Neumann's general equilibrium model, and structural equilibrium models (see dual linear program for details). [1] [2] [3] Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods.
Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances), by finding the eigenvector(s) associated with each eigenvalue. An equilibrium point is hyperbolic if none of the eigenvalues have zero
But this raises the question of how these prices and allocations have been arrived at, and whether any (temporary) shock to the economy will cause it to converge back to the same outcome that prevailed before the shock. This is the question of stability of the equilibrium, and it can be readily seen that it is related to the question of uniqueness.
[1] General equilibrium analysis, in contrast, begins with tastes, endowments, and technology being fixed, but takes into account feedback effects between the prices and quantities of all goods in the economy. The supply and demand model originated by Alfred Marshall is the paradigmatic example of a partial equilibrium model.
Instead of concluding that equilibrium was Pareto optimal, Edgeworth concluded that the equilibrium maximizes the sum of utilities of the parties, which is a special case of Pareto efficiency: It seems to follow on general dynamical principles applied to this special case that equilibrium is attained when the total pleasure-energy of the ...