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Envelope theorem. In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. [1] As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not ...
Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, denoting the indirect utility function as the Marshallian ...
Envelope (mathematics) Construction of the envelope of a family of curves. In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two ...
Shephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. [1] The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point of a given good ( ) with price is unique. The idea is that a consumer will buy a unique ideal ...
Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. Specifically, it states: The rate of an increase in maximized profits with respect to a price increase is equal to the net supply of the good.
Semi-continuity. In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real -valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively ...
Lagrange multiplier. In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
Baker–Campbell–Hausdorff formula. In mathematics, the Baker–Campbell–Hausdorff formula gives the value of that solves the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal ...