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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 ...
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. [1] It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision ...
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 ...
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 ...
The convex hull of the red set is the blue and red convex set. In geometry, the convex hull, convex envelope or convex closure[1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all ...
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 ...
Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: If we denote values at the optimum with a star (), then it can be shown that ( (,, …