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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 ( i {\displaystyle i} ) with price p i {\displaystyle p_{i}} is unique.
The GO cost function is flexible in the price space, and treats scale effects and technical change in a highly general manner. The concavity condition which ensures that a constant function aligns with cost minimization for a specific set of , necessitates that its Hessian (the matrix of second partial derivatives with respect to and ) being negative semidefinite.
If the utility function is strictly quasi-concave, there is the Shephard's lemma. ... Dual to the utility maximization problem is the cost minimization problem [2] [3]
Ronald William Shephard (November 22, 1912 – July 22, 1982) was professor of engineering science at the University of California, Berkeley. [ 1 ] He is best known for two results in economics, now known as Shephard's lemma and the Shephard duality theorem .
Roy's identity reformulates Shephard's lemma in order to get a Marshallian demand function for an individual and a good from some indirect utility function.. The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income in the indirect utility function (,), at a utility of :
This specification satisfies homogeneity of order 1 in prices, and is a second order approximation of any cost function. From this, demand equations are derived (using Shephard's lemma), but are however simpler to put in term of budget shares = (,) :
Whereas Marshallian demand comes from the Utility Maximization Problem, Hicksian Demand comes from the Expenditure Minimization Problem. The two problems are mathematical duals, and hence the Duality Theorem provides a method of proving the relationships described above. The Hicksian demand function is intimately related to the expenditure ...
Isocost v. Isoquant Graph. In the simplest mathematical formulation of this problem, two inputs are used (often labor and capital), and the optimization problem seeks to minimize the total cost (amount spent on factors of production, say labor and physical capital) subject to achieving a given level of output, as illustrated in the graph.