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Shephard's lemma is a 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.
Inverting this formula gives the indirect utility function (utility as a function of price and income): (,) = (),where is the amount of income available to the individual and is equivalent to the expenditure ((,)) in the previous equation.
The model formalises consumers' preferences for product variety by using a CES function. Previous attempts to provide a model that accounted for variety preference (such as Harold Hotelling 's location model ) were indirect and failed to provide an easily interpretable and usable form for further study.
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 :
A consumer's indirect utility (,) can be computed from their utility function (), defined over vectors of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector (,) by solving the utility maximization problem, and second, computing the utility ((,)) the consumer derives from that ...
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Image source: The Motley Fool. Microsoft (NASDAQ: MSFT) Q2 2025 Earnings Call Jan 29, 2025, 5:30 p.m. ET. Contents: Prepared Remarks. Questions and Answers. Call ...
Data Analysis Expressions (DAX) is the native formula and query language for Microsoft PowerPivot, Power BI Desktop and SQL Server Analysis Services (SSAS) Tabular models. DAX includes some of the functions that are used in Excel formulas with additional functions that are designed to work with relational data and perform dynamic aggregation.