indirect utility function การใช้

• The expenditure function is the inverse of the indirect utility function:
• The indirect utility function can be calculated by substituting the demand into the utility function.
• They also provide a complete characterization of substitutes preferences : Goods are substitutes if and only if the indirect utility function is submodular.
• Where v ( p, w ) is the indirect utility function ( which gives the utility level of having a given wealth under a fixed price regime ).
• An indirect utility function gives the optimal attainable value of a given utility function, which depends on the prices of the goods and the income or wealth level that the individual possesses.
• The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income w in the indirect utility function v ( p, w ), at a utility of u:
• This says that the indirect utility function evaluated in such a way that minimizes the cost for achieving a certain utility given a set of prices ( a vector p ) is equal to that utility when evaluated at those prices.
• To prove that the Engel curves of a function in Gorman polar form are linear, apply Roy's identity to the indirect utility function to get a Marshallian demand function for an individual ( i ) and a good ( n ):
• Where h ( \ mathbf { p }, u ) is the Hicksian demand and x ( \ mathbf { p }, w ) is the Marshallian demand, at the vector of price levels \ mathbf { p }, wealth level ( or, alternatively, income level ) w, and fixed utility level u given by maximizing utility at the original price and income, formally given by the indirect utility function v ( \ mathbf { p }, w ).