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In some cases, there is a unique utility-maximizing bundle for each price and income situation; then, (,) is a function and it is called the Marshallian demand function. If the consumer has strictly convex preferences and the prices of all goods are strictly positive, then there is a unique utility-maximizing bundle.
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 :
where (,) is the Hicksian demand and (,) is the Marshallian demand, at the vector of price levels , wealth level (or income level) , and fixed utility level given by maximizing utility at the original price and income, formally presented by the indirect utility function (,).
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 and a good ():
Let's say the utility function is the Cobb-Douglas function (,) =, which has the Marshallian demand functions [2] (,) = (,) =,where is the consumer's income. The indirect utility function (,,) is found by replacing the quantities in the utility function with the demand functions thus:
The consumer's demand is always to get the goods in constant ratios determined by the weights, i.e. the consumer demands a bundle (, …,) where is determined by the income: = / (+ +). [1] Since the Marshallian demand function of every good is increasing in income, all goods are normal goods .
[1]: 164 A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for , …, does not depend on wealth and is thus not subject to a wealth effect; [1]: 165–166 The absence of a wealth effect simplifies analysis [1]: 222 and makes quasilinear utility functions a common choice for modelling.
Hicksian demand is defined by : + + (+) (,) = .[1]Hicksian demand function gives the cheapest package that gives the desired utility. It is related to Marshallian demand function by and expenditure function by