Marshallian demand function

In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) specifies what the consumer would buy in each price and income or wealth situation, assuming it perfectly solves the utility maximization problem. Marshallian demand is sometimes called Walrasian demand (named after Léon Walras) or uncompensated demand function instead, because the original Marshallian analysis ignored wealth effects.

According to the utility maximization problem, there are L commodities with price vector p and choosable quantity vector x. The consumer has income I, and hence a set of affordable packages

where ${\displaystyle \langle p,x\rangle }$ is the inner product of the price and quantity vectors. The consumer has a utility function

The consumer's Marshallian demand correspondence is defined to be

${\displaystyle x^{*}(p,I)}$ is called a correspondence because in general it may be set-valued - there may be several different bundles that attain the same maximum utility. In some cases, there is a unique utility-maximizing bundle for each price and income situation; then, ${\displaystyle x^{*}(p,I)}$ is a function and it is called the Marshallian demand function.

...
Wikipedia