In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize their utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory (named after Léon Walras). 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 budget set of affordable packages where 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 Revealed preference Marshall's theory suggests that pursuit of utility is a motivational factor to a consumer which can be attained through the consumption of goods or service. The amount of consumer's utility is dependent on the level of consumption of a certain good, which is subject to the fundamental tendency of human nature and it is described as the law of diminishing marginal utility. As utility maximum always exists, Marshallian demand correspondence must be nonempty at every value that corresponds with the standard budget set. Uniqueness 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, 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. To prove this, suppose, by contradiction, that there are two different bundles, and , that maximize the utility. Then and are equally preferred. By definition of strict convexity, the mixed bundle is strictly better than . But this contradicts