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In
microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
, a consumer's Marshallian demand function (named after
Alfred Marshall Alfred Marshall (26 July 1842 – 13 July 1924) was an English economist, and was one of the most influential economists of his time. His book '' Principles of Economics'' (1890) was the dominant economic textbook in England for many years. I ...
) 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 In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for t ...
. It is a solution to the
utility maximization problem Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my u ...
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 In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentia ...
. Thus the change in quantity demanded is a combination of a
substitution effect In economics and particularly in consumer choice theory, the substitution effect is one component of the effect of a change in the price of a good upon the amount of that good demanded by a consumer, the other being the income effect. When a g ...
and a
wealth effect The wealth effect is the change in spending that accompanies a change in perceived wealth. Usually the wealth effect is positive: spending changes in the same direction as perceived wealth. Effect on individuals Changes in a consumer's wealth cause ...
. 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 Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist. He formulated the marginal theory of value (independently of William Stanley Jevons and Carl Menger) and pioneered the developmen ...
). 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 In economics, a budget set, or the opportunity set facing a consumer, is the set of all possible consumption bundles that the consumer can afford taking as given the prices of commodities available to the consumer and the consumer's income. Let the ...
of affordable packages :B(p, I) = \, where \langle p, x \rangle is the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of the price and quantity vectors. The consumer has a
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
:u : \textbf R^L_+ \rightarrow \textbf R. The consumer's Marshallian demand correspondence is defined to be :x^*(p, I) = \operatorname_ u(x).


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 In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a good or service describes how much pleasure or satisfaction is gained by consumers as a result of the increase or decrease in consumpti ...
. As utility maximum always exists, Marshallian demand correspondence must be nonempty at every value that corresponds with the standard budget set.


Uniqueness

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, x^*(p, I) is a function and it is called the Marshallian demand function. If the consumer has strictly
convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
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, x_1 and x_2, that maximize the utility. Then x_1 and x_2 are equally preferred. By definition of strict convexity, the mixed bundle 0.5 x_1 + 0.5 x_2 is strictly better than x_1 , x_2. But this contradicts the optimality of x_1 , x_2.


Continuity

The
maximum theorem The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematic ...
implies that if: * The utility function u(x) is continuous with respect to x, * The correspondence B(p, I) is non-empty, compact-valued, and continuous with respect to p,I, then x^*(p, I) is an
upper-semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...
correspondence. Moreover, if x^*(p, I) is unique, then it is a continuous function of p and I. Combining with the previous subsection, if the consumer has strictly convex preferences, then the Marshallian demand is unique and continuous. In contrast, if the preferences are not convex, then the Marshallian demand may be non-unique and non-continuous.


Homogeneity

The optimal Marshallian demand correspondence of a continuous utility function is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
with degree zero. This means that for every constant a>0, :x^*(a\cdot p, a\cdot I) = x^*(p, I). This is intuitively clear. Suppose p and I are measured in dollars. When a=100, ap and aI are exactly the same quantities measured in cents. When prices and wealth go up by a factor a, the purchasing pattern of an economic agent remains constant. Obviously, expressing in different unit of measurement for prices and income should not affect the demand.


Demand curve

Marshall's theory exploits that demand curve represents individual's diminishing marginal values of the good. The theory insists that the consumer's purchasing decision is dependent on the gainable utility of a goods or services compared to the price since the additional utility that the consumer gain must be at least as great as the price. The following suggestion proposes that the price demanded is equal to the maximum price that the consumer would pay for an extra unit of good or service. Hence, the utility is held constant along the demand curve. When the marginal utility of income is constant, or its value is the same across individuals within a market demand curve, generating net benefits of purchased units, or consumer surplus is possible through adding up of demand prices.


Examples

In the following examples, there are two commodities, 1 and 2. 1. The utility function has the Cobb–Douglas form: :u(x_1,x_2) = x_1^x_2^. The constrained optimization leads to the Marshallian demand function: :x^*(p_1,p_2,I) = \left(\frac, \frac\right). 2. The utility function is a CES utility function: :u(x_1,x_2) = \left \frac + \frac \right. Then x^*(p_1,p_2,I) = \left(\frac, \frac\right), \quad \text \quad \epsilon = \frac. In both cases, the preferences are strictly convex, the demand is unique and the demand function is continuous. 3. The utility function has the
linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
: :u(x_1,x_2) = x_1 + x_2. The utility function is only weakly convex, and indeed the demand is not unique: when p_1=p_2, the consumer may divide his income in arbitrary ratios between product types 1 and 2 and get the same utility. 4. The utility function exhibits a non-diminishing marginal rate of substitution: :u(x_1,x_2) = (x_1^ + x_2^), \quad \text \quad \alpha > 1. The utility function is not convex, and indeed the demand is not continuous: when p_1, the consumer demands only product 1, and when p_2, the consumer demands only product 2 (when p_1=p_2 the demand correspondence contains two distinct bundles: either buy only product 1 or buy only product 2).


See also

*
Hicksian demand function In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentia ...
*
Utility maximization problem Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my u ...
*
Slutsky equation The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky, relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed ...


References

* * * * * {{cite book , last1=Wong , first1=Stanley , title=Foundations of Paul Samuelson's revealed preference theory , date=2006 , publisher=Routledge , isbn=0-203-34983-0 , edition=Revised , url=http://www.library.fa.ru/files/Wong.pdf , access-date=19 April 2021 Demand