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In [[microeconomics]], a consumer's '''Marshallian demand function''' (named after [[Alfred Marshall]]) specifies what the consumer would buy in each price and 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 effect]]s.


According to the utility maximization problem, there are ''L'' commodities with prices ''p''.  The consumer has wealth ''w'', and hence a set of affordable packages


:<math>B(p, w) = \{x : \langle p, x \rangle \leq w\},</math>
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where <math> \langle p, x \rangle </math> is the [[inner product]] of the prices and quantity of goods. The consumer has a [[utility function]]  
 
:<math>u : \textbf R^L_+ \rightarrow \textbf R.</math>
 
The consumer's '''Marshallian demand correspondence''' is defined to be
 
:<math>x^*(p, w) = \operatorname{argmax}_{x \in B(p, w)} u(x).</math>
 
If there is a unique utility maximizing package for each
price and wealth situation, then it is called the '''Marshallian demand function'''.  See the [[utility maximization problem]] entry for a discussion of this definition.
 
==Example==
If there are two commodities, and the consumer has the utility function <math>U(x_1,x_2) = x_1^{0.5}x_2^{0.5}</math> (Cobb-Douglas), he chooses to spend half of its income on each commodity, and its Marshallian demand function is the following:
 
:<math>x(p_1,p_2,w) = \left(\frac{w}{2p_1}, \frac{w}{2p_2}\right).</math>
 
In general, an agent with Cobb-Douglas preferences, given by <math>U(x_1,x_2) = x_1^{a}x_2^{1-a}</math>, will use a constant share of his income in order to buy each of the two commodities, as follows:
 
:<math>x(p_1,p_2,w) = \left(\frac{aw}{p_1}, \frac{(1-a)w}{p_2}\right).</math>
 
If we are in a more general case, i.e. <math>U(x_1,x_2) = x_1^{a}x_2^{b}</math>, we have:
 
:<math>x(p_1,p_2,w) = \left(\frac{aw}{(a+b)p_1}, \frac{bw}{(a+b)p_2}\right).</math>
 
==See also==
* [[Hicksian demand function]]
* [[Utility maximization problem]]
 
==References==
*{{cite book |authorlink=Andreu Mas-Colell |last=Mas-Colell |first=Andreu |last2=Whinston |first2=Michael |lastauthoramp=yes |last3=Green |first3=Jerry |year=1995 |title=Microeconomic Theory |location=Oxford |publisher=Oxford University Press |isbn=0-19-507340-1 }}
[[Category:Demand]]
[[Category:Consumer theory]]

Revision as of 16:28, 4 March 2014


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