Excess demand function

In economics, the excess demand function is a mathematical function used to describe a market . For a given household, it indicates by how much its demand for each good exceeds its original equipment with the respective good (individual excess demand function).

Formal definition

Let be the vector of the prices of all n goods in the economy and be the initial endowment of consumer i with good k. The amount of consumers is . The individual equipment is then combined into an individual equipment vector. Keep going ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})\in \mathbb {R} _{+}^{n}}$ ${\displaystyle e_{k}^{i}}$ ${\displaystyle {\mathcal {I}}=\{1,\ldots ,I\}}$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})}$

${\displaystyle \mathbf {x} ^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})=[(x_{1}^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i}),\ldots ,x_{n}^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})]}$

the Marshallian demand of household i (with respect to all goods).

Definition: The (individual) excess demand function of household i , is given by ${\displaystyle \mathbf {z} ^{i}:\mathbb {R} _{+}^{n}\rightarrow \mathbb {R} ^{n}}$

${\displaystyle \mathbf {z} ^{i}(\mathbf {p} )\equiv \mathbf {x} ^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})-\mathbf {e} ^{i}}$

and the aggregate excess demand function of economics, by ${\displaystyle \mathbf {z} :\mathbb {R} _{+}^{n}\rightarrow \mathbb {R} ^{n}}$

${\displaystyle \mathbf {z} (\mathbf {p} )\equiv \sum _{i\in {\mathcal {I}}}\mathbf {z} ^{i}(\mathbf {p} )}$

given.

If the k th element is negative, there is an excess supply of good k in the economy . One can (somewhat less commonly) define a goods-specific excess demand function directly . This indicates for a given good k how high the related excess demand is in the economy; formally: ${\displaystyle \mathbf {z} (\mathbf {p} )}$ ${\displaystyle z_{k}:\mathbb {R} _{+}^{n}\rightarrow \mathbb {R} }$

${\displaystyle z_{k}(\mathbf {p} )\equiv \sum _{i\in {\mathcal {I}}}x_{k}^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})-\sum _{i\in {\mathcal {I}}}e_{k}^{i}}$.

properties

Properties of the aggregated excess demand function: Let the utility function of each consumer be continuous, strictly monotonically increasing and strictly quasi-concave; be further . Then: ${\displaystyle u^{i}}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \mathbf {p} \gg \mathbf {0} }$

1. ${\displaystyle \mathbf {z} (\mathbf {p} )}$ is steady.
2. ${\displaystyle \mathbf {z} (\mathbf {p} )}$is homogeneous from grade zero, i.e. for all . ${\displaystyle \mathbf {z} (\lambda \mathbf {p} )=\mathbf {z} (\mathbf {p} )}$ ${\displaystyle \lambda >0}$
3. ${\displaystyle \mathbf {z} (\mathbf {p} )}$satisfies the Walras law , that is, for everyone . ${\displaystyle \mathbf {p} \cdot \mathbf {z} (\mathbf {p} )=0}$ ${\displaystyle \mathbf {p} }$
4. There is a real one such that for all k and . ${\displaystyle s>0}$ ${\displaystyle z_{k}(\mathbf {p} )>-s}$ ${\displaystyle \mathbf {p} }$

Properties 1 and 2 follow directly from the properties of the Marshallian demand function. Grade zero homogeneity makes sense because the unit of prices is irrelevant for excess demand: If it is determined that there are no euros but only euro cents, the prices in the respective unit would increase by a factor of 100 nonetheless one does not expect that this will affect the quantities of goods that are offered or demanded. Property 3 applies because of

${\displaystyle \mathbf {p} \cdot \mathbf {z} (\mathbf {p} )=\mathbf {p} \cdot \left[\sum _{i\in {\mathcal {I}}}\mathbf {x} ^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})-\sum _{i\in {\mathcal {I}}}\mathbf {e} ^{i}\right]=\sum _{i\in {\mathcal {I}}}\left[\mathbf {p} \cdot \mathbf {x} ^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})-\mathbf {p} \cdot \mathbf {e} ^{i}\right]=0}$,

where the last equation follows because for every consumer the budget constraint is met with equality if the utility function is strictly monotonic. Intuitive: A utility-maximizing household will always have a demand in the amount of its entire wealth , which is why the value of its individual excess demand must be zero; as a result, however, the value of the aggregated excess demand of all households must also be zero. ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} ^{i}\leq \mathbf {p} \cdot \mathbf {e} ^{i}}$ ${\displaystyle \mathbf {p} \cdot \mathbf {e} ^{i}}$

literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 .
• James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also online: doi : 10.1007 / 978-3-540-32223-8 ).
• Hal Varian : Microeconomic Analysis. WW Norton, New York and London 1992, ISBN 0-393-95735-7 .

Remarks

1. ↑ See Mas-Colell / Whinston / Green 1995, pp. 580 f .; Moore 2007, p. 138; Varian 1992, p. 317. Deviating from Jehle / Reny 2011, p. 204, who define the aggregated excess demand function as a function (with the goods-specific excess demand function, see below). By employing it, one can of course easily convince oneself that both approaches lead to one and the same function . ${\displaystyle \mathbf {z} (\mathbf {p} )\equiv [z_{1}(\mathbf {p} ),\ldots ,z_{n}(\mathbf {p} )]}$ ${\displaystyle z_{k}(\mathbf {p} )}$ ${\displaystyle \mathbf {z} }$
2. ↑ See Jehle / Reny 2011, p. 204; implicitly in Mas-Colell / Whinston / Green 1995, p. 581.
3. ↑ See Mas-Colell / Whinston / Green 1995 p. 582 for all properties; Jehle / Reny 2011, p. 204 for 1-3.
4. ↑ For the proof of property 3 cf. Varian 1992, p. 318; so too, but on the basis of goods-specific excess demand functions, Jehle / Reny 2011, p. 204 f.