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2014-02-02T20:15:17Z

5.149.21.193: Remove dead reference.

In mathematics, the '''Christoffel–Darboux theorem''' is an identity for a sequence of [[orthogonal polynomials]], introduced by {{harvs|txt|authorlink=Elwin Bruno Christoffel|first=Elwin Bruno|last= Christoffel|year=1858}} and {{harvs|txt|authorlink=Jean Gaston Darboux|first=Jean Gaston|last= Darboux|year=1878}}. It states that

: <math> \sum_{j=0}^n \frac{f_j(x) f_j(y)}{h_j} = \frac{k_n}{h_n k_{n+1}} \frac{f_n(y) f_{n+1}(x) - f_{n+1}(y) f_n(x)}{x - y}</math>

where ''f''''j''(''x'') is the ''j''th term of a set of [[orthogonal polynomials]] of squared norm ''h''''j'' and leading coefficient ''k''''j''.

==See also==
*[[Turán's inequalities]]
*[[Sturm Chain]]

==References==

*{{Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6 | mr=1688958 | year=1999 | volume=71}}
*{{Citation | last1=Christoffel | first1=E. B. | title=Über die Gaußische Quadratur und eine Verallgemeinerung derselben. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002150239 | language=German | doi=10.1515/crll.1858.55.61 | year=1858 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=55 | pages=61–82}}
*{{Citation | last1=Darboux | first1=Gaston | title=Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série | language=French | jfm=10.0279.01 | year=1878 | journal=Journal de Mathématiques Pures et Appliquées | volume=4 | pages=5–56, 377–416}}

{{DEFAULTSORT:Christoffel-Darboux formula}}
[[Category:Orthogonal polynomials]]
[[Category:Functional analysis]]

Fresnel integral

2014-01-29T14:19:05Z

5.149.21.193: typo --- dx

{{Refimprove|date=December 2009}}
[[File:Budget constraint.svg|thumb|250px|Budget constraint, where <math>A=\frac{m}{P_y}</math> and <math>B=\frac{m}{P_x}</math>]]

A '''budget constraint''' represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a [[budget]] [[constraint (mathematics)|constraint]] and a [[preference map]] to analyze consumer choices. Both concepts have a ready [[Consumer Theory|graphical representation]] in the two-good case.

== Uses ==
=== Individual choice ===
[[File:Indifference curves showing budget line.svg|thumb|right|An individual should consume at (Qx, Qy).]]
Consumer behaviour is a maximisation problem. It means making the most of our limited resources to maximise our utility. As consumers are insatiable, and utility functions grow with quantity, the only thing that limits our consumption is our own budget.<ref>http://www.policonomics.com/budget-constraint/</ref>

An individual [[consumer]] should choose to consume goods at the point where the most preferred available [[indifference curve]] on their [[indifference curve#Map_and_properties_of_indifference_curves|preference map]] is [[tangent]] to their budget constraint. That is, the indifference curve tangent to the budget constraint represents the maximum utility obtained utilizing the entire budget of the consumer. The tangent point (the xy coordinate) represents the amount of goods x and y the consumer should purchase to fully utilize their budget to obtain maximum utility.<ref>Lipsey (1975). p 182.</ref> A line connecting all points of tangency between the indifference curve and the budget constraint is called the [[expansion path]].<ref name="Salvatore">Salvatore, Dominick (1989). ''Schaum's outline of theory and problems of managerial economics,'' McGraw-Hill, ISBN 978-0-07-054513-7</ref>

All two dimensional budget constraints are generalized into the equation:

<math>P_x x+P_y y=m</math>

Where:
* <math>m=</math> money income allocated to consumption (after saving and borrowing)
* <math>P_x=</math> the price of a specific good
* <math>P_y=</math> the price of all other goods
* <math>x=</math> amount purchased of a specific good
* <math>y=</math> amount purchased of all other goods

The equation can be rearranged to represent the shape of the curve on a graph:

<math>y= (m/P_y)-(P_x/P_y) x</math>, where <math>(m/P_y)</math> is the y-intercept and <math>(-P_x/P_y)</math> is the slope, representing a downward sloping budget line.

The factors that can shift the budget line are a change in income (m), a change in the price of a specific good (<math>P_x</math>), or a change in the price of all other goods (<math>P_y</math>).

=== International economics ===
[[File:Production Possibilities Frontier Curve.svg|thumb|right|Point X is unobtainable given the current "budget" constraints on production.]]

A [[production-possibility frontier]] is a budget constraint presented by the limitation of available [[factors of production]]. Under [[autarky]] this is also the limitation of consumption by individuals in the country. However, the benefits of [[international trade]] are generally demonstrated through allowance of a shift in the [[consumption-possibility frontier]]s of each trade partner which allows access to a more appealing indifference curve.
On "toolbox", Hecksher-Ohlin and Krugman models of international trade, the budget constraint of the economy (its CPF) is determined by the terms-of-trade (TOT) as a downward-sloped line with slope equal to those TOTs of the economy (The TOTs are given by the price ratio Px/Py, where x is the exportable commodity and y is the importable).

== Many goods ==
While low level demonstrations of budget constraints are often limited to two good situations which provide easy graphical representation, it is possible to demonstrate the relationship between multiple goods through a budget constraint.

In such a case, assuming there are <math>n\,</math> goods, called <math>x_i\,</math> for <math>i=1,\dots,n\,</math>, that the price of good <math>x_i\,</math> is denoted by <math>p_i\,</math>, and if <math>\,W\,</math> is the total amount that may be spent, then the budget constraint is:
:<math>\sum_{i=1}^np_ix_i\leq W.</math>

Further, if the consumer spends his income entirely, the budget constraint binds:
:<math>\sum_{i=1}^np_ix_i=W.</math>
In this case, the consumer cannot obtain an additional unit of good <math>x_i\,</math> without giving up some other good. For example, he could purchase an additional unit of good <math>x_i\,</math> by giving up <math>p_i/p_j\,</math> units of good <math>x_j.\,</math>

== Notes ==
{{Reflist}}

== See also ==
*[[Choice modelling]]
*[[Contingent valuation]]
*[[Guns versus butter model]]
*[[Heckscher–Ohlin theorem]] on country level budget constraints called resource endowments
*[[Opportunity cost]]
*[[Scarcity]]
*[[Trade-off]]

== References ==
* {{Cite book|title=An introduction to positive economics|edition=fourth|pages=214–7|first=Richard G.|last=Lipsey|year=1975|publisher=Weidenfeld & Nicolson|isbn=0-297-76899-9}}

{{DEFAULTSORT:Budget Constraint}}
[[Category:Consumer theory]]
[[Category:Budgets|Constraint]]
[[Category:Economics curves]]

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