Budget constraint

As a budget constraint (also: budget constraint ) is called in economics is a common condition in mathematical models, which ensures that the actors do not spend more money than them is available. If only two goods are considered, the budget restriction can be represented graphically by means of a straight line in a two-goods diagram; this is then commonly referred to as the budget line (also: household line, balance line ). Those bundles of goods that meet the budget restriction form a set , the so-called budget amount (also: consumption possibilities amount ).

Basic concept

Two-goods case

A particularly simple budget constraint is used in the standard model of household theory. There one assumes a fixed budget and a finite amount of purchasable goods that can be bought at given prices.

Consider two goods, apples (good 1) and pears (good 2). The number of apples in demand is labeled with and the number of pears in demand with . The price of the two goods is given; an apple costs exactly (this variable can stand for “1 euro”, for example), a pear . The consumer has an income of . Then the budget restriction is ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {1}}$ ${\ displaystyle p_ {2}}$ ${\ displaystyle y}$

{\ displaystyle p_ {1} \ cdot x_ {1} + p_ {2} \ cdot x_ {2} \ leq y}

.

In words: The total expenditure on apples (price of an apple times the number of apples) and the total expenditure on pears (price of a pear times the number of pears) must together be at most as high as the disposable income. Casually: you can't spend more money than you have.

Graphic construction (two-goods case)

Budget straight

Again, consider only apples (good 1) and pears (good 2) with the demand for apples and the demand for pears. An apple costs 4 euros ( ), a pear 6 euros ( ). The consumer has a budget of 24 euros ( ). Then the budget restriction is: ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {1} = 4}$ ${\ displaystyle p_ {2} = 6}$ ${\ displaystyle y = 24}$

{\ displaystyle 4x_ {1} + 6x_ {2} \ leq 24}

.

The “limit” of your budget condition can be found by assuming that the consumer is fully exhausting his budget, that is, that the budget restriction is binding (that is, with equality applies), and so

{\ displaystyle 4x_ {1} + 6x_ {2} = 24}

.

The budget line delivers:

{\ displaystyle x_ {2} = 4 - {\ frac {2} {3}} x_ {1}}

.

This is shown in the figure. The number of apples consumed (good 1) is plotted on the horizontal axis and the number of pears consumed (good 2) on the vertical axis of a two-dimensional coordinate system . The budget line now runs through the point on the vertical axis at which the person in question spends their entire budget on good 2, and through the point on the horizontal axis at which the person spends their entire budget on good 1. Every point that is on the budget line just barely fulfills the budget restriction. Points A and B mark extreme points in the budget allocation: in A the consumer spends his entire fortune on pears, in B his entire fortune on apples. The colored area below (and with) the budget line is the budget amount. It contains all the combinations of quantities that the consumer can afford. Points inside this area represent bundles of goods, the consumption of which does not exhaust the wealth. In point C, for example, two apples and two pears are bought, which leads to expenses of just 20 euros. Bundles of goods outside the budget cannot be reached. Point D, for example, stands for the consumption of 2 pears (12 euros) and 4 apples (16 euros), which is obviously not feasible with the given budget.

The slope of the budget line (also limit rate of transformation , GRT for short) is generally , corresponds to the negative ratio of the two goods prices. ${\ displaystyle -p_ {1} / p_ {2}}$

General

You can see the budget constraint described on the multi-goods case generalize: Label with demanded by a certain amount of consumer goods , and summarize the vector demand respect of all goods together. The price of every good is strictly positive, for everyone , and one can agree as the price vector of the economy. Then the budget restriction is ${\ displaystyle x_ {i} \ geq 0}$ ${\ displaystyle i}$ ${\ displaystyle i = 1, \ ldots, n}$ ${\ displaystyle \ mathbf {x} = (x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {\ mathbb {R}} _ {+} ^ {n}}$ ${\ displaystyle p_ {i}> 0}$ ${\ displaystyle i = 1, \ ldots, n}$ ${\ displaystyle \ mathbf {p} = (p_ {1}, \ ldots, p_ {n}) \ in \ mathbb {R} _ {++} ^ {n}}$

{\ displaystyle \ mathbf {p} \ cdot \ mathbf {x} = p_ {1} \ cdot x_ {1} + \ ldots + p_ {n} \ cdot x_ {n} \ leq y}

.

The budget amount is accordingly the amount of all goods bundles that meet the budget restriction, that is ${\ displaystyle {\ mathcal {B}}}$

{\ displaystyle {\ mathcal {B}}: = \ left \ {\ mathbf {x} \ in \ mathbb {R} _ {+} ^ {n} | \ mathbf {p} \ cdot \ mathbf {x} \ leq y \ right \}}

.

Intertemporal budget restriction

Budget restrictions can also be formulated for intertemporal problems. In the simplest case, for example, a consumer in a two-period model may be faced with the decision to spend his income in period 1, or to save and only consume in period 2. Denote the consumption expenditure in the period and the income in the period . If you also allow saved money to be invested at one interest rate and borrowed at the same interest rate, the following intertemporal budget restriction can be set up: ${\ displaystyle x_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle y_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle r> 0}$

{\ displaystyle x_ {2} \ leq y_ {2} + (y_ {1} -x_ {1}) + r \ cdot (y_ {1} -x_ {1}) = y_ {2} + (1 + r ) \ cdot (y_ {1} -x_ {1})}

In words: In period 2, you can only consume as much as you earned in period 2, plus that part of the income from period 1 that was not spent (i.e. saved) in period 1, as well as the interest accrued on it. If so, the household saved in period 1, but if so, it went into debt. In order to see that the budget restriction found really also covers the case of debt in period 1, one can formulate: ${\ displaystyle (y_ {1} -x_ {1})> 0}$ ${\ displaystyle (y_ {1} -x_ {1}) <0}$

{\ displaystyle x_ {2} \ leq y_ {2} - (x_ {1} -y_ {1}) - r \ cdot (x_ {1} -y_ {1})}

,

which is obviously equivalent to the condition found above. In words: In period 2, you can only consume as much as you earn in period 2, minus the amount that was spent (i.e. borrowed) beyond the available income in period 1, as well as the interest accrued on it.

A simple intertemporal budget restriction for the case of an unlimited number of periods is used, for example, in the standard variants of the overlapping generation models .

Web links

Commons : Budget restriction - collection of images, videos and audio files

Balance line - definition in the Gabler Wirtschaftslexikon
The budget straight line - Article at mikrooekonomie.de

literature

Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
Hal Varian : Fundamentals of Microeconomics. Translated from the American by Reiner Buchegger (Original title: Intermediate Microeconomics. A Modern Approach. ). 8th edition Oldenbourg, Munich 2011, ISBN 978-3-486-70453-2 .
Robert S. Pindyck and Daniel L. Rubinfeld: Microeconomics. 6th edition. Pearson studies, Munich a. a. 2005, ISBN 3-8273-7164-3 .
Susanne Wied-Nebbeling and Helmut Schott: Fundamentals of microeconomics. Springer, Heidelberg a. a. 2007, ISBN 978-3-540-73868-8 .

Individual evidence

↑ See e.g. Breyer 2011, p. 127; Harald Wiese: Microeconomics. An introduction. 4th edition. Springer, Heidelberg a. a. 2005, ISBN 978-3-642-11599-8 , p. 23 f .; Wied-Nebbeling / Schott 2007, p. 18.
↑ See e.g. Breyer 2011, p. 153.
↑ See for example Breyer 2011, p. 118; Harald Wiese: Microeconomics. An introduction. 4th edition. Springer, Heidelberg a. a. 2005, ISBN 978-3-642-11599-8 , p. 23.
↑ Cf. for example Ferry Stocker and Kerstin M. Strobach: Mikroökonomik. Revision course and exercises. 4th edition Oldenbourg, Munich 2012, ISBN 978-3-486-70777-9 , p. 92.
↑ is the set of all tuples of real numbers with ; the set of all tuples of real numbers with . ${\ displaystyle \ mathbb {R} _ {+} ^ {n}}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i} \ geq 0}$ ${\ displaystyle \ mathbb {R} _ {++} ^ {n}}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i}> 0}$

[1] See e.g. Breyer 2011, p. 127; Harald Wiese: Microeconomics. An introduction. 4th edition. Springer, Heidelberg a. a. 2005, ISBN 978-3-642-11599-8 , p. 23 f .; Wied-Nebbeling / Schott 2007, p. 18.

[2] See e.g. Breyer 2011, p. 153.

[3] See for example Breyer 2011, p. 118; Harald Wiese: Microeconomics. An introduction. 4th edition. Springer, Heidelberg a. a. 2005, ISBN 978-3-642-11599-8 , p. 23.

[4] Cf. for example Ferry Stocker and Kerstin M. Strobach: Mikroökonomik. Revision course and exercises. 4th edition Oldenbourg, Munich 2012, ISBN 978-3-486-70777-9 , p. 92.

[5] s the set of all tuples of real numbers with ; the set of all tuples of real numbers with . ${\ displaystyle \ mathbb {R} _ {+} ^ {n}}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i} \ geq 0}$ ${\ displaystyle \ mathbb {R} _ {++} ^ {n}}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i}> 0}$