Income effect

Slutsky decomposition: income effect and substitution effect

As income effect in which is Mikroökonomik the change of the demand for a good denotes that arises due to a change of the (real) income. More precisely, one speaks of an income effect when the real income of an actor changes due to a price change in a market; this change in real income then in turn causes the said change in demand.

When looking at the consequences of a price change, one differentiates from the income effect a second occurring effect, the so-called substitution effect . Depending on the scenario, the effects are mutually reinforcing or opposing each other.

The so-called Slutsky decomposition is often used to consider the income and substitution effects in isolation ; In the graphical analysis in particular, the Hicks decomposition (which is analytically less productive) is often used .

Formal definition

The income effect results directly from the Slutsky equation (for the proof and explanation of the individual functions and variables, reference is made to the article Slutsky decomposition )

{\ displaystyle \ underbrace {\ frac {\ partial x_ {i} (\ mathbf {p}, y)} {\ partial p_ {j}}} _ {\ text {overall effect}} = \ underbrace {\ frac {\ partial x_ {i} ^ {h} (\ mathbf {p}, {\ overline {u}})} {\ partial p_ {j}}} _ {\ text {substitution effect}} \ underbrace {-x_ {j} (\ mathbf {p}, y) {\ frac {\ partial x_ {i} (\ mathbf {p}, y)} {\ partial y}}} _ {\ text {income effect}}}

.

It is handled differently in the literature, whether the income effect is defined as above or the minus sign is excluded from the expression, which then only stands for the income effect. This is of course essential for the application, because the signs of the effect are reversed in this case (so that, for example, the income effect would always be positive for normal goods, see below). We use the former definition throughout the following. ${\ displaystyle x_ {j} (\ mathbf {p}, y) \ cdot \ left [\ partial x_ {i} (\ mathbf {p}, y) / \ partial y \ right]}$

For , the term to the right of the equal sign indicates the effect of a change in the price of a good on the demand for the same good (self-price case). ${\ displaystyle i = j}$

properties

Ambiguity of the sign

The income effect of a price change is said to be positive if the signs of the change in income and demand are different. Otherwise one speaks of a negative income effect. (Note that, on the other hand, the substitution effect is referred to as positive if the signs of the price change and the [compensated] demand change are identical.)

Example 1 (price increase)

Consider a situation in which there are only two goods: good 1 and good 2. One has to examine how the demand for these two goods changes when the price of good 1 increases. The price increase does two things:

On the one hand, the price increase of good 1 leads to good 1 becoming relatively more expensive and therefore less attractive for the consumer compared to good 2. The household will therefore ask for fewer units of good 1 and correspondingly more of good 2. Because good 1 is replaced (substituted) by good 2, this is referred to as the substitution effect .

On the other hand, because the price of good 1 rises, the household has a lower income available overall, so that it restricts its consumption of both goods (this is of course based on the assumption that both goods are normal, see below). This effect is known as the income effect .

Overall effect: In this example, both the substitution effect for good 1 (price increase → decrease in demand) and the income effect for good 1 (price increase → decrease in income → decrease in demand) are negative. Both effects add up to a negative overall effect - the demand for good 1 consequently decreases. With regard to good 2, however, the substitution effect (positive) and the income effect (negative) work against each other. A clear statement about the overall effect is not possible here. If the substitution effect outweighs the income effect in terms of amount, consumption of good 2 increases; however, if the income effect outweighs it, it decreases.

Example 2 (price reduction)

The reasoning is reversed in part, when one takes a price increase price reduction considered of good first This does two things:

On the one hand, the price reduction of good 1 means that good 1 is relatively cheaper and therefore more attractive for the consumer compared to good 2. The household will therefore ask for more units of good 1 and correspondingly fewer of good 2. Because good 2 is replaced (substituted) by good 1, this is referred to as the substitution effect .

On the other hand, the fact that the price for good 1 falls means that the household has a greater income overall, from which it can request further goods. The household then uses the increased income to consume more units of both goods (this is of course based on the assumption that both goods are normal, see below). This effect is known as the income effect .

Overall effect: In this example, both the substitution effect and the income effect for good 1 are positive (price decrease → increase in demand or income increase → increase in demand). Both effects add up to a positive overall effect - the demand for good 1 consequently increases. With regard to good 2, however, the substitution effect (negative) and the income effect (positive) work against each other. A clear statement about the overall effect is not possible here. If the substitution effect outweighs the income effect in terms of amount, the consumption of good 2 decreases; however, if the income effect outweighs it, it increases.

generalization

In fact, this is a general insight. While the substitution effect due to a change in the price of a good with convex indifference curves is always negative for the demand for this good (a higher relative price for good 2 always leads to a decrease in the demand for this good for a consumer who minimizes expenditure), the sign of the income effect is revealed (and thus also the overall effect) only from further assumptions. Often the question of which of the effects outweighs, even with the given equipment of the actors, depends on which area of the respective budget line you are in.

Income effect for different types of goods

The designations of goods as inferior and normal each refer to the direction of the overall effect on demand in the event of a change in income (while prices remain constant). According to a common classification, a good is called inferior when an increase in income leads to a decrease in demand for the good and as normal when an increase in income leads to an increase in demand. It can therefore be seen that the income effect is always negative for normal goods, but positive for inferior goods. This follows directly from the formal definition, because according to the definition, is with normal goods and with inferior goods. ${\ displaystyle \ partial x_ {i} (\ mathbf {p}, y) / \ partial y> 0}$ ${\ displaystyle \ partial x_ {i} (\ mathbf {p}, y) / \ partial y <0}$

literature

Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
Alfred Endres and Jörn Martiensen: Microeconomics. An integrated presentation of traditional and modern concepts in theory and practice. Kohlhammer, Stuttgart 2007, ISBN 978-3-17-019778-7 .
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 0-195-07340-1 .
Hal Varian : Microeconomic Analysis. WW Norton, New York and London 1992, ISBN 0-393-95735-7 .
Hal Varian : Intermediate Microeconomics. A modern approach. 8th edition. WW Norton, New York and London 2010, ISBN 978-0-393-93424-3 .
Susanne Wied-Nebbeling and Helmut Schott: Fundamentals of microeconomics. Springer, Heidelberg a. a. 2007, ISBN 978-3-540-73868-8 .

Remarks

↑ See Breyer 2011, p. 145; Varian 2010, pp. 137, 141 ff .; Wied-Nebbeling / Schott 2007, p. 64 ff.
^ With Breyer 2011, p. 148; Endres / Martiensen 2007, p. 138; Anton Barten and Volker Böhm: Consumer Theory. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 382-429, here p. 417; Jochen Schumann, Ulrich Meyer and Wolfgang Ströbele: Basics of the microeconomic theory. 9th edition. Springer, Heidelberg u. a. 2011, ISBN 978-3-642-21225-3 , p. 85; Harald Wiese: Microeconomics. An introduction. Springer, Heidelberg a. a. 2010, ISBN 978-3-642-11599-8 (also online: doi : 10.1007 / 978-3-642-11600-1 ), p. 110 f .; Thorsten Hens and Paolo Pamini: Fundamentals of analytical microeconomics. Springer, Heidelberg a. a. 2008, ISBN 978-3-540-28157-3 (also online: doi : 10.1007 / 978-3-540-28158-0 ), p. 58. Against Wied-Nebbeling / Schott 2007, p. 75; Nolan H. Miller: Notes on Microeconomic Theory. online ( Memento from December 15, 2011 in the Internet Archive ) (PDF; 1 MB), p. 65, accessed on January 2, 2015; Carl P. Simon and Lawrence Blume: Mathematics for Economists. WW Norton, New York and London 1994, ISBN 0-393-95733-0 , p. 556.
↑ See Breyer 2011, p. 146; Varian 2010, p. 143 f.
↑ Illustrative Wied-Nebbeling / Schott 2007, p. 68 f.
↑ The terminology is chronically inconsistent in the literature (especially in German). The distinction here follows in any case Varian 2010, p. 143 ff .; Varian 1992, p. 117; Jehle / Reny 2011, p. 56; Mas-Colell / Whinston / Green 1995, p. 25. For terminological heterogeneity cf. Wied-Nebbeling / Schott 2007 (for which only those goods are normal whose share of total household expenditure decreases with increasing income, while it increases in the case of luxury goods , whereby normal and luxury goods are both combined to form superior goods , namely those in which demand rises with increasing income; on the other hand, they put the inferior goods, the demand of which falls in absolute terms with increasing income): “Unfortunately, there is no agreement here in the literature. Sometimes a distinction is only made between inferior and superior goods, sometimes only between inferior and normal goods [as in this Wikipedia article, note v. Wikipedia], whereby the second category comprises all goods for which the demand increases with increasing income. Other authors also label our normal goods as relatively inferior and our inferior goods as absolutely inferior. ”(P. 49)
↑ Cf. u. a. Harald Wiese: Microeconomics. An introduction. Springer, Heidelberg a. a. 2010, ISBN 978-3-642-11599-8 (also online: doi : 10.1007 / 978-3-642-11600-1 ), p. 110; Thorsten Hens and Paolo Pamini: Fundamentals of analytical microeconomics. Springer, Heidelberg a. a. 2008, ISBN 978-3-540-28157-3 (also online: doi : 10.1007 / 978-3-540-28158-0 ), p. 58. Note the definition of the income effect above (section “Formal definition”). If one uses the alternative definition described, exactly the opposite applies here.

[1] See Breyer 2011, p. 145; Varian 2010, pp. 137, 141 ff .; Wied-Nebbeling / Schott 2007, p. 64 ff.

[2] With Breyer 2011, p. 148; Endres / Martiensen 2007, p. 138; Anton Barten and Volker Böhm: Consumer Theory. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 382-429, here p. 417; Jochen Schumann, Ulrich Meyer and Wolfgang Ströbele: Basics of the microeconomic theory. 9th edition. Springer, Heidelberg u. a. 2011, ISBN 978-3-642-21225-3 , p. 85; Harald Wiese: Microeconomics. An introduction. Springer, Heidelberg a. a. 2010, ISBN 978-3-642-11599-8 (also online: doi : 10.1007 / 978-3-642-11600-1 ), p. 110 f .; Thorsten Hens and Paolo Pamini: Fundamentals of analytical microeconomics. Springer, Heidelberg a. a. 2008, ISBN 978-3-540-28157-3 (also online: doi : 10.1007 / 978-3-540-28158-0 ), p. 58. Against Wied-Nebbeling / Schott 2007, p. 75; Nolan H. Miller: Notes on Microeconomic Theory. online ( Memento from December 15, 2011 in the Internet Archive ) (PDF; 1 MB), p. 65, accessed on January 2, 2015; Carl P. Simon and Lawrence Blume: Mathematics for Economists. WW Norton, New York and London 1994, ISBN 0-393-95733-0 , p. 556.

[3] See Breyer 2011, p. 146; Varian 2010, p. 143 f.

[4] Illustrative Wied-Nebbeling / Schott 2007, p. 68 f.

[5] The terminology is chronically inconsistent in the literature (especially in German). The distinction here follows in any case Varian 2010, p. 143 ff .; Varian 1992, p. 117; Jehle / Reny 2011, p. 56; Mas-Colell / Whinston / Green 1995, p. 25. For terminological heterogeneity cf. Wied-Nebbeling / Schott 2007 (for which only those goods are normal whose share of total household expenditure decreases with increasing income, while it increases in the case of luxury goods , whereby normal and luxury goods are both combined to form superior goods , namely those in which demand rises with increasing income; on the other hand, they put the inferior goods, the demand of which falls in absolute terms with increasing income): “Unfortunately, there is no agreement here in the literature. Sometimes a distinction is only made between inferior and superior goods, sometimes only between inferior and normal goods [as in this Wikipedia article, note v. Wikipedia], whereby the second category comprises all goods for which the demand increases with increasing income. Other authors also label our normal goods as relatively inferior and our inferior goods as absolutely inferior. ”(P. 49)

[6] Cf. u. a. Harald Wiese: Microeconomics. An introduction. Springer, Heidelberg a. a. 2010, ISBN 978-3-642-11599-8 (also online: doi : 10.1007 / 978-3-642-11600-1 ), p. 110; Thorsten Hens and Paolo Pamini: Fundamentals of analytical microeconomics. Springer, Heidelberg a. a. 2008, ISBN 978-3-540-28157-3 (also online: doi : 10.1007 / 978-3-540-28158-0 ), p. 58. Note the definition of the income effect above (section “Formal definition”). If one uses the alternative definition described, exactly the opposite applies here.