Slutsky decomposition

Slutsky decomposition: income effect and substitution effect

The Slutsky decomposition is a method to derive the basically unobservable Hicksian demand function for the price from the potentially observable Marshallian demand ; the resulting equation is called the Slutsky equation .

It turns out that by means of the Slutsky decomposition the change in demand for a good caused by a price change can be broken down into a substitution and an income effect.

The method is named after the mathematician and economist Yevgeny Slutsky .

presentation

Let the Marshallian demand for a good depend on a price vector and individual income . (The Marshallian demand results from the utility maximization problem of the household and indicates the quantity of goods - depending on the goods prices - that is required to achieve the highest possible level of utility with a given income ). ${\displaystyle x_{i}(\mathbf {p} ,y)}$ ${\displaystyle i}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle y}$ ${\displaystyle y}$

Furthermore, one can agree as a Hicksian (also: compensated) demand for the good , whereby here stands for the level of utility to be achieved. (Hicks's demand results from the household's problem of minimizing expenditure and indicates the quantity of goods - depending on the goods prices - that is required to achieve a given level of utility as cheaply as possible ). ${\displaystyle x_{i}^{h}(\mathbf {p} ,{\overline {u}})}$ ${\displaystyle i}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle {\overline {u}}}$

Then:

Slutsky equation:

${\displaystyle \underbrace {\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}} _{\mathrm {Gesamteffekt} }=\underbrace {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}} _{\mathrm {Substitutionseffekt} }\underbrace {-x_{j}(\mathbf {p} ,y){\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial y}}} _{\mathrm {Einkommenseffekt} }}$

In words, the equation (read from left to right) answers the question of how the demand for a good changes if the price of a good changes with constant income . The answer is that the change corresponds to the difference between the substitution and income effects. The substitution effect corresponds to the change in the compensated demand for as a result of the change in the price of ; subtracted from this is an expression that indicates how the change in income affects the demand for , modified with the total demand for . ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle i}$ ${\displaystyle j}$

interpretation

Every change in price is accompanied by a change in real income . As the income the demand influenced the change in demand, which is solely due to the price change (is substitution effect ) by the income effect on the empirical observation falsified . The Slutsky decomposition simulates the price change with constant real income. It turns out that the demand for a normal good must decrease in the event of a price increase, if real income is kept constant ( law of demand ).

The Hicks decomposition is methodologically somewhat different , but it basically comes to the same result. Here, it is not real income but the utility (index value) of the household that is kept constant. The Hicks decomposition therefore gives the household just the amount that is necessary so that it can reach the original indifference curve again.

All possible cases at a glance

Here: Hicks decomposition

Assumption: (price decrease of good 1, cp ) ${\displaystyle p_{1}^{*}<p_{1}}$

The substitution effect (SE) is therefore always positive for good 1 and always negative for good 2. The direction and strength of the income effect (EE) and thus also the overall effect (GE) depend on the type of goods. An overview of each possible case is therefore given below.

Explanation of the graphic: P is the original optimum point ( light blue : original budget line) and P * the hypothetical optimum point with an adjusted budget line ( green ). Points A to E on the new budget line ( blue ) represent examples of every possible new, optimal bundle - depending on the type of goods and thus on the course of the associated indifference curves ( red ).

	Good 1		Good 2		Overall effect
	Art	EE	Art	EE	Good 1	Good 2
A.	Giffen-Gut	-	superior	++	-	+
B.	inferior good	-	superior	++	+	+
C.	normal	+	superior	++	+	+
D.	normal	+	normal	+	+	-
E.	normal	+	inferior	-	+	-

proof

From the duality of Marshall's and Hicks 'demand it follows first (see the article Hicks' demand function ). It then differentiates both sides using the chain rule for the price of a commodity , and obtains ${\displaystyle x_{i}^{h}(\mathbf {p} ,{\overline {u}})=x_{i}(\mathbf {p} ,e(\mathbf {p} ,{\overline {u}}))}$ ${\displaystyle j}$ ${\displaystyle p_{j}}$

${\displaystyle {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}={\frac {\partial x_{i}(\mathbf {p} ,e(\mathbf {p} ,{\overline {u}}))}{\partial p_{j}}}+{\frac {\partial x_{i}(\mathbf {p} ,e(\mathbf {p} ,{\overline {u}}))}{\partial e(\mathbf {p} ,{\overline {u}})}}\cdot {\frac {\partial e(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}}$.

It is (see the article Indirect utility function ) and also (because, according to the assumption, a consumer at prices and income achieves maximum utility ), which together implies. In addition, ( Shephard's Lemma ) applies , so that the above equation can be rewritten as ${\displaystyle e(\mathbf {p} ,v(\mathbf {p} ,y))=y}$ ${\displaystyle v(\mathbf {p} ,y)={\overline {u}}}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle y}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle e(\mathbf {p} ,{\overline {u}})=y}$ ${\displaystyle \partial e(\mathbf {p} ,{\overline {u}})/\partial p_{j}=x_{j}^{h}(\mathbf {p} ,{\overline {u}})}$

${\displaystyle {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}={\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}}+{\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial e(\mathbf {p} ,{\overline {u}})}}\cdot x_{j}^{h}(\mathbf {p} ,{\overline {u}})}$.

Repeated application of the initially postulated property of duality delivers

${\displaystyle {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}={\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}}+{\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial e(\mathbf {p} ,{\overline {u}})}}\cdot x_{j}(\mathbf {p} ,e(\mathbf {p} ,{\overline {u}}))}$,

and still there (see above) too ${\displaystyle e(\mathbf {p} ,{\overline {u}})=y}$

${\displaystyle {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}}={\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}}+{\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial e(\mathbf {p} ,{\overline {u}})}}\cdot x_{j}(\mathbf {p} ,y)}$,

what was to be shown.

Slutsky matrix

If you rearrange the Slutsky equation according to the substitution effect, the expression appears on the right-hand side

${\displaystyle \underbrace {\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}} _{\mathrm {Gesamteffekt} }+\underbrace {x_{j}(\mathbf {p} ,y){\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial y}}} _{\mathrm {Einkommenseffekt} }}$

This gives the -th entry of a matrix, the so-called Slutsky matrix (also: Slutsky substitution matrix ) : ${\displaystyle (ij)}$ ${\displaystyle (n\times n)}$ ${\displaystyle S}$

${\displaystyle S(\mathbf {p} ,y)\equiv \left({\begin{array}{ccc}{\frac {\partial x_{1}(\mathbf {p} ,y)}{\partial p_{1}}}+x_{1}(\mathbf {p} ,y){\frac {\partial x_{1}(\mathbf {p} ,y)}{\partial y}}&\cdots &{\frac {\partial x_{1}(\mathbf {p} ,y)}{\partial p_{n}}}+x_{n}(\mathbf {p} ,y){\frac {\partial x_{1}(\mathbf {p} ,y)}{\partial y}}\\\vdots &\ddots &\vdots \\{\frac {\partial x_{n}(\mathbf {p} ,y)}{\partial p_{1}}}+x_{1}(\mathbf {p} ,y){\frac {\partial x_{n}(\mathbf {p} ,y)}{\partial y}}&\cdots &{\frac {\partial x_{n}(\mathbf {p} ,y)}{\partial p_{n}}}+x_{n}(\mathbf {p} ,y){\frac {\partial x_{n}(\mathbf {p} ,y)}{\partial y}}\end{array}}\right)}$

It shows the corresponding substitution effect for any two goods.

It can be shown that it is symmetric and negative semidefinite . ${\displaystyle S}$

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 .
• Hal Varian : Microeconomic Analysis. WW Norton, New York and London 1992, ISBN 0-393-95735-7 .

Web links

• Income and substitution effect - Article including Slutsky decomposition at mikrooekonomie.de

Remarks

1. ↑ Note that agreed as usual . ${\displaystyle p_{j}>0\;\forall j}$
2. ↑ See the largely identical evidence in Mas-Colell / Whinston / Green 1995, p. 71; Jehle / Reny 2011, p. 53 f. [there in more detail] and 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.
3. ↑ For the proof, see Jehle / Reny 2011, p. 59.