Slutsky equation

The Slutsky decomposition is a method to determine the derivation of the principle unobservable Hicksian demand function for the price of the potentially observable Marshallian demand; the resulting equation is called the Slutsky equation.

It turns out that using the Slutsky decomposition caused by a price change demand change can be decomposed for a commodity into a substitution and an income effect.

The method is named after mathematician and economist Eugene Slutsky.

Representation

Be the Marshallian demand for a good in response to a price vector and the individual income. ( The Marshallian demand results from the utility maximization problem of the household and are the volume of goods - depending on the freight fares - that is required to achieve a given income the highest possible level of utility ).

Furthermore, one compatible as Hicksian (also: compensated ) demand for the good, which here represents the achievable level of utility. ( The Hicksian demand results from the output minimization problem of the budget and are the volume of goods - in Abgängigkeit of freight fares - which is necessary to the lowest possible cost to achieve a given level of utility ).

Then:

Slutsky equation:

In words, the equation answered ( read from left to right) the question of how the demand for a good changes when adjusting for income the price of good changes. The answer is that the change in the sum of the substitution and income effect corresponds. The substitution effect is the change of the compensated demand due to the change in the price of; is subtracted from an expression that specifies how the change in income on the demand for impact modified with the total demand.

Interpretation

Each price change is accompanied by a change in real income. As the income influences demand, demand will change that is attributable solely to the change in price ( substitution effect ), distorted in the empirical observation by the income effect. The Slutsky decomposition simulates the price change at constant real income. It turns out that the demand for a normal good must go back to a price increase, if real income is held constant ( law of demand ).

Essential for something different the Hicks decomposition, however, comes to basically the same result. Here not the real income, but the benefits (index value) of the budget is kept constant. The Hicks decomposition is the budget that is just the amount that is necessary to enable it to achieve the original indifference curve again.

Evidence

It is (see the article Indirect utility function ) and also ( because after acceptance achieved a consumer on price and income with just a maximum of the benefits ), which implies together. In addition, ( Shephards lemma) applies, so that one can rewrite the above equation to

But supplies Repeated application of the above postulated duality property

And there still (see above) also

What was to be shown.

Slutsky matrix

If you compare the Slutsky equation after the substitution effect to, is on the right side of the expression

Thus, the th entry of a matrix is given, the so-called Slutsky matrix (also: Slutsky substitution matrix ):

It shows the associated substitution effect for any two goods.

It can be shown that is symmetric and negative semi-definite.