In microeconomics, the Slutsky equation (or Slutsky identity), named after Eugen Slutsky, relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed level of utility.
Slutsky derived this formula to explore a consumer's response as the price of a commodity changes.
When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease.
In contrast, if the price decreases, the budget set moves outward, which leads to an increase in the quantity demanded.
is the Marshallian demand, at the vector of price levels
The right-hand side of the equation equals the change in demand for good i holding utility fixed at u minus the quantity of good j demanded, multiplied by the change in demand for good i when wealth changes.
Still, it can be calculated by referencing the other two observable terms in the Slutsky equation.
The equation above is helpful because it demonstrates that changes in demand indicate different types of goods.
The substitution effect is negative, as indifference curves always slope downward.
The reverse holds when the price increases and purchasing power or income decreases.
When consumers run low on money for food, they purchase instant noodles; however, the product is not generally considered something people would normally consume daily.
This is due to money constraints; as wealth increases, consumption decreases.
In any case, the substitution effect or income effect are positive or negative when prices increase depending on the type of goods: However, it is impossible to tell whether the total effect will always be negative if inferior complementary goods are mentioned.
While there are several ways to derive the Slutsky equation, the following method is likely the simplest.
is the expenditure function, and u is the utility obtained by maximizing utility given p and w. Totally differentiating with respect to pj yields as the following: Making use of the fact that
by Shephard's lemma and that at optimum, one can substitute and rewrite the derivation above as the Slutsky equation.
The Slutsky equation can be rewritten in matrix form: where Dp is the derivative operator with respect to prices and Dw is the derivative operator with respect to wealth.
is the maximum utility the consumer achieves at prices
is exactly equal to the corresponding component of the Hicksian substitution matrix
Rearrange the Slutsky equation to put the Hicksian derivative on the left-hand-side yields the substitution effect: Going back to the original Slutsky equation shows how the substitution and income effects add up to give the total effect of the price rise on quantity demanded: Thus, of the total decline of
One can check that the answer from the Slutsky equation is the same as from directly differentiating the Hicksian demand function, which here is[3] where
The derivative is so since the Cobb-Douglas indirect utility function is
when the consumer uses the specified demand functions, the derivative is: which is indeed the Slutsky equation's answer.
The Slutsky equation also can be applied to compute the cross-price substitution effect.
rises, the Marshallian quantity demanded of good 1,
Again rearranging the Slutsky equation, the cross-price substitution effect is: This says that when
When there are two goods, the Slutsky equation in matrix form is:[4] Although strictly speaking, the Slutsky equation only applies to infinitesimal price changes, a linear approximation for finite changes is standardly used.
[5] In the extreme case of income inferiority, the size of the income effect overpowers the size of the substitution effect, leading to a positive overall change in demand responding to an increase in the price.
Slutsky's decomposition of the change in demand into a pure substitution effect and income effect explains why the law of demand doesn't hold for Giffen goods.
Intermediate microeconomics : a modern approach (Ninth edition.).