Slutsky equation
Encyclopedia
The Slutsky equation in economics
, named after Eugen Slutsky
(1880-1948), relates changes in Marshallian demand to changes in Hicksian demand. It demonstrates that demand changes due to price changes are a result of two effects:
Each element of the Slutsky matrix is given by
where
is the Hicksian demand and 
is the Marshallian demand, at price level p, wealth level w, and utility level u. The first term represents the substitution effect, and the second term represents the income effect.
The above equation is also known as the Hicksian decomposition of demand. The left hand side of the equation represents how demand for one good changes in response to price changes for another good. The right hand side of the equation says that this change is equal to the change in demand holding expenditure fixed at w* plus the change in demand when income changes multiplied by how much income has to change to keep utility constant.
The same equation can be rewritten in matrix form as
where Dp is the derivative operator with respect to price and Dw is the derivative operator with respect to wealth.
The matrix
is known as the Slutsky matrix, and given sufficient smoothness conditions on the utility function, it is symmetric, negative semidefinite, and the Hessian of the expenditure function.
where 
is the expenditure function
. Differentiating the above equation yields the following.
Making use of the fact that
and
where 
is the indirect utility function
, we can substitute and rewrite the derivation above as the Slutsky equation.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, named after Eugen Slutsky
Eugen Slutsky
Evgeny "Eugen" Evgenievich Slutsky was a Russian/Soviet mathematical statistician, economist and political economist.-Slutsky's work in economics:...
(1880-1948), relates changes in Marshallian demand to changes in Hicksian demand. It demonstrates that demand changes due to price changes are a result of two effects:
- a substitution effect, the result of a change in the exchange rate between two goods; and
- an income effectIncome effectIn economics, the consumer's preferences, money income and prices play an important role in solving the consumer's optimization problem...
, the effect of a change in price results in a change of the consumer's purchasing power.
Each element of the Slutsky matrix is given by
where
is the Hicksian demand and is the Marshallian demand, at price level p, wealth level w, and utility level u. The first term represents the substitution effect, and the second term represents the income effect.The above equation is also known as the Hicksian decomposition of demand. The left hand side of the equation represents how demand for one good changes in response to price changes for another good. The right hand side of the equation says that this change is equal to the change in demand holding expenditure fixed at w* plus the change in demand when income changes multiplied by how much income has to change to keep utility constant.
The same equation can be rewritten in matrix form as
where Dp is the derivative operator with respect to price and Dw is the derivative operator with respect to wealth.
The matrix
is known as the Slutsky matrix, and given sufficient smoothness conditions on the utility function, it is symmetric, negative semidefinite, and the Hessian of the expenditure function.Derivation
While there are several ways to derive the slutsky equation, the following method is likely the simplest. Begin by noting the relationship where is the expenditure functionExpenditure function
In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices....
. Differentiating the above equation yields the following.
Making use of the fact that
and where is the indirect utility functionIndirect utility function
In economics, a consumer's indirect utility functionv gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over market conditions....
, we can substitute and rewrite the derivation above as the Slutsky equation.