digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky (1880–1948), 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. The equation demonstrates that the change in the demand for a good, caused by a price change, is the result of two effects:

The Slutsky equation decomposes the change in demand for good i in response to a change in the price of good j:

${\displaystyle {\partial x_{i}(\mathbf {p} ,w) \over \partial p_{j}}={\partial h_{i}(\mathbf {p} ,u) \over \partial p_{j}}-{\partial x_{i}(\mathbf {p} ,w) \over \partial w}x_{j}(\mathbf {p} ,w),\,}$

where ${\displaystyle h(\mathbf {p} ,u)}$ is the Hicksian demand and ${\displaystyle x(\mathbf {p} ,w)}$ is the Marshallian demand, at the vector of price levels ${\displaystyle \mathbf {p} }$, wealth level (or, alternatively, income level) ${\displaystyle w}$, and fixed utility level ${\displaystyle u}$ given by maximizing utility at the original price and income, formally given by the indirect utility function ${\displaystyle v(\mathbf {p} ,w)}$. The right-hand side of the equation is equal to 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.

The first term on the right-hand side represents the substitution effect, and the second term represents the income effect.[1] Note that since utility is not observable, the substitution effect is not directly observable, but it can be calculated by reference to the other two terms in the Slutsky equation, which are observable. This process is sometimes known as the Hicks decomposition of a demand change.[2]

The equation can be rewritten in terms of elasticity:

${\displaystyle \epsilon _{p,ij}=\epsilon _{p,ij}^{h}-\epsilon _{w,i}b_{j}}$

where εp is the (uncompensated) price elasticity, εph is the compensated price elasticity, εw,i the income elasticity of good i, and bj the budget share of good j.

The same equation can be rewritten in matrix form to allow multiple price changes at once:

${\displaystyle \mathbf {D_{p}x} (\mathbf {p} ,w)=\mathbf {D_{p}h} (\mathbf {p} ,u)-\mathbf {D_{w}x} (\mathbf {p} ,w)\mathbf {x} (\mathbf {p} ,w)^{\top },\,}$

where Dp is the derivative operator with respect to price and Dw is the derivative operator with respect to wealth.

The matrix ${\displaystyle \mathbf {D_{p}h} (\mathbf {p} ,u)}$ 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 identity ${\displaystyle h_{i}(\mathbf {p} ,u)=x_{i}(\mathbf {p} ,e(\mathbf {p} ,u))}$ where ${\displaystyle e(\mathbf {p} ,u)}$ is the expenditure function, and u is the utility obtained by maximizing utility given p and w. Totally differentiating with respect to pj yields the following:

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

Making use of the fact that ${\displaystyle {\frac {\partial e(\mathbf {p} ,u)}{\partial p_{j}}}=h_{j}(\mathbf {p} ,u)}$ by Shephard's lemma and that at optimum,

${\displaystyle h_{j}(\mathbf {p} ,u)=h_{j}(\mathbf {p} ,v(\mathbf {p} ,w))=x_{j}(\mathbf {p} ,w),}$ where ${\displaystyle v(\mathbf {p} ,w)}$ is the indirect utility function,

one can substitute and rewrite the derivation above as the Slutsky equation.

## References

1. ^ Nicholson, W. (2005). Microeconomic Theory (10th ed.). Mason, Ohio: Thomson Higher Education.
2. ^ Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton.

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Slutsky_equation — Please support Wikipedia.
 786 videos foundNext >
 SlutskyEquationHow to apply the Slutsky equation to calculation substitution and income effects of a price change. 11a. The Slutsky Equation and Demand CurvesIn this video, I offer a derivation of the Slutsky Equation (an equation that decomposes the Marshallian demand curve's price effect into income and substitution ... A.9 Income and substitution effects | Consumption - MicroeconomicsLearn more: http://www.policonomics.com/slutskys-equation/ This video explains what the income and substitution effects are, and how to analyse them in order ... Microeconomics Theory I - Lecture 23 (ECON - 203)Lecture 23: Slutsky Equation. slutsky hicks.movBrian O'Roark from Robert Morris University illustrates the income and substitution effects through the Slutsky and Hicks approaches. Income and Substitution Effects (Mathematically)Chapter - Income and Substitution Effects (Mathematically) This microeconomics series by EurekaWow is aligned with the textbook Hal Varian (baby version) as ... Slutsky equation of labour supply Income and Substitution EffectsWhat are Income and Substitution Effects? * How do they work? * How do they add up to the total price effect? 8-(:-) Check out more at www.vibedu.com. Slutsky Equation and Demand CurvesSlutsky Equation. Example Income and Subsitution Effects For Normal and Inferior GoodsTutorial on understanding the income and substitution effects for normal and inferior goods when the price of a good rises and income and substitution effects for ...
 786 videos foundNext >

We're sorry, but there's no news about "Slutsky equation" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter