# Slutsky equation

The Slutsky equation (or Slutsky identity) in economics, 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. There are two parts of the Slutsky equation, namely the substitution effect, and income effect. In general, the substitution effect is negative. He designed this formula to explore a consumer's response as the price changes. When the price increases, the budget set moves inward, which causes the quantity demanded to decrease. In contrast, when the price decreases, the budget set moves outward, which leads to an increase in the quantity demanded. The equation demonstrates that the change in the demand for a good, caused by a price change, is the result of two effects:

• a substitution effect: the good becomes relatively cheaper, so the consumer purchases other goods as substitutes
• an income effect: the purchasing power of a consumer increases as a result of a price decrease, so the consumer can now afford better products or more of the same products, depending on whether the product itself is a normal good or an inferior good.

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

${\partial x_{i}(\mathbf {p} ,w) \over \partial p_{j}}={\partial h_{i}(\mathbf {p} ,u) \over \partial p_{j}}-{\partial x_{i}(\mathbf {p} ,w) \over \partial w}x_{j}(\mathbf {p} ,w),\,$ where $h(\mathbf {p} ,u)$ is the Hicksian demand and $x(\mathbf {p} ,w)$ is the Marshallian demand, at the vector of price levels $\mathbf {p}$ , wealth level (or, alternatively, income level) $w$ , and fixed utility level $u$ given by maximizing utility at the original price and income, formally given by the indirect utility function $v(\mathbf {p} ,w)$ . 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. 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.

The equation can be rewritten in terms of elasticity:

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

$\mathbf {D_{p}x} (\mathbf {p} ,w)=\mathbf {D_{p}h} (\mathbf {p} ,u)-\mathbf {D_{w}x} (\mathbf {p} ,w)\mathbf {x} (\mathbf {p} ,w)^{\top },\,$ where Dp is the derivative operator with respect to price and Dw is the derivative operator with respect to wealth.

The matrix $\mathbf {D_{p}h} (\mathbf {p} ,u)$ 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 $h_{i}(\mathbf {p} ,u)=x_{i}(\mathbf {p} ,e(\mathbf {p} ,u))$ where $e(\mathbf {p} ,u)$ 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:

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

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

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

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

## Giffen goods

A Giffen good is a product that is in greater demand when the price increases, which are also special cases of inferior goods. In the extreme case of income inferiority, the size of income effect overpowered 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.