Wikipedia

# 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:

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

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.

## 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.

## Example

A Cobb-Douglas utility function (see Cobb-Douglas production function) with two goods and income ${\displaystyle w}$ generates Marshallian demand for goods 1 and 2 of ${\displaystyle x_{1}=.7w/p_{1}}$ and ${\displaystyle x_{2}=.3w/p_{2}.}$ Rearrange the Slutsky equation to put the Hicksian derivative on the left-hand-side yields the substitution effect:

${\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}={\frac {\partial x_{1}}{\partial p_{1}}}+{\frac {\partial x_{1}}{\partial w_{1}}}x_{1}=-{\frac {.7w}{p_{1}^{2}}}+{\frac {.7}{p_{1}}}{\frac {.7w}{p_{1}}}=-{\frac {.21w}{p_{1}^{2}}}}$

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:

${\displaystyle {\frac {\partial x_{1}}{\partial p_{1}}}={\frac {\partial h_{1}}{\partial p_{1}}}-{\frac {\partial x_{1}}{\partial w}}x_{1}=-{\frac {.21w}{p_{1}^{2}}}-{\frac {.49w}{p_{1}^{2}}}}$

Thus, of the total decline of ${\displaystyle .7w/p_{1}^{2}}$ in quantity demanded when ${\displaystyle p_{1}}$ rises, 21/70 is from the substitution effect and 49/70 from the income effect. Good 1 is the good this consumer spends most of his income on (${\displaystyle p_{1}q_{1}=.7w}$), which is why the income effect is so large.

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]

${\displaystyle h_{1}(p_{1},p_{2},u)=.7p_{1}^{-.3}p_{2}^{.3}u,}$

where ${\displaystyle u}$ is utility. The derivative is

${\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}=-.21p_{1}^{-1.3}p_{2}^{.3}u,}$

so since the Cobb-Douglas indirect utility function is ${\displaystyle v=wp_{1}^{-.7}p_{2}^{-.3},}$ and ${\displaystyle u=v}$ when the consumer uses the specified demand functions, the derivative is:

${\displaystyle {\frac {\partial h_{1}}{\partial p_{1}}}=-.21p_{1}^{-1.3}p_{2}^{.3}(wp_{1}^{-.7}p_{2}^{-.3})=-.21wp_{1}^{-2}p_{2}^{0}=-{\frac {.21w}{p_{1}^{2}}},}$

which is indeed the Slutsky equation's answer.

The Slutsky equation also can be applied to compute the cross-price substitution effect. One might think it was zero here because when ${\displaystyle p_{2}}$ rises, the Marshallian quantity demanded of good 1, ${\displaystyle x_{1}(p_{1},p_{2},w),}$ is unaffected (${\displaystyle \partial x_{1}/\partial p_{2}=0}$), but that is wrong. Again rearranging the Slutsky equation, the cross-price substitution effect is:

${\displaystyle {\frac {\partial h_{1}}{\partial p_{2}}}={\frac {\partial x_{1}}{\partial p_{2}}}+{\frac {\partial x_{1}}{\partial w}}x_{2}=0+{\frac {.7}{p_{1}}}{\frac {.3w}{p_{2}}}=-{\frac {.21w}{p_{1}p_{2}}}}$

This says that when ${\displaystyle p_{2}}$ rises, there is a substitution effect of ${\displaystyle -.21w/(p_{1}p_{2})}$ towards good 1. At the same time, the rise in ${\displaystyle p_{2}}$ has a negative income effect on good 1's demand, an opposite effect of the exact same size as the substitution effect, so the net effect is zero. This is a special property of the Cobb-Douglas function.

## Changes in Multiple Prices at Once: The Slutsky Matrix

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.

When there are two goods, the Slutsky equation in matrix form is:[4]

${\displaystyle {\begin{bmatrix}{\frac {\partial x_{1}}{\partial p_{1}}}&{\frac {\partial x_{1}}{\partial p_{2}}}\\{\frac {\partial x_{2}}{\partial p_{1}}}&{\frac {\partial x_{2}}{\partial p_{2}}}\\\end{bmatrix}}={\begin{bmatrix}{\frac {\partial h_{1}}{\partial p_{1}}}&{\frac {\partial h_{1}}{\partial p_{2}}}\\{\frac {\partial h_{2}}{\partial p_{1}}}&{\frac {\partial h_{2}}{\partial p_{2}}}\\\end{bmatrix}}-{\begin{bmatrix}{\frac {\partial x_{1}}{\partial w}}x_{1}&{\frac {\partial x_{1}}{\partial w}}x_{1}\\{\frac {\partial x_{2}}{\partial w}}x_{2}&{\frac {\partial x_{2}}{\partial w}}x_{2}\\\end{bmatrix}}}$

Although strictly speaking the Slutsky equation only applies to infinitesimal changes in prices, it is standardly used a linear approximation for finite changes. If the prices of the two goods change by \Delta p_1 and \Delta p_2, the effect on the demands for the two goods are:

${\displaystyle {\begin{bmatrix}\Delta x_{1}\\\Delta x_{2}\end{bmatrix}}\approx {\begin{bmatrix}{\frac {\partial h_{1}}{\partial p_{1}}}&{\frac {\partial h_{1}}{\partial p_{2}}}\\{\frac {\partial h_{2}}{\partial p_{1}}}&{\frac {\partial h_{2}}{\partial p_{2}}}\\\end{bmatrix}}\cdot {\begin{bmatrix}\Delta p_{1}\\\Delta p_{2}\end{bmatrix}}-{\begin{bmatrix}{\frac {\partial x_{1}}{\partial w}}x_{1}&{\frac {\partial x_{1}}{\partial w}}x_{1}\\{\frac {\partial x_{2}}{\partial w}}x_{2}&{\frac {\partial x_{2}}{\partial w}}x_{2}\\\end{bmatrix}}\cdot {\begin{bmatrix}\Delta p_{1}\\\Delta p_{2}\end{bmatrix}}}$

Multiplying out the matrices, the effect on good 1, for example, would be

${\displaystyle \Delta x_{1}\approx \left({\frac {\partial h_{1}}{\partial p_{1}}}\Delta p_{1}+{\frac {\partial h_{1}}{\partial p_{2}}}\Delta p_{2}\right)-\left({\frac {\partial x_{1}}{\partial w}}\right)\left(x_{1}\Delta p_{1}+x_{2}\Delta p_{2}\right)}$

The first term is the substitution effect. The second term is the income effect, composed of the consumer's response to income loss times the size of the income loss from each price's increase.

## 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.[5] 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.