Definitions

# Slutsky equation

The Slutsky equation (or Slutsky identity) 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:

• a substitution effect, the result of a change in the exchange rate between two goods; and
• an income effect, the effect of price results in a change of the consumer's purchasing power.

Each element of the Slutsky matrix is given by

$\left\{partial x_i\left(p, w\right) over partial p_j\right\} = \left\{partial h_i\left(p, u\right) over partial p_j\right\} - \left\{partial x_i\left(p, w\right) over partial w \right\} x_j\left(p, w\right),,$

where $h\left(p, u\right)$ is the Hicksian demand and $x\left(p, w\right)$ 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 same equation can be rewritten in matrix form and is called the Slutsky matrix

$D_p x\left(p, w\right) = D_p h\left(p, u\right)- D_w x\left(p, w\right) x\left(p, w\right)^top,,$

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