Slutsky equation

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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.

The equation in matrix is called the Slutsky matrix and takes the form

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

or a given element of the matrix is

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

where $h (p, u)$ is the Hicksian demand and $x (p, w)$ is the Marshallian demand. The first term represents the substitution effect, and the second term represents the income effect.