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:

Each element of the Slutsky matrix is given by

{\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, 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.[1]

The above equation is also known as the Hicksian decomposition of demand. The left hand side of the equation represents how demand for one good changes in response to price changes for another good. The right hand side of the equation says that this change is equal to the change in demand holding expenditure fixed at w* plus the change in demand when income changes multiplied by how much income has to change to keep utility constant.[2]

The same equation can be rewritten in matrix form as

D_p x(p, w) = D_p h(p, u)- D_w x(p, w) x(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 D_p h(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 relationship h_i(p,u) = x_i (p, e(p,u)) where  e(p,u) is the expenditure function. Differentiating the above equation yields the following.

 \frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial x_i(p,e(p,u))}{\partial p_j}  %2B \frac{\partial x_i (p,e(p,u))}{\partial e(p,u)} \times \frac{\partial e(p,u)}{\partial p_j}

Making use of the fact that  e(p,u) = w and

 \frac{\partial e(p,u)}{\partial p_j} = h_j(p,u) = h_j(p, v(p,w)) = x_j(p,w) where v(p,w) is the indirect utility function, we can substitute and rewrite the derivation above as the Slutsky equation.

See also

References

  1. ^ Nicholson, W. 2005. Microeconomic Theory. 10th edn. Mason, Ohio. Thomson Higher Education.
  2. ^ Varian, H. 1992. Microeconomic Analysis. 3d edn. New York, N.Y. W.W Norton & Company, Inc