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The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky (1880–1948), 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. The equation demonstrates that the change in the demand for a good, caused by a price change, is the result of two effects:

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.[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 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[edit]

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.

See also[edit]

References[edit]

  1. ^ Nicholson, W. (2005). Microeconomic Theory (10th ed.). Mason, Ohio: Thomson Higher Education. 
  2. ^ Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton. 

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Slutsky_equation — Please support Wikipedia.
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4 news items

The Conversation AU

The Conversation AU
Wed, 19 Nov 2014 11:41:15 -0800

There are two broad reasons for this – one involving use of the phrase “Slutsky equation”, so let's get to that first: it sounds fun. The noisiest concern with the consequences of the ABC cuts seems to be focusing on the ABC itself, such as all the ...
 
Harvard Crimson
Tue, 04 Nov 2014 18:31:28 -0800

But in the realm of theory, outside of kinked demand curves and the Slutsky equation, economists have largely avoided romance. The economics of fertility is a growing and important field, but there has been comparatively little study of the process ...
 
Register
Wed, 08 Sep 2010 22:31:30 -0700

The reporter also pointed out in blacklisting "Slutsky," Google isn't displaying real-time results for her great uncle, Russian mathematical economist Eugen Slutsky, known for his eponymous economic model the Slutsky Equation. And though Google wouldn ...
 
Business Insider
Tue, 09 Aug 2011 01:55:52 -0700

This is formalized in the Slutsky equation (as with 'homoskedasticity' and 'fat tails', guaranteed to make the class snicker), but the bottom line is that the effects go in different directions. For normal goods, the income effect is positive, higher ...
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