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Concept
Slutsky equation
Summary
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 can be negative for consumers as it can limit choices. He designed this formula to explore a consumer's response as the price changes. When the price increases, the budget set moves inward, which also 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 substitution effect is due to the effect of the relative price change while the income effect is due to the effect of income being freed up. 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: when the price of good changes, as it becomes relatively cheaper, if hypothetically consumer's consumption remains same, income would be freed up which could be spent on a combination of each or more of the goods.
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:
where is the Hicksian demand and is the Marshallian demand, at the vector of price levels , wealth level (or, alternatively, income level) , and fixed utility level given by maximizing utility at the original price and income, formally given by the indirect utility function . 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.
Official source
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