Slutsky equation

In microeconomics, the Slutsky equation (or Slutsky identity), 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 is negative. Slutsky derived this formula to explore a consumer's response as the price of a commodity changes. When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease. In contrast, if 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 a good changes, as it becomes relatively cheaper, consumer consumption could hypothetically remain unchanged. If so, income would be freed up, and money could be spent on one or more goods.
• an income effect: the purchasing power of a consumer increases as a result of a price decrease, so the consumer can now purchase other products or more of the same product, depending on whether the product(s) 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:

${\displaystyle {\partial x_{i}(\mathbf {p} ,w) \over \partial p_{j}}={\partial h_{i}(\mathbf {p} ,u) \over \partial p_{j}}-{\partial x_{i}(\mathbf {p} ,w) \over \partial w}x_{j}(\mathbf {p} ,w),\,}$

where ${\displaystyle h(\mathbf {p} ,u)}$ is the Hicksian demand and ${\displaystyle x(\mathbf {p} ,w)}$ is the Marshallian demand, at the vector of price levels ${\displaystyle \mathbf {p} }$, wealth level (or income level) ${\displaystyle w}$, and fixed utility level ${\displaystyle u}$ given by maximizing utility at the original price and income, formally presented by the indirect utility function ${\displaystyle v(\mathbf {p} ,w)}$. The right-hand side of the equation equals 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. Note that since utility is not observable, the substitution effect is not directly observable. Still, it can be calculated by referencing the other two observable terms in the Slutsky equation. This process is sometimes known as the Hicks decomposition of a demand change.

The equation can be rewritten in terms of elasticity:

${\displaystyle \varepsilon _{p,ij}=\varepsilon _{p,ij}^{h}-\varepsilon _{w,i}b_{j}}$

where ε_p is the (uncompensated) price elasticity, ε_p^h is the compensated price elasticity, ε_w,i the income elasticity of good i, and b_j the budget share of good j.

Overall, the Slutsky equation states that the total change in demand consists of an income effect and a substitution effect, and both effects must collectively equal the total change in demand.

${\displaystyle \Delta x_{1}=\Delta x_{1}^{s}+\Delta x_{1}^{l}}$

The equation above is helpful because it demonstrates that changes in demand indicate different types of goods. The substitution effect is negative, as indifference curves always slope downward. However, the same does not apply to the income effect, which depends on how income affects the consumption of a good.

The income effect on a normal good is negative, so if its price decreases, the consumer's purchasing power or income increases. The reverse holds when the price increases and purchasing power or income decreases.

An example of inferior goods is instant noodles. When consumers run low on money for food, they purchase instant noodles; however, the product is not generally considered something people would normally consume daily. This is due to money constraints; as wealth increases, consumption decreases. In this case, the substitution effect is negative, but the income effect is also negative.

In any case, the substitution effect or income effect are positive or negative when prices increase depending on the type of goods:

However, it is impossible to tell whether the total effect will always be negative if inferior complementary goods are mentioned. For instance, the substitution effect and the income effect pull in opposite directions. The total effect will depend on which effect is ultimately stronger.