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 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:

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

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, alternatively, income level) ${\displaystyle w}$, and fixed utility level ${\displaystyle u}$ given by maximizing utility at the original price and income, formally given by the indirect utility function ${\displaystyle v(\mathbf {p} ,w)}$. 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 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, in simple words, the Slutsky equation states the total change in demand consists of an income effect and a substitution effect and both effects collectively must equal the total change in demand.

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

The equation above is helpful as it represents the fluctuation in demand are indicative of different types of good. The substitution effect will always turn out negative as indifference curves are always downward sloping. However, the same does not apply to income effect as it depends on how consumption of a good changes with income.

The income effect on a normal goods is negative, and if the price decreases, consequently purchasing power or income goes up. The reverse holds when price increases and purchasing power or income decreases, as a result of, so does demand.

Generally, not all goods are "normal". While in an economic sense, some are inferior. However, that does not equate quality-wise that they are poor rather that it sets a negative income profile - as income increases, consumers consumption of the good decreases.

For example, consumers who are running low of money for food purchase instant noodles, however, the product is not generally held as something people would normally consume on a daily basis. This is due to the constrains in terms of money; 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 depends on the type of goods:

However, whether the total effect will always be negative is impossible to tell 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.