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The Slutsky's Equation breaks down a change in demand due to price change into the substitution effect and the income effect. The equation takes the form:
The term on the left is the change in demand when price changes, where x is the (Marshallian) demand for a good and p is the price. The term h is the Hicksian or the compensated demand. The term measures the substitution effect. The term is the income, and measures the income effect. See below for more explanations and the derivation of the equation.
We can make sense of the substitution and the income effects by this intuitive story. Suppose a consumer is consuming the optimal amount of two goods x and y, given his income and suddenly the price of x drops. The consumer will respond to this price change in two ways. First, as x becomes relatively cheaper the consumer will shift some of his consumption of y to x (assume x and y are not perfect complement). Second, as the price of x drops, even if the consumer does not make any consumption shift from y to x, he has more purchasing power because of the savings that results from the price drop in x. This savings allows the consumer to buy more goods (x or y). The shift in consumption from y to x is the substitution effect, and the increase in purchasing power due to the savings is the income effect.
When we read the Slutsky's equation, the term dh/dp is the substitution effect. This is because the compensated depend h(p_{1}, p_{2}, u) fixes the consumer's utility level, and when the consumer's purchasing power remains constant, the term dh/dp only measures the shift in consumption when the price changes. On the other hand, the income effect depends on the amount of good the consumer is consuming (x), and the consumer's reaction to an income change that comes from the "savings". Thus the term measures the income effect.
Notice that in equilibrium, the (Marshallian) demand and the compensated demand are the same. That is,
, where v is the value function of the utility maximization problem. To simplify notation, we write
, a fixed level of utility, and we write the budget constraint as
.
Now equate the two demands as above,
Without loss of generality, differentiate with respect to p_{1},
Note that the budget constraint in the Marshallian demand depends on p, so we have to use total derivative when differentiating the left side of the equation. The second term is merely an application of the chain rule. The term is the derivative of the budget constraint
with respect to ,
ie, .
Substitute this in and the equation becomes the Slutsky's equation
When the consumer is endowed with the goods instead of a fixed income, the budget constraint is . Writing the budget constraint this way, and by differentiating the budget constraint with respect to , it is easy to see that the Slutsky's equation becomes
In other words, the substitution effect remains the same, but the income effect applies to the excess demand rather than the demand itself.
Given the two examples and the derivation above, we can see that the Slutsky's equation always has the same format, and each format is different only because the budget constraint is different. Students can try deriving Slutsky's equations for other situations, such as one with Labour Supply, or Intertemporal Choices.
back to the consumer.
The last part above can be visualized intuitively if we assume the existence of indifference curves. If the consumer has a Cobb-Douglas indifference curve, x' is different from and worse than x*. However, if x_{1} and x_{2} are perfect complement of each other (L-shaped indifference curves), then the consumer will choose the same bundle before and after the tax.