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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:

${\frac {dx}{dp}}={\frac {dh}{dp}}-x{\frac {dx}{dm}}$ 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 $dh/dp$ measures the substitution effect. The term $m$ is the income, and $x(dx/dm)$ measures the income effect. See below for more explanations and the derivation of the equation.

## Intuition

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(p1, p2, 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 $dx/dm$ that comes from the "savings". Thus the term $x(dx/dm)$ measures the income effect.

## Derivation

Notice that in equilibrium, the (Marshallian) demand and the compensated demand are the same. That is,

$x(p_{1},p_{2},m)=h(p_{1},p_{2},v(p_{1},p_{2},m))\$ , where v is the value function of the utility maximization problem. To simplify notation, we write

$u=v(p_{1},p_{2},m)\$ , a fixed level of utility, and we write the budget constraint as

$p_{1}x_{1}+p_{2}x_{2}-m=0\$ .

Now equate the two demands as above,

$x(p_{1},p_{2},m)=h(p_{1},p_{2},u)\$ Without loss of generality, differentiate with respect to p1,

${\frac {dx}{dp_{1}}}+{\frac {dx}{dm}}\times {\frac {dm}{dp_{1}}}={\frac {dh}{dp_{1}}}$ 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 $dm/dp_{1}$ is the derivative of the budget constraint

$p_{1}x_{2}+p_{1}x_{2}-m\$ with respect to $p_{1}$ ,

ie, ${\frac {dm}{dp_{1}}}=x_{1}$ .

Substitute this in and the equation becomes the Slutsky's equation

${\frac {dx}{dp_{1}}}+{\frac {dx}{dm}}\times x_{1}={\frac {dh}{dp_{1}}}$ ## Endowment income effect

When the consumer is endowed with the goods instead of a fixed income, the budget constraint is $p_{1}x_{1}+p_{2}x_{2}-p_{1}w_{1}-p_{2}w_{2}=0$ . Writing the budget constraint this way, and by differentiating the budget constraint with respect to $p_{1}$ , it is easy to see that the Slutsky's equation becomes

${\frac {dx}{dp_{1}}}={\frac {dh}{dp_{1}}}-{\frac {dx}{dm(x_{1}-w_{1})}}$ In other words, the substitution effect remains the same, but the income effect applies to the excess demand rather than the demand itself.

## Other Slutsky equations

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.

## Exercise

• Q* Explain why imposing a tax on a certain good can hurt the consumer, even when the tax revenue collected by the government is rebated via a direct income transfer

back to the consumer.

• A* First we have to build a model for which the question can be made sense. Consider, in a two-good world, a consumer facing an exogenous income m and prices p1 and p2. The consumer's budget constraint is $p_{1}x_{1}+p_{2}x_{2}=m$ , and let the optimal bundle under that budget constraint be $x*=(x_{1}*,x_{1}*)$ . Now consider an excise tax t applied to good 1, and the tax revenue being rebated exogenously to the consumer. Then the consumer's budget constraint becomes $(p_{1}+t)x_{1}+p_{2}x_{2}=m+R$ , where $R=t\times x_{1}'$ and $x'=(x_{1}',x_{2}')$ denotes the optimal bundle under the new budget constraint. Now we substitute the new optimal bundle into its budget constraint and it becomes $p_{1}x_{1}'+p_{2}x_{2}'=m$ . As the equation has the same prices and income as the old budget constraint, it demonstrates that the new optimal bundle was affordable before the application of the tax. However, the bundle x* was chosen under the budget constraint before tax. By revealed preference, x* must be at least as good as x'. When is x* strictly better than x' so that the tax is hurting the consumer? One necessary condition is that the marginal rate of substitution at x* is between p and p + t. When that's the case, facing a tax leads to a change in the optimal bundle and the consumer is strictly worse off.

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 x1 and x2 are perfect complement of each other (L-shaped indifference curves), then the consumer will choose the same bundle before and after the tax.